ABSTRACT
Title of Dissertation: NON-EQUILIBRIUM THERMODYNAMICS
OF CYTOSKELETAL SELF-ORGANIZATION
Carlos Floyd
Doctor of Philosophy, 2021
Dissertation Directed by: Professor Garegin A. Papoian
Department of Chemistry and Biochemistry
The actin-based cytoskeleton is a polymer network that plays an essential role in cell
biology. By self-organizing into various local architectures, the cytoskeleton performs
physiological functions that allow the cell to physically interact with its environment.
It is also an example of biological active matter, consuming chemical free energy at a
local scale to produce directed motion and do mechanical work. While it is well-known
that cytoskeletal free energy transduction occurs, it has been a challenge to say anything
quantitative about this far-from-equilibrium process due to the difficulty of making the
necessary experimental measurements. This lack of methodology to quantify cytoskele-
tal energetics significantly hinders our understanding of the self-organization process
underlying the cytoskeleton?s physiological functionality. To address this research gap,
we develop in this thesis an explicit computational method to quantify chemical and
mechanical free energy changes during simulated cytoskeletal self-organization using the
software package MEDYAN (Mechanochemical Dynamics of Active Networks). We then
apply this tool in several studies to advance our understanding of the self-organization
process and its thermodynamic characteristics. For instance, we analyze the thermo-
dynamic efficiency of mechanical stress generation and the network?s time-dependent
dissipation rates under a range of network conditions. We also investigate the recent
experimentally discovered phenomenon of cytoskeletal avalanches, which we identify in
simulation as anomalous mechanical dissipation events. Our analysis clarifies the phe-
nomenology and underlying mechanism of these avalanche events, which we propose may
play an important role in cellular information processing. The in silico method devel-
oped in this thesis provides a new perspective on cytoskeletal self-organization and may
be extended to investigate other biological active matter systems.
NON-EQUILIBRIUM THERMODYNAMICS
OF CYTOSKELETAL SELF-ORGANIZATION
by
Carlos Floyd
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2021
Advisory Committee:
Professor Garegin A. Papoian, Co-Chair/Co-Advisor
Professor Christopher Jarzynski, Co-Chair/Co-Advisor
Professor Michelle Girvan
Professor Sergei Sukharev
Professor Arpita Upadhyaya
?c Copyright by
Carlos Floyd
2021
ii
Acknowledgments
This work would not have been possible without the people whom I?ve known over
the past six years. First, I would like to express my sincere gratitude toward my advisors,
Garyk Papoian and Chris Jarzynski. I have benefited from many helpful discussions with
you on scientific topics, developing research ideas, navigating career decisions, and life
in general. I?m very fortunate to have spent this time in graduate school working with
such great scientists and mentors, and I hope there will be plenty of chances to interact
again down the road. My committee of Michelle Girvan, Sergei Sukharev, and Arpita
Upadhyaya also deserve my sincere thanks. Your work in guiding this thesis has been of
great help, and I appreciate the opportunities I?ve had along the way to learn from each
of you.
I would like to thank the members of the Papoian and Jarzynski labs, who have
been colleagues and friends throughout my time in graduate school. The members of
the Papoian lab, including Mary Pitman, Hao Wu, Haiqing Zhao, and the MEDYAN
developers Aravind Chandrasekaran, James Komianos, Haoran Ni, and Qin Ni have
made it a pleasure to come to lab everyday. I am especially indebted to the MEDYAN
developers for our close working relationship - tracking down segfaults would have been a
much more annoying without your help. I am also grateful to the members and associates
of the Jarzynski lab, including Andrew Smith, Wade Hodson, Debankur Bhattacharyya,
Guilherme De Sousa, Long Him Cheung, Joshua Chiel, and Kanupriya Sinha. Discussing
aspects of non-equilibrium and quantum physics with all of you has been a very rewarding
part of my graduate school experience. To the members of both labs, thanks for making
the day-to-day work of research more enjoyable.
iii
I would also like to thank the University of Maryland organizations I have been a
member of, including the Biophysics program, the Institute for Physical Science and
Technology, the Physics of Living Systems network, and the Computation and Mathe-
matics of Biological Networks (COMBINE) program. The faculty and students in these
organizations have greatly enriched my time in graduate school. It?s been eye-opening
to engage with the broad interdisciplinary research being done at this and other univer-
sities through seminars and discussions with the members of these organizations. I am
particularly grateful for the training and financial support from the COMBINE program.
By exposing me to ideas in network science and machine learning and by providing clear
advice for professional development, the COMBINE program has had a large positive
impact on my time here.
I would like to acknowledge our co-workers at the University of Oxford, Radek Erban
and Ravinda Gunaratne. While it isn?t presented in this thesis, I have learned a lot
from our work together. I have also benefited from a productive collaboration with
Herbert Levine developing the work on cytoskeletal avalanches. I wish to thank my
undergraduate advisor, Rosa Tamara Branca, as well, for guiding me through my first
research experience and helping with my graduate school applications.
Finally, I wouldn?t have been able to reach this milestone in my life without the
support of my family and friends. My parents, Jim and Katie Floyd, my brother and
sister-in-law, Cam and Kim Floyd, and my girlfriend, Michelle Swanson (and our cat
Bowie), are a constant source of strength for which I am deeply grateful. To my friends
in Maryland, North Carolina, and elsewhere, I thank you as well.
iv
Contents
Abstract
Acknowledgments ii
Table of Contents iv
List of Figures vii
List of Tables ix
List of Abbreviations x
1 Introduction 1
1.1 The cytoskeleton as biological active matter . . . . . . . . . . . . . . . . . 3
1.1.1 Functions and components of the cytoskeleton . . . . . . . . . . . . 3
1.1.2 Active matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Computational modeling of the cytoskeleton . . . . . . . . . . . . . 9
1.2 Non-equilibrium thermodynamics of active matter . . . . . . . . . . . . . 11
1.2.1 Overview of non-equilibrium theories . . . . . . . . . . . . . . . . . 11
1.2.2 Using non-equilibrium thermodynamics to explain active matter
self-organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Outline of chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Low-dimensional manifold of actin polymerization dynamics 22
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Brooks-Carlsson model . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2 Quasi-steady state approximation . . . . . . . . . . . . . . . . . . . 28
2.2.3 Constant tip approximation . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 Eigenvalues of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.2 Steady state concentrations . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Gibbs free energy change of a discrete chemical reaction event 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.1 ?G of reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.2 Thermodynamics of a reaction-diffusion compartment grid . . . . . 55
3.2.3 ?G of diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
v
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.1 ?G of reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.2 ?G of diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Quantifying dissipation in actomyosin networks 68
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.1 Measuring dissipation in MEDYAN . . . . . . . . . . . . . . . . . . 72
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.1 Total dissipation rates of disordered networks do not increase . . . 77
4.3.2 More compact networks are more efficient . . . . . . . . . . . . . . 85
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 Graph representations, motor stalling, and volume scaling effects 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.1 Graph mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.2 Volume scaling study . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.1 Self-organization produces percolated, coalesced networks . . . . . 98
5.3.2 Dissipation rates are extensive and depend on motor stalling . . . . 102
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6 Understanding cytoskeletal avalanches using mechanical stability anal-
ysis 111
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3.1 Asymmetric heavy-tailed distributions and correlations of ?U . . . 115
6.3.2 Distinguishing features of cytoquake events . . . . . . . . . . . . . 119
6.3.3 Testing different concentrations and system sizes . . . . . . . . . . 121
6.3.4 Connection to experiments . . . . . . . . . . . . . . . . . . . . . . 124
6.3.5 Normal mode decomposition probes network?s mechanical state . . 128
6.3.6 Cytoquake forecasting . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3.7 Cytoquakes have enhanced displacement along soft modes . . . . . 134
6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7 Summary and future directions 138
A Description of MEDYAN simulation platform 144
A.1 Simulation protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.2 Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.3 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
A.4 Mechanochemical coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
vi
B Supporting information for Chapter 2 152
B.1 Details of BC model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
B.2 Steady state concentrations in the BC model . . . . . . . . . . . . . . . . 157
C Supporting information for Chapter 3 160
C.1 Discrete variables and the relation to the Gibbs-Duhem equation . . . . . 160
C.2 Accuracy of the approximations . . . . . . . . . . . . . . . . . . . . . . . . 162
C.3 ?G of solvent fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
D Supporting information for Chapter 4 165
D.1 Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
D.2 Mean-field model of treadmilling dissipation . . . . . . . . . . . . . . . . . 173
E Supporting information for Chapter 6 181
E.1 Weibull plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
E.2 Computing filament displacements . . . . . . . . . . . . . . . . . . . . . . 181
E.3 Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
E.4 Shannon entropy of tension distribution . . . . . . . . . . . . . . . . . . . 187
E.5 Machine learning implementation details . . . . . . . . . . . . . . . . . . . 188
vii
List of Figures
1.1 STORM image of the actin cytoskeleton . . . . . . . . . . . . . . . . . . . 5
1.2 Local cytoskeletal architectures . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Actomyosin network components . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Scales of computational models of the cytoskeleton . . . . . . . . . . . . . 10
2.1 Chemical species and reactions of actin filament . . . . . . . . . . . . . . . 26
2.2 Comparison of BC, QSSA, and CT models . . . . . . . . . . . . . . . . . . 30
2.3 Visualized trajectory of CT model . . . . . . . . . . . . . . . . . . . . . . 41
3.1 Schematic drawing of a RDME simulation volume . . . . . . . . . . . . . . 45
3.2 Comparison of ?G formulas for discrete case . . . . . . . . . . . . . . . . 64
3.3 Comparison of ?G formulas for continuous case . . . . . . . . . . . . . . . 66
4.1 Energy level diagram of MEDYAN simulation cycle . . . . . . . . . . . . . 74
4.2 Trajectories of various dissipation contributions . . . . . . . . . . . . . . . 78
4.3 Snapshot of actomyosin network . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4 Pie chart of different reactions? contributions to ?Gchem . . . . . . . . . . 81
4.5 Distributions of various dissipation contributions . . . . . . . . . . . . . . 82
4.6 A simple model to rationalize ?Gstress . . . . . . . . . . . . . . . . . . . . 84
4.7 Bar plots of ?G+ +chem, dissipated and ?Gmech, dissipated . . . . . . . . . . . . . 86
4.8 Dependency of efficiency on network contraction . . . . . . . . . . . . . . 88
4.9 Flow chart of free energy transduction in actomyosin networks . . . . . . . 91
5.1 Schematic depiction of graph mapping pipeline . . . . . . . . . . . . . . . 96
5.2 Visualization of five different actomyosin network sizes . . . . . . . . . . . 98
5.3 Plots of ?(t) for different conditions . . . . . . . . . . . . . . . . . . . . . . 100
5.4 Plots of ?b(t) for different conditions . . . . . . . . . . . . . . . . . . . . . 101
5.5 Plots of ?(t) for different conditions . . . . . . . . . . . . . . . . . . . . . . 103
5.6 Dissipation rate as a function of the system size . . . . . . . . . . . . . . . 105
5.7 Plots of myosin energy consumption for different conditions . . . . . . . . 106
5.8 Myosin energy consumption density across system sizes . . . . . . . . . . . 107
6.1 Snapshot of actomyosin network showing vibrational normal mode . . . . 114
6.2 Statistics of ?U(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3 Distributions of ?U across simulation parameters . . . . . . . . . . . . . . 118
6.4 Qualitative feature of cytoquakes . . . . . . . . . . . . . . . . . . . . . . . 120
6.5 Rank-size distributions of filament displacements . . . . . . . . . . . . . . 121
6.6 Non-Gaussian parameter across accessory protein concentrations . . . . . 123
6.7 Finite-size scaling of ?U . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
viii
6.8 Finite-size scaling of filament displacements . . . . . . . . . . . . . . . . . 127
6.9 Dependence of negative eigenvalues on FT . . . . . . . . . . . . . . . . . . 130
6.10 Level spacing statistics of vibrational modes . . . . . . . . . . . . . . . . . 132
6.11 Trends of metrics defined on Hessian eigen-decomposition . . . . . . . . . 133
6.12 Scatter plot of ?U and ?P . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.1 Mechanical energy potentials used in MEDYAN . . . . . . . . . . . . . . . 148
B.1 Comparison of MEDYAN and BC model for non-tip species . . . . . . . . 155
B.2 Comparison of MEDYAN and BC model for tip species . . . . . . . . . . . 156
B.3 Visualization of the BC model Jacobian . . . . . . . . . . . . . . . . . . . 158
D.1 Diagrams of reaction loops used to determine ?G0 . . . . . . . . . . . . . 167
D.2 Power-stroke cycle of a myosin head . . . . . . . . . . . . . . . . . . . . . 170
D.3 Trajectories of mean-field model of treadmilling dissipation . . . . . . . . 177
D.4 Integrated and steady state treadmilling dissipation . . . . . . . . . . . . . 178
D.5 Steady state concentrations across Nfil . . . . . . . . . . . . . . . . . . . . 179
D.6 Contributions of various reactions to steady state treadmilling dissipation 180
E.1 Weibull plots for |?U?| . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
E.2 Weibull plots for ?U+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
E.3 Schematic illustration of how filament displacements are computed . . . . 185
E.4 Results from machine learning model of cytoquake prediction . . . . . . . 189
E.5 PR and ROC curves for original data . . . . . . . . . . . . . . . . . . . . . 194
E.6 PR and ROC curves for shuffled data . . . . . . . . . . . . . . . . . . . . . 195
ix
List of Tables
B.1 Rate constants used in the BC model . . . . . . . . . . . . . . . . . . . . . 154
D.1 Chemistry simulation parameters used in Chapter 4 . . . . . . . . . . . . . 171
D.2 Remaining simulation parameters used in Chapters 4 . . . . . . . . . . . . 172
E.1 Simulation parameters used in Chapters 5 and 6 . . . . . . . . . . . . . . 186
x
List of Abbreviations
ADP Adenosine diphosphate
ATP Adenosine triphosphate
AUC Area under the curve
BC Brooks-Carlsson
CCDF Complementary cumulative distribution function
CT Constant tip
F-actin Polymerized actin
FPR False positive rate
G-actin Unpolymerized actin
GTP Guanosine triphosphate
in silico ?in silicon?
in vitro ?in the glass?
in vivo ?in the living?
MEDYAN Mechanochemical dynamics of active networks
ML Machine learning
MTP Main treadmilling pathway
NMIIA Non-muscle myosin II A
ODE Ordinary differential equation
PCA Principal component analysis
PCM Parallel cluster model
PDF Probability density function
Pi Inorganic phosphate
QSS Quasi-steady state
QSSA Quasi-steady state approximation
RDME Reaction-diffusion master equation
ROC Receiver operating characteristic
SOC Self-organized criticality
STORM Stochastic optical reconstruction microscopy
TPR True positive rate
1
Chapter 1
Introduction
The fundamental unit of life on earth is the cell, and inside every cell is a structural pro-
tein network called the cytoskeleton [1]. Like the musculoskeletal system in our bodies,
the cytoskeleton supports the body of the cell and is involved in dynamic physiological
processes that require it to exert coherent mechanical forces [2?4]. This allows the cell
to physically negotiate with its surroundings. These dynamic physiological processes,
which we give examples of below, result from the cytoskeleton consuming chemical fuel
to emergently and collectively self-organize into specific local architectures tailored to
the physiological task. Whereas some hard-shelled organisms get by with no internal
skeleton, there are no ?cellular invertebrates? that lack a cytoskeleton; it is present in
some form in the cells of all three domains of life (eukaryotes, bacteria, and archaea)
[5?7]. Understanding in detail how cytoskeletal systems self-organize to perform their
physiological functions is therefore an important piece of understanding life in general.
Much of the underlying chemistry and physics relevant to cytoskeletal networks have
already been mapped out, as experimental investigations have uncovered a wealth of
knowledge about the structural and mechanical properties of cytoskeletal proteins as
well as the chemical reactions in which those proteins participate [8?19]. As with other
complex systems, though, knowing about the components of cytoskeletal networks does
not fully explain their emergent behaviors, and it is a challenge to synthesize this ex-
perimental knowledge to address questions about the collective dynamics of many inter-
acting cytoskeletal components [20?22]. An important set of such questions pertains to
2
the thermodynamics of cytoskeletal self-organization, i.e. the changes in the network?s
chemical and mechanical energy and the exchange of heat with its surroundings [23, 24].
Thermodynamics is a pillar of classical physics and a natural framework through which
to analyze the engine-like transduction of chemical fuel into mechanical work that en-
ables cytoskeletal self-organization and its physiological functionality [25?27]. While
experiments can probe the energetics of processes involving individual cytoskeletal com-
ponents, it is nearly impossible to experimentally access the energetics of entire cytoskele-
tal networks [28?31].1 As a result, very little is known about system-wide free energy
transduction in cytoskeletal networks, a vital research gap that hinders our theoretical
understanding of cytoskeletal self-organization.
As with other complex systems, a powerful methodological approach to understand-
ing the emergent behaviors of cytoskeletal networks is computer simulation [37, 38].
Mesoscopic simulations allow one to reproduce the complicated collective behaviors ob-
served in living cytoskeletal networks while also giving access to ample amounts of infor-
mation generally not available through experimental measurements [39?42]. Although
the computational modeling of cytoskeletal dynamics is becoming a mature and well-
developed technique, it has not yet been effectively used to monitor cytoskeletal ther-
modynamics at the network level. Doing so requires combining in a consistent way the
biophysical models and protocols used in simulation with a thermodynamic treatment
of mesoscopic non-equilibrium systems.
In this thesis, we develop such a method to computationally quantify the thermody-
namics of cytoskeletal networks using the simulation platform MEDYAN (Mechanochem-
ical Dynamics of Active Networks), and apply it to investigate open problems in the
theory of cytoskeletal self-organization. In the introduction, we first describe in greater
detail the cytoskeleton, its interpretation as an active matter system, and options for
1There is an interesting partial exception to this limitation of experimental methods. Recent the-
oretical work has illustrated that the presence of non-equilibrium driving in complex systems can be
detected as non-zero probability currents in a low-dimensional projection of the system?s phase space
[32, 33]. This work was extended to not just detect but actually quantify the approximate entropy
production of cytoskeletal networks [34, 35]. This method underestimates the total entropy production
by only analyzing currents in the low-dimensional projection, but certain interesting trends can still be
observed using this approach [36]. We compare our explicit computational method to quantify entropy
production with this indirect experimental method in Chapter 4.
3
modeling it computationally. We then we give an overview of the thermodynamics of
mesoscopic systems and how it may help to explain active matter self-organization, and
at the end we outline the subsequent chapters of the thesis.
1.1 The cytoskeleton as biological active matter
1.1.1 Functions and components of the cytoskeleton
The cytoskeleton plays a central role in the life of a cell. It is an interconnected network of
polymers and accessory proteins which extends throughout the cell, adopting varied local
architectures in different regions (Figures 1.1 and 1.2).2 As the cell?s ?skeleton? it struc-
turally supports the cell membrane like a scaffold, and it further participates in numerous
processes which generally result in the production of useful mechanical work through
the consumption of an out-of-equilibrium supply of chemical fuel, typically adenosine
triphosphate (ATP) or guanosine triphosphate (GTP) [2, 43, 44]. Specific cellular pro-
cesses which rely on cytoskeletal activity include: endocytosis, in which extracellular
material is brought inside the cell via the cytoskeleton?s coordinated mechanical manip-
ulations of the cell membrane [45?47]; cell migration, in which the leading edge of the
cell is pushed forward through active polymerization of actin fibers while the rear of the
cell is pulled through the contraction of actin stress fibers (Figure 1.2) [48, 49]; cytoki-
nesis, in which the cytoskeleton cinches the cell along the equatorial plane during cell
division [50, 51]. These examples begin to illustrate the variety of function and structure
exhibited by the cytoskeleton, a variety which is perhaps nowhere more evident than in
neurons [52]. By selectively activating accessory proteins which interact with the poly-
mer network in various ways, the cell can finely control the behavior of the cytoskeletal
machinery. These accessory proteins fall into different classes based on their function
[43, 53, 54]. These classes include: capping proteins which inhibit polymerization of
filaments; cross-linking proteins which bind nearby filaments together; severing proteins
2Although a cytoskeleton is found in the cells of all domains of life, we will focus throughout this
thesis on eukaryotes. Thus the particular protein components we mention are specific to the eukaryotic
cytoskeleton.
4
which cut filaments into smaller pieces; sequestering proteins which remove monomers
from the available pool and slow down polymerization; nucleators which promote the
creation of new filaments. From in vivo experiments it is becoming clear which acces-
sory proteins are implicated in which physiological processes, although in many cases
the detailed mechanisms are not well understood [55?62].
Even though living cells utilize an array of accessory proteins to control cytoskeletal
behavior, a rich variety of interesting dynamical behaviors can be observed using just a
subset of these many components. For instance, reconstituted systems of only cytoskele-
tal filaments and molecular motors are sufficient to produce several qualitatively different
dynamic behaviors, including orientationally coherent flocking, rotating vortices, travel-
ing plane waves, moving nematic defects, and large-scale contractility [67?69]. In this
thesis we primarily focus on a parsimonious subset of cytoskeletal proteins constituting
a system called an actomyosin network (Figure 1.3.A). We will be concerned with three
proteins: actin filaments, which are ? 8 nm in diameter semi-flexible polar polymers of
varying length that act as mechanical scaffold (Figure 1.3.B) [8, 55]; non-muscle myosin
II A (NMIIA) minifilaments, which are ? 200 nm long aggregates of tens of myosin
heads that bind to actin filaments and walk in a directed manner toward the filament plus
ends (Figure 1.3.C) [19, 70]; ?-actinin, a cross-linking protein ? 35 nm long that stably
binds to adjacent filaments to mechanically connect them (Figure 1.3.D) [71, 72]. We will
elaborate on additional aspects of these molecules as they become relevant throughout
the thesis.
1.1.2 Active matter
The field of active matter reaches across many scales and concerns the collective behavior
of interacting agents that can individually convert energy into directed motion [73].
Active matter research is an exciting and recently established field that mixes condensed
matter physics, non-equilibrium statistical mechanics, chemistry, and other disciplines to
understand and even control the dynamics of driven collectives [74?78]. Active matter
systems include swarming flocks of birds [79?81], groups of robots programmed with
5
Figure 1.1: Images using stochastic optical reconstruction microscopy
(STORM) of two layers of the actin cytoskeleton in COS-7 fibroblast
cells. Using this technique, individual actin filaments can be resolved
in three-dimensional space, showing a remarkable heterogeneous network
structure. Actin filaments are colored according to their vertical position
in the cell. The left two panels show the ventral part of two cells (which
are shown on the top and bottom rows), and the central two panels show
the dorsal parts. The rightmost panels show a slice along the dotted lines
gray lines indicating more clearly the vertical actin positions. Part of the
cell imaged on the top was a bulged structure that was excluded from the
imaging range used. Image reprinted from Ref. 63 with permission from
Springer Nature.
6
Figure 1.2: A schematic illustration of the various local architectures
adopted by the cytoskeleton during cell migration. The cortical cytoskele-
ton, shown in gray, sits underneath and supports the cell membrane.
Stable bundles of actin filaments form stress fibers, shown in dark blue,
which produce a strong contractile force that pulls on focal adhesions at-
tached to the underlying substrate, moving the cell along its polarity axis.
Filopodial structures, shown in orange, are also formed from bundles of
filaments and protrude from the edge of a cell to detect changes in the
surrounding chemical environment, guiding cell migration. The lamel-
lipodium, shown in purple, is a highly branched network which protrudes
on the cell?s edge through directed polymerization to push it forward.
This combined process of pushing at the front and pulling up the rear
requires a coordination of cytoskeletal self-organization by chemical sig-
naling that differentially activates associated proteins in different regions
of the cell. Image reprinted from Ref. 64 with permission from Faculty
of 1000.
7
Figure 1.3: Composite image showing the primary components of ac-
tomyosin networks considered in this thesis. A) A snapshot from a ME-
DYAN simulation is shown illustrating how the different components co-
localize in space in a self-organized actomyosin network. The color label-
ing of each protein is indicated by the text color underneath the images.
B) An image of the polar, double helical actin filament is shown along with
the secondary protein structure of a G-actin monomer. The binding loca-
tion of adenosine nucleotides is shown as a small green molecule located
in the monomer?s minus-end cleft and as colored spheres in the filament
image. Signaling molecules such as profilin can bind to unpolymerized
monomers and control their polymerization kinetics. Image reprinted
from Ref. 65 with permission from IOP Publishing. C) The secondary
structure of a bare ?-actinin molecule is shown along with two snapshots
of its configuration bound to pairs of actin filaments. Signaling molecules
can bind to the ends of the dumbbell-like molecule, shown in red, which
activate its actin-binding functionality. Image reprinted from Ref. 66
with permission from Elsevier. D) Cartoons of the non-muscle myosin II
minifilament and dimers are shown along with the secondary structure of
an individual myosin head. The primary features of the myosin molecule
including its head, rotating lever arm, and coiled coil tail region are indi-
cated. Near the lever arm are regulatory regions which allow for selective
activation through various cell signaling pathways. Tens of myosin heads
aggregate to form a myosin minifilament, which has enhanced processiv-
ity compared to individual heads. Image reprinted from Ref. 65 with
permission from IOP Publishing.
8
simple rules [82, 83], bacterial colonies [84], and actomyosin networks, the subject of
our work [85, 86]. We highlight the fact that actomyosin networks are an active matter
system because it helps to generalize the work presented in this thesis beyond the specifics
of the cytoskeleton; a similar set of questions to those addressed in this thesis could be
asked about a broad class of other systems as well.
Like other biological active matter systems, actomyosin networks couple the con-
sumption of a chemical potential energy source to drive certain processes ?uphill? away
from their equilibrium [26]. This free energy transduction process can produce confor-
mational displacements against a tension, so that mechanical work is done as a result of
a spontaneous chemical reaction [87]. Free energy transduction occurs via two molecular
mechanisms in actomyosin networks: the power-stroke cycle of myosin motors [88?90]
and actin filament polymerization [91, 92].
The power-stroke cycle of myosin is a sequence of chemical states whose net effect is
the movement of the myosin head one binding site forward on the actin filament and the
hydrolysis of one solvated ATP molecule [70, 90, 93]. Thus, the thermodynamic driving
force of an ATP molecule?s chemical potential is coupled through the power-stroke cycle
to the generation of tension along the myosin minifilament. The force production by
myosin walking is transmitted throughout the cytoskeleton partly through the binding
of passive (i.e. not ATP-consuming) cross-linking elements, e.g. ?-actinin. Recent work
has shown that cross-linkers are in fact necessary to produce robust force generation and
contractility in actomyosin networks [94?97].
Actin polymerization involves the hydrolysis of ATP and subsequent release of inor-
ganic phosphate by polymerized monomers in the actin filament, converting ATP-bound
polymerized monomers to ADP-bound monomers [14]. ATP-bound monomers tend to
accumulate near the filament plus ends while ADP-bound monomers tend to accumu-
late near and depolymerize from the minus ends [98]. The chemical reaction network
describing this process has a steady state in which monomers polymerize at the plus end
at the same rate that they depolymerize from the minus end, causing the filament to
move forward without its length changing, a situation termed ?treadmilling? [99]. The
9
thermodynamic driving force of ATP hydrolysis in this active polymerization process
can be coupled via a ratcheting mechanism to push against a resistive load, such as the
cell membrane during lamellipodial protrusions in migrating cells [100?102]. In this way,
the free energy of ATP is transduced through actin polymerization into cellularly useful
mechanical work.
Another important (though not ATP consuming) facet of actomyosin network dy-
namics are the force-sensitive kinetic reaction rates controlling cross-linker and myosin
filament unbinding as well as myosin filament walking. At high tension, cross-linkers
will unbind more quickly (?slip-bond?) whereas motors will unbind and walk less quickly
(?catch-bond? and ?stalling?) [72, 87, 103]. Additionally, the polymerization rate of an
actin filament decreases as the load against which it polymerizes is increased [101]. In-
tuitively, these force-sensitive reactions can be understood as having binding energy
landscapes whose shapes change when a force is applied to the molecule, altering the
kinetic rates and either stabilizing or destabilizing the bound state. The mathematical
forms for these nonlinear dependencies on force are provided in Appendix A. These force-
sensitive reactions control the actomyosin network connectivity, which in turn determines
the ability of the network to globally distribute stress [104]. Thus, the force-sensitive
feedback introduces nonlinear coupling between the stress sustained by an actomyosin
network and the network?s ability to reorganize in response to that stress.
1.1.3 Computational modeling of the cytoskeleton
The complexity of cytoskeletal networks makes it difficult to study their emergent dy-
namics using only mathematical analysis, although certain phenomena such as nonlinear
rheological properties and glassy dynamics have been partly rationalized using mathe-
matical models [105?109]. Luckily, however, these emergent properties can be studied
in great detail using computer simulations. A key consideration in choosing a simu-
lation method for studying cytoskeletal dynamics is identifying the characteristic time
and length scales of the behaviors one is interested in. While highly resolved atomistic
simulations can distinguish, for example, the mechanism by which tensile forces alter an
10
Figure 1.4: The computational modeling the actin cytoskeleton can be
done at several length and time scales, and the choice of a model depends
on the types of behavior one is interested to study. All-atom models
of actin monomers and short filament segments can be coarse-grained,
assigning one bead to each domain of the actin monomer. Another coarse-
graining step can be taken to assign one bead to each monomer. Further
steps eventually yield a model for many interacting filaments, allowing
collective network dynamics to be studied. Using the MEDYAN model
one can simulate lengths of several micrometers and times of thousands
of seconds. Image reprinted from Ref. 112 with permission from Springer
Nature.
actin monomer?s catalytic activity, such simulations could not describe many interact-
ing filaments on a time scale where interesting collective behaviors are known to occur
[110, 111]. A hierarchy of simulation methods at different scales is therefore needed to
run the gamut of multiscale cytoskeletal dynamics, as illustrated in Figure 1.4. In this
thesis we are interested in the emergent dynamics of many cytoskeletal proteins occurring
at length scales of up to hundreds of micrometers and time scales of thousands of sec-
onds, the scales at which physiological cytoskeletal processes occur. We therefore require
a coarse-grained simulation platform for studying semi-flexible polymer networks.
Several such platforms for studying network-level dynamics are available, including
AFiNeS, CytoSim, the model of Kim and coworkers, and MEDYAN [39?42]. Each of
11
these allows simulating fascinating emergent cytoskeletal phenomena while striving to
preserve realistic microscopic physics. These models use similar spatial resolutions to
represent an actin filament, assigning ? 10 ? 100 actin monomers to a single discrete
element. Additionally, the mechanical energy of the filaments is modeled similarly in
these platforms, accounting for the elastic stretching of and bending between the discrete
elements. While the mechanical potentials are comparable, the ways that these platforms
treat the chemical reaction dynamics are quite different. The chemistry model used in
MEDYAN is the most detailed, accounting for the spatial distribution and local reaction
propensities of all chemical species in the system rather than assuming a globally well-
mixed reservoir. To study the thermodynamics of active cytoskeletal networks we will
need to track the chemical energetics at high resolution, for which MEDYAN is the best
suited.3 Indeed, we will find in Chapter 4 that a large contribution to the system?s free
energy comes the diffusive flux of solvated species, which could not be accounted for in
other platforms. In Appendix A, we provide an overview of the MEDYAN simulation
platform for reference.
1.2 Non-equilibrium thermodynamics of active matter
1.2.1 Overview of non-equilibrium theories
As with our choice of a computational modeling approach, our choice of a thermody-
namic description for cytoskeletal self-organization should reflect the appropriate time
3One technical drawback of using MEDYAN to study thermodynamics is that the mechanical dynam-
ics do not obey the principle of microscopic reversibility, which roughly says that if the probability of a
transition in one direction (A ? B) is non-zero then the probability of the reverse transition (B ? A)
should also be non-zero [113, 114]. In MEDYAN, the chemical dynamics are stochastic, but for a given
chemical state and network configuration the new configuration is obtained deterministically through
mechanical equilibration. In other words, the mechanical dynamics occur at zero temperature, and
reverse transition probabilities are zero when their forward counterparts are non-zero. This approach
has greater computational efficiency than, for example, numerically integrating a stiff Langevin equa-
tion at small time steps, and it could in principle be straightforwardly amended by including thermal
fluctuations in the network?s configuration [40]. The methods for tracking thermodynamic quantities
presented in this thesis should after some minor modifications still apply for such an amended model,
and we would expect only small quantitative corrections to the results presented here. This is because
the active fluctuations from ATP-consuming processes which MEDYAN accounts for should dominate
over the thermal fluctuations which MEDYAN neglects. This issue is nonetheless an avenue for future
work.
12
and length scales, leveraging where possible the advantages of coarse-graining. Sev-
eral non-equilibrium theories useful at different scales have been developed, and we first
review these theories before discussing which is best suited for tracking cytoskeletal
thermodynamics using the MEDYAN platform.
Ignoring the quantum level (whose thermodynamic description encompasses its own
growing field of research; see Refs. 115?117), the most fine-grained, microscopic descrip-
tion of a non-equilibrium system coupled to a reservoir tracks the probability density
over phase space of all of the system and reservoir?s classical degrees of freedom, which
obey Hamiltonian dynamics [118?120]. The Liouville equation describes the probability
density?s time evolution and requires knowing the explicit form for the Hamiltonian.
Thermodynamic quantities, such as the work done by the varying an external control
parameter and the system and reservoir ensemble entropies, can be defined as functions
of their degrees of freedom and their occupied phase space volume (although some tech-
nical details, such as whether the system is ergodic and mixing, need to be considered;
see Ref. 121).
Assuming weak coupling between the system and reservoir can facilitate calculations
within the Liouville framework, but for many systems of interest this interaction and
the many degrees of freedom needed to track for the reservoir hopelessly complicate
this microscopic approach. This difficulty motivates a coarse-graining step in which the
deterministic Hamiltonian dynamics are replaced by stochastic dynamics reflecting the
statistical properties of the system-reservoir interaction. In this step the interaction
with the reservoir is renormalized into an effective diffusion constant that enters into the
either the Fokker-Planck equation describing the time evolution of the system?s proba-
bility density over a continuous state space, or the Langevin equation describing a single
stochastic system trajectory [122, 123]. One could also use a jump process to model the
stochastic evolution of the system?s microstates interacting with the implicit reservoir.
The reservoir?s dynamics are assumed to be sufficiently fast for it to instantaneously be
at equilibrium for a given system microstate, allowing the system?s dynamics to be as-
sumed Markovian. For a single fluctuating trajectory through the system?s microstates,
13
the work done on the system by an external driving protocol or non-conservative force as
well as the heat exchanged with the implicit reservoir can obtained using functions of the
system?s degrees of freedom [124, 125]. The heat exchanged with a constant temperature
reservoir can be unambiguously associated with the change in the reservoir entropy, but
the change in the entropy of the system itself is less clearly defined in the Langevin
description since the entropy is typically interpreted as a property of an ensemble, not
one trajectory. This issue was resolved through a consistent definition of a fluctuating
system entropy which is equal to the usual ensemble entropy when averaged over the
time-dependent probability density obtained from the Fokker-Planck solution [126, 127].
These definitions for the thermodynamic quantities of a Markovian system interacting
with an implicit reservoir are the basis for the field of ?stochastic thermodynamics? [128].
Some physical processes can be well-approximated as transitions through a discrete
set of locally equilibrated ?mesostates? rather than through the system?s full set of mi-
crostates. The transitions are still stochastic, with rates that can be derived from the
underlying Langevin or jump dynamics [36, 129]. The energy landscapes for which this
approximation is realistic can be thought of as ?hilly,? characterized by local minima
separated by relatively high barriers. For such systems, the microstate dynamics can
be replaced by a jump process through the discrete mesostate space in another coarse-
graining step [130, 131]. The hilly landscapes allowing for the partitioning of state space
into a discrete set of local minima typically also allow for a time scale separation between
the long time spent occupying a given minimum and the short time spent transitioning
over the barriers between minima [129]. If the relaxation timescale within the minimum
is also short, then the system loses memory of its previous location quickly enough to
assume that the jump dynamics are Markovian. Additionally, this time scale separation
can allow one to assume that when the system occupies a given minimum the system
explores a locally equilibrated ensemble of configurations before transitioning out, so
that a well-defined free energy may be assigned to the mesostate [132]. This notion of a
locally equilibrated ensemble of system microstates associated with a given energy mini-
mum is a key part of our thermodynamic description of the cytoskeleton, a system whose
14
dynamics are driven by slow chemical reactions and fast solvent relaxation with a hilly,
time scale separated energy surface as just described [109, 133]. We elaborate on this
time scale separation in Chapter 3. The state space used in MEDYAN is partly discrete
(to describe the copy numbers and locations of diffusing chemical species) and partly
continuous (to describe the positions of the network?s mechanical elements), and can be
viewed as mesoscopic in scale. Thus, our computational thermodynamic description of
the cytoskeleton will be at this mesoscopic level, in which the system moves stochasti-
cally through a series of locally equilibrated states to which well-defined thermodynamic
free energies may be assigned [134].
There is a further set of coarse-graining steps which may be taken to give a macro-
scopic description of the system. If the system is large enough that fluctuations about
the mean behavior are negligible, then one can replace the stochastic dynamics with
another deterministic dynamics [25, 135]. Unlike the Hamiltonian dynamics, however,
which gives the deterministic evolution of the microscopic degrees of freedom, these
give the deterministic evolution of macroscopic thermodynamic forces and fluxes such as
chemical concentrations, chemical potential, temperature, and heat. Much research has
been dedicated to the macroscopic regime in which the thermodynamic driving forces are
small, allowing one to obtain linear transport equations such as Fick?s law of diffusion
and Fourier?s law of heat flow, as well as Onsager?s reciprocity relations describing how
different types of transport coefficients are interdependent [25].
Finally we mention that, in general, a system may be driven out of thermodynamic
equilibrium by two different methods [136]. In the first, the dynamics of the system
obey the principle of detailed balance but the system is externally driven through the
time-dependent variation of some externally manipulated control parameter (such as the
position of a piston wall). By definition of the principle of detailed balance, if the control
parameter came to rest then the system would eventually settle to equilibrium with no
net probability currents in its phase space. In the second method, there is no externally
manipulated control parameter to drive the system, but there is an externally maintained
non-zero thermodynamic force on the system (such as out-of-equilibrium chemostatted
15
concentrations of chemical reactants and products), so that the dynamics of the system
do not obey detailed balance. This produces a steady state with non-zero net probability
currents in phase space. The second method applies to actomyosin networks, which are
driven out-of-equilibrium due to a large chemical potential difference between ATP and
its hydrolysis products. Throughout this thesis this chemical potential will be fixed for a
given simulated network due to an assumption of chemostatted concentrations of ATP,
ADP, and Pi.
1.2.2 Using non-equilibrium thermodynamics to explain active matter
self-organization
As described above, a defining feature of active matter systems is the consumption and
dissipation of free energy by the system?s individual constituents, such as myosin motors
and actin monomers. Because they are driven, active matter systems can be analyzed
using non-equilibrium thermodynamics. Having described the different scales at which
the thermodynamics of a non-equilibrium system can be analyzed, we next review several
proposed principles that offer ways to explain the self-organization of a non-equilibrium
system in terms of its thermodynamic quantities.
In the 1960?s, Prigogine proposed that non-equilibrium conditions can allow for the
spontaneous emergence of ordered (i.e. seemingly low entropy) spatiotemporal patterns,
which he called ?dissipative structures.? The organization of these structures facilitates
the eventual relaxation to equilibrium, but they can persist at steady state if the non-
equilibrium boundary conditions are held fixed [23, 137]. In Prigogine?s description,
dissipative structures represent a broken symmetry of the system. They emerge at the
point when certain boundary condition parameters pass a threshold that causes the
equilibrium organization of the system to become unstable with respect to thermal fluc-
tuations. A classic example is the nonlinear, auto-catalytic chemical reaction network
called the ?Brusselator? [138]. As the chemostatted concentrations of the input reactants
are increased, a Hopf bifurcation occurs and the steady state concentrations of the dy-
namical species approach a limit cycle of oscillating concentrations, which is understood
16
a dissipative structure that breaks time translation symmetry. We can view biological
organisms and their material parts, such as actomyosin networks, as complex dissipative
structures operating within the boundary conditions of their environment. This is an
interesting interpretation that places living organisms in the same category as hurricanes
and circulating convective fluid patterns called Rayleigh-B?rnard cells, but it is does not
have much predictive power and we mention it mainly out of completeness [139, 140].
Among the most provocative developments in macroscopic non-equilibrium ther-
modynamics are the various extremal principles characterizing non-equilibrium steady
states. These principles claim that the manifested steady states of a non-equilibrium sys-
tem are those for which the entropy production rate is optimized. In these arguments the
entropy production rate is formulated as a functional of time-varying yet deterministic
thermodynamic forces and fluxes, and whether it is minimized or maximized depends on
the boundary conditions and how far away from equilibrium (in the linear or nonlinear
regime) the system is. Examples of such principles have been put forward by Helmholtz,
Onsager, Prigogine, Gyamarti, and Ziegler [135, 141?144]. For instance, Prigogine?s
minimum entropy production principle says that the steady state of a non-equilibrium
system in the linear response regime corresponds to a minimization of the entropy pro-
duction over all choices of the unconstrained thermodynamic forces and fluxes. While
seeming to offer a powerful unifying approach to understanding non-equilibrium systems,
the applicability and accuracy of these principles are heavily debated in the literature,
with some researchers pointing to counter-examples as evidence that no such principle
could apply generally to all systems [145?149]. Due to these complications it is not clear
a priori whether we should expect actomyosin networks to obey an extremal principle.
Rather than delve into the technical details of these theories, we will take an empirical
approach to assess whether an extremal principle applies in our case.
A recent result in non-equilibrium statistical mechanics allows one to express the
irreversibility of transitions between the macrostates of fluctuating systems (defined as
arbitrary groups of microstates) in terms of the entropy production accompanying the
17
set of microstate trajectories which connect the two macrostates. In principle this al-
lows one to make predictions about the system?s likely macrostate transitions in terms
of its dissipation. The basis for these results is the Crooks fluctuation theorem, which
applies to a system whose dynamics obey the principle of detailed balanced but are
non-autonomous due to the variation of some work parameter by some external agent
[114, 150, 151]. In 2013, England derived an extension of this principle by averaging
Crooks? relation over all paths connecting some beginning and final microstates, and
then averaging the resulting relation over all microstates belonging to some beginning
and ending macrostates. The implication of this result, which England has termed ?dis-
sipative adaptation?, is that the irreversibility of macrostate transitions (rather than just
the microstate trajectories used in Crooks? relation) increases with the dissipated work
accompanying typical system trajectories [152?155]. The fact that high work absorption
and dissipation should be associated with high irreversibility is expected, since the work
absorption facilitates the traversal of high kinetic barriers, and if the absorbed work is
dissipated then returning over the barrier is unlikely, which makes the barrier traversal
more irreversible. Actomyosin networks and their architectures are most conveniently de-
scribed in terms of macrostates, i.e. according to some coarse property such as its radius
of gyration, rather than microstates, i.e. the specific arrangement of each polymer in the
network. This result of England?s is then at a coarse enough level for analyzing transi-
tions between different types of actomyosin architectures whose probabilities of emerging
we would like to understand. However, the result of England?s is intractable to exactly
test for non-?toy model? systems, as it requires knowing both the forward and reverse
probabilities of a macrostate transition. The reverse transitions will be exceedingly rare
and difficult to measure the probability of. This result also makes no claims about the
steady state behavior of a system, instead claiming that the transient dynamics have an
irreversibility related to the dissipated work. Due to these issues, it is not clear whether
the dissipative adaptation principle is useful to understand cytoskeletal self-organization.
A slightly more operative principle, called ?low rattling,? was proposed by Chvykov
18
and England in 2018 [83, 156]. It does not actually make any reference to the thermo-
dynamic quantities of a system (such as its entropy production), but focuses instead on
the system?s kinetic rates to make predictions about the typical steady sate behavior.
The intuition behind this principle is that a system will tend to end up in a place it
that it is slow to leave from, i.e. in a state where the kinetic rates leading the system
away are small. While non-equilibrium systems can be contrived which do not obey this
principle (because for instance the states with small exit rates also have small entrance
rates), it has been shown that a ?typical? system obeys the principle, which can thus be
interpreted as a tendency rather than a definite law [156]. In actomyosin networks, the
force-sensitive reaction rates (slip and catch-bonds and motor stalling) set the stage for
kinetics that depend on the system?s configuration, and the low rattling principle may
well apply in these systems.
Finally, we consider the non-equilibrium dynamical phenomenon of avalanches, which
can appear as heavy tails in the distributions of dissipation event sizes. Avalanche-like
processes occur in many systems throughout the natural sciences when there is a com-
bination of slow driving on the system and fast, configuration-dependent dissipation
[157]. For certain systems (and evidently in the earth) the event sizes tend to power-
law distributions for large systems, a phenomenon called self-organized criticality (SOC)
[158?162]. The name comes from the fact that power-law distributions also characterize
the fluctuations of order parameters at the critical points of equilibrium statistical sys-
tems, i.e. when the system?s tuning parameters are at particular critical values [163]. In
the non-equilibrium case, though, no parameter fine-tuning is needed to bring about the
critical point. As long as the system is slowly driven, rapidly dissipates energy to the
environment, and conservatively transfers energy internally, then the critical point is a
dynamical attractor towards which the system spontaneously self-organizes [164]. SOC
can be observed in many disparate systems and may apply to cytoskeletal networks if
the conditions of cytoskeletal driving and dissipation are allowing [165?167]. Even if not
technically SOC, cytoskeletal networks may exhibit avalanche-like dynamics in which a
19
cascading dissipation of stored energy results from a comparatively slower energy accu-
mulation process, leading to heavy-tailed (if not power-law) distributions of event sizes.
However, the connection between cooperative, avalanche-like non-equilibrium dynamics
and cytoskeletal self-organization remains to be established. There is strong experimental
motivation for understanding this connection, as in vivo studies have discovered heavy-
tailed cytoskeletal displacement statistics, and these may explain the unpredictability of
measured cellular traction forces [168?170].
1.3 Outline of chapters
Our aim in this thesis is to develop a computational method for quantifying the non-
equilibrium thermodynamics of the cytoskeleton and apply it to study open questions
in the theory of cytoskeletal self-organization. Some of these open questions will be
about determining which of the above principles for non-equilibrium systems apply to
cytoskeletal networks. The chapters cover topics of both method development and appli-
cation and are presented in a logically dependent order, such that earlier chapters should
help to understand later chapters. We next outline these chapters and how they relate
to each other.
In Chapter 2 we analyze the mean-field chemical dynamics of actin filament polymer-
ization, aging, and depolymerization. This chemical system is active due to the hydrolysis
of ATP by the actin subunits and has a non-equilibrium steady state with non-zero net
polymerization and depolymerization rates. We present analytical work in which the
fast and slow time scales of the chemical dynamics are identified and their separation
leveraged to simplify the model. This allows new insights into the transient and steady
state behavior of the system. The model of the filament (de)polymerization chemistry in
this chapter is later used in more complete simulations of cytoskeletal self-organization
based on the MEDYAN platform.
In Chapter 3 we examine how to monitor the Gibbs free energy of a system of
solvated chemical species whose dynamics are modeled using the reaction diffusion master
20
equation (RDME). In this approach the copy numbers of the chemical species are discrete
quantities and can be comparable in size to the stoichiometric coefficients of the reactions
in which the species participate. We find that in this regime certain textbook expressions
for the Gibbs free energy change of a reaction are highly inaccurate, requiring us to
derive new expressions from first principles. The derivations and formulas presented in
this chapter are crucial for tracking cytoskeletal thermodynamics in MEDYAN, which
uses the RDME to propagate the chemical dynamics.
Chapter 4 is the central methodological component of this thesis. In this chapter
the results of Chapters 2 and 3 as well as additional new modeling considerations are
synthesized within the MEDYAN platform to develop a method for tracking the ther-
modynamics of cytoskeletal self-organization. The method complies with the built-in
MEDYAN simulation protocol and accounts for chemical energy consumption as well as
the accumulation and dissipation of mechanical energy. We discuss how the method,
which could be used for other biological active matter systems, is parameterized specif-
ically for actomyosin networks. The method is then applied in this chapter to study
the thermodynamics of actin filament treadmilling, which we compare to the mean-field
model predictions of Chapter 2, as well as the efficiency of the transduction of chemical
to mechanical energy by motors, which depends on accessory protein concentrations.
We generally find dissipation rates to reach a minimum at steady state, in apparent
agreement with the minimum entropy production principle mentioned above.
In Chapter 5 we explore several interrelated aspects of cytoskeletal structure and
dissipation: the effect of the system size, the role of force-sensitive motor stalling, and the
description of cytoskeletal architectures using a graph representation. Using community-
detection routines on the graph representation allows for a systematic way to identify
higher-order topological features (i.e. filament bundles) in cytoskeletal networks. The
system size scaling of the graph metrics and dissipation rate are investigated. We also
show that motor stalling can significantly decrease the dissipation rate during the course
of cytoskeletal self-organization, which we suggest agrees with the low rattling principle
discussed above.
21
Chapter 6 is dedicated to exploring anomalously large events in cytoskeletal dis-
placements, which have been termed ?cytoquakes? by experimenters. Understanding
these cooperative mechanical relaxation events and their underlying cause is an open
problem, which may have significant biological importance due to a plausible role that
cytoquakes might play in enhancing cellular information processing. Our free energy
tracking method is applied in this chapter together with several additional new analysis
techniques to characterize cytoquake phenomenology and its causes. One new technique
is normal mode decomposition of the network, whose trends across different experimen-
tal conditions we tangentially explore. Our main results indicate that cytoquakes can
indeed be explained as an avalanche-like process of slow energy accumulation punctu-
ated by occasional large, cascading energy dissipation events, although the system is not
technically SOC.
Finally, in Chapter 7 we summarize the work presented in this thesis and suggest
future research directions. Appendix A presents a self-contained description of the ME-
DYAN simulation platform for the reader?s reference, and Appendices B-E have support-
ing information for Chapters 2, 3, 4, and 6.
22
Chapter 2
Low-dimensional manifold of actin
polymerization dynamics
This chapter and Appendix B are adapted from: Floyd, C., Jarzynski, C., & Papoian,
G. (2017). Low-dimensional manifold of actin polymerization dynamics. New Journal
of Physics, 19(12), 125012.
2.1 Introduction
Actin filaments are an integral part of the cytoskeleton of eukaryotes and are involved in
functions such as controlling cell shape, cell motility, organelle redistribution, and me-
chanical coupling with the cellular environment. These filaments are formed of globular
subunits which polymerize in a non-equilibrium process that in vivo is modulated by an
array of accessory proteins. They are helical and polar, with distinct plus (?barbed?)
and minus (?pointed?) ends at which subunits have different rates of association and
dissocation [1]. Hydrolysis of cellular ATP leads to filament ?treadmilling?, in which
there are equal and opposite rates of polymerization at the barbed and pointed ends
of the filaments, which drives the polymerization process away from equilibrium and
allows actin networks to be responsive to cellular signals on relatively fast timescales
[171?173]. Each actin subunit molecule is bound to a nucleotide, which can be in one of
several hydrolysis states: adenosine triphophate (ATP), adenosine diphosphate (ADP),
or an intermediate state ADP-Pi, in which ADP is still bound to a hydrolyzed inorganic
23
phosphate molecule. Release of inorganic phosphate by ADP-Pi converts it to ADP
[174]. The hydrolysis state of the bound nucleotide has dramatic effects on the kinetic
polymerization and depolymerization rate constants of the globular subunit [11]. In ad-
dition, these hydrolysis states affect the binding affinity of accessory proteins as well as
structural properties such as filament persistence length [175, 176]. Thus it is of interest
to be able to predict the hydrolysis state of the nucleotide bound to each actin subunit
in a filament, or at least the fraction of actin subunits bound to nucleotides in a certain
hydrolysis state.
Over several decades a variety of models describing actin polymerization dynamics
have been put forward, and these models have evolved alongside the growth of exper-
imental knowledge about the nature of actin. Some early models tracked the number
of filaments with a certain degree of polymerization under different assumptions about
the filament polarity, geometry, and the cooperativity of polymerization, among other
factors [14, 16, 27, 177?179]. Polymerization and depolymerization rate constants for
ATP and ADP-bound actin were measured for the first time in 1986 [11]. A subset of
more recent models have investigated aspects such as the effects of accessory proteins
on actin polymerization via tracking the time-varying concentration of actin subunits
distinguished by their polymerization state and by the hydrolysis state of the nucleotide
they are bound to. A variable is assigned to the concentration of each species and com-
plexes between certain species, and equations of motion in terms of mean-field kinetic
rate constants are written for each. In this context, ?mean-field? refers to the assumption
that the the solution of actin subunits and accessory proteins is homogeneous and obeys
mass-action kinetics. The resulting coupled ordinary differential equations (ODEs) are
solved numerically, and the effects of varying parameters such as reservoir ATP/ADP
disequilibrium, total filament concentration, fraction of capped plus ends, free actin
concentration, and profilin concentration are investigated [180, 181]. One point of con-
tention is whether transitions between hydrolysis states of polymerized subunits occurs
in a random fashion, in which hydrolysis states of a subunit?s neighbors do not affect the
hydrolysis rate of that subunit, or in a vectorial fashion, in which an ATP-bound subunit
24
will only hydrolyze ATP if its neighbor towards the minus end is ADP-Pi bound, leading
to a contiguous ATP-bound cap at the plus end. Recent models suggest that the truth
is in the middle, such that coupling exists in ATP cleavage rates between neighboring
polymerized subunits, but not such that the process is truly vectorial [92, 182]. Most
mean-field models make the assumption of random ATP hydrolysis for simplicity.
An important disadvantage of such mean-field models aimed at resolving the roles
of accessory proteins is that their level of detail inhibits analytical solutions to the time
courses and, hence, obscures deeper insights into dynamical behaviors of these systems.
While this approach has successfully allowed modelers to, for example, rule out certain
mechanisms of profilin?s action on critical concentrations [98], one might ask what is the
simplest such model that reproduces time courses from more detailed models. This is
the aim of the present chapter. The model reduction here is based on a 2009 model
by Brooks and Carlsson (BC) [183], which presents a system of differential equations
that admits only numerical solution but does not include extra detail by accounting for
accessory proteins. It is useful for predicting the process of polymerization when a pool
of subunits are added to an initial concentration of seed filaments, and is sufficiently
simple to be incorporated into larger-scale cellular models without too much additional
computational cost.
In this chapter, we report on two successive reduction schemes of the 11-dimensional
BC model: a quasi-steady-state approximation that leverages fast dynamics of the fila-
ment tips, leading to a 5-dimensional system of ODEs, and a subsequent linearization
approximation. The latter equations admits an analytical solution whose implications
we investigate, revealing interesting features of actin polymerization process projected on
the slow dynamical manifold. Our analytical model reduction approaches show excellent
agreement with the results obtained from stochastic simulations of the full BC model
and also when compared with diffusion mapping analyses of stochastic trajectories.
25
2.2 Methods
2.2.1 Brooks-Carlsson model
The BC model of actin polymerization is an 11-dimensional system of ODE?s tracking
the concentrations of non-tip actin subunits in different states as well as the concentra-
tions of filament tip subunits in different states [183]. It is assumed that the number
concentration of filaments N remains constant, implying an absence of filament nucle-
ation, splitting, or joining. Additionally, the total concentration of actin subunits M is
assumed to remain constant, such that actin subunits are not created or destroyed in
any reaction. Since there are typically many actin subunits belonging to a given actin
filament, we have N M . All species are assumed to be well-mixed and in large enough
quantities to be treated effectively via a mean-field description. In other words, the size
of the stochastic fluctuations is negligible compared to the concentrations of the species.
Unpolymerized (globular) actin subunits are referred to as G-actin, while polymerized
(filamentous) actin subunits are referred to as F-actin. Actin filaments are helical, but
they are more easily modeled as linear chains, which is a realistic approximation if one
assumes that the reaction propensities of a given F-actin subunit are determined only
by the state of the nucleotide bound by that subunit and not by the subunit?s neigh-
bors. Such a chain is displayed in Figure 2.1, along with some of the reactions allowing
interconversion between subunit types. The variables representing the concentrations of
these actin species are superscripted by the hydrolysis state of the bound nucleotide (for
what follows we refer more simply to a subunit being in a certain hydrolysis state as
opposed to the nucleotide attached to a subunit as being in that state). The hydrolysis
states are ATP, ADP-Pi, and ADP, denoted T, Pi, and D, respectively. The tip subunits
are denoted T and are further subscripted according to which tip they are on. Thus, for
example, the concentration of tip subunits at the plus end bound to ADP-Pi is denoted
TPi+ . Because inorganic phosphate rapidly dissociates from G-actin, GPi is taken to be
0 and is not tracked. With the 3 hydrolysis states of each of the 2 tips, the 3 states of
the F-actin and the 2 states of G-actin, the number of tracked variables is 11.
26
Figure 2.1: A linear actin filament, with subunits colored according to
hydrolysis state. Random, as opposed to vectorial, hydrolysis is assumed
here. Some of the reactions are slightly misleading as drawn: for example
a hydrolysis reaction converting FT to FPi would happen at a single lo-
cation in the filament, i.e. the subunit would not change into its neighbor
as shown here. Also, the polymerization of GT onto the minus end would
convert TD T? into T? , and a similar statement applies to GD polymerizing
to the plus end. The depolymerization of TPi? is not shown.
The different subunit types interconvert through chemical reactions. These reactions
can be classified as polymerization/depolymerization reactions which change G actin to F
actin and vice versa, or as reactions in which the hydrolysis state of the subunit changes.
The evolution of the concentrations of a given tip subunit hydrolysis states in principle
depends on the hydrolysis state of the subunit adjacent to the tip, which itself depends
on the hydrolysis state of the next subunit in the filament, and so on. Every subunit in
the filament then requires explicit tracking, causing the dimensionality of the model to
be roughly equal to the degree of polymerization of a filament, which typically contains
hundreds of subunits. The major accomplishment of the BC model is to truncate this
set of recursion equations by assuming that the hydrolysis state of the subunit adjacent
to the tip depends only on the hydrolysis state of the tip subunit. Using the results
of stochastic simulations of a more complete model, they write empirical equations to
capture these relationships, and in doing so they close off an 11-dimensional subset of the
original hundreds of equations. They find close agreement between their truncated model
and the full stochastic simulation over a wide and realistic range of parameters. The
equations of motion for the 11 variables in the BC model can be written as a nonlinear
27
system of ODE?s:
x? = f(x) (2.1)
where x is a vector of the concentrations of the 11 species, and f is a nonlinear vector-
valued function of x. This system of equations must be solved numerically, but the
results match well with simulations that accurately model experimental data [184]. The
steady state vector xss satisfying f(xss) = 0 is unique for given values of N and M and
under the condition that all concentrations be real and non-negative, and it is attracting.
We give more details of the BC model in Appendix B, where we list the 11 ODE?s.
A separation of timescales exists between the dynamics of the non-tip subunit
states and those of tip subunit states: the latter evolve much more rapidly than
the former. Thus we partition the vector x into slow and fast variables: xs ?
(GT , GD, F T , FPi, FD)?, x ? (T T Pi D T Pi D ?f + , T+ , T+ , T? , T? , T? ) , where the sub-
scripts ?s? and ?f? refer to ?slow? and ?fast?. Terms of comparable magnitude appear in
the equations of motion for both xf and xs, but the elements in xf are typically much
smaller than those in xs because N  M , and so xf reaches equilibrium more rapidly
since the distance to equilibrium is not as large as it is for xs. To see the separation
timescales concretely, we refer the reader to the end of Section 2.3.1, where we show the
spectrum of the Jacobian matrix of the BC model evaluated at steady state. With the
new collective variables xs and xf, Equation 2.1 can be usefully rewritten as follows:
x?s = Axs + Bxf (2.2)
x?f = h(xs, xf) (2.3)
where A and B are matrices whose off-diagonal elements are combinations of kinetic
rate constants and whose columns sum to zero due to conservation of actin, and h is
a nonlinear vector-valued function containing terms that are up to cubic products of
variables.
The BC model provides a computationally accessible model of the dynamics of actin
polymerization, however we might ask for several other features in such a model. We ask
28
that it: i) be low-dimensional, ii) capture the interesting and important timescales, and
iii) be exactly solvable. To meet these goals, we make the decision to track only the vector
xs explicitly. If we were to track the concentration of the tip subunit states, i.e. the
elements in xf, we would automatically increase the dimensionality of the model, and we
will discuss reasons why we may assume that the tip subunits are evolving in such a way
that keeping track of them explicitly is unnecessary. We make two approximations for
how to treat xf in Equation 2.2, first utilizing the fact that a separation of timescale exists
between the dynamics of xf and xs, and then utilizing the fact that |Bxf|  |Axs| except
when the system is near its steady state, at which point these terms have comparable
magnitudes. This second approximation follows since xf contains terms up to O(N) and
xs contains terms up to O(M).
It is also possible to demonstrate that a low-dimensional description of the slow dy-
namics is a valid approximation through the use of diffusion mapping on a stochastically
generated data set based on the BC model. This analysis is described in the supporting
information of Ref. 99.
2.2.2 Quasi-steady state approximation
The quasi-steady state approximation (QSSA) relies on a separation of timescales be-
tween fast and slow variables. This separation allows one to assume that the fast variables
xf are always in equilibrium with respect to the slow variables xs, and therefore that the
values of the slow variables determine the values of the fast variables at any moment. We
can imagine that xf is effectively being ?dragged around? by the values of the elements
in xs. So, we can solve for functions x
eq
f (xs) relating the quasi-equilibrated fast variables
in terms of the slow variables by imagining holding xs fixed and finding the equilibrium
values of xf. This amounts to the condition h(x
eq
f ; xs) = 0. The functions x
eq
f (xs)
are then substituted in the equations of motion for the slow variables giving the closed
system of equations
x? = Ax + Bxeqs s f (xs). (2.4)
29
This subsystem is lower-dimensional, though it is nonlinear since xeqf (xs) is nonlinear,
and it describes the evolution of the system on the slow timescales.
In the BC model, the condition h(xeqf ; xs) = 0 implies the following algebraic
systems of equations (see Appendix B):
dT T? = 0
dt
dTPi? = 0 (2.5)
dt
dTD? = 0.
dt
Only four of these six equations are linearly independent due to the conservation of
number of plus and minus end filament tips, so we use the following supplementary
equations to find a solution of the combined systems of equations:
T T Pi D? + T? + T? = N. (2.6)
The system of Equations 2.5, 2.6 can be solved numerically resulting in tabulated func-
tions of the forms T T T D Pi T D D T D? (G , G ), T? (G , G ), and T? (G , G ). These functions
do not depend on F T , FPi, and FD because these variables do not enter into the the
function h. Subsequently, the nonlinear (slow) system described by Equation 2.4 is nu-
merically integrated. Figure 2.2 displays a comparison of the QSSA approximation to
the original 11-dimensional BC model.
This approximation succeeds in reducing the dimensionality of the model to 5. We
note, however, that the dynamics of this model lie on a 4-dimensional submanifold of the
full 5-dimensional manifold due to the fact that A and B are both singular, corresponding
to conservation of actin. This model also captures the interesting timescales describing
polymerization events and dynamics of the hydrolysis states of F and G actin, while it
neglects the fast dynamics corresponding to such events taking place at the tips. However
the model is still nonlinear and analytically unsolvable, so we propose an alternative
model reduction scheme.
30
( )
+
+
+
-
-
-
( )
Figure 2.2: The numerical integration of the BC model is displayed as
solid colors, the numerical integration of the QSSA model (Section 2.2) as
a long dashed gray line, and the exact solution of the CT model (Section
2.3) as a short dashed black line. The time courses of the concentrations
of the various species in xs (top) and xf (bottom) are shown. The CT
and QSSA models have the same steady state behavior as the BC model,
and the QSSA model approximately retains the dependence of the tip
subunit states on the slow non-tip subunit states.
( ) ( )
31
2.2.3 Constant tip approximation
In addition to having sufficiently fast dynamics as to be effectively described as in quasi-
equilibrium with respect to xs, the vector xf is also small in magnitude compared to
xs, and this fact can be utilized to drastically simplify the QSSA model. Although
the tip dynamics are fast, one can profitably assume that the concentration of filament
tip subunit states, that is, the elements in xf, are constant in time. We refer to this
assumption as the constant tip (CT) approximation. Certainly this assumption is not
realistic since the tips have fast dynamics, but the effect of this error on the equations of
motion of the other actin species is small in the regime where the number concentration
of filaments is much smaller than the total amount of actin in the system, i.e. when
N  M , or when the tip subunit states quickly attain their limiting values. This
assumption allows the model to be reduced to a 5-dimensional linear system of equations
that can be solved analytically. The procedure is to replace the dynamical variables T ij
by constants N?ij , chosen such that the steady state of the CT model coincides with
the steady state of the BC model. We define ?ij as the fraction of the filament tips at
the j (plus or minus) end that are in the i (T, Pi, D) hydrolysis state. These constants
determine the rate of depolymerization reactions, and replacing the variables T ij with
them causes the term Bxf to be a constant source and sink term in Equation 2.2, which
as a result becomes linear.
The steady states of the BC and CT models will be the same if
N?T? = lim T
T
? (t)
t??
N?Pi? = lim T
Pi(t) (2.7)
t?? ?
N?D? = lim T
D
? (t).
t??
In other words, the constant tip subunit state fractions in the CT model should be chosen
as the steady state values of the tip subunit state fractions in the BC model. As a result,
only the approach to steady state will be different between the two models. These
32
limiting values can be found by numerically solving the algebraic system of equations
f(xss) = 0
T T? + T
Pi
? + T
D
? = N (2.8)
GT +GD + F T + FPi + FD = M.
where xss is the 11-dimensional steady state vector of concentrations in the BC model,
and taking the real non-negative solution. Equation 2.8 determines the values of ?ij
which will be unique for a given N and M .
We define
b ? Bxssf (2.9)
where xssf contains the constants N?
i
j instead of the variables T
i
j . The equation of motion
for xs is
x?s = Axs + b (2.10)
where
? ?
????a? b c 0 0 0 ????? b ?c? d 0 0 0 ???A = ??? a 0 ?e f 0 ???? (2.11)?? 0 0 e ?f ? g h ??
? 0? ?
d 0 g ?h
?? i? ???
?
j + k?
b = ???
??
?i
?? ? ?
??? (2.12)
k ??
?j
33
with
a = N(kTon, + + k
T
off, +) g = kphos
b = knex h = krephos
c = k T Trenex i = N(?+koff, + + ?
T T
?koff, -)
. (2.13)
d = N(kD + kD ) j = N(?D Don, + off, + +koff, + + ?
D D
?koff, -)
e = khyd k = N(?
PikPi P i P i+ off, + + ?? koff, -)
f = krehyd
Equation 2.10 is a linear system of differential equations which can be solved exactly.
One point of difficulty in solving this system is that the columns of A sum to zero due
to conservation of actin, causing A to be singular. In chemical reaction network theory,
one often has such systems with linear conservation laws. If the system is linear, with
only first order or pseudo-first order reactions, then such a singular system can be solved
cleanly by a method using the Drazin inverse AD of the matrix A. We believe that
this method has certain advantages over other approaches to solving singular systems of
differential equations. If A has Jordan decomposition
?? ??J1 0?A = V ?V?1 (2.14)
0 J0
where J1 and J0 correspond to the non-zero and zero eigenvalues respectively, then? ?
?1
AD ??J1 0= V ??V?1. (2.15)
0 0
In the supporting information of Ref. 99 the details of solving Equation 2.10 as well as
the advantages of this method are discussed [185]. The solution is
( )
x = ?AD + eAts G b. (2.16)
The matrix
1
G = x ? D
| |2 s
(0)b + A (2.17)
b
34
encodes information about the initial conditions xs(0).
2.3 Results
2.3.1 Eigenvalues of A
Perhaps the most important benefit of an exactly solvable model is the ability to formally
determine the eigenvalues governing the linear dynamics. These eigenvalues characterize
the relaxation times of the components of a perturbation from equilibrium, in the basis
of the eigenvectors. The expressions for the eigenvalues thus shed light on the timescales
that describe the different chemical processes. In our modeling we have included the
reverses of kinetically dominant forward reactions, and we have set the rate constants
of these reactions to be equal to something on the order of the corresponding forward
reaction rate constants multiplied by a small parameter . Thus b = b?, where b? ? c,
f = f?, where f? ? e, and h = h?, where h? ? g. By writing reverse rate constants
this way, we can Taylor expand the expressions for the eigenvalues around the point
 = 0 to simplify the result. Doing so to first order in , we have
?1 = 0( ? ) (2.18)1
?2 = ? a? b? ? c? d+ (a? d+ b? ? c)2 + 4b?c
2
?
? ?(c? b cd+  ? ) (2.19)a? c? d1
?3 = ? a? b? ? c? d? (a? d+ b? ? c)2 + 4b?c
2
?
? ?(? b (a? d)a  ? ) (2.20)a? c? d1
?4 = ?(e? f? ? g ?) h? + (e? g + f? ? h?)2 + 4f?g2 ?
? ?(g ?
gf
h? ?  (2.21)
e? g ? )1
?5 = ? e? f? ? g ? h? ? (e? g + f? ? h?)2 + 4f?g
2
?
? ?e? ef . (2.22)
e? g
35
The fact that the nonzero eigenvalues are negative implies the stability of the steady
state. These eigenvalues have no dependence on M , the total concentration of actin
subunits, but they do depend on N , the number concentration of filaments, since this
term appears in the expressions for a and d. For the parameterization used here and
with N = 1 nM , the eigenvalues can be ordered by magnitude as follows:
|?5| > |?3| > |?2| > |?4| > |?1|. (2.23)
Equations 2.18-2.22 are in terms of reaction rate constants and can be interpreted
as representing certain collective subprocesses in the chemical system corresponding to
combinations of those reactions. The ordering indicates the comparative rates of those
subprocesses. We interpret the zeroth order terms of the eigenvalues as follows:
? ?1 = 0 because actin subunits are conserved in this system, causing A to be sin-
gular. Equivalently, one can say that the dynamics unfold on a 4-dimensional sub-
manifold of the 5-dimensional manifold, and that this submanifold is determined
by M .
? ?2 represents the combination of the forward nucleotide exchange reaction (GD ?
GT ) and the polymerization ofGD. Both of these reactions convertGD into another
species, so this eigenvalue represents the subprocess of GD depletion.
? ?3 represents the polymerization of GT .
? ?4 represents the release of phosphate by FPi to form FD.
? ? represents the hydrolysis of ATP converting F T to FPi5 .
Whether the magnitudes of these eigenvalues are increased or decreased due to the
presence of reversible reactions (i.e. when  > 0) depends on the parameterization, since
the sign of the first order terms depend on the comparative sizes of certain parameters.
36
In the full BC 11-dimensional model, one can numerically evaluate the Jacobian
matrix of f(x) at steady state: ?
? ? ?f
?
J ?? . (2.24)?x x=xss
We find that, for the same parameterization, the smallest four non-zero eigenvalues
of J? are exactly equal to the non-zero eigenvalues of A. This implies that the CT
model has captured the slowest dynamics of the BC model by ignoring the processes
involving the tip subunits. The main benefit of the CT approximation is that these slow
linear dynamics can now be easily analyzed. These dynamics provide information about
the polymerization process, the nucleotide composition of the filaments, and nucleotide
exchange of globular actin. For most purposes, these aspects are of primary interest,
and the processes at the tips are of lesser importance.
To quantitatively judge the separation of timescales of the BC model, we list the
nonzero numerically evaluated eigenvalues of J? when N = 1 nM and  = 0, in decreas-
ing magnitude: (?5.49, ?1.41, ?0.833, ?0.393, ?0.300, ?0.0130, ?0.0129, ?0.002).
For comparison, we do the same with the eigenvalues of A:
(?0.300, ?0.0130, ?0.0129, ?0.002). While there is no large spectral gap, there
is certainly a wide range of timescales, and it would suffice to even keep the small-
est 3 non-zero eigenvalues. We later show how, by combining certain species, the
dimensionality can be reduced to 3.
2.3.2 Steady state concentrations
One might expect that if we increase the amount of actin subunits in the system by a
different choice of initial conditions, the concentrations of the different species at steady
state would change. An interesting feature of this system is that this is true only for
some species, and which species it is true for depends on whether or not we have included
reversible reactions (if  > 0). Additionally, this can be shown to be true in both the CT
model and the BC model. We demonstrate it first in the CT model.
37
We find the steady state vector of concentrations xsss by taking the limit of Equation
2.16 as t??:
xsss ? lim xs(t) = ?ADb + CGb (2.25)
t??
where
C ? lim eAt. (2.26)
t??
We eigendecompose A as UDV? and use it in the expression for C:
C = U lim eDtV?. (2.27)
t??
With the exception of ?1, which is zero, the eigenvalues of A are negative, so we have
C = ?U diag(1, 0, 0, 0, 0) V???? 0 0 0 0 0? ?
?
?
?? 0 0 0 0 0? ?
?
= ? ??fh
?
0 0 0 0????V (2.28)? eg? hg 0 0 0 0??
1 0 0 0 0
where (0, 0, fh h ?eg , g , 1) is the eigenvector corresponding to ?1. Thus the top two rows of
C are zero, and the top four rows would be zero if we exclude reversible reactions. This
is also true of the term CGb. Now, the matrix G is the only place where the initial
conditions appear in Equation 2.25. So if the top two rows CGb are zero, then the first
two elements of xsss cannot have any dependence on initial conditions. In other words,
GD and GT always reach the same concentrations at steady state no matter what the
initial concentrations of any of the species are. If we have no reversible reactions, the
same is also true for the species F T and FPi.
Consider the following thought experiment, assuming for simplicity  = 0. We start
with some initial amount M of actin in any form and let the system come to steady
state. We then add an amount ?M more actin to the system, in any form, and wait
38
until the system has reached steady state again. We would find that the only difference
between the two steady states is that the concentration of FD had increased by ?M .
If  > 0, then we would find that the steady state values of F T , and FPi, FD had all
increased, and the sum of these changes would be ?M .
This lack of dependence of the steady state concentrations of some species on M is
not an artifact of the CT approximation; it is also the case in the BC model. It can not
be shown to be true by taking the limit t ? ? as is the case here, but it can instead
be shown by observing that a subset of the system of algebraic equations f(xss) = 0 is
closed, and that a solution for the subset could be obtained without specifying M . This
implies that the steady state concentrations of the species represented by that solution
have no dependence on M . We give the details of this argument in Appendix B.
2.4 Discussion
We have argued that the dynamics of the polymerization of actin subunits into filaments
can effectively be subdivided as follows: fast nonlinear dynamics govern the states of the
filament tip subunits, and slow linear dynamics govern the change in polymerization and
hydrolysis states of non-tip subunits. One cannot completely separate the tip subunit
dynamics from the non-tip subunit dynamics because they are coupled; the tip subunit
states depend on the concentrations of GT and GD through polymerization reactions,
and the non-tip subunit states depend on the tip subunit states via depolymerization
reactions. However, because of the typical size of N compared toM , the non-tip subunit
states depend only comparatively weakly on the tip subunit states during most of a
typical trajectory. We have shown two ways to approximate this coupling to achieve a
significant reduction in dimensionality of the system. First, in the QSSA model, it is
assumed that, on the slow timescale, the number of tips in a certain hydrolysis state
depends only on GT and GD. This allows one to write a closed system of equations of
motion of the non-tip subunits, describing evolution of the entire system on the slow
5-dimensional submanifold of the full 11-dimensional space. This model is physically
39
realistic and quite accurate because it preserves the dependence of the tip subunit states
on the concentration of the non-tip subunits, however the resulting equations of motion
are analytically intractable.
In the CT model, we make the seemingly unrealistic assumption that the tip subunits
have no dependence on the non-tip subunits and in fact do not evolve at all, but remain
fixed for all times at their steady state values. In this sense we turn the tip subunit
concentrations from variables into constants, and the equations of motion for the non-tip
subunits become 5-dimensional and linear with the depolymerization terms involving the
tip subunits entering as a non-homogenous term b. By choosing the fixed values of the
tip subunits as the resting values, we ensure that the steady states of the two models will
be the same. The CT assumption is valid because of the weak dependence of the non-tip
subunit states on the tip subunit states. In other words, b is comparatively small, and
the discrepancy between trajectories of the CT and BC model due to b not containing
realistic values for all times is not pronounced. In exchange for the cost of this error,
there is an important benefit, which is the ability to write symbolic expressions for the
eigenvalues of the matrix A. We note that these eigenvalues agree with the numerically
calculated smallest nonzero eigenvalues of J? of the full 11-dimensional model, indicating
that indeed the linear non-tip subunit dynamics are the slowest of all of the processes
in the BC model. In addition, qualitative results about the nature of the dependence of
the steady state concentrations on the initial conditions agree for the full and reduced
model. Dimensionality reduction is also possible by diffusion map analysis of a simulated
trajectory of the BC model (supporting information of Ref. 99)). The results of this
analysis indicate that fewer than 6 or 7 dimensions suffice to faithfully reproduce the
polymerization dynamics.
Eigenvalue analysis of A allows one to understand the timescales that govern the lin-
ear non-tip subunit dynamics as well as how these timescales depend on the parameters.
These timescales approximately represent the following processes: depletion of GD via
polymerization and conversion to GT , polymerization of GT , hydrolysis of ATP convert-
ing F T to FPi, and phosphate release converting FPi to FD. As might be expected, the
40
timescales involving polymerization depend on N , and their magnitude compared to that
of the other timescales may change significantly. We have treated the presence of the
reverses of some nearly irreversible reactions essentially as perturbations by regarding
the rate constants of backward reactions as equal to the rate constant of the correspond-
ing forward reaction multiplied by a small parameter . As shown above, the inclusion
of these reactions introduce small corrections to the eigenvalues. These timescales allow
one to understand a typical trajectory of the system. Such a trajectory can be visualized
in three dimensions by combining multiple species into a single species and choosing to
not to visualize a variable whose value is determined by the other three due to conser-
vation of actin. In Figure 2.3, we combine F T and FPi together, since they have similar
structural properties in the filament, and we do not visualize FD. Thus we further re-
duce dimensionality to 4 (of which only 3 dimensions are independent) by neglecting the
timescale corresponding to ATP hydrolysis, i.e. ?5. In the trajectory depicted, GT and
GD are quickly made small via polymerization and nucleotide exchange reactions. The
polymerization of GT causes F T+Pi to increase, and when GT has become small, F T+Pi
converts to FD via the slow process of phosphate release, and at the end, nearly all of
the actin is in the form FD.
Conclusion
The main results of this chapter are the elucidation of the degree to which not explicitly
accounting for tip subunit state dynamics during actin polymerization is a passable as-
sumption, and the resulting insight into the hierarchy of processes involved in the slow
linear dynamics of the non-tip subunit states. The CT model is overly simple but use-
ful to understand basic features of the polymerization process. In other more detailed
models, actin related proteins are incorporated either by introducing new variables repre-
senting the proteins and the protein-subunit complexes, or by including new parameters
that multiply certain terms in the equations of motion [98, 180, 181]. These adaptations
could be straightforwardly included in the modeling done here. Additionally, the effect
41
Figure 2.3: Visualization of a 3,000 s trajectory of the CT model be-
ginning from 1 ?M of GT and 1 ?M of GD, with N = 1 nM and with
FD not visualized. The curve is colored according to time, with pink
representing early times. All units are ?M. Vectors are drawn, not to
scale, and labeled to represent the direction that certain reactions pull
the trajectory and at which point in the trajectory those pulls become
dominant.
42
of different concentrations of solvated ATP, ADP-Pi, and ADP could be investigated by
changing the values of the pseudo-first order reaction rate constants, or even by regard-
ing those reactions as second order and tracking the concentrations of the nucleotide
species as separate variables. These modifications run counter to the goal here of model
reduction, however. Different choices in modeling are of course suited to different pur-
poses, and the choices made here address a desire to have a simple mental picture of an
otherwise obscured and complex process.
43
Chapter 3
Gibbs free energy change of a
discrete chemical reaction event
This chapter and Appendix C are adapted from: Floyd, C., Papoian, G. A., & Jarzynski,
C. (2020). Gibbs free energy change of a discrete chemical reaction event. The Journal
of chemical physics, 152(8), 084116.
3.1 Introduction
In recent years the coarse-grained computational modeling of intracellular environments
has enjoyed significant advances. An important paradigm shared by many such models is
to treat the evolution of reacting chemical species? copy numbers and spatial distributions
by simulating a reaction-diffusion master equation (RDME) [186]. In this approach, the
system volume is divided into compartments, each with local values of the copy numbers
and chemical potentials of the different chemical species (Figure 3.1). The RDME is
a differential equation describing the evolution of the probability P (N, t) of observing
the vector of copy numbers N = {Ni,A}i?L,A?? of chemical species i in compartment A
at time t, where L is the set of solute species and ? is the set of compartments in the
system. The RDME can be written schematically as
dP (N, t) ( )
= M? + D? P (N, t) (3.1)
dt
44
where M? and D? represent operators describing chemical reactions and inter-compartment
diffusion, respectively [187]. An in-depth description of the RDME approach can be
found in Refs. 188, 189, where the forms of the operators are discussed. Rather than
directly solving Equation 3.1, one often simulates trajectories of the vector N obey-
ing the stochastic dynamics encoded in the RDME, using a variant of the Gillespie
algorithm [190]. The sizes of the compartments are commonly determined by the Ku-
ramoto lengths, the mean free diffusional path for a species before it participates in a
chemical reaction. Within each compartment, the spatial distributions of the reacting
species are assumed to be homogeneous, allowing the use of mass-action kinetics with
the compartment?s local values of the species? concentrations to describe the stochastic
chemical reaction propensities. Molecules can additionally jump between adjacent com-
partments in ?diffusion events? (whose propensities also depend on the compartments?
local concentrations of species) to give rise to concentration gradients on the scale of the
compartment length. This modeling approach is appropriate when the Kuramoto length
is small compared to the system size (i.e. the assumption of homogeneity over the system
volume fails), yet large compared to the intermolecular distance scale [186]. Examples of
simulation platforms employing such an approach include Virtual Cell [191, 192], lattice
microbes [193, 194], and MEDYAN [42, 195].
One important aspect of simulating non-equilibrium biological systems is the compu-
tation of thermodynamic forces that drive the observed flux on the network of chemical
reactions [26, 196, 197]. Determining these forces can allow for the quantification of
entropy production in chemically reactive systems [198, 199]. In several research groups,
measuring entropy production in biological active matter has been a recent goal [34, 152].
For instance, in Chapter 4 we quantify the entropy production rates of self-organizing
non-equilibrium actomyosin networks in MEDYAN using the expressions derived here as
a first step [195]. The ability to measure dissipation in active matter systems will allow
to test the applicability of different physical organizing principles relating the production
of entropy to the likelihood of observing certain trajectories [200, 201]. For isothermal,
isobaric, chemically reactive solutions, which include most mesoscopic biological systems,
45
Figure 3.1: An example of a cubic compartment grid used in the simu-
lation of a RDME. Each compartment, labeled with letters A,B, ..., has
local values of the quantitiesNR, NG, referring to the copy numbers of the
red and green molecules in the compartment, and of ?R, ?G, referring to
the chemical potentials of those molecules. Molecules can react with each
other within compartments (long dashed arrow), as well as hop between
adjacent compartments, representing diffusion (short dashed arrow).
46
measuring the total entropy production amounts to determining the change in Gibbs free
energy that accompanies chemical reactions and diffusion down concentration gradients
[23, 25, 202]. A ubiquitous textbook expression for the change in Gibbs free energy G
accompanying a chemical reaction is
?G = kBT logKeq + kBT logQ (3.2)
where Keq is the equilibrium constant, Q is the reaction quotient (we give definitions of
these quantities below), kB is Boltzmann?s constant, and T is the temperature [1, 44].
At equilibrium, Q = K?1eq and as a result ?G = 0. However, in this chapter, we argue
that Equation 3.2 is a biased approximation to the exact value of ?G accompanying a
chemical reaction which holds when the copy numbers of the reacting molecules are large,
such as on the order of Avogadro?s number. When the system is small, such as when copy
numbers are on the order of 100 as is often the case in RDME simulations of intracellular
environments, then thermodynamic expressions such as Equation 3.2 require corrections
[203, 204].
As a simple motivating example, consider a mixture of an even number of two chem-
ical species, red and green, which inter-convert at equal rates. At equilibrium, the copy
numbers of these molecules will be equal, and Q = K?1eq = 1. Now if a reaction were to
occur at equilibrium to produce an additional red molecule and one fewer green molecule,
then we would expect that the Gibbs free energy of the system had increased, since we
have left equilibrium where the free energy attains its minimum. However the predic-
tion of Equation 3.2 is that ?G = 0 for this reaction. The assumption whose violation
leads to Equation 3.2 being incorrect is that the copy number of chemical species is a
continuous quantity. When these variables are considered as discrete, then a different
expression for ?G must be used to give correct behavior.
Similarly for diffusion between adjacent compartments, a common expression for the
change in Gibbs free energy accompanying the jump of a molecule i from compartment
47
A with copy number Ni,A, to compartment B where its copy number is Ni,B, is
Ni,B
?G = kBT log . (3.3)
Ni,A
Imagine however we have a situation where Ni,A = Ni,B, and a molecule jumps from
compartment A to B. The Gibbs free energy should have increased since we have de-
parted from the highest entropy distribution of the molecules over the two compartments,
however Equation 3.3 will predict that ?G = 0.
In this chapter we derive exact expressions for the change in Gibbs free energy ac-
companying chemical reactions within compartments and diffusion events between com-
partments, and we further show how these expressions relate to the familiar textbook
formulas Equations 3.2 and 3.3 through a series of approximations. We also discuss the
assumptions involved in defining the Gibbs free energy of a grid of homogeneously mixed
compartments which can exchange energy and particles, such as that used in a simulation
of a RDME. Finally we present numerical simulations using MEDYAN to demonstrate
the need to use these more exact expressions for ?G in order to obtain sensible results
when copy numbers are treated as discrete variables. Only these more exact expressions
will give correct, unbiased behavior when measuring entropy production in mesoscopic
in silico studies of biological non-equilibrium systems that rely on the RDME formalism.
3.2 Methods
3.2.1 ?G of reactions
Here we make successive approximations to the formula for ?G accompanying a chemical
reaction, and our notation reflects the level of approximation in which certain quantities
are being used: when appropriate, we subscript quantities with a parenthesized number,
i.e. ?G(0), where increasing numbers represent more approximate versions. The symbol? will indicate that the quantities of chemical species are being represented by mole
fractions ?i, rather than concentrations Ci. In this section we treat the case that our
48
system comprises a single closed compartment of a homogeneous dilute solution in which
a chemical reaction has occurred and derive an expression for the change in the Gibbs
free energy. In this system the number of solvent molecules is fixed and the solute
molecules participate in chemical reactions, causing their copy numbers to change. In
the next sections we consider a system with multiple weakly interacting compartments,
each of which represents a homogeneous solution with local copy numbers of solvent and
solutes and between which both solvent and solutes can diffuse. The nearly exact result
Equation 3.21 obtained in this section will also apply to those systems, as we argue
below.
Before restricting to the case of a single closed compartment, we establish notation
for properties of the chemical species in a compartment grid. The chemical potential for
species i in compartment A can be expressed either as depending upon the mole fraction,
?i,A, or upon the concentration, Ci,A, of that species in the compartment:
? 0 0i,A = ??i (T, p) + kBT log?i,A = ?i (T, p) + kBT logCi,A (3.4)
where kB is Boltzmann?s constant, ??0i (T, p) is the standard state chemical potential at
temperature T and pressure p when working with the dimensionless ? , and ?0i,A i (T, p) is
the same when working with Ci,A. Ci,A and ?i,A both play the role of the ?composition
variable? leading to these different, yet equivalent expressions for the chemical potential
[205]. We make the distinction between dependence upon copy number and dependence
on concentration in order to establish parameters that can be used in simulation, which
commonly works with copy numbers, based on those given in the literature, which typi-
cally use units of concentration. Here we make the assumption of an ideal-dilute solution
and neglect the coefficient of activity of the solute species [206]. The mole fraction can
be written
Ni,A ? Ni,A?i,A = = (3.5)
NA j?M Nj,A
where Ni,A is the copy number of species i in compartment A, NA is the total copy
number of molecules in compartment A, and M is the set of all species including the
49
solvent in the system. Similarly, the concentration can be written
Ni,A
Ci,A = (3.6)
?A
where
?A = NAvVA (3.7)
is a conversion factor, NAv is Avogadro?s number, and VA is the compartment volume
(we assume constant pressure and that the fluctuations in volume are negligible for these
liquid systems, allowing the use of Gibbs free energy). Using Equations 3.4, 3.5, and 3.6,
we can write1
??0
NA
i (T, p) = ?
0
i (T, p) + kBT log . (3.8)?A
Using standard arguments concerning thermodynamic extensivity, it is possible to
establish that the Gibbs free energy of the solution in compartment A can be written as
a weighted sum over the chemical potentials of the species:
?
GA(NA) = Ni,A?i,A(?i,A) (3.9)
i?M
where NA = {Ni,A}i?M is the vector of species copy numbers Ni,A, and where we have
explicitly indicated the dependency of ?i,A upon mole fraction ?i,A via Equation 3.5
[202, 207]. One may be concerned that Equation 3.9 fails to apply when the copy numbers
of solutes are small. As explained below, we make the assumption that boundary effects
are still negligible, which allows us to treat G as a first order homogenous function of the
number of copies of the system [204]. This is the necessary property to establish Equation
3.9, so the small copy numbers of solutes does not render this approach invalid. We rely
on Equations 3.4 and 3.9 to derive changes in Gibbs free energy accompanying chemical
reactions and inter-compartment diffusion.
1The argument of the logarithm, NA , has dimensions of concentration, however it is understood that
?A
such quantities are taken with reference to some standard concentration, typically 1 M .
50
Consider a reaction of the general form
k+
?1X1 + ?2X2 + . . . ????? ?1Y1 + ?2Y2 + . . . (3.10)
k?
where Xi represent reactants, Yj represent products, ?i and ?j are stoichiometric coeffi-
cients, and the rate of reaction is k+ to the right and k? to the left. We have dropped
the subscript A indicating the compartment in which the reaction takes place, and now
restrict to the case that our system is a single compartment. When this reaction has
occurred once to the right, the copy numbers of reactants have changed Ni ? Ni ? ?i,
and those of the products have changed Nj ? Nj+?j . We calculate the change in Gibbs
free energy accompanying this reaction by considering it as resulting from these finite,
discrete changes in copy numbers [204], not from infinitesimal changes. Using Equations
3.4, 3.5, and 3.9, the Gibbs free energy before the reaction has occurred can be written
as
? ( ) ? ( )
Ginitial
Ni Nj
= N ??0( i i + kBT log ) + N
0
j ??j + kBT logN N
i?R j?P
N
0? s+Ns ??s + kBT log (3.11)N
where R is the set of reactants, P is the set of products, the subscript s refers to the
solvent, and where we have dropped the dependence of the standard state chemical po-
tential on T and p. ??0i and ??
0
j describe the chemical potential at a reference concentration
of the solute in the solvent (also referred to as the solute standard state), whereas ??0?s
describes the chemical potential at a reference state of pure solvent (also referred to as
the solvent standard state) [206]. We assume here for simplicity and without loss of
generality that there are no solute species which have not participated in the reaction
(i.e. spectator solute species). These species would also contribute terms to Equation
3.11 but, when we subtract the initial from the final Gibbs free energy, their inclusion
51
would not give a different result. The final Gibbs free energy is
? ( ) ? ( )
Gfinal = (N ? ? ) ??0 Ni ? ?i Nj + ?j( i i i + kBT lo)g + (Nj + ?j) ??
0
j + kBT logN + ? N + ?
i?R j?P
+N ??0?
Ns
s s + kBT log (3.12)N + ?
where ? ?
? = ?j ? ?i (3.13)
j?P i?R
is what we refer to as the ?stoichiometric difference?, or the amount by which the total
species copy number N has changed. As described in Refs. 205, 208, when ? 6= 0 it is
important to account for the solvent species in the calculation of free energy differences.
This is because the free energy of the solvent, which is the most numerous species in
the reaction volume, will not be the same after the reaction has taken place since its
mole fraction will change as N changes to N + ?. Neglecting the solvent species when
? 6= 0 leads to expressions for ?G that are off by an amount ?kBT [205]. Whereas
the authors of Refs. 205, 208 describe the appearance of this erroneous term while
formulating the Gibbs free energy as a function of a continuous degree of advancement
of reaction d? = ?dNi/?i = dNj/?j , here we treat the extent of reaction as a discrete
quantity. In the limit that ?i/Ni ? 0, ?j/Nj ? 0 for each reactant and product, the
discrete case passes into the continuum case, however under the assumption of small
copy numbers we do not satisfy this limit. In Appendix C we discuss further differences
between the continuum treatment and the discrete treatment, as well as the relation to
the Gibbs-Duhem Equation. ? ?
After some algebra (using the fact that i?RNi+ j?P Nj +Ns = N), we can write
the change in Gibbs free energy as
?G final initial(0) = G ?G ? (N ? ? )Ni??i ?i i (Nj + ? )Nj+?jj NN
= ??G0 + kBT log (3.14)
NNi NN j (N + ?)
N+?
i?R i j?P j
52
where ? ?
??G0 = ? ??0j j ? ? 0i??i . (3.15)
j?P j?R
Equation 3.14 is exact, but we would like to avoid specifying N in simulation since the
solvent is typically not modeled explicitly, as we elaborate on in the next section. We
would also like to determine ??G0 from literature values of ?G0. To these ends we first
rewrite Equation 3.14 as
NN
?G(0) = ??G0 + kBT log Q?(0) + kBT log (3.16)(N + ?)N+?
where ? (Ni ? ? )Ni??i ? (N + ? )Nj+?ji j j
Q?(0) = . (3.17)
NNi Nj
i?R i j?P Nj
From Equations 3.8 and 3.15 we can write
N
??G0 = ?G0 + ?kBT log (3.18)
?
where ? ?
?G0 = ? ?0 ? ? ?0j j i i . (3.19)
j?P j?R
Inserting this to Equation 3.16, we have
( )? N
?G = ?G0
N N
(0) + kBT log + kBT log Q?? (N + ?()N+? ) (0)N+?
= ?G0 ? N?kBT log ? + kBT log + kBT log Q?(0). (3.20)N + ?
We now make the approximation that N  ?, which is certainly reasonable for most
realistic parameterizations of the compartment grid (in the example of a 0.125 ?m3
compartment filled with water, N ? 109 while ? ? 1). With this we can write the third
53
term in Equation 3.20 as ??kBT ,2 giving
?G 0(1) = ?G ? ?kBT log ?? ?kBT + kBT log Q?(1), (3.21)
where Q?(1) = Q?(0) (we update the subscript to indicate the use of this quantity in
a more approximate version of the formula for ?G). We recommend using Equation
3.21 in simulation because it allows to incorporate literature values for ?G0 and avoids
specification of N . To understand the term ??kBT in Equation 3.21 and to understand
the relationship between Equations 3.21 and 3.16 and the textbook expression for ?G,
we make further approximations leveraging the large sizes of the solute copy numbers
compared to their stoichiometric coefficients. First, we rewrite Q?(1) as
?( )? Ni ?( )Nji ?j
Q?(1) = 1? (Ni ? ? ??i ?ji) 1 + (Nj + ?j) . (3.22)N N
i? i jR j?P
Assuming Ni  ?i and Nj  ?j , and using the limit
( y)x
lim 1? = e?y (3.23)
x?? x
we obtain
? ?
kBT log Q?(1) ? ?kBT + kBT log (Ni ? ? ??i ?ji) (Nj + ?j) . (3.24)
i?R j?P
Inserting this into Equation 3.21, canceling the term ?kBT , gives
?G 0(2) = ?G ? ?kBT log ? + kBT log Q?(2), (3.25)
where ? ?
Q? ??i ?j(2) = (Ni ? ?i) (Nj + ?j) . (3.26)
( ) i?R j?P2We have ? N+? (N+?) ( )? ? (N+?)log = log 1 + ? ?? for large N .
N N
54
Finally, introducing ? ? ?Q? = N ?i ?j(3) i Nj (3.27)
i?R j?P
and discarding terms in v?G that scale like ?i or j(2) N N , we arrive ati j
?G 0(3) = ?G ? ?kBT log ? + kBT log Q?(3) = ?G0 + kBT logQ (3.28)
where ? ? ?Q = C ?i ?ji Cj . (3.29)
i?R j?P
We see that Equation 3.28 is obtained from Equation 3.25 upon making the approxima-
tions Ni ? ?i ? Ni and Nj + ?j ? Nj . This final result in Equation 3.28 is a standard
textbook expression for the change in Gibbs free energy [1, 44].
We thus have four expressions for ?G of chemical reactions:
? Equation 3.14 for ?G(0) is exact, however it requires specifying N .
? Equation 3.21 for ?G(1) uses the approximation N  ?. We recommend the use
of this expression because it is the most exact expression for which we need not
specify N , and since it is written in terms of ?G0, for which literature values can
be found.
? Equation 3.25 for ?G(2) uses the approximations Ni  ?i and Nj  ?j .
? Equation 3.28 for ?G(3) uses the approximations Ni  ?i and Nj  ?j again.
In Appendix C, we provide expressions for the accuracy of these approximations. Note
that, without specifying the copy number of solvents Ns, we can only approximately
compute changes in Gibbs free energy, and not the instantaneous Gibbs free energy of
the system. Typically only the changes are of interest.
The definition of Gibbs free energy states that
?G = ?H ? T?S. (3.30)
55
Comparing this expression to Equation 3.28, we identify
?G0 = ?H ? T?S0, (3.31)
and
? T?Smix = kBT logQ, (3.32)
where ?H is the enthalpy of reaction, and ?S0 and ?Smix represent the changes in
entropy due to the molecular conformations and the translational motions, respectively:
?S = ?S0 + ?Smix. (3.33)
It is instructive to realize that the discrepancies between the various expressions for ?G
are due to how the ?Smix term in Equation 3.33 is written.
When employing any of the above expressions that involve the logarithms of products
of copy numbers that are raised to the power of other copy numbers, we recommend
splitting the logarithm of products into a sum of logarithms as well as using log xy =
y log x in order to prevent overflow resulting from computing very large numbers. The
results of this section do not assume a system consisting of multiple, weakly-interacting
compartments, which we discuss next.
3.2.2 Thermodynamics of a reaction-diffusion compartment grid
Now that we have treated the scenario of a reaction event occurring within a single
compartment, we want to generalize to the case of a grid of compartments. For this we
develop an argument based on timescales which will allow us to track the copy numbers
of the reactive solutes while ignoring those of the inert solvent. This is a necessary
modeling feature due to the computational infeasibility of tracking the solvent copy
numbers in each compartment. After describing the thermodynamic framework for a
grid of compartments, in the next section we derive the change in G resulting from
diffusion of solutes between adjacent compartments.
56
In simulating a RDME, one commonly treats diffusion between adjacent compart-
ments and chemical reactions within compartments using an augmented set of all species
and reactions in the system that treats species as distinct if they belong to separate com-
partments. Thus if there are |L| reacting species and |?| compartments, where L and
? are the sets of solute species and compartments respectively, then in the augmented
set there are |L||?| species tracked. The solvent species is not tracked explicitly, as we
discuss below. The number of reactions in the augmented system, including r chemical
reactions per compartment and roughly z|L| diffusion events per compartments (where
z is the assumed constant number of neighbors of each compartment, ignoring boundary
compartments), is |?|(r+z|L|). This augmented set of species and reactions is then sim-
ulated using the Gillespie algorithm, in which the reaction propensities are appropriately
scaled according to the compartment volumes [189].
Crucial to the justification of this strategy to simulate a RDME is the assumption
that within each compartment, the reacting species can be considered homogeneously
distributed, so that one may use mass-action kinetics to determine the propensities.
This assumption amounts to the condition that the timescale describing diffusion within
compartments, ?D, is much less than the timescale of chemical reactions, ?C :
?D  ?C . (3.34)
This comparison should be done for each diffusing and reacting species, and the timescale
of the fastest reaction (taken as the inverse of the propensity) for each species should
be used. If the condition holds, then the process of intra-compartmental diffusion will
homogenize the solution faster than chemical reactions occur, so the assumption of mass-
action kinetics holds. Let the dimension of the space be d, the length of the (here assumed
cubical) compartments be h, and the diffusion constant of a species be D. Then
2
?D ?
h
. (3.35)
2dD
57
The Kuramoto length is given by
?
lK = 2dD?C , (3.36)
so one can see that the condition ?D  ?C is equivalent to the condition lK  h, and
thus one can enforce this condition by choosing a smaller compartment size h. For fixed
total volume, there is a trade-off between h and |?|, which determines the size of the
augmented system, and therefore the computational efficiency. The timescale of intra-
compartment diffusion ?D is approximately equal to the timescale of inter-compartment
diffusion (which can be given as the inverse of kD = D2 ), so a third way of describing thish
condition is that the frequency of jumps between adjacent compartments is much greater
than the frequency of chemical reactions inside the compartments, which can be checked
empirically in simulation [186, 189]. We note that in the literature these expressions may
differ up to a constant coefficient depending on the reference.
In order to approximately describe the thermodynamics of this system, we distinguish
between the inert solvent and the dilute, chemically reactive solutes. We assume here
that the system is impermeable to the flow of either kind of species to the exterior.
Within the system, all species are permeable (though local diffusion constants may be
incorporated for each species [186]). We treat the solutes explicitly (at the level of
their compartment concentrations), whereas we model the solvent implicitly through an
appropriate limit as in the steps leading Equation 3.21 above. To this end we assume the
existence of a laboratory timescale ?l whose purpose is to define the temporal resolution
of our measurements, such that any changes occurring on a timescale longer than ?l will
be measured. We assume that ?l is much longer than the timescale describing the local
compartment fluctuations in the solvent copy numbers ?s, yet shorter than but on the
order of the timescale describing the diffusion of the solutes ?D. On this timescale ?l, then
in the time between the chemical reactions and diffusion events involving the solutes,
the system is quasi-equilibrated with respect to fluctuations involving the fast processes
of solute and solvent homogenization, and thus the system may be assigned well-defined
58
values of Gibbs free energy [132]. This hierarchy of timescales can be written as
?s  ?l . ?D  ?C . (3.37)
Typical ratios of the diffusion constants for solute to solvent are in the range of 1/10 to
1/100, placing ?D/?s in the range of 10 to 100 [209].
Thus on the timescale ?l there is enough temporal resolution to track the diffusion and
chemical reactions of the solutes, while allowing averaging over the fluctuations of the
solvent. To describe activity occurring over the large grid of compartments for extended
systems, we introduce new timescales ? gx , where x refers to any of the timescales defined
above. If we hold the compartment size h and the chemical concentrations fixed and
add more compartments to our system, the rates of solute diffusion events and chemical
reactions occurring anywhere in the system will scale as |?|, the number of compartments,
and thus the timescales needed to describe them scale as |?|?1, i.e. ? gl ? ?l/|?|. One
might be concerned that ? gl will be less than ?s for large systems, ostensibly violating
our requirement that we be able to average over the solvent fluctuations to define quasi-
equilibrated states. However, the timescale of solvent fluctuations across the whole
grid, ? gs , will also scale inversely with |?|, so the condition for being able to average
over solvent fluctuations expressed in Equation 3.37 does not ultimately depend on the
number of compartments. In other words, for systems with many compartments, as
long as Equation 3.37 holds for one compartment, we can be sure that our laboratory
timescale that describes the whole grid, ? gl , will be short enough to describe activity
of the solutes while long enough to allow averaging over the fluctuations of the solvent
occurring locally in each compartment.
We assume that the exterior of the system acts as a reservoir for the thermodynamic
variables p and T . The volume V of the system also remains constant, however under
the assumption of the solvent being an incompressible liquid, the change in quantity pV
is approximately zero, and for each reaction the change in Gibbs free energy is equal
to that of the Helmholtz free energy. Thus it is inconsequential whether we consider
59
p or V to be reservoir variables, and we choose p in order to speak of the Gibbs free
energy of the system. We can write the Gibbs free energy of the system as G(N, p, T ),
where N = {{Ni,A}i?M}A?? represents the set of copy numbers of all solute and solvent
species in each compartment in the grid. We further assume the compartments to be only
weakly-interacting; that is, they can exchange energy and particles, but the interaction
of the two subsystems does not contribute a term to the Gibbs free energy of the system.
This is equivalent to assuming that the Gibbs free energy of the compartments GA are
linearly additive: ?
G = GA (3.38)
A??
without any terms of the form GAB. To justify this, we first note that the interaction free
energy between two adjacent compartments will primarily be due to the interaction of
the solvent at the interface. This interfacial free energy will, even for mesoscopically sized
compartments, be small compared to the free energy due to the bulk of the compartment.
Further, our main interest will be in changes in the Gibbs free energy of the system due
to solute diffusion between compartments and chemical reactions, neither of which will
significantly affect the interfacial free energy, so all terms of the form GAB will drop out
of the expression for ?G.
In our approach of averaging over fluctuations in the solvent amounts and taking
the limit that this average is large compared to the copy numbers of the solutes, we are
choosing to neglect the changes in Gibbs free energy of the system owing to the solvent
fluctuations. We justify this with an argument that these fluctuations are small compared
to those resulting from the activity of the solute molecules; it also because it arises out
of necessity due to the computational intractability of tracking the solvent fluctuations.
On the laboratory time scale ?l, each compartment has an average copy number of
solvent molecules, Ns,A. The fluctuations in this quantity will have a standard deviation
on the order of 1/2Ns,A [132]. As indicated previously (to arrive at Equation 3.21), we
avoid specifying Ns,A by taking the limit that it is much larger than the number of solute
species, and thus fluctuations in this quantity don?t matter when computing nearly exact
60
changes in the Gibbs free energy accompanying reactions involving the solutes. We need
to establish that the change in Gibbs free energy of a compartment due to a fluctuation
in the solvent copy number on the order of 1/2Ns,A is small compared to the change
in Gibbs free energy accompanying chemical reactions and inter-compartment solute
diffusion events, and thus that the fluctuations in this quantity are not outweighing the
changes that we are measuring. In Appendix C we show that if the concentrations of
solutes are very different in the two compartments, then this Gibbs free energy change
is on the order of 1/2Ns,A kBT , where  is the ratio of solutes to solvent. We show that
this quantity is typically much less in magnitude than the change in Gibbs free energy
accompanying a solute diffusion event. If the concentrations are nearly equal, then the
Gibbs free energy is on the order of kBT , and is thus negligibly small. We make the
modeling choice to ignore these changes in the Gibbs free energy resulting from solvent
fluctuations because they tend to be small and they do not represent the processes we
are interested in which involve the activity of the solute molecules.
To summarize, we track the changes in the Gibbs free energy of the system by
computing a value of ?G whenever a chemical reaction occurs within a compartment or
a solute diffusion event occurs between adjacent compartments. By the linear additivity
of the compartments? free energies, any change in the free energy of a single compartment
is equal to the change in free energy of the whole system (i.e. ?GA = ?G). We assume
that the activity of the solvent only contributes small, fluctuating, unbiased changes to
the free energy which we ignore; we also assume that the amount of solvent is so large
that one can neglect the fluctuations in this quantity when computing the changes in
free energy for processes involving the solute copy numbers.
3.2.3 ?G of diffusion
To describe the change in Gibbs free energy accompanying a solute diffusion event be-
tween neighboring compartments we use a similar approach to the one used above for
chemical reactions. The key difference here is that as opposed to multiple species being
61
involved in a reaction taking place in a single compartment, we now have a single species
involved in a diffusion event taking place between two compartments.
Consider species i diffusing from compartment A, where its initial copy number is
Ni,A, to compartment B, where its intial copy number is Ni,B. The total copy numbers
of molecules in the compartments A and B are NA and NB. Assume there is just one
spectator species constituting the solvent, labeled s with copy numbers Ns,A and Ns,B
(we remove our uncertainty in the exact values of these numbers by taking a limit later).
As a result of the diffusion event, we have the following changes in these quantities:
Ni,A ? Ni,A ? 1, Ni,B ? Ni,B + 1, Ns,A ? Ns,A, Ns,B ? Ns,B, NA ? NA ? 1, and
NB ? NB + 1. The initial Gibbs free energy is
( ) ( )
Ginitial = N (??0
Ni,A Ni,B
i,A i + kBT log )+Ni,B ??0i + kBT logNA ( NB )
Ns,A
+N ??0,? + k T log +N ??0,?
Ns,B
s,A s B s,B s + kBT log , (3.39)
NA NB
and the final Gibbs free energy is
( ) ( )
final Ni,A ? 1 Ni,B + 1G = (Ni,A(? 1) ??0 + k 0i BT log ) + (N + 1) ?? + k T logNA ? ( i,B i B1 ) NB + 1
+N ??0,?
Ns,A 0,? Ns,B
s,A s + kBT log +Ns,B ??s + kBT log . (3.40)
NA ? 1 NB + 1
The difference of these two expressions leads to an exact formula, similar to Equation
3.16, which we omit here. Analogously to the approximation N  ? made in the context
of chemical reactions, here we assume that NA, NB  1, which amounts to setting
NA ? 1 ? NA and NB + 1 ? NB in Equations 3.39 and 3.40. Using this approximation,
we can express the change in Gibbs free energy as
(N ? 1)(Ni,A?1) (N (Ni,B+1)i,A i,B + 1)
?G = kBT log (3.41)N
N i,A
N
N i,Bi,A i,B
62
We see that we, analogously to making the approximation Ni  ?i above, if we here
take Ni,A, Ni,B  1, then Equation 3.41 reduces to the common expression
Ni,B
?G = kBT log . (3.42)
Ni,A
The right hand side of Equation 3.42 will always be less than that of Equation 3.41,
which, although the difference is typically very slight, can lead to systematically biased
calculations, as we show next.
3.3 Results
Here we perform stochastic simulations to illustrate the effects of approximating ?G for
reactions and diffusion when copy numbers are considered small and discrete. We use
MEDYAN, a simulation platform designed to study active networks at high resolution,
which is equipped with a RDME simulation engine of the type described above [42].
With this, we report on two different simulation set-ups, to test the discrepancy between
the nearly exact and approximate forms of ?G corresponding to reactions and to diffu-
sion. We compare the nearly exact Equations 3.21 and 3.41 for reactions and diffusion,
respectively, with their approximate counterparts, Equations 3.28 and 3.42. We observe
that only the nearly exact expressions result in sensible behavior, i.e. that the rate of
change of Gibbs free energy on average is 0 kBT/s when the system is at equilibrium.
3.3.1 ?G of reactions
To test the effect of approximation for ?G of reactions, we consider the simple reaction
scheme
k
A+B ???
+
?? C (3.43)
k?
where the rate constants to the right and left are k+ = 0.05 ?M?1s?1 and k? = 0.01
s?1 respectively, giving an equilibrium constant of Keq = k?/k+ = 0.2 ?M . We perform
stochastic simulations with the Next Reaction Method in MEDYAN [210], using a single
63
compartment of size 0.125 ?m3 (i.e. in this example there is no diffusion). To employ
Equation 3.21 in simulation, we first compute ?G0 ? ?kBT log ? ? ?kBT = 3.71 kBT
for the forward reaction and ?3.71 kBT for the reverse reaction, and then when each
reaction fires during simulation, the quantity Q?(1) is computed from the instantaneous
values of the copy numbers to determine ?G(1). A similar approach is taken to employ
Equation 3.28. We begin with NA = 100, NB = NC = 50, and repeat a simulation of
100 s duration 3,000 times to obtain averages of the trajectory of rates ?t?G(t) resulting
from the forward and reverse reactions (the notation ?t?G(t) implies that the measured
rates vary smoothly as a function of time, however this quantity is calculated as the
total change in Gibbs free energy resulting from discretely timed chemical events during
1 second-long windows). Figure 3.2 displays the results of these simulations. Note how,
while the two trajectories bear close similarity, the equilibrium value of ?t?G(t) centers
around 0 kBT/s for the nearly exact formulation Equation 3.21, yet erroneously centers
around ? ?0.08 kBT/s for the approximate version, Equation 3.28.
3.3.2 ?G of diffusion
To test the effect of approximating ?G for diffusion in a compartment-based reaction-
diffusion scheme, we next employed MEDYAN to simulate diffusion of 1,000 molecules
with diffusion constant 20 ?m2s?1 in a 2 ? 2 ? 2 grid of compartments, each a cube
with volume 0.125 ?m3. The initial distribution of molecules is uniformly random over
the compartment grid, and thus the system begins near equilibrium and is then allowed
to stochastically evolve for 100 s, i.e. the molecules hop randomly between adjacent
compartments. For each diffusion event, the value of ?G is determined using the nearly
exact Equation 3.28 for one run, and in another run the approximate Equation 3.42
is used. The difference in the trajectories of ?t?G for these simulations is stark, as
displayed in Figure 3.2. While the trajectory centers around 0 kBT/s for the nearly exact
formulation of ?G, it erroneously centers around ? ?1, 930 kBT/s for the approximate
version. Diffusion events are very frequent in this system, occurring around 240,000
times per second (this number can be calculated from the parameters of the system and
64
A) B)
C) D)
Figure 3.2: Numerical results illustrating the difference between nearly
exact and approximate formulations of ?G for reactions and diffusion. A)
Averages over 3,000 simulations of the chemical scheme A+B 
 C, using
the nearly exact Equation 3.21. The blue curve represents the trajectory
of ?t?G resulting from the forward reaction, the red curve represents
that from the reverse reaction, and the black curve represents their sum.
Shaded regions represent the standard deviation over the 3,000 repeated
trials. The inset displays a blow-up of the black curve once the system
has gotten close to equilibrium. B) The same as just described, but using
the approximate Equation 3.28. C) A single trajectory of the diffusion
of 1,000 molecules over a 1 ?m3 cubic grid of 8 compartments, beginning
from a random initial spatial distribution. Values of ?G are calculated
using Equation 3.41. D) The same as just described, however values of
?G are calculated using Equation 3.42.
65
is also observed during simulations). Thus, since Equation 3.42 is always less than the
nearly exact quantity, even by a small amount on the order 0.05 kBT for this system, this
systematic bias is amplified by the frequency of diffusion events to produce significant
differences from expected behavior, necessitating the use of a more exact formula for
?G. Lastly, we performed a simulation involving both reactions and diffusion across
multiple compartments. We found again that only when the nearly exact formulas were
used did the rate of change of Gibbs free energy center on 0 kBT/s at equilibrium. It is
not additionally illuminating to show the data, so we do not display it here.
3.4 Discussion
We have argued that when the copy numbers of the reactants and products are treated
as small, discrete quantities, then certain approximations leading to the textbook for-
mulas for ?G of reaction and diffusion, Equations 3.2 and 3.3, break down and lead to
biased results. We emphasize that this is true only when the copy numbers and reaction
occurrences are treated as discrete; when they are treated as continuous, one should use
the textbook formulas. This can be shown by considering a continuous version of the
chemical system described by Equation 3.43. The time evolution of the concentrations
of the chemical species is obtained by solving a system of ordinary differential equations
that employ mass-action kinetics, and from this solution, the rates ?t?G(t) resulting
from the forward and reverse reactions are computed using the instantaneous values of
the species? copy numbers which enter into Equations 3.21 and 3.28. Figure 3.3 dis-
plays the results of these calculations. Here, the total rate of ?t?G(t) only approaches
0 kBT /s, as it must at equilibrium, when Equation 3.28 is used.
When the copy numbers of the chemical species are treated as continuous, there is
no notion of a single occurrence of a reaction; instead the evolution of the system is
parameterized by the continuous variable ? which quantifies the extent of advancement
of the reaction [211]. In this framework, which is adopted in classical thermodynamics,
the copy number of any species never jumps instantaneously from Ni to Ni + ?i, and
66
A) B)
Figure 3.3: Analytical results illustrating that, when copy numbers are
treated as continuous variables, the textbook formula for ?G of reaction
should be used. A) Calculation of the rates ?t?G(t) using Equation 3.21
resulting from the forward (blue curve) and reverse (red curve) reactions,
as well as their sum (black curve), as described in the main text. The inset
shows a blow-up of these curves as the system approaches equilibrium.
B) The same as just described, but using Equation 3.28.
thus the premise of the derivation presented above leading to Equation 3.16 falls apart.
This explains why the seemingly more accurate Equation 3.21 is wrong when applied to
chemical dynamics that are described using continuous variables to represent the copy
number of species. When the chemical dynamics are modeled this way, the textbook
expression Equation 3.28 for the change in Gibbs free energy is valid. Another way to
understand the difference in these expressions is to view Equation 3.28 as giving the
instantaneous slope of G with respect to the degree of advancement of the reaction
when these quantities are viewed as continuous variables [205, 208], whereas Equation
3.16 effectively integrates the continuously varying quantity G through a single discrete
reaction occurrence.
If the copy numbers of chemical species are not large compared to the stoichiometric
coefficients (i.e. Ni is not much greater than ?i), it is best to describe the chemical dy-
namics by treating the copy numbers as discrete variables participating in stochastically
timed chemical reactions. This is the philosophy adopted by several recent models of
intracellular environments, where copy numbers of molecules of interest are sometimes
quite small. In these cases, adoption of the more exact expression for ?G can not only
67
improve precision, but can ensure correct behavior. If concentration gradients are ex-
pected to be a strong source of entropy production (e.g. diffusion of monomeric actin
along the lengths of filopodia, see Refs. 212?214), then using the nearly exact formula
presented here can ensure that the resulting measurements of dissipation are not strongly
biased.
68
Chapter 4
Quantifying dissipation in
actomyosin networks
This chapter and Appendix D are adapted from: Floyd, C., Papoian, G. A., & Jarzyn-
ski, C. (2019). Quantifying dissipation in actomyosin networks. Interface focus, 9(3),
20180078.
4.1 Introduction
The actin-based cytoskeleton is a dynamic supramolecular structure that, by sustaining
and releasing mechanical stress in response to various physiological cues, mediates the
exertion of force by cells both on their environments and within their bodies [2, 215].
These cytoskeletal structures are composed of long (on the order of 1 ?m in vivo [216])
actin polymers which are interconnected by various cross-linkers, as well as by myosin
motor filaments, resulting in a three-dimensional network-like organization referred to
as an ?actomyosin network? [217, 218].1 Part of the intricacy of actomyosin network
dynamics is due to the mechanosensitive kinetic reaction rates controlling cross-linker
and myosin filament unbinding as well as myosin filament walking: at high tension, cross-
linkers will unbind more quickly (slip-bond) whereas motors will unbind and walk less
quickly (catch-bond and stalling). [72, 87, 103]. These reactions control the actomyosin
1In our terminology, we will distinguish between ?cross-linking proteins,? which will include both
active (e.g. myosin filaments) and passive (e.g. ?-actinin and fascin) proteins that bind to adjacent
actin filaments, and ?cross-linkers,? which refer exclusively to passive cross-linking proteins.
69
network connectivity, which in turn determines the ability of the network to globally
distribute stress [104]. Thus the mechanosensitive feedback introduces nonlinear coupling
between the stress sustained by an actomyosin network and the network?s ability to
reorganize in response to that stress. In order to be responsive to physiological cues, the
dynamics of these systems occur far from thermodynamic equilibrium; the hydrolysis
of an out-of-equilibrium concentration of ATP molecules fuels a) the stress-generating
activity of the myosin motor filaments, and b) filament treadmilling [88?92]. Filament
treadmilling is a steady state situation in which the polymerization at the plus end
of the filament is compensated by the depolymerization at the minus end, resulting in
the filament moving forward without its length changing. As a result of these these
local free energy-consuming processes, actomyosin networks constitute an interesting
and biologically important example of soft active matter. Active matter is composed of
agents that individually transduce free energy from some external source, in this case
the chemical potential energy of many ATP molecules [26, 73, 86]. Dissipation in these
systems results when the free energy consumed ?G is greater than the quantity of work
W done by the system on its environment, with the remainder ?G ? W serving to
increase the total entropy.
The viewpoint of actomyosin networks as active matter systems has been fruitfully
adopted in recent theoretical and experimental studies, yet a lack of ability to quan-
tify the rates of free energy transduction by these systems has hindered development of
some of these lines of study. The emergence of distinctive dynamical states (for instance
pulsing actin waves or vortices) during the self-organization of actomyosin systems has
been documented in several in vitro experiments [67, 219, 220]. These emergent patterns
depend sensitively on the concentrations of myosin filaments and cross-linkers: myosin
filament concentration controls the level of active stress generation, and cross-linker con-
centration controls the degree of mechanical coupling of actin filaments, which has been
described using the language of percolation theory [104, 221]. While these emergent dy-
namic patterns have been characterized in detail, a general mechanism explaining why
70
these patterns emerge under given conditions has not yet been proven. It might be ex-
pected, given that these systems operate away from thermodynamic equilibrium, that
the quantity of free energy dissipated during a system?s evolution is optimized, similar
to the principle of minimum entropy production in the near-equilibrium theory of ir-
reversible thermodynamics [198]. However, this minimum entropy production principle
breaks down in the far-from-equilibrium, nonlinear-response regime, where many active
matter systems including actomyosin networks operate [201]. It has recently been pro-
posed that another optimization principle applies arbitrarily far from equilibrium. This
idea, referred to as dissipative adaptation, suggests that, in general, a coarse-grained
trajectory of some non-equilibrium system will be more likely than all alternative tra-
jectories if the amount of free energy absorbed and dissipated along that trajectory is
maximal [152, 155, 200]. This organizing principle has been borne out in model sys-
tems [153, 154], yet has also been shown to have certain counter examples [222]; it
remains actively debated. In the case of actomyosin systems, it has not yet been tested
because of the difficulty in measuring dissipated free energy using most experimental
approaches. In this chapter we take the first steps toward such a test, by developing a
simulation methodology allowing the quantification of dissipated free energy during the
self-organization of actomyosin networks.
In addition to being of interest in the field of active matter systems, dissipation in
actomyosin networks has also been an important factor in recent experimental stud-
ies of cell mechanics. Rheological properties of actomyosin networks largely determine
rheological properties of the whole cell, and it has been discovered that the dissipative
component of the cell?s viscoelastic response to mechanical oscillations (called the loss
modulus) is partly attributable to the ATPase activity of myosin motor filaments in ac-
tomyosin cytoskeleton [223?225]. In the context of cell mechanics, myosin filaments have
several roles: they produce mechanical stress by pulling on actin filaments, they dissipate
mechanical stress by disassembling actin filaments and higher order stress-sustaining fil-
ament structures, and they consume chemical energy through the hydrolysis of ATP
molecules. It has been proposed that a lack of detailed understanding of the effects
71
of these processes, and of dissipation in actomyosin systems more generally, underlies
inconsistent, widely variable traction force microscopy measurements of cell migration
[170]. Progress along this line is hindered by an absence of methods to study dissipation
in actomyosin networks directly and at sufficiently high spatio-temporal resolution.
To address these needs, we introduce a computational approach to measure dis-
sipation during simulations of actomyosin network self-organization using the simula-
tion platform MEDYAN [42, 226]. MEDYAN simulations marry stochastic reaction-
diffusion chemistry algorithms with detailed mechanical models of actin filaments, cross-
linkers, myosin motor filaments, and other associated proteins, and it also accounts for
mechanosensitive reaction rates. This combination of simulation features makes this soft-
ware uniquely capable of probing the complexity of actomyosin network dynamics. For
instance, past studies utilizing MEDYAN have investigated the dependence of network
collapse on myosin filament and cross-linker concentrations, as well as the origin of local
contractility in actomyosin networks [42, 94]. We refer the reader to the paper describing
MEDYAN for a detailed discussion of the various aspects of the simulation platform [42],
while here we describe an extension of that platform that allows for calculation of the
energetics of the chemical and mechanical events occurring during simulation. We utilize
these new capabilities to characterize the dissipation resulting from filament treadmilling,
for which we further introduce a mean-field model, as well as from myosin filament walk-
ing. We study both the time-dependence and the distributions of dissipation rates as
concentrations of cross-linkers and myosin filaments are varied, observing that transduc-
tion of chemical energy to stored mechanical energy is more efficient at denser network
organizations. For these simulations, we first explore systems with ?plain? myosin fila-
ments and cross-linkers that are not mechanosensitive, in order to simplify the overall
dynamics. We then introduce their mechanochemical coupling to understand its effect
on the observed trends. We end by discussing how this new methodology can provide a
valuable technique to advance the studies of actomyosin networks mentioned above.
72
4.2 Methods
4.2.1 Measuring dissipation in MEDYAN
We first give a brief overview of the MEDYAN simulation platform. MEDYAN em-
ploys a stochastic chemical evolution algorithm in conjunction with mechanical repre-
sentations of polymers and cross-linking proteins to simulate the dynamics of networks
with active components, including but not limited to actomyosin networks. The simu-
lation space comprises a grid of reaction-diffusion compartments, inside which chemical
species (e.g. unpolymerized subunits or cross-linking proteins) are assumed to be ho-
mogeneously distributed without specified locations, and which participate in reactions
(e.g. (de)polymerization or (un)binding) according to mass-action kinetics; the species
can additionally jump between compartments in diffusion events. When an unpolymer-
ized subunit polymerizes to or nucleates a filament, it becomes part of the mechanical
subsystem, gaining location coordinates in the simulation volume and becoming subject
to mechanical potentials depending on its interaction with other mechanical elements.
Through chemical reactions such as myosin filament binding and walking, the mechanical
energy of the system changes, and the new net forces are then periodically relaxed in a
mechanical equilibration phase, using conjugate gradient minimization. We fill in salient
details of the above overview as they become relevant below. A user provides input
data including system size and simulation length, mechanical parameters (e.g. stretch-
ing and bending constants and excluded volume cutoff distances), size of the polymer
subunits, energy minimization algorithm parameters, chemical simulation algorithm pa-
rameters, choices for the modeling of force-sensitive reaction rates, initial conditions of
the filaments (either specified or randomly generated), a list of reacting species and their
associated parameters, a list of reactions involving those species and their associated
parameters, and a list of desired output information. The output of a simulation is a set
of trajectory files containing information at each time point, which can include positions
of the mechanical network elements, tensions on the elements, and copy numbers of the
chemical species, among other things. In Appendix D we discuss parameterization of
73
the simulations analyzed in this chapter. MEDYAN is extensible in that it is possible to
implement new types of outputs, depending on the experimental needs; in this chapter
we describe a novel output that reports the changes in the Gibbs free energy of the
system.
As a MEDYAN simulation progresses, the Gibbs free energy of the system continu-
ally changes due to occurrences of chemical reactions and structural rearrangements of
the polymer network. These processes are driven by an out-of-equilibrium concentra-
tion of ATP which fuels filament treadmilling and myosin filament walking. Dissipation
measurement in MEDYAN works by calculating running totals of the chemical and me-
chanical energy changes. The running totals can then be converted into instantaneous
rates by taking the numerical derivative at each time point using the forward difference
quotient. The algorithm for tracking these energy changes is compatible with the follow-
ing sequence of consecutive procedures that make up one iterative cycle of a MEDYAN
simulation [42]:
1. Evolve system with stochastic chemical simulation for time tmin.
2. Calculate the changes in the mechanical energy resulting from the reactions in Step
1.
3. Mechanically equilibrate the network based on the new stresses calculated in Step
2.
4. Update the reaction rates of force-sensitive reactions based on the new forces.
Dissipation tracking is done by calculating for each of these four steps a change in
free energy, and then adding these free energy changes to determine the total change
in free energy resulting from each cycle, ?Gdissipated. Since, at least in this study, the
actomyosin network is not mechanically coupled to any work reservoir external to the
simulation volume (i.e. W = 0), ?Gdissipated is indeed dissipated energy [227]. This
methodology could be straightforwardly extended to account for work exchanged with
an external system in future studies, however. Step 4 in the MEDYAN simulation cycle
does not result in a change in free energy, as explained in Appendix D. The notation
74
1 - chemical simulation
? 2 - mechanical deformation
? 1 3 - mechanical equilibration
4 - update reaction rates
?
? ?2
? ?
?
?
? 3 ?
? ?
Figure 4.1: Energy level diagram indicating which changes in free en-
ergy are tabulated during the 4 procedures constituting one cycle in ME-
DYAN simulation. Step 4, mechanochemical update of reaction rates,
does not result in a free energy change. Dotted lines represent energy
levels that are intermediate during the iterative cycle, and solid lines
represent the energies at the beginning and end of one cycle.
chosen for these free energy changes, as well as the direction in which the energy is
changing during each procedure, is illustrated in Figure 4.1.
From this picture, we have the following relations:
?Gdissipated = ?Gchem + ?Gmech (4.1)
?Gdissipated = ?Gchem, dissipated + ?Gmech, dissipated (4.2)
?Gchem, dissipated = ?Gchem + ?Gstress (4.3)
?Gmech, dissipated = ?Gmech ??Gstress (4.4)
For the depicted relative position of energy levels, the sign convention is such that all
values of ?G except for ?Gstress will be negative (indicated by the arrow?s direction),
since G refers to the free energy of the system, not of its environment, and will therefore
be tend to be negative as the system moves down the free energy landscape. The usual
intuition that the total dissipation is positive can be stated
?G+dissipated = ??Gdissipated > 0, (4.5)
?
?
?
?
?
?
75
where the superscript ?+? indicates the positive change in the total entropy.
Equation 4.3 says that, given some change in the system?s chemical potential energy,
?Gchem, resulting from reactions occurring during Step 1, a portion of that energy is
used to deform the polymer network (e.g. via myosin filament pulling on actin filaments).
This increases the mechanical energy in the network by an amount ?Gstress. Only the
portion of ?Gchem which has not gone into ?Gstress has been dissipated as heat. In Step
3 the network is mechanically equilibrated, resulting in relaxation of net forces (though
not of all stresses) and updating of the network elements? positions. We refer to the
decrease in mechanical energy resulting from this relaxation as ?Gmech, dissipated.
The calculation of ?Gstress and ?Gmech, dissipated is based on a set of mechanical
potentials describing interactions between elements of the actomyosin network. Polymers
are modeled as a sequence of thin, unbendable, yet extensible cylinders that are joined at
their ends by beads whose positions define the polymer?s configuration. The structural
resolution of MEDYAN is at the level of the cylinders, which in this study are 27 nm
long and have effective diameters of approximately 5 nm, however the diameter is not a
parameter of the simulation, being instead effectively determined by the strength of the
excluded volume interaction between cylinders. Cross-linking proteins (e.g. ?-actinin
and myosin filaments) are modeled as Hookean springs connecting these cylinders by
attaching to discrete binding sites. Included among the mechanical potentials are various
modes of filament deformation, excluded volume interactions, and stretching of cross-
linkers and myosin filaments. Mechanical equilibration is accomplished by constrained
minimization of the mechanical energy with respect to the positions of the network
elements. A full description of the mechanical potentials and equilibration protocols
is given in Ref. 42. Determining ?Gstress and ?Gmech, dissipated requires evaluating the
instantaneous total mechanical energy of the system at certain points during the iterative
simulation cycle and taking the difference of those values.
The calculation of ?Gchem and ?Gchem, dissipated is accomplished by incrementing a
running total of the chemical energy Gchem whenever a reaction stochastically occurs
during Step 1, and finding the accumulated change at the end of the protocol. Chemical
76
stochastic simulation in MEDYAN uses a Gillespie-like reaction-diffusion algorithm over
a grid of compartments which constitutes the simulation volume. Diffusing species are
assumed to be homogeneous (i.e. obey mass-action kinetics) inside the compartments,
and can jump between the compartments leading to concentration gradients at the scale
of the compartment length (taken to be roughly the Kuramoto length of diffusing G-
actin, following [226]). The evolving polymer network is overlaid on this compartment
grid, with each piece of a polymer reacting with diffusing species according to the concen-
trations in its local compartment. Again, we refer the reader to [42] for a more detailed
description of the chemical dynamics. For the present purpose of measuring dissipation,
we introduced into this simulation protocol a precise formula for the change in Gibbs
free energy corresponding to the occurrence of various reactions as a function of the in-
stantaneous compartment concentrations. This formula was derived in Chapter 3. The
set of chemical reactions used to describe actomyosin networks in this study is based
on a previous model of actin polymerization dynamics that explicitly treats hydrolysis
states of the nucleotide bound to each actin subunit [99, 183]. This level of detail allows
to quantify the dissipation resulting from ATP hydrolysis during filament treadmilling.
To increase computational efficiency, we neglect the dynamics of nucleotide hydrolysis
states of the tips of the filaments. This has been shown in previous work to be a valid
approximation to the full dynamics which includes the states of the filament tips [99].
We refer to the resulting set of reactions describing actin polymerization dynamics as the
Constant Tip (CT) model. However unlike in the original CT model, here we explicitly
include G-actin bound to ADP-Pi as a reacting species, for completeness and since the
extra computational strain of doing so is small. The actin subunit species tracked in
this model are distinguished by their polymerization state and by the hydrolysis states
of the nucleotide to which they are bound. We notate a species as G or F , to represent
globular (un-polymerized) or filamentous (polymerized) actin respectively, superscripted
by T , Pi, or D to represent that it is bound to ATP, ADP-Pi, or ADP, respectively; thus
for instance filamentous actin bound to ADP-Pi is notated FPi. We also include reac-
tions describing cross-linker (un)binding and myosin filament (un)binding and walking.
77
We exclude filament nucleation, severing, destruction, and annealing reactions, thus the
number of filaments is constant throughout the simulation trajectories. In Appendix D
we describe how we compute the change in Gibbs free energy for each reaction in this
set, as well as how we parameterize the simulations.
Lastly, we developed a mean-field model to describe just the dissipation resulting
from reactions in the CT model, i.e. excluding cross-linkers and myosin filaments. We
describe the model and present its results in Appendix D. We find, among other things,
that the steady state dissipation rate from filament treadmilling counter-intuitively does
not depend on the total amount of actin, but only on the number of filaments.
4.3 Results
4.3.1 Total dissipation rates of disordered networks do not increase
We studied dissipation rates accompanying the process of myosin-driven network self-
organization. We first excluded in these simulations the force-sensitivity of the reaction
rates describing cross-linker unbinding and myosin filament unbinding and walking. This
allowed us to understand a simplified version of the dynamics (we later discuss the
the effect of including mechanochemical feedback). We analyzed the trajectories of the
quantities ?Gdissipated, ?Gchem, dissipated, ?Gmech, dissipated, ?Gchem, and ?Gmech over
a set of simulations with identical initial concentrations but different random filament
distributions. In Figure 4.2, we display these trajectories averaged over 10 runs. We
used a simulation volume of 1 ?m3, divided into 8 compartments, and initial conditions
of equal amounts (10 ?M each) of GT and GD actin in a 0.08 ?M pool of seed filaments
containing F T , as well as 0.2 ?M myosin and 1.0 ?M cross-linkers. Similar trajectories
for other conditions are shown in the supporting information of Ref. 195. Points lying
outside 3 median absolute deviations (MAD) have been excluded for this visualization
[228].2
2For certain statistical aggregations in this chapter we use the MAD since it is the estimate of
scale most robust against outliers, having a breakdown point of 50%. However, we use the unscaled
version of the MAD and, as a result, do not claim that this statistic is a consistent estimator of the
standard deviation [229]. The distributions of most of the quantities of interest are too pathological
78
Figure 4.2: Top Left : Combined trajectories of 5 quantities tracked
during a MEDYAN simulation, averaged over 10 separate runs. The
color coding is indicated by the remaining panels. Note the close overlap
between ?G+ +dissipated and ?Gchem. In the remaining panels, the individual
trajectories are visualized with their standard deviations at each time
point over the 10 runs visualized as lighter curves above and below the
main curve. In the plots for ?G+ +dissipated and ?Gmech, the full range is
visualized, however this range is cropped in the other plots to aid visibility.
For each tracked quantity, there is an initial transient phase (lasting a few tens of
seconds) followed by fluctuations around a roughly steady value; we do not observe in
this set of simulations any slow approach to a significantly different total dissipation
rate, which might correspond to large reorganizations of the network. However for high
concentrations of cross-linkers (CCL = 5.0 ?M) and myosin (CM = 0.4 ?M), we observe
that the contribution of ?G+ +mech, dissipated to ?Gdissipated tends to increase relative to
that of ?G+chem, dissipated (see the supporting information of Ref. 195). It is known that
network percolation and collapse occurs only under certain conditions of myosin and
cross-linker concentrations [42, 221], and under these conditions which result in collapse
we observe an increase in ?G+ relative to ?G+mech, dissipated chem, dissipated (Figure 4.8).
to allow for a straightforward choice of scaling factor, so we simply us the unscaled version, defined as
MAD = mediani|xi ?medianjxj |, where xi are elements in the data set.
79
This indicates that more mechanical stress is being created by myosin filament walking
as the network collapses and becomes more densely cross-linked. The transient phase
corresponds to the initial polymerization of the seed filaments followed by the initial
mechanical coupling of filaments by cross-linkers and myosin filaments. Following the
transient phase, the networks in this study are generally disordered (Figure 4.3). The
dissipation rate corresponding to the initial polymerization of the seed filaments is much
larger than the chemical dissipation resulting from myosin filament activity. However,
following the transient phase, after which the treadmilling dissipation has reached its
steady state (Appendix D), the contribution from myosin filament activity outweighs
the dissipation resulting from filament treadmilling. Tracking the instantaneous rate of
change in ?G+chem resulting from each reaction separately, we observe myosin filament
walking to contribute the majority to ?G+chem after the transient phase, however the
amount depends on CCL and CM . The integrated contributions of each reaction to
?G+chem for different conditions are shown in Figure 4.4 and the supporting information
of Ref. 195. Interestingly, diffusion contributes an appreciable fraction to ?G+chem; plus
and minus ends of actin filaments tend to localize together (Figure 4.3) and through
treadmilling deplete the concentrations of GT and GD in certain reaction compartments
relative to others, leading to significant diffusion gradients on the scale of the com-
partment length. This suggests that the establishment of concentration gradients is an
important driving force in actomyosin self-organization, as has been noted in other work
[212, 230]. In the initial transient phase, the mechanical energy changes appreciably
as the filaments grow and are initially coupled to each other. Following this, however,
the rate of ?G+mech is on average near zero. This indicates that, despite the process
of mechanical stress generation through myosin filament activity and treadmilling, the
resulting stress is dissipated through fast relaxation such that, on a slower timescale, the
mechanical energy of the system does not change in a significant, persistent way.
80
Figure 4.3: Snapshot of a percolated actomyosin network in MEDYAN
under conditions CCL = 5.0 ?M and CM = 0.4 ?M . Actin filaments
are drawn in red, cross-linkers are drawn in green, myosin filaments are
drawn in blue, and the plus ends of filaments are drawn as black spheres.
81
Figure 4.4: Integrated contributions of each reaction in MEDYAN to
the total ?Gchem along a simulation trajectory with CCL = 1.0 ?M and
CM = 0.2 ?M .
82
Figure 4.5: Histograms and fitted probability distribution functions for
6 tracked quantities. For each histogram, the full trajectory for each of
10 runs is combined into a single data set. A log-normal distribution
was used to fit the histograms of ?G+ +dissipated and ?Gchem, a generalized
normal distribution was used to fit ?G+mech, and the rest were fit with
gamma distributions. All distributions are fit using the SciPy package to
determine shape, scale, and location parameters [231]. Quantities were
made positive or negative in order to produce the best fits.
Aggregating each of the 2000 s of the 10 trajectories into a collective data set for each
condition of CM and CCL, we next analyzed the distribution of the instantaneous rates
of ?G+dissipated, ?G
+ + + + +
mech, dissipated, ?Gchem, dissipated, ?Gstress, ?Gmech, and ?Gchem. In
Figure 4.5 we plot histograms of these 6 quantities for the conditions CCL = 1.0 ?M ,
CCL = 0.2 ?M . In the supporting information of Ref. 195 the same plots for other
conditions are displayed.
Each distribution contained heavy tails which, for the purpose of fitting, we sup-
pressed by excluding data lying outside 5 MAD?s of the median. No distributions were
sufficient to cleanly fit the full set of data, so we focus here on the center of the distribu-
tion and include a qualitative discussion of the heavy tails below. With the exception of
?G+mech which was fit with a generalized normal distribution, each distribution exhibited
83
significant skew and could be fit reasonably well with a log-normal or a gamma distri-
bution. At higher concentrations of myosin and cross-linkers, the histograms were less
cleanly fit by any standard distributions (see the supporting information of Ref. 195).
Log-normal distributions are fairly ubiquitous across different fields and systems
[232]. For instance, it has been shown that the distributions of concentrations of species
in a chemical reaction network are log-normal [233]. Gamma distributions are similarly
common and often difficult to discriminate from log-normal distributions [234]. A spec-
ulative explanation for the gamma distribution of ?Gstress is as follows: ?Gstress can be
viewed as resulting from a number of myosin filament steps that, according to the cen-
tral limit theorem, is approximately normally distributed given a sufficiently long time
between simulation snapshots tsnap (these stepping events are not truly i.i.d., but to a
first approximation we may assume they are). The main effect of each of these steps is to
increase the harmonic stretching potential on the myosin filaments as well as on the actin
filament cylinders by a roughly fixed stepping distance. Thus the increase in mechanical
energy, ?Gstress, is approximately a quadratic function of the normally distributed num-
ber of myosin filament steps per tsnap. As shown in Figure 4.6, the resulting distribution
of ?Gstress is well-fit by a gamma distribution and bears qualitative similarity to the
histogram of ?Gstress in Figure 4.5. In Figure 4.5, ?Gstress and ?G+mech, dissipated have
similar distributions, indicating that, for each MEDYAN cycle, almost all the stress accu-
mulated following Step 1 is then immediately relaxed. The remainder goes into ?G+mech,
whose distribution is centered on zero with little skew. Furthermore, the distribution of
?G+mech, dissipated has particularly heavy tails, indicating infrequent yet large relaxation
events. This tendency has some precedent in avalanche-prone systems, whose hallmarks
are self-organized criticality, intermittency, and scale-invariance in their distribution of
avalanche event sizes [161, 235?237]. While we do not observe true scale-invariance in
this set of simulations (i.e. power-laws cannot fit these distributions cleanly), it will be
interesting to continue to explore actomyosin networks dynamics in the framework of
self-organized criticality [166].
84
Figure 4.6: A simple model to rationalize the distribution for the
?Gstress rate at for the condition of CCL = 0.1 ?M , CM = 0.1 ?M .
Left : Histogram of the number of myosin filament steps per second mea-
sured over a set of 10 trajectories of 2000 s each. The fitted curve shows
the normal approximation to this distribution. Middle: The fitted nor-
mal distribution approximating the number of myosin filament steps per
second (? = 9.22, ? = 3.13) is sampled from 20,000 times. Each sample
of the number of myosin steps per second is squared and multiplied by
a constant ? ? 1, giving a sample of ?Gstress for a one second window.
This procedure models the increase in mechanical stress by a spring that
increases its length beyond equilibrium by a normally distributed number
of fixed-length steps each second. The resulting histogram is well fit by a
gamma distribution. Right : For comparison, the measured histogram of
?Gstress is shown. This drastically oversimplified model of stress genera-
tion in actomyosin networks thus captures qualitative features describing
the shape of this distribution.
85
4.3.2 More compact networks are more efficient
We simultaneously varied concentrations of myosin filaments CM , and cross-linkers, CCL,
which is known to produce a range of network architectures [42, 238]. Using cross-linker
concentrations of 0.1, 1.0, and 5.0 ?M , and myosin concentrations of 0.1, 0.2, and 0.4
?M , we studied the effects on ?G+ + +dissipated, ?Gchem, dissipated, and ?Gmech, dissipated. The
concentrations of actin subunits and filaments are the same as described above. For
each of the 9 conditions, we ran 10 simulations of 2000 s. In Figure 4.7, we display
the median values of ?G+ +chem, dissipated and ?Gmech, dissipated along each trajectory and
over each repeated trajectory for these conditions. The median is used here because of
its insensitivity to outliers such as are found in the heavy-tailed distributions of these
quantities [229].
We find that as CM is increased, the median rates of ?G+ +dissipated, ?Gchem, dissipated,
and ?G+mech, dissipated all tend to increase. Further, the value of of ?G
+
mech, dissipated
relative to ?G+chem, dissipated increases. As CCL is increased with CM fixed, then
?G+chem, dissipated tends to decrease. Increased concentrations of myosin filaments ob-
viously have a strong effect on the dissipation simply because they are the chief active
agents in the system once treadmilling has reached its steady state. We can define a
measure of efficiency for the present purpose as:
?G+ + +?Gstress mech, dissipated ??Gmech ?G? mech, dissipated? = + = + + + (4.6)?Gchem ?Gdissipated ??Gmech ?Gdissipated
where the approximation follows since, as illustrated in Figure 4.2, the average of ?G+mech
is close to zero. Thus it is evident that, perhaps surprisingly, as CM increases, ? increases:
the more motors are in our system, the more efficiently can chemical energy be con-
verted into mechanical stresses. Further, as CCL increases, ? tends to increase because
?Gchem, dissipated decreases relative to ?Gmech, dissipated. At higher levels of cross-linking,
86
Figure 4.7: Bar plot representing the contributions of ?G+chem, dissipated
and ?G+ +mech, dissipated to the total, ?Gdissipated. The letters in the abscissa
labels designate ?low,? ?medium,? and ?high.? The first letter represents
the concentration of myosin, CM : ?L? = 0.1 ?M , ?M? = 0.2 ?M , ?H?
= 0.4 ?M , and the second letter represents the concentration of cross-
linkers, CCL: ?L? = 0.1 ?M , ?M? = 1.0 ?M , ?H? = 5.0 ?M . The median
of each quantity is taken over 10 runs of 2000 s, and error bars represent
1 MAD.
87
which introduce mechanical constraints on the filaments, the walking of myosin motor fil-
aments will produce more stress compared to at low levels of cross-linking, when filament
sliding can result from myosin filament walking, producing less stress.
We repeated the above experiments with the inclusion of mechanochemical feedback
on the reaction rates controlling cross-linker unbinding and myosin filament unbinding
and walking (see the supporting information of Ref. 195). We observed, somewhat
surprisingly, no qualitative differences compared to the results described for the case of
no feedback, however there were quantitative differences in the stress production and
radius of gyration for certain conditions. These quantitative differences result from the
fact that, for the concentrations CM and CCL used above, we did not observe significant
collapse of the actomyosin network when mechanochemical feedback was included due
to the stalling and catching of the myosin filaments. As a result the networks were less
densely cross-linked, and the efficiency was lower. The total dissipation rates are largely
unaffected by the inclusion of mechanochemical feedback; it was primarily the degree
to which chemical energy had been converted to stress that was different for certain
conditions.
It is worthwhile to mention how these results compare with in vitro studies of dis-
sipation in actomyosin systems. While the computational approach described in this
chapter allows uniquely highly-resolved and direct measurement of free energy changes,
other experimental methods have produced qualitatively similar results to those obtained
here. Rheological experiments have determined that a single cell?s response to compres-
sion is similar in nature to a muscle?s response to increasing load, suggesting that the
actomyosin network underlies the cell?s mechanical responsiveness [224]. Further, this
responsiveness is modulated by blebbistatin, a myosin ATPase inhibitor, highlighting
myosin?s role in negotiating how the network rearranges in response to the sustained
stress. Using the metric of mechanical dissipation to measure the degree of structural
rearrangements and release of stress, we confirm that these processes indeed sensitively
88
Figure 4.8: Networks which tend to collapse also tend to increase their
ability to transduce chemical energy into mechanical stress. The change in
the radius of gyration averaged over the last 100 s of a 2000 s simulation,
Rg,f divided by the radius of gyration averaged over the initial 100 s,
Rg,i measures the degree of network collapse over the trajectory. The
same ratio is formed for the efficiency ?, by taking the median over the
same 100 s windows. These quantities are measured for each of the 9
conditions, and are averaged over the 10 repetitions of each. The error
bars indicate the standard deviations over th?ese s?ets of 10 measurements.
The radius of gyration is defined as 1 nRg = 2n i=1(ri ? rGC) , where
rGC is the geometric center of the actin network and ri is the position of
the ith bead in the network. The efficiency is defined as ? = ?Gstress .
?G+chem
89
depend on myosin activity, which we control here through its concentration. Additional
rheological studies have probed the mechanical dissipation of actin cortices more directly,
using the loss modulus as a readout. These have also indicated that inhibiting myosin
activity reduces the mechanical dissipation of the system, causing it to be behave more
elastically [223]. Lastly, we mention a recent study that quantified mechanical dissipa-
tion of actin filaments using a novel experimental method [34]. By measuring the flow
through a low-dimensional phase space defined by the amplitudes of the filament bend-
ing modes [32], they determine the entropy production of fluctuating actin filaments in
different phases of contractility. They relate the entropy production to the degree of
transverse bending of the filaments, as opposed to sarcomeric filament sliding, caused by
myosin filament walking. Similarly, we here relate the hindrance of filament sliding due to
cross-linker density to increased mechanical dissipation rates. Quantitative comparisons
of these two approaches to mechanical dissipation measurement will be an interesting
future direction. We note finally that a unique capability of quantifying dissipation using
MEDYAN is the ability to simultaneously measure the energetics of chemical reactions
in addition to the changes in the mechanical energy, which is not currently available
using in vitro methods.
4.4 Discussion
We have introduced a methodology for tracking the energetics of chemical and me-
chanical events during a MEDYAN simulation, allowing us to probe the properties of
actomyosin networks as dissipative active matter systems. The distinction between dissi-
pation?s mechanical and chemical origins is natural in the context of MEDYAN?s iterative
simulation procedure which carries out chemical stochastic simulation, mechanical de-
formation, and mechanical relaxation at separate times. As explained in Ref 42, this
procedure exploits a separation of timescales between the characteristic mechanical re-
laxation times of actomyosin networks [239] compared to typical waiting time between
reactions that introduce mechanical stresses, such as myosin filament walking [88] or
90
filament growth [12]. Ultimately the source of all dissipation is the chemical potential
of ATP molecules driving treadmilling and myosin filament activity. This is reflected by
the near equality of ?G+chem and ?G
+
dissipated in Figure 4.2. On a fast timescale (that of
the characteristic mechanical relaxation time), however, the free energy of chemical re-
actions cause small force deformations of the network which are then quickly and almost
fully relaxed. Thus, for a myosin filament stepping event, only a portion of the chemical
free energy ?G+chem, dissipated is immediately dissipated as heat, with the rest going into
temporarily increased mechanical energy of the actomyosin network, ?Gstress. The fast
relaxation of this new mechanical stress constitutes what we refer to as mechanical dissi-
pation, ?G+mech, dissipated, and the small residual stress after this relaxation has balanced
all net forces acting on the system results in a change of the mechanical energy of the
system on a slow timescale, ?G+mech.
One interpretive framework which is useful to understand the flow of free energy in
actomyosin networks is illustrated in Figure 4.9. We can think of the different forms of
free energy storage, including chemical potentials, mechanical stress, concentration gradi-
ents (which could also be considered as arising from chemical potential differences across
compartments), and dissipated energy, as nodes on a directed graph, where edges repre-
sent transduction of energy from one form of storage to another. The weights of these
edges represent the amount of free energy flowing through them. In this picture, we can
describe the process of network percolation, which occurs at increasing concentrations
of cross-linkers and myosin filaments, as widening the edge flowing from chemical po-
tential into mechanical stress, while thinning the edge from chemical potential directly
to dissipation. The edge weights corresponding to the establishment of concentration
gradients and the resulting diffusive dissipation will not be affected dramatically by the
onset of percolation except to the extent that percolated networks might lead to the
formation of more bundles, and therefore more significant concentration gradients. At
steady state, we have a stationary current on the graph, fueled by the chemical poten-
tial of the assumed limitless supply of ATP. Of course, at this level of description, we
91
Mechanical 
Stress
Mechanical 
Myosin Relaxation
Walking
Heat of 
Chemical      Reactions Dissipation
Potential
Treadmilling 
Diffusion
Concentration 
Gradients
Figure 4.9: A simple schematic illustrating the flow of free energy in
actomyosin network systems. Blue compartments represent forms of free
energy storage, arrows represent the transfer of free energy from one form
to another, and the green compartment indicates dissipation as ultimate
destination of all free energy flows in this non-equilibrium system. Arrows
are labeled with the mechanisms by which these free energy transduction
from one form to another are achieved. In this depiction, the sizes of
compartments and the widths of the arrows indicating the magnitudes of
the represented quantities are not to scale.
have coarse-grained away the details of the specific chemical reaction networks and me-
chanical potentials which constitute the system, however by doing so we gain a clearer
understanding of how percolation of the network alters the flows of free energy.
The new capabilities of MEDYAN allowing detailed energetics computations should
provide a way to address outstanding issues in the the fields of active matter systems
and of cell mechanics. In this study we observe dissipation rates to stay at a low fluc-
tuating steady state following a high initial transient phase, apparently in contrast with
the predictions of the dissipative adaptation hypothesis. However we also do not observe
slow reorganization of the actomyosin network into different higher order structures such
92
as bundles (for the concentrations of components used here we observe disordered net-
works only), and as a result we can not rule out that such reorganizations correspond to
marked changes in the dissipation rates. A dedicated test of the hypothesis of dissipa-
tive adaptation is in principle possible using this methodology. For instance, a simulated
gliding assay, which has been shown in vitro to lead to diverse dynamical patterns [67],
may indicate that these emergent patterns correspond to an optimization of the chemical
dissipation from myosin filament walking, as argued in Ref. 152. In the context of cell
migration, studies investigating the mechanically dissipative activity of myosin filaments
are also feasible if one incorporates in simulation an external substrate against which the
actomyosin network pulls. In this type of study, it should be straightforward to determine
how the amount of stress which is sustained against the substrate is altered by myosin
activity given the energetics calculations in the methodology described here. It is also
feasible to do simulated measurements of the dynamic shear modulus by compressing
the actomyosin network at different frequencies. It should then be possible to directly
observe the degree to which the dissipation of elastically stored energy is attributable
to myosin walking. In fact, this last research question has already been investigated to
some extent using an alternate computational model to the one presented here [240]. We
hope some of these potential future experiments will shed light on outstanding questions
in the studies of actomyosin networks.
93
Chapter 5
Graph representations, motor
stalling, and volume scaling effects
5.1 Introduction
In Chapter 4, we studied dissipation in actomyosin networks across a range of cross-
linker and motor concentrations which had a strong effect on the efficiency of chemical
to mechanical energy transduction. An important parameter whose impact we have not
yet discussed in much detail is the sensitivity of myosin motor stalling, which controls
the coupling between the motors? ATP consumption rate and the tension sustained by
the motors [87, 90]. Additionally, we have not yet discussed the role of the system size
in determining dissipation rates [25]. For larger systems, it is useful to introduce a more
detailed way to describe the structure of an actomyosin network than by, for example,
the radius of gyration. One approach to describing structure is to develop a graph rep-
resentation of the actomyosin network, allowing the use of metrics from the network
science literature to describe cytoskeletal structure [241]. With these considerations in
mind, the purpose of this chapter is three-fold: i) to describe a method for constructing
graph objects from actomyosin networks; ii) to show how motor stalling in percolated
networks leads to an appreciable decrease in the energy consumption rate; iii) to dis-
cuss how the graph objects and dissipation rates depend on the system volume. These
issues are interrelated, as a key determinant of whether significant motor stalling occurs
94
is the frustration of filament sliding caused by bound cross-linkers which is naturally
characterized using metrics defined on the graph representation. The system boundary
can also affect the graph metrics and dissipation rates, causing these quantities to scale
inhomogeneously with the system volume.
5.2 Methods
5.2.1 Graph mapping
Recent decades have seen an explosion in the development of network science, partly mo-
tivated by an increasing awareness of the prevalence of graph objects occurring through-
out the natural sciences [241?243]. These include the Internet, social networks, gene
regulatory networks, and many others [244?246]. Network science offers a way to sum-
marize key features of these different complex systems in a unified way, by abstracting
away the specific nature of the interaction between components to focus instead on how
the presence of these interactions (called edges) is distributed among the components
(called nodes). To use network science to describe actomyosin networks, we first need
a protocol which takes the actomyosin network?s configuration (i.e. positions and bind-
ing information of the filaments and associated proteins) and produces a graph object
G(E ,V) (i.e. a collection of nodes V and edges E). Any such protocol can be viewed as an
onto but not one-to-one (and hence not invertible) mapping, as we are inevitably discard-
ing some information from the full configuration in producing the graph representation.
This implies that the mapping is not unique, and indeed we can construct several differ-
ent protocols which capture different types of information about the full configuration
in the graph representation, as discussed below. In the mapping we use here, the nodes
V = {vi}
Nf
i=1 are the actin filaments (where Nf is the number of filaments), and they
are connected by un-directed weighted edges E N= {e fi,j}i,j=1 whose weights Ei,j = Ej,i
represent the number of passive cross-linkers (e.g. ?-actinin molecules) bound to the pair
of filaments i and j. By representing the pattern of cross-linker binding, this mapping
captures the topological structure of the actomyosin network and contains information
95
on the degree of filament frustration caused by mechanical constraints in the form of
bound cross-linkers [104].
A ubiquitous higher-order structural feature of actomyosin networks is a bundle of
several nearby and highly inter-connected (cross-linked) filaments [2, 247]. One could
even interpret these bundles as being the elementary objects in a coarse-grained repre-
sentation of the original actomyosin network. To express this idea in the language of
network science, we can identify bundles as communities (i.e. highly inter-connected
groups of nodes) in the graph representation [248, 249]. We can systematically deter-
mine these bundles using community detection methods such as the Girvan-Newman,
Clauset-Newman-Moore, or Louvain algorithms [248, 250, 251]. These methods produce
a partitioning of the nodes CK = {{v Ki}i?I }k=1, where the K communities are definedk
as the sets of node indices Ik. This is done in a hierarchical fashion over the range of
K values from 1 to Nf , producing a dendrogram of partitions. Each CK is chosen to
approximately maximize over all choices of K partitions the modularity
K ( )
Q 1
? ? kikj
K = Ei,j ? (5.1)
2m 2m
k=1 i,j?Ik ?
where m is the sum of all edge weights in the network, and ki = j Ei,j is the total edge
weight belonging to node i [249, 251]. The modularity reflects the amount of edge weight
contained within the communities relative to the total edge weight, including edges
between communities, in the graph. The value K? is then chosen as that which gives the
maximum modularity over all partitions the dendrogram, i.e. K? N= arg maxK{Q fK}K=1.
To implement community detection in our graphs representing actomyosin networks we
use the Louvain algorithm, a fast method which performs greedy optimization to locally
maximize QK over the K partitions.
With a partition CK? it becomes possible to create a coarse-grained network G?(E? , V?)
by collapsing the identified communities Ik into nodes v?k, which now have node weights
V?K = |Ik| (5.2)
96
Figure 5.1: Schematic illustration of the pipeline mapping an acto-
myosin network to a graph G, on which pruning, community detection,
and coarse-graining is performed to produce a new network G?. The color
of the nodes in G indicates the node?s community, and this color is pre-
sereved in the drawing of G?. The colors of the filaments in the actomyosin
networks also indicate the identified communities (bundles) of the fila-
ment, but these colors do not correspond to those in the drawings of G
and G?. To aid visibility of the bundles, a smaller actomyosin network is
shown on the left than the one used to produce G and G?.
given by the number of filaments in the bundle. The new edges e?k,l have weights E?k,l
which sum over those between the members of the corresponding communities:
??
E?k,l = Ei,j . (5.3)
i?Ik j?Il
We can also optionally prune the networks G by removing edges with a weight below
some threshold ET , which we usually take to be 5. This excludes connections which
correspond to very weakly cross-linked filaments and which aren?t mechanically coupled
in the way we consider bundled filaments to be. The pipeline summarizing the graph
mapping methods described above is schematically illustrated in Figure 5.1.
5.2.2 Volume scaling study
To study the effect of volume size on the structural properties and dissipation rates of
actomyosin networks, we held the protein concentrations fixed and varied the side length
of a cubic system in increments of 0.5 ?m from 0.5 ?m to 5 ?m. This corresponds to
volumes ranging from 0.125 ?m3 to 125 ?m3, covering four orders of magnitude (Figure
97
5.2). We also tested two conditions of myosin stalling force Fstall (i.e. the force at
which the motor heads stop walking; see Appendix A) and two concentrations of cross-
linkers CCL. The concentrations used are 12 ?M for actin monomers, 0.02 ?M for
actin filaments, 0.015 ?M for myosin minifilaments, and either 0.25 ?M (low) or 5 ?M
(high) for ?-actinin. With these concentrations of actin monomers and filaments the
mean filament length is ? 1.6 ?m. The two stall forces tested are 15 pN (easily stalled)
and 300 pN (hard to stall). We performed 7 trials for each of the 40 conditions of size,
cross-linker concentration, and stall force. Larger systems take longer to run and not all
systems completed the full 1000 s of requested simulation time, but interesting scaling
behaviors can still be observed.
To aid computational efficiency, we increased the cylinder size used to discretize the
filament backbones from 27 nm, used for system side lengths up to 1.5 ?m, to 108 nm,
used for system side lengths 2.0 ?m and greater. This switch occurs when the side
length exceeds the mean filament length. Because of this, the detailed mechanical prop-
erties of the networks are not comparable across the system sizes, and we instead focus
on topological and chemical scaling behaviors. In our investigations of the anomalous
mechanical energy fluctuations described in the Chapter 6 we perform a similar scaling
study over a smaller range of sizes, and in that study we do control for the cylinder size.
We also note for the remainder of the thesis the model of filament treadmilling is sim-
plified and does not distinguish between different nucleotide-bound actin subunits. This
aids in computational efficiency, and the mean-field model of treadmilling dissipation
described in earlier chapters already provides an essentially complete understanding.
The (de)polymerization rate constants of this simplified model, along with the other
simulation parameters used in this chapter, are displayed in Table E.1.
98
Figure 5.2: Visualization of 5 actomyosin networks with cubic bound-
aries whose side lengths range from 1 to 5 ?m in increments of 1 ?m. 5
additional network sizes than visualized here with non-integer side lengths
were also used in the volume scaling study.
5.3 Results
5.3.1 Self-organization produces percolated, coalesced networks
There are many available metrics in the network science literature which can be used
to describe various properties of graphs. Here we report on just a few such metrics
which display interesting scaling behavior with the system volume, however the mapping
described above opens the door for a much broader array of network features to be used
to characterize actomyosin network structure.
We first show in Figure 5.3 the degree of network percolation, as quantified by the
proportion ? of total filaments belonging to the largest connected component of the un-
pruned graph G. The trajectory of ?(t) is a sigmoidal curve across the range of conditions
tested, however ?(t) only saturates at 1, indicating full network percolation, when the
concentration of cross-linkers is high. Furthermore, for low concentration of cross-linkers
?(t) robustly saturates around 0.9 only for system volumes above 3.375 ?m3. Below this
volume the fluctuations appear to more significantly affect the trajectory which does not
reach a definite steady state value. We elaborate on the scaling of the fluctuations as a
function of system size below, when we discuss fluctuations of the dissipation rate. We
note that this separation in behavior at 3.375 ?m3 could be due to the change in the
99
cylinder length, but this it not likely to be the case since for other conditions and other
reported trends we do not see a similar separation at this volume. Finally, we observe
that there is minimal dependence of ?(t) on the stall-force of myosin.
Next we show that the number of identified communities normalized by the system
size, or the bundle density ?b, appears to reach a limiting value for large system sizes
at steady state (Figure 5.4). We refer to the aggregation of filaments into a small set of
bundles as coalescence of the network. As an intensive quantity, the density of bundles
should not depend on the system size as long as the system is large enough that boundary
effects become negligible. It should depend on protein concentrations, and we observe
the limiting density to be greater for the low cross-linker concentration than for the
high cross-linker concentration. This is expected since the filaments are less able to
coalesce into bundles at low cross-linker concentrations. The trajectories of ?b(t) for
large systems are non-monotonic, with an initial increase to a maximum ??b during the
initial polymerization of seed filaments and first cross-linking events based on random
filament proximity. The trajectories ?b(t) agree for all volumes up to this point, after
which they separate. Larger systems show a decrease after this due to the coalescing
of bundles together into a smaller number of communities. We conjecture that for the
smallest system size this coalescence does not occur (and ?b(t) continues to increase)
because too much of the actin mass is at the simulation boundary, where it cannot
interact with enough other filaments to form bundles. We also observe that for high
concentration of cross-linkers this coalescence process occurs at a faster rate compared
to low concentration. Finally, we also observe minimal dependence of these trends on
the stall force of myosin.
We next discuss the mean node connectivity of the coarse-grained graph G?, which
has an interesting but slightly ambiguous scaling behavior (Figure 5.5). Given a pair
of nodes vi and vj in a graph G, the connectivity between these nodes is defined as
100
Figure 5.3: Plots of the proportion ?(t) of total filaments belong-
ing to the largest connected component of the graph G(t). A) Av-
erages of trajectories over 7 trials of each system size are shown for
Fstall = 15 pN and CCL = 0.25 ?M . The uncertainty in these trajec-
tories is not visualized to aid visibility. The smallest volume 0.125 ?m3
is not visualized as its fluctuations are so large that it impedes visibil-
ity of the other trajectories. B) Averages of trajectories are shown for
Fstall = 300 pN and CCL = 0.25 ?M . C) Averages of trajectories for
Fstall = 15 pN and CCL = 5 ?M . C) Averages of trajectories are shown
for Fstall = 15 pN and CCL = 5 ?M . D) Averages of trajectories are
shown for Fstall = 300 pN and CCL = 5 ?M .
101
Figure 5.4: Plots of the density of bundles ?b(t). A) Averages of trajec-
tories over 7 trials of each system size are shown for Fstall = 15 pN and
CCL = 0.25 ?M (c.f. Figure 5.3). B) Averages of trajectories are shown
for Fstall = 300 pN and CCL = 0.25 ?M . C) Averages of trajectories for
Fstall = 15 pN and CCL = 5 ?M . C) Averages of trajectories are shown
for Fstall = 15 pN and CCL = 5 ?M . D) Averages of trajectories are
shown for Fstall = 300 pN and CCL = 5 ?M .
102
the minimum number of edges that is necessary to delete in order to remove all paths
connecting the pair, not accounting for edge weights. The mean node connectivity ? is
defined as the average of the node connectivity over all pairs of nodes in G. In the context
of actomyosin networks, ? can be interpreted as the number of force chains connecting
the typical pair of filaments or bundles. It is an expensive metric to calculate for large
graphs, but for the coarse-grained graphs G? it is easy to determine and informs on the
number of redundant paths at the level of filament bundles. ?(t) is generally observed to
monotonically increase and then plateau for all conditions and sizes. For an un-percolated
network, we expect that as the network size increases ? obtains a limiting value related
to the typical size of the connected graph components. This appears to be borne out in
the data for the low cross-linker concentration conditions. For a percolated network, we
expect that ? continues to increase as some function of the network size, since additional
nodes added to the percolated network create additional paths contributing to ?. This
expected behavior is observed up at least up to intermediate system sizes, after which
it is unclear whether the trajectories ?(t) begin to collapse on one another or continue
to increase with system size. The limiting value of ?(t) for percolated networks thus
appears to have a non-trivial dependence on the system size, but our investigations did
not find a definite empirical form for this scaling that holds for all sizes. This issue could
benefit from additional future research. Finally, we note again that there is minimal
observed dependence of these trends on the stall force of myosin motors.
5.3.2 Dissipation rates are extensive and depend on motor stalling
In previous chapters we introduced how the dissipation rate D? ? ?t?Gdissipated(t) is
dynamically calculated in simulation. Here we discuss how these rates depend on the
system size and the experimental conditions of stall force and cross-linker concentration.
We first show that the dissipation rate behaves like a first-order homogeneous function
of the system size, even for small systems. This means that D?(?V ) = ?D?(V ) for any
positive scaling parameter ?, i.e. D? ? V . The linear dependence on system size is
103
Figure 5.5: Plots of the mean node connectivity ?(t) for the coarse-
grained graph G?. A) Averages of trajectories over 7 trials of each system
size are shown for Fstall = 15 pN and CCL = 0.25 ?M (c.f. Figure
5.3). B) Averages of trajectories are shown for Fstall = 300 pN and
CCL = 0.25 ?M . C) Averages of trajectories for Fstall = 15 pN and
CCL = 5 ?M . C) Averages of trajectories are shown for Fstall = 15 pN
and CCL = 5 ?M . D) Averages of trajectories are shown for Fstall =
300 pN and CCL = 5 ?M .
104
illustrated in Figure 5.6 for data collected across all system sizes and averaged over the
time interval from 200 to 250 s and across the 7 trials. The standard deviation ?D?
pooled over the trials is displayed in the inset of Figure 5.6. We find that ?D? scales with
the square root of the system size, i.e. ? 1/2D? ? V . This behavior is familiar from the
theory of equilibrium statistical mechanics. For example, the relative fluctuations ?E/N
of the system energy E for an equilibrium canonical ensemble scales with the number of
particles N like N?1/2, so that ? ? N1/2E [132, 163]. It is interesting to find that the
same scaling is observed for the non-equilibrium thermodynamic quantity D?.
D? is an extensive quantity, and we could consider a corresponding density d? ? D?/V .
For this intensive quantity, we expect the fluctuations to decrease with the system size,
since ? = ? /V ? V ?1/2d? D? . For intensive quantities such as the energy consumption
rate density (Figure 5.8), bundle density (5.4), and the proportion of filaments in the
connected cluster (Figure 5.1), we indeed find the V ?1/2 scaling of the fluctuations to
be approximately obeyed (data not shown).
It is evident from Figure 5.6 that the condition corresponding to Fstall = 15 pN and
CCL = 0.25 ?M has a lower mean dissipation rate than the other conditions. We next
show that this results from the stalling of the myosin motors in percolated networks. In
Figure 5.7 we display the trajectories of chemical energy consumption by myosin motors
for the 27 ?m3 system, averaged over the 7 trials. The chemical energy consumption by
motors directly depends on their walking rate, which is slowed for motors with low stall
force under large tensions. Clearly, the low stall force causes the energy consumption
rate to decrease as the network self-organizes for this condition. Interestingly, the low
stall force alone is not sufficient to produce a decrease in the energy consumption rate,
as filament frustration from bound a high concentration of cross-linkers is also needed
to cause the typical tension experienced by motors to be large enough to cause stalling.
That the energy consumption rate only appears to decrease implies that the long-time
limit behavior corresponds to motors that tend to be maximally stalled for the given
105
Figure 5.6: The dissipation rate D? is shown as a function of the system
size and across the four experimental conditions, averaged as described
in the main text. The error bars show the standard deviation. A small
horizontal offset is added to the scatter plot data to aid visibility. Func-
tions of the form aV are fit to determine a for each condition and shown
as dotted lines. The inset shows the standard deviations as a function of
the system size for each condition. Functions of the form aV 1/2 are fit to
determine a for each condition and shown as dotted lines.
106
Figure 5.7: The chemical energy consumption rate by myosin motors
is shown for the 27?m3 system and for the four experimental conditions,
averaged over the 7 trials. The uncertainty in these trajectories is not
substantial for this system size and is omitted to aid visibility.
conditions. Finally, we show in Figure 5.8 how this trend depends on the system size
by plotting the chemical energy consumption rate normalized by the system volume
for the condition mentioned above. All trajectories of this rate collapse onto a single
curve, indicating that the system volume plays no appreciable role in the stalling-induced
decrease in the energy consumption rate.
107
Figure 5.8: The density of the chemical energy consumption rate by
myosin motors is shown for the condition Fstall = 15 pN and CCL = 5 ?M
as system size is varied, averaged over the 7 trials (c.f. Figure 5.3).
108
5.4 Discussion
The graph mapping protocol we have introduced can be placed in the context of other
possible mappings given the configuration of an actomyosin network. We briefly list
some other mappings:
1. Nodes: Filament plus or minus ends. Edges: Binary, representing whether nodes
are less than a distance cutoff from each other other.
2. Nodes: Filaments. Edges: Weighted, representing the total tension magnitudes of
cross-linkers or motors connecting the nodes.
3. Nodes: Myosin motors. Edges: Binary, representing whether nodes are less than
a distance cutoff from each other other.
4. Nodes: Filaments. Edges: Binary, representing whether the minimal distance
along the filament lengths is less than a distance cutoff [252].
The choice of a mapping reflects which property of the full configuration one wishes
to study. For instance mapping 1) allows studying the co-localization of filament end
points during self-organization, whereas mapping 2) allows to study more quantitatively
the force chains spreading throughout the network [104, 195]. The mapping discussed
used this chapter is useful to study the frustration of filaments due to the mechanical
constraints from bound cross-linkers. Thus, graph representations offer a flexible ap-
proach to describe abstract features of cytoskeletal networks and may be of use more
broadly in the study of soft-matter systems. One important consideration is the amount
of information available when constructing a graph representation. For instance, exper-
imental methods using fluorescence-based imaging will not be able to easily determine
the binding information of all cross-linking molecules. Such methods could determine
the proximity of various elements, though, allowing the use of mappings 1), 3), and 4).
Our volume scaling study was inspired by the powerful analysis tool of finite-size
scaling, which has been used to study the near-critical behavior of lattice-based statistical
systems [253, 254]. However, it is important to note that a 125 ?m3 cubic box of
109
actomyosin is of interest in more of a theoretical sense rather than an applied biological
one. In real cells, most of the actomyosin mass is concentrated at the cell cortex, a shell
of thickness 0.1? 1 ?m that sits underneath the cell membrane [255]. If we assume that
a typical cell is a spherical cow with a diameter of 20 ?m, then the volume of actomyosin
is in the range 124 ? 1135 ?m3, with an outer and inner surface area of 1256 ?m2 and
1231 ?m2 respectively. [209]. The surface area-to-volume ratio for the cortex is therefore
? 2?20 ?m?1, whereas for the cubic system it is 1.2 ?m?1. In our volume scaling study
we considered the extrapolation to an infinite bulk system, for instance to argue that
density of bundles approaches a fixed value (Figure 5.4). We argued that for small to
intermediately sized systems the boundary plays an appreciable role in preventing the
coalescence of bundles. We can therefore expect the boundary to play an important role
in the structure of cell cortices and conjecture that cortical actomyosin is less coalesced
than an equal amount of bulk actomyosin.
The observed dependence of the chemical energy consumption rate on motor stalling
can be placed in the context of several developments in the study of active matter sys-
tems. First, a recent study has highlighted how the force-sensitive catch-bond behavior
(in which unbinding is slower at high tension) of myosin motors in breaking the de-
tailed balance of motor binding reactions [256]. This was then argued to give rise to
network-level effects, such that highly force-sensitive motors produced a jammed, solid-
like phase with small actin flow compared to the liquid-like phase produced by less
force-sensitive or force-insensitive motors. This tendency of force-sensitivity to lead to a
actomyosin state characterized by diminished motor activity and associated actin move-
ment is also reflected in our results showing that myosin energy consumption steadily
decreases for highly sensitive motor stalling. In contrast, however, we do not find that
motor stalling sensitivity (which was not included in the study referenced above) is a
strong determinant for the topology of cross-linker binding, as evidenced by the minimal
change in the trends of the graph representation metrics. We note that in our study
the catch-bond force-dependence is held fixed at an experimentally validated value and
only the stalling sensitivity is varied. Second, a recent theory of active matter assemblies
110
makes predictions for the typical steady state behavior purely in terms of kinetic factors
[83, 156]. This ?low rattling? theory formalizes the basic intuition that a system will
tend to stay for a long time in a place that it is slow to leave from. Thus the typical
non-equilibrium steady state configuration of a system should be the state that has the
smallest associated kinetic factors leading the system out of it. The monotonic increase
in cross-linker-induced filament frustration and concomitant motor stalling implies a de-
crease in the motor walking kinetics. Thus the eventual actomyosin state is one which
the system will be the slowest to leave, at least via motor-induced transitions. This
seemingly agrees with the low-rattling idea. Future work could further elucidate to what
degree actomyosin dynamics can be understood in the context of this new theory.
111
Chapter 6
Understanding cytoskeletal
avalanches using mechanical
stability analysis
6.1 Introduction
The actin-based cytoskeleton is an active biopolymer network that plays a central role
in cellular physiology, providing the cell with a means to control its shape and produce
mechanical forces during processes such as migration and cytokinesis [2, 3, 257, 258].
These cellular-level forces arise from the collective non-equilibrium activity of molecular
motors interacting with the actin filament scaffold, enabling dynamic, driven-dissipative
cytoskeletal remodeling [106, 259, 260]. Recent experimental efforts have uncovered
a remarkable phenomenon exhibited by cytoskeletal networks in vivo: these networks
undergo large, sudden structural rearrangements significantly more frequently than pre-
dicted by a Gaussian distribution [168, 169]. Heavy-tailed distributions of event sizes
are well-known in seismology, where the Gutenberg-Richter law describes the power-law
relationship between the energy released by an earthquake and such an earthquake?s fre-
quency [162, 261]. Due to this analogy the term ?cytoquake? which we adopt here, has
been coined by experimenters to describe large cytoskeletal remodeling events. In previ-
ous work we have reported the first in silico observations of this phenomenon, appearing
112
as heavy tails in the distributions of mechanical energy released by cytoskeletal networks
[195]. These findings suggest that avalanche-like processes may play a fundamental role
in cytoskeletal dynamics.
The physics underlying cytoquakes is not well understood, as current explanations
based on experimental data are mostly speculative and rely on qualitative comparisons to
systems amenable to computational study which similarly exhibit non-exponential relax-
ation, such as jammed granular packings and spin glasses [168, 169, 262, 263]. In addition,
in previous studies little emphasis has been given to the possible biological roles played by
cytoquakes. We propose one such role, that, as suggested by the fluctuation-dissipation
theorem, these large mechanical fluctuations are concomitant with a large susceptibil-
ity, allowing the cytoskeleton to be highly sensitive to physiological cues arriving via
various cell signaling pathways [264]. Dynamic instability is already an acknowledged
feature of certain cytoskeletal components such as microtubules and filopodia [265]. A
similar design principle may also apply to larger cytoskeletal structures to allow fast
remodeling. For instance, avalanche-like dynamics may serve a useful purpose in the
lamellipodia of migrating cells, which probe local chemical gradients and must quickly
collapse protrusions in unsuccessful search directions [3]. However, to investigate such
possible biological roles we first need a more detailed account of cytoquake physics, which
is the subject of this chapter.
6.2 Methods
We study a subsystem of the full cytoskeleton called an actomyosin network. This con-
sists of semi-flexible actin filaments and associated proteins, including active molecular
motors (e.g. minifilaments of non-muscle myosin IIA) and passive cross-linkers (e.g.
?-actinin). The actin filaments hydrolyze ATP molecules in a directed polymerization
process which reaches a steady state called treadmilling. The myosin minifilaments
(?200 nm in length) transiently bind to pairs of actin filaments and also hydrolyze ATP
as fuel to walk along the filaments, generating motion and mechanical stresses. These
113
active process drive the network away from equilibrium. The cross-linkers (?35 nm)
bind more stably to nearby filaments, serving to transmit the force produced by motors
and to both store and through unbinding dissipate the resulting energy, heating the
environment [4, 71, 88, 93, 94, 170, 266]. Dissipation of stored mechanical energy also
occurs as filaments relax out of strained configurations, in a manner which can depend
on constraints from bound cross-linkers and motors. Additionally, the rates of motor
walking and unbinding as well as of cross-linker unbinding depend exponentially on the
forces sustained by these molecules, giving rise to nonlinear coupling between the me-
chanical state of the network and its chemical propensities [87, 267]. These means by
which the ability of the network to mechanically relax depends on its current state set the
stage for avalanche-like dynamics. An actomyosin network as represented in simulation
is visualized in Figure 6.1.
To computationally study cytoskeletal networks at high spatio-temporal resolution,
we use the simulation platform MEDYAN [42, 195, 247, 268, 269]. MEDYAN simulations
combine stochastic chemical dynamics with a mechanical representation of filaments
and associated proteins. Simulations proceed iteratively in a sequence of four steps:
1) stochastic chemical simulation for a time ?t (here 0.05 s), 2) computation of the
resulting new forces, 3) quasi-equilibration via minimization of the mechanical energy,
and 4) updating of force-sensitive reaction rates. Extensions to the MEDYAN platform
presented in Chapter 4 allow calculation of the change in the system?s Gibbs free energy
during each of these steps [195, 270], originally applied to study the thermodynamic
efficiency of myosin motors in converting chemical free energy to mechanical energy under
various conditions of cross-linker and motor concentration. We employ this methodology
here and focus on the statistics of the system?s mechanical energy U as it self-organizes.
We performed MEDYAN simulations of small cytoskeletal networks consisting of 50
actin filaments in 1 ?m3 cubic boxes with varying concentrations of ?-actinin cross-linkers
([?]) and of NMIIA minifilaments ([M ]). MEDYAN simulations combine stochastic
chemical dynamics with a mechanical representation of filaments and associated pro-
teins. Simulations proceed iteratively in a sequence of four steps: 1) stochastic chemical
114
Figure 6.1: A snapshot from a MEDYAN trajectory of an actomyosin
network in a 1 ?m3 box for the condition C3,3. Actin filaments are shown
in red, ?-actinin is shown in green, and myosin motors are shown in blue.
Beads representing the joined points (i.e. hinges) of thin cylinders (here
54 nm long) are visualized as red spheres. The cyan filaments represent
motion of the network corresponding to a soft, delocalized vibrational
mode determined from Hessian analysis, as described below. In the inset
we zoom in on part of the network and exclude associated proteins to
show greater detail of this vibrational motion.
115
simulation for a time ?t (here 0.05 s), 2) computation of the resulting new forces, 3) equi-
libration via minimization of the mechanical energy, and 4) updating of force-sensitive
reaction rates such the as slip-bonds of cross-linkers, catch-bonds of motors, and mo-
tor stalling. The boundaries of the box exert an exponentially repulsive force against
the filaments with a short screening length of 2.7 nm. Five concentrations of ?-actinin
(ranging from 0.17 to 5.48 ?M) and five concentrations of myosin miniflaments (ranging
from 0.003 to 0.08 ?M) were used with a constant G-actin monomer concentration of
13.3 ?M , in the regime of physiological concentrations [209]. This led to a steady state
filament length distribution with mean 0.48 ?m and standard deviation 0.26 ?m. We
label these conditions Ci,j , where i = 1, ..., 5 represents the rank of the cross-linker con-
centration and j = 1, ..., 5 represents the rank of the myosin motor concentration. The
length of the simulation cycle ?t was chosen as 0.05 s, although we explore dependence
on this parameter below (Figure 6.3). Five runs of each condition Ci,j were simulated.
We omit here other associated proteins, such as the branching agent Arp2/3, finding
that our minimal system is sufficient to produce heavy-tailed distributions of event sizes,
although it has recently been discovered that branching acts to enhance avalanche-like
processes [271].
6.3 Results
6.3.1 Asymmetric heavy-tailed distributions and correlations of ?U
We first characterize the observed occurrence of avalanche-like dynamics in these simu-
lations. The simulations begin with short seed filaments that quickly polymerize (tens
of seconds) to their steady state lengths. Following this, the slower process (hundreds of
seconds) of dominantly myosin-driven self-organization occurs which for most conditions
results in geometric contraction to a percolated network (see Movie 1) [94, 272]. The me-
chanical energy U(t) fluctuates near a quasi-steady state (QSS) value, which we analyze
as a stochastic process. In Figure 6.2.A we display the trajectory of U(t) for condition
C3,3 (with ?-actinin concentration [?] = 2.82 ?M , and motor concentration [M ] = 0.04
116
?M . We tracked the net changes of the mechanical energy ?U(t) = U(t + ?t) ? U(t)
resulting from each complete cycle of simulation steps 1) - 4).1 For the purpose of analyz-
ing the observed asymmetric heavy tails in the distribution of ?U , we treat the negative
increments ?U? (energy release) and positive increments ?U+ (energy accumulation) as
samples from separate distributions with semi-infinite domains. The complementary cu-
mulative distribution functions (CCDFs or ?tail distributions?, the probability P (X ? x)
of observing a value of the random variable X above a threshold x, as a function of x) of
the observed samples collected from all five runs at QSS are illustrated in Figure 6.2.B.
Both distributions display striking heavy tails relative to a fitted half-normal distribu-
tion. The CCDFs are better fit by stretched exponential (Weibull) functions of the form
P (X ? x) = e?((x/?)k) [273]. We justify this choice of distribution using Weibull plots,
as discussed in Appendix E. We find k = 0.60? 0.06 for |?U?| and k = 0.83? 0.07 for
?U+ with uncertainty taken over the five runs, indicating shallower tails for energy re-
lease compared to energy accumulation. We also measured the non-Gaussian parameter
? = ?x
4?
? 2?2 ? 1, where ?x
m? is the mth moment about zero; for a half-normal distribution
3 x
? = 0, and ? > 0 quantifies heavy-tailedness. We find ? = 11.37 ? 5.37 for |?U?| and
? = 1.96 ? 0.58 for ?U+. This, along with the shallower tails of the fitted stretched
exponential functions, indicates greater deviation from Gaussianity for energy release
compared to energy accumulation. These results support the picture that typically en-
ergy accumulates comparatively slowly and is released via large avalanche-like events.
The heavy-tailed distributions of |?U?|, the magnitudes of the negative energy in-
crements which are the chief subject of this chapter, may have strong dependence on
certain key parameters governing the mechanical equilibration protocol. To ensure that
these distributions are not artifacts of simulation we investigate whether changing the
parameters FT and ?t alters the qualitative properties of the distributions. In Figure 6.3
we compare these distributions using 3 runs for each parameter choice. Only weak depen-
dence on FT is observed. We find strong dependence on ?t, however for each parameter
1In previous chapters the symbol ?Gmech was used for ?U , the net change of mechanical energy
around the simulation cycle. ?U will be used throughout this chapter to keep the notation simpler. We
also use here the bracket notation for concentrations (e.g. [?]) which in previous chapters were notated
as CCL. It is hoped that the notation is clear from context and will not cause confusion.
117
A
2200
4000
2000
3000 1800
1600
2000 1400 1401
1000
0
0 250 500 750 1000 1250 1500 1750 2000
Time (s)
B
100
10 1
10 2
10 3
U < 0 kBT
U > 0 kBT
10 4
101 102
| U| (kBT)
C D
100 105
10 1
10 2 104
10 3
10 1 100 101 10 1 100 101
f (Hz) (s)
Figure 6.2: Statistics of ?U . A: Trajectory of the network?s mechanical
energy U(t) for condition C3,3. Inset: A blow-up of the trajectory to
show instances of rare events (|?U | > 100 kBT ) of energy release (blue)
and accumulation (green). B: CCDFs of |?U?| (blue) and ?U+ (green)
collected from five runs when the system is at QSS after 1000 s. Dotted
lines in lighter colors represent fits to the data of a half-normal CCDF, and
dashed lines represent fits of stretched exponentials. C: The normalized
power spectral density of U(t) for a single run at QSS from which the
spectral exponent ? = 1.72 is determined by fitting a power-law, shown
offset in red. D: The semivariogram obeys the scaling relationship ? ?
?2Ha over the scaling range.
S(f) CCDF U (kBT)
118
choice heavy tails exist and thus we may conclude that the cytoquake phenomenon is
not an artifact despite their frequency and magnitude having dependence on ?t. While
a smaller choice for FT and ?t should correspond more closely to reality, we find that for
the smallest of the tested values for these parameters the simulations did not complete
in the allotted computer wall time of 2 weeks. Thus our choices for these parameters
used in this chapter are chosen to be small while still allowing us to run full 2,000 s
simulations.
A
0
10
1
10
2
10 FT (pN)
0.1
3 1
10
5
10
4
10
0 1 2 3
10 10 10 10
B ?U (kBT)
0
10
1
10
2
10 ?t (s)
0.01
3 0.05
10
0.1
0.2
4
10
1 2 3 4 5
10 10 10 10 10
?U/?t (kBT/s)
Figure 6.3: A: Complementary cumulative distribution functions of
the negative increments |?U?| at QSS for various choices of the force
tolerance parameter FT plotted against fitted half-normal CCDFs. For
these runs condition C3,3 is used with ?t = 0.05 s. A: Complementary
cumulative distribution functions of the negative increments |?U?| at
QSS for various choices of the time between minimization, ?t. The energy
increments are normalized by ?t for more direct comparison between these
curves. For these runs condition C3,3 is used with FT = 2 pN .
We next analyze the temporal correlations of U(t) at QSS. A self-affine stochastic
time series G(t), for which G(t) and |?|HaG(t/?) have the same statistics for any scaling
parameter ?, has a power spectral density S(f) exhibiting a power-law dependence on
CCDF CCDF
119
frequency f : S(f) ? f??, where the spectral exponent ? is the persistence strength,
related to the color of the signal [274, 275]. We find ? = 1.72?0.02 for U(t) as shown in
Figure 6.2.C, painting U(t) as a pinkish brown signal; thus U(t) is non-stationary and
has temporally anti-correlated increments ?U . Self-affine time series further obey the
theoretical relationship ? = 2Ha+1 when 1 ? ? ? 3, whereHa is the Hausdorff exponent
determined from the scaling of the semivariogram ?(?) = 12(G(t+ ?)?G(t))
2 ? ?2Ha ,
and where the bar represents temporal averaging [159, 276]. We find that this rela-
tionship is satisfied by U(t), as shown in Figure 6.2.D, yielding Ha = 0.36 ? 0.01 and
confirming that U(t) is self-affine. Such non-Markovian and self-affine time series and
spatial patterns commonly arise in various complex geophysical processes (e.g. the tem-
poral variation of river bed elevation), further supporting the connection between the
cytoskeleton and earth systems [277, 278].
6.3.2 Distinguishing features of cytoquake events
We find that cytoquakes are correlated with several changes in the state of the network.
In Figure 6.4 we show that rare large events of energy accumulation correspond to a
greater than usual number of myosin motor steps whereas rare large events of energy
release correspond to greater than usual total displacement of the actin filaments and
a slightly greater number of linker unbinding events. These large total displacements
do not come from highly localized motions, and instead depend on many filaments each
being displaced more than usual (Figure 6.5). This agrees with the notion of cytoquakes
as collective structural rearrangements of the network.
We also observe cytoquakes to induce a spatial homogenization of the tension sus-
tained by the network during large events of energy release, as qu?antified by changes in
the Shannon entropy of the spatial tension distribution H(t) = ? ijk Pijk(t) lnPijk(t).
The tension distribution Pijk is constructed by discretizing the simulation volume of
1 ?m3 into a grid of 103 voxels indexed by i, j, k, and computing the proportion of the
total network tension belonging to the mechanical elements (filament cylinders, cross-
linkers, and motors) inside each voxel; additional details can be found in Appendix E.
120
A B
140 *** *** 12 * ***- *** *** ***
120 10
100 8
80
6
60
4
40
20 2
0 0
( , 100) ( 100, 0) (0, 100) (100, ) ( , 100) ( 100, 0) (0, 100) (100, )
C D
20.0 ** ***
*** ******
** ***
17.5 - 0.2
15.0
0.1
12.5
10.0
0.0
7.5
5.0
0.1
2.5
0.0
( , 100) ( 100, 0) (0, 100) (100, ) ( , 100) ( 100, 0) (0, 100) (100, )
Figure 6.4: A: Differences in the total filament displacement between
simulation cycles for which ?U is less than ?100 kBT , cycles for which
?U ? (?100 kBT, 0 kBT ), cycles for which ?U ? (0 kBT, 100 kBT ),
and finally cycles for which ?U is greater than 100 kBT . To compare
these distributions, the two-sided p-value of the Wilcoxon rank-sum test
between pairs of cycle types is reported as being either not significant:
- (p ? 0.05), significant at level 1: * (p < 0.05), at level 2: ** (p <
0.01), or at level 3: *** (p < 0.001). The total filament displacement is
computed as the sum over all filaments of the distance between a filament
at time t and that filament at time t + ?t. The calculation of distance
between filaments is described in Figure E.3. Since there many more
simulation cycles for the categories ?U ? (?100 kBT, 0 kBT ) and ?U ?
(0 kBT, 100 kBT ), we took a random sub-sample (? 300 each) of all events
for these categories to be roughly equal to the number of events for which
?U < ?100 kBT and for which ?U > 100 kBT . In these combination
violin and box-and-whisker plots, the red circle represents the mean, the
red bar represents the median, and the notches in the box represent the
95% confidence interval of the median. B: Differences in the number of
motor walking events between the different cycle types as just described.
C: Differences in the number of ?-actinin unbinding events between the
different cycle types. D: Differences in the Shannon entropy of the spatial
tension distribution of network tension.
Linker Unbinding Events Total Filament Displacemnt (nm)
H (nats) Motor Walking Events
121
4.0
?U Range
3.5 ( , 100)
( 100, 0)
3.0 (0, 100)
(100, )
2.5
2.0
1.5
1.0
0.5
0.0
0 10 20 30 40 50
Rank
Figure 6.5: Rank-size distribution of the displacements experienced by
each of the 50 filaments during simulation cycles when ?U is in different
ranges, in units of kBT . For each cycle, the filaments are ranked ac-
cording to their displacement and these ranks are plotted against the
corresponding displacement. The average and standard deviation of
these rank-displacement curves are taken over each cycle in a given cat-
egory. The curves for the categories ?U ? (?100 kBT, 0 kBT ) and
?U ? (0 kBT, 100 kBT ) are almost exactly coincident. This data is
collected from one run of condition C3,3 at QSS.
6.3.3 Testing different concentrations and system sizes
We next discuss how these results generalize to different concentrations of associated
proteins and different system sizes, and then connect our results to existing experiments.
Five concentrations of ?-actinin (ranging from 0.17 to 5.48 ?M) and five concentrations
of myosin miniflaments (ranging from 0.003 to 0.08 ?M) were tested with a constant G-
actin monomer concentration of 13.3 ?M , in the regime of physiological concentrations
[209]. At the lowest concentrations of cross-linkers and motors, the network did not
contract, representing a very different actomyosin phase which we omit comparisons
to. For all of the conditions producing contracting networks, we found that asymmetric
Displacement (nm)
122
heavy-tailed distributions of ?U persist, with large values of the non-Gaussian parameter
for |?U?| (? ? 5 ? 20) and ?U+ (? ? 2 ? 5), although ? for negative increments was
observed to decrease with the motor concentration (Figure 6.6). We conclude that the
avalanche-like energy fluctuations discussed above are not highly sensitive to associated
protein concentrations. These fluctuations may depend on the parameters of the force-
sensitive reaction rates (which are taken here to correspond to experimental values), but
we leave this interesting question for future work.
To check the robustness of anomalous energy fluctuations in cytoskeletal networks,
we measured the non-Gaussian parameters ? for |?U?| and ?U+ at a range of con-
centrations of cross-linkers and myosin motors. For the lowest concentrations of these
proteins, the networks did not contract (conditions C1,j and Ci,1, i, j = 1 . . . 5), and we
omit them from the comparison. The results are displayed in Figure 6.6. Despite a trend
toward smaller ? for |?U?| with increasing motor concentration, we may conclude that
the asymmetric heavy tails of ?U persist across these conditions.
123
[ ] ( M)
20 1.49
2.82
4.15
5.48
15
10
5
0
0.03 0.04 0.05 0.06 0.07 0.08
[M] ( M)
Figure 6.6: Plots of the non-Gaussian parameter ? for the distributions
of |?U?| (solid lines) and of ?U+ (dashed lines) at QSS for various con-
centrations of myosin motor ([M ]) and ?-actinin cross-linkers ([?]). The
mean and standard deviation is shown over five runs of each condition.
A small horizontal offset is added to the points to ease visibility.
We performed a finite-size scaling study by holding the concentrations of condition
C3,3 fixed (with ?-actinin concentration [?] = 2.82 ?M and motor concentration [M ] =
0.04 ?M) and varying the system volume V . Holding the concentrations of all chemical
species for condition C3,3 fixed, we performed simulations using cubic volumes with side
lengths ranging from 0.5 to 2.5 ?m. Larger systems reach QSS at later times, and our
simulations of larger systems did not reach QSS in the allotted computational time. As
a result, we collected samples of |?U?| for these systems on the approach to QSS, from
300 to 800 s, once the networks had all nearly fully percolated (i.e. nearly all filaments
belonged to a single component connected by cross-linkers), trusting that the relevant
scaling behavior could still be observed. Stretched exponential functions approximately
fit the distributions of ?U+ and |?U?| for all system sizes (see Figure 6.7.A for the fits
124
of |?U?|). Larger systems displayed steeper tails as indicated by the observed power-
law decay of ? for |?U?| and ?U+ (Figure 6.7.B), although interestingly ? for |?U?|
is larger than that for ?U+ by a constant factor of roughly 3 for all systems sizes.
The steeper tails are also evidenced by the slow growth of the Kohlrausch exponents
k with V (Figure 6.7.C). Thus, the distributions of energy release and accumulation
across the entire network become narrower and more Gaussian for large systems. This,
in contrast to driven-dissipative systems that exhibit self-organized criticality (SOC),
suggests the existence of some intrinsic and finite scale for avalanche-like releases of
energy in cytoskeletal networks. By summing over many local energy fluctuations of this
finite scale, the distribution of the fluctuations in the total energy U becomes increasingly
Gaussian for large systems owing to the central limit theorem. This intrinsic scale may
be partly determined by the non-conservative transfer (dissipation) of mechanical energy
as it spreads through the network during avalanches [4, 164].
6.3.4 Connection to experiments
Existing experimental definitions of cytoquakes define them as large local displacements
of the cytoskeleton probed using transmembrane attached microbeads or flexible micro-
post arrays, rather than as large changes in the cytoskeleton?s total energy U as done
here [168, 169]. To compare our results to experiments, we make the corresponding local
measurements of the displacements of individual filaments, finding that the resulting dis-
tributions are heavy-tailed with values of the non-Gaussian parameter for most filaments
in the range ? ? 1? 5, in semi-quantitative agreement with in vivo measurements (Fig-
ure 6.8) [169]. We also find our estimate of typical actin filament displacement speeds
(? 10 nm/s) to be consistent in order of magnitude with separate in vitro measure-
ments of contractile networks [69]. In addition, we find that the heavy tails of these local
measurements persist even for large system sizes, whereas global measurements such as
the total energy or total filament displacement become increasingly Gaussian for large
systems. This distinction between local and global measurements may be important in
interpreting future studies of anomalous statistics in cytoskeletal self-organization. We
125
A
100
10 1
10 2
V ( m3)
0.125
10 3
1.0
3.375
8.0
15.625
10 4
10 1 100 101 102 103
U/V (k T/ m3B )
B C
102 U < 0 kBT 1.4
U > 0 kBT
1.2
101 1.0
0.8
100
0.6
0.4
10 1
10 1 100 101 0 5 10 15
V ( m3) V ( m3)
Figure 6.7: A: CCDFs of |?U?| normalized by the system volume V
collected for 5 runs for increasing system sizes plotted against the fitted
stretched exponential functions. B: The non-Gaussian parameter ? for
|?U?| and ?U+ are plotted with uncertainty taken over the different
runs. C: The Kohlrausch exponents k for |?U?| and ?U+.
CCDF
k
126
finally mention in connection to experiments that it has recently been argued that more
detailed understanding of mechanical dissipation by cytoskeletal networks should help
to more precisely control traction-based measurements of cellular force production [170].
Previous in vivo studies have framed cytoquakes as large local displacements of the
cytoskeleton, which were measured by tracking the motion of apically attached mi-
crobeads as well as the deflection of micropost arrays under the basal surface [168, 169].
Here, we have leveraged the information accessible from simulation to frame cytoquakes
as large releases of stored mechanical energy, which we show typically correspond to large
global displacements. This energy-centered point of view deepens the analogy between
cytoquakes and earthquakes, which are thought to result from an avalanche-like process
of energy accumulation and release in the earth?s crust that manifests as occasional large
displacements [159, 276]. Our measurements of displacements in Figures 6.4 Figure 6.5
are global in the sense of being summed for each simulation cycle over all filament in the
network, capturing more modes of network buckling than the local displacement mea-
surements obtained experimentally, e.g. using attached microbeads. However, we can
also make a similar local measurement to compare to experiments. Rather than sum-
ming over all filaments, we track each filament individually and measure the set { }N? ff f=1,
where Nf = 50 is the number of filaments, of the non-Gaussian parameter corresponding
to each filament f ?s distribution of displacements from 300 to 800 s. Displacements are
calculated according to Figure E.3. The displacements experienced by a filament should
depend on the filament?s location in the network, but we observe for nearly all filaments
a heavy-tailed distribution characterized by ?f > 0, as shown in Figure 6.8. In addition,
we find the distribution of ?f itself to be heavy-tailed, which appears to agree with ex-
periments using micropost arrays [169]. Finally, we observe similar values of ?f across
simulation volumes V .
These local measurements, obtained by tracking each filament individually, can be
compared to global measurements, obtained by summing over every filament to get the
total displacement. The distribution of total displacements is closer to Gaussian, charac-
terized by ? ? 0 for most volumes tested (Figure 6.8). As with the increasing Gaussianity
127
of ?U for large systems, this can be attributed to the central limit theorem since many
filaments were summed over to determine the total displacement. We therefore conclude
that in large systems, metrics can be heavy-tailed when measured locally but Gaussian
when measured globally.
25
20
15
10
5
0
0 5 10 15
V ( m3)
Figure 6.8: Plots of the distributions of non-Gaussian parameter ?f
for the distributions of displacements of individual filaments, for different
simulation volumes V . For a single run of each V , the displacement of
each filament from time point to time point in the system is tracked and,
from these time series, ?f is determined for each filament f . The box and
whisker plots summarize the distribution of ?f for all filaments in the
system, with the median shown as the red bar, the box extending from
the first to the third quartiles, and the whiskers extending across the range
of ?, omitting outliers. The diamonds indicate the value of ? obtained
when, instead of tracking each filament?s displacement individually, the
total summed displacement of all filaments from time point to time point
is tracked. These global measurements of displacement are more Gaussian
(with ? ? 0) than the corresponding local measurements obtained from
tracking filaments individually.
128
6.3.5 Normal mode decomposition probes network?s mechanical state
Having described the statistics of the increments ?U , we next aim to connect the occur-
rence of cytoquakes, defined as large values of |?U?|, to the cytoskeletal network?s me-
chanical stability. To this end we implemented a method to compute the Hessian matrix
H of the mechanical energy function U . The eigen-decomposition of H is ? = {?k}3Nk=1,
where 3N is the number of mechanical degrees of freedom in the system, which comprises
N ?beads? that are used to discretize the actin filaments. ? is related to the mechani-
cal stability of the cytoskeletal network: the eigenvectors vk are the normal vibrational
modes of the network, and the eigenvalues ?k indicate the stiffness (|?k|) and stability
(sgn(?k)) of the corresponding mode. An example vibrational mode is illustrated in Fig-
ure 6.1. We draw inspiration for studying ? in the current context from several sources:
in single-molecule molecular dynamics studies, the saddle-points of U (i.e. points in
the landscape with some imaginary frequencies) are associated with transition states
[280, 281]; studies of polymer networks show that internal stresses produce non-floppy
vibrational modes even below the isostatic threshold [282]; in simulations of glass-forming
liquids, the instantaneous normal mode spectrum allows inference about proximity to
the glass transition and determination of incipient plastic deformation regions [283?285];
in deep learning models for predicting earthquake aftershock distributions, it was found
that certain metrics also related to stability (e.g. the von-Mises criterion) are informa-
tive model inputs [286, 287]. We describe some interesting observed trends on metrics
defined on ? below.
In MEDYAN, semi-flexible filaments are represented as a connected sequence of thin
cylinders whose joined endpoints (i.e. hinges) are called beads. The set of potentials
defining the mechanical energy of the filaments and associated proteins is outlined below.
The mechanical energy U is a function of these beads? positions, and elements of the
Hessian matrix are defined as
?2H U ??Fi?i?,j? = = = ?
?Fj?
, (6.1)
?xi??xj? ?xj? ?xi?
129
where xi? is the ?th Cartesian component of the position of the ith bead. We have
? = x, y, z and i = 1, ..., N where N is the number of beads in the network, so H is
a square symmetric 3N -dimensional matrix. The number of beads N(t) will change as
filaments (de)polymerize; in these simulations, at QSS a single filament of length 0.5 ?m
comprises ? 10 cylinders (11 beads), each ? 50 nm in length. After each mechanical
minimization, H(t) is constructed by numerically computing the derivatives on the right
of Equation 6.1. The derivative ?Fi??x is found using a second-order central differencej?
approximation by moving the jth bead in the ?? directions by a small amount and
determining the changes in the force component Fi? [288]. Due to issues of numerical
accuracy, we do not assume the symmetry of the matrix H, but instead directly compute
each component Hi?,j? and then symmetrize the result: 1(H?2 + H)?H.
We next describe some interesting observed trends of metrics defined on ?. We
distinguish between unstable, stable, soft, and stiff modes: for unstable modes ?k < 0,
for stable modes ?k ? 0, for soft modes 0 ? ?k < ?T , and for stiff modes ?k ? ?T ,
where we define the threshold ?T = 40 pN/nm to discriminate between the twin peaks
in the density of states (Figure 6.11.B). The set {? }3Nk k=1 is visualized with these modes
labeled in Figure 6.11.A for a QSS time point of condition C3,3. A very small number of
unstable modes persist after each minimization cycle, later iterations stopping once the
maximum force on any bead is below a threshold FT (here 1 pN). Thus the minimized
configurations are in fact saddle-points of U ; this is expected as it is known from the
theory of minimizing loss functions that the ratio of saddle-points to true local minima
increases exponentially with the dimensionality of the domain [289]. We expect that
in the space of all possible network topologies (i.e. patterns of cross-linkers and motors
binding to filaments), the energy landscape will be rugged, leading to the well-appreciated
glassy dynamics of non-equilibrium cross-linked networks [109, 133]. For a fixed topology,
however, which is the result of the chemical reactions occurring during step 1) of the
iterative simulation cycle, the energy landscape should be smooth (i.e. non-rugged)
with respect to the beads? positions, with a single nearby local minimum being sought
during mechanical minimization in step 3). The residual unstable modes are therefore
130
thought to be an unimportant artifact of thresholded stopping in the conjugate-gradient
minimization routine, and not representative of some physical feature of cytoskeletal
networks. The observed quantitative dependence of the number of residual unstable
modes on FT supports this conclusion and is illustrated in Figure 6.9.
0.0200
0.0175
0.0150
0.0125
0.0100
0.0075
0.0050
0.0025
0.0000
0 2 4 6 8 10
FT (pN)
Figure 6.9: Scatter plot showing the fraction of negative eigenvalues
remaining after mechanical minimization when different choices of the
parameter FT are used. The data is collected from QSS for 3 runs of
C3,3, with the standard deviation taken over time and over the runs.
We quantify the number of degrees of freedom involved in a given normalized eigen-
vector vk using the inverse participation ratio [283]:
???? ???1N 3rk = (v 4k,i?) . (6.2)
i=1 ?=1
If the eigenmode involves only one degree of freedom, then one component of vk will
be one and the rest will be zero, and rk = 1. On the other hand, if the eigenmode is
evenly spread over all 3N degrees of freedom, then each component vk,i? = (3N)?1/2,
and rk = 3N . In Figure 6.11.B we plot rk for the unstable, soft, and stiff modes along
Fraction Negative Eigenvalues
131
with the density of states, showing that the soft modes involve many degrees of freedom
while the stiff and unstable modes are comparatively localized.
We find that the mean value ?rk? varies non-monotonically with myosin motor con-
centration [M ] and ?-actinin concentration [?] (Figure 6.11.C). To understand this trend
we implemented a mapping from the cytoskeletal network into a graph and measured its
mean node connectivity, a purely topological measure of network percolation. We con-
struct a graph to represent the cross-linker binding topology of cytoskeletal networks.
Nodes in the graph correspond to actin filaments, and weighted edges (which may be
thresholded and converted to binary edges in an unweighted graph) correspond to the
number of cross-linkers connecting the pair of filaments. The mean node connectivity
is defined as the average over all pairs of nodes in the unweighted graph of the number
of edges necessary to remove in order to disconnect them, thus quantifying the typical
number of force chains between filaments, or equivalently the extent of network perco-
lation [104, 290]. Revealingly, the mean node connectivity correlates closely with ?rk?
for the stable modes across the various conditions Ci,j (Figure 6.11.D). We also find the
number of connected components of H and of the graph?s adjacency matrix to match
for most time points, supporting this connection between network topology and stable
mode delocalization. Intermediate concentrations of myosin motors enhance the network
percolation, but as [M ] continues to increase the motors act to disconnect cross-linked
network structures causing the mean node connectivity and ?rk? to decrease.
We observe that as a network contracts and becomes percolated during the process
of myosin-driven self-organization, the stable modes steadily delocalize (?rk? increases)
and stiffen (the geometric mean ??k?g increases), as shown in Figures 6.11.E and Figures
6.11.F. During this process we also witness a qualitative change in the level spacing
statistics of the very soft and delocalized modes (?k < 10 pN/nm, rk > 100) from
a Poisson to a Wigner-Dyson distribution (Figure 6.10). This indicates that in the
percolated state these vibrational modes interact and exhibit level repulsion, similar to
soft particles near the jamming transition [169, 262, 291]. Future studies may reveal
further similarities between these systems and other marginally stable solids [108, 133].
132
t = 15 s
1.2
t = 2000 s
1.0
0.8
0.6
0.4
0.2
0.0
0 1 2 3 4 5
/
Figur?e 6.10: Histograms of the level spacings ?? = ?k+1 ? ?k, where
?k = ?k are ordered so that ?k increases as k increases, normalized
by their average ?? for the very soft (?k < 10 pN/nm) and delocalized
(rk > 100) vibrational modes at different times of a run of condition
C3,3. The Poisson distribution p(??/??) = e???/?? and the Wigner-
? 2
Dyson distribution p(??/??) = ?2 (??/??)e
? 4 (??/??) are plotted as
red and blue solid lines. This transition in distributions signifies that in
the percolated network at 2000 s the frequencies of these modes are no
longer randomly spaced and begin to interact, exhibiting level repulsion
for small ??/??.
p( / )
133
A B
400
8000
Unstable
350 0.07
7000 Soft
6000 Stiff 300 0.06
5000 250 0.05
4000 200 0.04
3000 150 0.03
2000
100 0.02
1000
50 0.01
0
0 0.00
0 500 1000 1500 4 2 0 2 410 10 10 10 10
C Rank D |?k| (pN/nm)
140
8
120
100 6
80 [?]  (?M)
4
1.49
60 2.82
2 4.15
40 5.48
0.00 0.02 0.04 0.06 0.08 0.00 0.02 0.04 0.06 0.08
E [M] (?M) F [M] (?M)
160 12
140
10
120
100
8
80
60 6
40
4
20
0 500 1000 1500 2000 0 500 1000 1500 2000
Time (s) Time (s)
Figure 6.11: Metrics defined on Hessian eigen-decomposition. A: Or-
dered eigenspectrum {? }3Nk k=1 at a QSS time point for condition C3,3.
B: Scatter plot of the pairs |?k|, rk (circles) plotted against the density
of states (solid lines), i.e. the proportion of eigenvalues between ? and
?+d?. C: The mean value at QSS of ?rk? for the stable modes for various
conditions Ci,j . The conditions C1,j with low linker concentrations are
not visualized as these networks did not percolate and obscure visualiza-
tion for the remaining conditions. The mean is taken over the last 500 s
and over different runs. D: The mean value of the mean node connectiv-
ity for various conditions. E: Trajectories of ?rk? of the stable modes as
the network self-organizes for the conditions C2,3, C3,3, C4,3, and C5,3,
with the mean and standard deviation taken over the different runs. F:
Similar trajectories of ??k?g of the stable modes.
? ? ? ?
r r ?k (pN/nm)k k
? ?
?k g (pN/nm)
rk
Mean Node Connectivity
Density of States
134
6.3.6 Cytoquake forecasting
Can the eigen-decomposition of the Hessian matrix be used to forecast cytoquake oc-
currence? Intuition suggests that, in analogy with the connection between imaginary
frequencies (i.e. unstable modes) and molecular transition states, the vibrational modes
of the cytoskeletal network may contain information that a large structural rearrange-
ment is poised to occur [280, 281]. To test this idea, and without detailed a priori
knowledge about which features in ? would be informative, we implemented a machine
learning model using the eigen-decomposition as the input and outputting the predicted
probability of observing a large event of energy release (?U < ?100 kBT ) occurring
within the next 0.15 s. As detailed in Appendix E, we found that, indeed, the Hessian
eigenspectrum ? contains sufficient information to forecast cytoquake occurrence with
significant accuracy compared to a random model. We first reduced the dimensionality
of ?(t) using principal component analysis, finding that 30 dimensions sufficed to ex-
plain > 95% of the variance across time points, and then used the reduced input in a
three layer feed-forward neural network. We validated our model using receiver operat-
ing characteristic curves, achieving an area under the curve (AUC) of 0.70 when using
data from five runs of condition C3,3. This improvement in prediction performance over
a random model (which would have an AUC of 0.5) implies that mechanical instability,
as encoded in the Hessian eigenspectrum, precedes the occurrence of cytoquakes.
6.3.7 Cytoquakes have enhanced displacement along soft modes
To further study the connection between cytoquakes and mechanical stability, we mea-
sured the projections of the network?s displacements onto the vibrational normal modes
{v 3Nk}k=1. Network displacements d(t) were found by tracking the movement of each
of the N(t) beads during simulation cycles. As a working approximation, beads that
depolymerized during a cycle were assigned a displacement of 0, and beads that newly
polymerized were not assigned elements in d(t). The 3N -dimensional displacement vec-
tors d were then normalized to have unit l?ength. We define dk = d ?vk as the projections
of d onto the eigenmodes vk, which obey d2k k = 1 owing to the normalization of d and
135
vk. Thus the quantity d2k is the weight of the dis?placement d along the kth eigenmode.
With this we define the effective stiffness ?P = k d
2
k?k as the displacement-weighted
average of the eigenvalues. In Figure 6.12 we display a scatter plot of the pairs ?U(t),
?P (t) measured during QSS for a run of condition C3,3, along with a kernel density
estimate of their joint probability density function (PDF). We distinguish between soft
(0 ? ?k < ?T ) and stiff (?k ? ?T ) eigenmodes, where ?T = 40 pN/nm separates the
twin peaks in the density of states (Figure 6.11). The structure of the joint PDF is
markedly asymmetric about ?U = 0 and shows that ?P during cytoquake events is
almost always soft, whereas for all other simulation cycles ?P could be soft or stiff with
2
similar probabilities. We also consider dn = kk r as the weight of the displacement alongk
eigenmode k per degree of freedom involved in the eigenmode, where rk is the inverse
participation ratio [283]. We define nsoft and nstiff as the mean of nk over the soft and
stiff subsets. Values of nsoft/nstiff for different simulation cycle types are displayed in
the inset of Figure 6.12, showing that nsoft < nstiff typically only during cytoquakes.
Based on this analysis, we conclude that during the large collective rearrangements cor-
responding to cytoquakes, cytoskeletal networks exhibit enhanced displacement along
the soft vibrational modes. We qualify these results by observing that, since cytoquakes
involve particularly large network displacements, it may be inappropriate to interpret
them using the local harmonic approximation to U implicit in Hessian analysis [285]. In
addition, changes in network topology from linker and motor (un)binding cannot be cap-
tured using normal mode decomposition of instantaneous network configurations. The
eigenspectrum ?(t) still informs on the stability of the energy minimized configuration
before a cytoquake, but caution should be used in interpreting the cytoquake motion
from t to t + ?t as decomposing cleanly into non-interacting motions along the normal
modes vk. We leave a detailed analysis of the anharmonicity of cytoquake deformations
to future work.
136
2
10
5
4
3
10
3
2
3
10
1
0
2
00) , 0)10 1 00 , 100
) )
, ( 1
,
(0 (100
(
4
10
1
10
0 5
10 10
800 600 400 200 0 200 400
?U (kBT)
Figure 6.12: Scatter plot of the pairs ?U, ?P measured during QSS
for a run of condition C3,3. From these points, a Gaussian kernel density
estimate of the joint PDF (treating ?P on a log-scale) is constructed and
shown as a contour plot. Red guidelines demarcate regions of interest.
Inset: Combination violin and box-and-whisker plots showing the ratio
nsoft/nstiff for different categories of simulation cycles, c.f. Figure 6.4.
The inset is not blocking any of the scatter plot data.
?P (pN/nm)
nsoft/nstiff
Probability Density
137
6.4 Discussion
We have shown evidence supporting the following picture of active cytoskeletal network
self-organization: cytoskeletal networks explore a rugged mechanical energy landscape
in a stochastic process characterized by occasional, sudden jumps out of metastable
configurations [109, 133]. These jumps entail non-Gaussian dissipation of mechanical
energy and are accomplished by an avalanche-like process of spreading destabilization,
resulting in a collective structural rearrangement and a homogenization of tension. These
collective motions have large projections along the soft, delocalized vibrational modes,
and furthermore, properties of these modes can be used to predict when such relaxation
events will occur. Future research could help elucidate the role of force-sensitive reactions
in controlling collective relaxation as well as the importance of anharmonicites in the
energy landscape of cytoskeletal networks.
138
Chapter 7
Summary and future directions
We have presented in this thesis a computational method for quantifying the energetics
of non-equilibrium cytoskeletal processes and applied it to gain new insights into the
nature of cytoskeletal self-organization. As described in Chapter 1, this self-organization
under different concentrations of accessory proteins is responsible for the diverse physi-
ological functionality of the cytoskeleton. Thus, the new insights into cytoskeletal self-
organization have biological implications. To conclude this thesis, we next summarize
some of our key results and their biological implications and suggest future research
directions.
We observed a generic tendency of cytoskeletal networks under fixed chemical condi-
tions to decrease their dissipation rates over time, in both the active processes of actin
subunit hydrolysis (Chapters 2 and 4) and motor walking (Chapters 4 and 5). Cy-
toskeletal conditions can be finely controlled by the cell by activating accessory proteins
to varying degrees, but for a given concentration of activated proteins the total dissi-
pation rate reaches a minimum in the steady state. This observation can be used as a
predictive principle for steady state organizations of cytoskeletal structures. The mini-
mization of dissipation from ATP hydrolysis by actin subunits appears consistent with
the minimum entropy production principle of Prigogine applying to chemically reactive
systems which, however, was only proved by Prigogine in the near-equilibrium regime
[135]. In our simulations of contractile networks, polymerizing filaments only minimally
interact with a boundary, so that the Brownian ratchet-like decrease in polymerization
139
rates due to forces on the filament tips is not a major effect [101]. Polymerization against
the leading edges of cells in lamellipodia may be strongly affected by this force-sensitivity,
however, and it would be interesting to see how our results extend to that scenario. Pre-
sumably the net decrease in polymerization rates would only drive the average steady
state dissipation rates lower, but recent work has shown that the collective interactions
of branched filaments with a flexible boundary can give rise to scale-free fluctuations
in the membrane shape, which could manifest as non-trivial distributions of dissipation
rates [166, 292]. The decrease in dissipation from motor walking results from motors
stalling and catching in networks with increasing typical tensions, such that the steady
state is the most kinetically arrested. This agrees with the low rattling principle pro-
posed by Chvykov and England [83, 156]. It would be interesting to study the how the
typical time scale of attaining minimized dissipation rates compares with the time scales
on which cells modulate the activation of accessory proteins. The transient behavior of
the network likely plays an important role in cellular processes like chemotaxis or elec-
trotaxis, where the cell is actively responding to sometimes rapidly changing external
cues [293, 294]. It is also possible to experimentally manipulate myosin activation using
light, which can allow one to, for example, control the dynamics of nematic defects in
actomyosin gels [69, 77]. Computationally studying transient dissipation dynamics under
time varying protein activation would thus be a promising future research direction.
An additional insight gained from the studies presented in this thesis pertains to the
nature of mechanical relaxation by cytoskeletal networks. It was argued in Chapter 6
that mechanical relaxation can be highly collective, with large events involving many
filaments, rather than local, with events involving just a few filaments. The cooperativ-
ity of mechanical relaxation can lead to a wide range of relaxation event sizes and may
be a root cause of the observed variability in traction force measurements of migrating
cells [170]. As mentioned in Chapter 6, it is plausible that this cooperativity allows for
enhanced mechanical responsiveness to a changing cellular environment, which could be
a significant physiological advantage. Several open questions remain about these cyto-
quake dynamics, though. One question regards the characteristic time and length scales
140
of cytoquakes. Our study was at a fixed time resolution of 20 Hz and measured energy
fluctuations summed over the whole 1 ?m3 network, so the cytoquakes (or ?cytoquake-
like events?) we observed were limited to these scales. In a larger system, though, a
cytoquake may have an extended lifetime as it cascades throughout the network, open-
ing the door to a broader range of spatiotemporal scales than analyzed here. Current
work underway in Reich?s group is investigating this question in vivo, and this work
could be complemented using MEDYAN simulations of large networks that spatially re-
solve the release of mechanical energy [169]. Another question is about the degree to
which cooperative mechanical relaxation depends on force-sensitive reactions. Cooper-
ativity should result from just the structural properties of a network with insensitive
reaction rates, where the ability of the network to relax depends on its configuration and
the mutual mechanical constraints filaments impose on each other. However, we expect
cooperativity to be amplified when the reactions controlling network connectivity sensi-
tively depend on the network forces. The degree to which avalanche-like dynamics can be
controlled through the parameters of force-sensitive reaction remains to be investigated.
One possible application of the dissipation tracking method presented here would
be to compare it with various indirect methods for quantifying entropy production
based solely on observations of the system?s trajectory or even snapshots of its struc-
ture [32, 34, 35, 130, 295, 296]. A benefit of these indirect methods is that they can
be applied to experimentally accessible low-resolution representations of the system tra-
jectory, although as a result it underestimates the true entropy production [36]. One
could compare these calculations of entropy production in cytoskeletal networks with
our explicit computational approach to assess the accuracy of the indirect estimates in
real biological systems.
An additional principle potentially useful for understanding cytoskeletal self-
organization via its non-equilibrium thermodynamics is the set of thermodynamic un-
certainty relations [297]. Under various assumptions on the system?s state space and
dynamics, these relations provide bounds on the uncertainty in an arbitrarily defined
current of a non-equilibrium system in terms of the system?s dissipation rate, such that a
141
larger dissipation rate implies a smaller uncertainty in the current. Originally established
within a limited range of applicability, they have since been quite broadly generalized and
appear to hold in some form for many systems of interest [130, 298?301]. These results
have been applied to determine a bound on the efficiency of a discrete-state molecular
motor moving on a linear track, and it is reasonable to expect that they also have an
implication for higher-order cytoskeletal self-organization (in which many such motors
collectively cause changes in network morphology) [302]. For instance, the network con-
traction rate may have an uncertainty that depends on the dissipation of consumed ATP
energy by molecular motors. This could be relevant to cells with low concentrations
of ATP, leading not just to slower physiological processes but a greater uncertainty in
such processes. Testing the applicability of the thermodynamic uncertainty relations for
cytoskeletal self-organization thus remains a promising future research direction using
our computational method.
A theoretical question which would be of interest beyond cytoskeletal networks is
about how entropy production varies as a function of length scale for active matter
systems, where energy is injected at small scales. By comparison one can consider a
turbulent fluid driven by out-of-equilibrium boundary conditions. This system has a
so-called Kolmogorov cascade of energy flow, in which energy is transferred down from
large scale coherent fluid motions, or eddies, to progressively smaller eddies until it
is finally dissipated due to viscous interactions at the smallest length scale, lost in the
incoherent degrees of freedom of molecular vibrations and disorganized motion [303, 304].
In contrast, energy in active matter systems is injected not at the global level (e.g.
through boundary conditions) but at the local level (e.g. by molecular motors). Does
a similar energy cascade picture apply in this case? Recent work on coarse-graining
non-equilibrium chemical reaction networks suggests that a cascade-like picture does
apply, such that dissipation occurs maximally at small scales where energy is injected,
and dissipation decreases when measured using coarser reaction network representations
[305]. It would be interesting to investigate how the spatial scale at which one is looking
affects the measured dissipation in cytoskeletal networks. This topic may be relevant
142
to the graph coarse-graining procedure for describing bundled cytoskeletal architectures
discussed in Chapter 5.
A final future research direction that we would like to mention, and which is cur-
rently underway, is to drastically improve the realism of the mechanical model used in
MEDYAN and to apply the improved model to study collective chiral symmetry break-
ing phenomena in cytoskeletal systems. Experiments in vitro and in vivo have revealed
the existence of remarkable rotating dynamical phases of active networks [67, 306, 307].
Emerging at micrometer length scales, these collective swirling motions display a sensi-
tive dependence on experimental conditions such as cross-linker concentration, which can
even reverse the direction of the rotation. Their emergence may be explainable in terms
of the dissipation minimization and low rattling principles mentioned earlier, and hence
it is of interest to study such phenomena computationally using our free energy tracking
method. To model such phenomena using MEDYAN, however, filament torques must be
accounted for. We are addressing this by developing a new, finite-width filament model
based on the Cosserat theory of elastic rods [308, 309]. This model will allow to more
closely study how mechanical energy is stored in cytoskeletal networks (i.e. in shearing,
stretching, bending, or twisting deformations), as well as to investigate novel rotational
phenomena of cytoskeletal systems. Preliminary applications of the model have for in-
stance shown that motorized filaments undergo axial rotations in a stochastic process
similar to biased diffusion. This ?active rotational diffusion? may be closely connected
to dissipation by myosin motors and remains to be further investigated.
In addition to the future projects mentioned above, which will likely require some
appreciable additional modeling effort, there are also some relatively straightforward
foreseeable applications. Several accessory proteins known to strongly modulate cy-
toskeletal architecture, such as capping and branching molecules, have not been consid-
ered throughout this thesis [268, 271]. Studying entropy production in systems including
these additional agents may allow to test the generality of the observed dissipation mini-
mization, and they could additionally affect the cooperative mechanical relaxation lead-
ing to cytoquakes. Certain common cytoskeletal structures, such as stable cross-linked
143
stress fibers and the cortical actomyosin shell, could also be reproduced in simulation
and the concomitant dissipation signatures of these structures investigated [247]. Our
computational method to track cytoskeletal thermodynamics is available for public use
in the currently released version of MEDYAN, and we hope that continuation of the
work presented in this thesis may provide new insights into cytoskeletal self-organization
and its physiological role in the life of a cell.
144
Appendix A
Description of MEDYAN simulation
platform
A detailed description and justification of the MEDYAN model can be found in Ref.
42, and additional extensions and applications of MEDYAN to study the dynamics of
actomyosin networks are described in Refs. 94, 195, 247, 268?270. Here we briefly outline
the relevant aspects of MEDYAN to facilitate understanding the results in this thesis,
and direct the reader to the above references for a more thorough introduction.
A.1 Simulation protocol
A MEDYAN simulation proceeds by iterating a cycle of four steps which propagate the
chemical and mechanical dynamics forward while maintaining a tight coupling between
the two. The steps are as follows:
1. Evolve system using stochastic chemical simulation for a time ?t.
2. Compute the changes in the mechanical energy resulting from the reactions that
occurred in step 1).
3. Mechanically equilibrate the network in response to the new stresses from step 2).
4. Update the reaction rates of force-sensitive reactions based on the new tensions
from step 3).
145
This protocol reflects a separation of timescales between the slow chemical dynamics
and the fast mechanical response, such that the mechanical subsystem is assumed to
always remain near equilibrium and to adiabatically follow the chemical changes in the
network. As argued in Ref. 42, supported using experimental evidence from Refs. 12, 88,
239, this timescale separation holds for typical cytoskeletal networks which experience
localized force deformations with fast relaxation times compared to the typical waiting
time between myosin motor walking steps and filament growth-induced deformations.
A.2 Chemistry
In MEDYAN, diffusing chemical species are represented with discrete copy numbers be-
longing to several compartments, which form a regular grid comprising the simulation
volume. The compartment size is chosen so that it may be assumed that inside the com-
partments the diffusing species are well-mixed, allowing the use of mass-action kinetics
to determine their instantaneous propensities to participate in chemical reactions within
compartments and diffusion events between adjacent compartments. The minimum Ku-
ramoto length (i.e. the mean free diffusional path length of a reactive species before
it participates in a chemical reaction) among the species sets this compartment size to
ensure that the well-mixed assumption holds [310]. The diffusing chemical species may
participate in local chemical reactions according to the copy numbers of the reactants
in its compartment, or else they may jump to an adjacent compartment in a diffusion
event with a propensity determined by its copy number in the original compartment
[189]. The algorithm for stochastically choosing which event (including local reactions
or jumps between compartments) will occur next is the Next Reaction Method, an ac-
celerated variant of the Gillespie algorithm [189, 190]. These are Monte Carlo methods
which randomly select both the time to any next event and which event will occur at
that time in accordance with each event?s instantaneous propensity.
The user specifies the different chemical species and the reactions that they partici-
pate in. Several types of reactions are possible. Regular reactions involve only diffusing
146
species (e.g. the conversion of ADP-bound to ATP-bound G-actin monomer). Poly-
merization reactions result in the subtraction of a diffusing monomer from the local
compartment and its conversion into a filament species, and depolymerization reactions
do the opposite. Filaments in MEDYAN?s have definite spatial coordinates, rather than
the compartment-level description of the diffusing species? positions. This network of
spatially resolved filaments is overlaid on the compartment grid, so that sections of fil-
aments are able to react with diffusing species according to their local copy numbers.
In addition, filaments have mechanical properties which will be discussed in the next
section. A filament may react with a diffusing species such as a cross-linker (e.g. ?-
actinin), branching (e.g. Arp2/3), or molecular motor (e.g. NMIIA). Binding reactions
involve a discrete set of binding sites along the filament, and they stochastically occur
as chemical reaction events according to the number of those binding sites and the local
copy number of diffusing binding molecules. A bound molecular motor may participate
in a walking reaction, which causes it to move one of its ends to an adjacent binding site,
stretching the motor and generating forces. Other reactions not used in this thesis but
encompassed by MEDYAN include filament nucleation, filament destruction, filament
severing, and filament branching reactions.
A.3 Mechanics
The mechanical energy U of networks in MEDYAN is a function of the positions of the
filament beads and the lengths of the molecules bound to the filaments. There are also
potentials describing a branched filament?s energy which are not included in this thesis.
Filament beads mark the joined end points (i.e. hinges) of the cylinders comprising the
filament. Individual cylinders can stretch but not bend, but a bending energy term is
included for pairs of adjacent cylinders. The energy term for the stretching of cylinders
is
1
U 2str = Kfil,str(l ? l0) , (A.1)
2
147
where l = ||ri+1 ? ri|| is the length of the cylinder whose beads are at positions ri+1
and ri, l0 is the cylinder?s equilibrium length, and Kstr is the spring constant of this
harmonic potential. The energy term for the bending of adjacent cylinders is
Ubend = bend (1? cos(?i,i+1)) , (A.2)
where bend parameterizes the strength of the interaction and ?i,i+1 is the angle between
the cylinders. Molecules bound to pairs of filaments (e.g. ?-actinin and NMIIA) of
stretched length lbound have a harmonic stretching energy term:
1
U 0 2bound,str = Kbound,str(lbound ? lbound) , (A.3)2
where the subscript ?bound? indicates that the variables and parameters are specific to
the bound molecule. An excluded volume interaction is included to prevent cylinders
from overlapping. The analytical formula for this interaction is complicated but can be
expressed as a double integral over the two lengths of the participating cylinders i and
j: ? 1 ? 1 dsdt
Uvol,ij = Kvol , (A.4)
0 0 ||ri(s)? rj(t)||4
where ri(s) = ri+s(ri+1?ri) is the position along the i cylinder, which is parameterized
by a variable s running from 0 to 1 along the cylinder?s length. These positions ri(s) are
also therefore functions of the cylinders? bead positions, ri and ri+1. See Ref. 311 for
a detailed derivation of the excluded volume potential. Finally, an exponentially decay-
ing boundary repulsion term prevents the filaments from poking outside the simulation
volume:
U ?di/?boundary = boundarye , (A.5)
where boundary parameterizes the interaction strength, di is the distance from the bound-
ary to the nearest endpoint of cylinder i, and ? parameterizes the interaction screening
length.
At the end of each chemical evolution cycle, the positions of the bound molecules
148
Figure A.1: Depiction of the mechanical potenetials used in MEDYAN.
The double helical structure of filaments is coarse-grained into a sequence
of cylinders (shown in red) whose connected hinges are called beads
(shown as black spheres). Cylinders expereince a stretching potential
(Equation A.3), a bending poential (Equation A.2), and an excluded vol-
ume repulsion (Equation A.4). Myosin minifilaments (shown in blue) and
?-actinin cross-linkers experience a stretching potential (Equation A.3).
Note that the mathematical notation differs slightly from what it used
in the main text. Image reprinted from Ref. 247 with permission from
PLoS.
and the filament beads will have changed due to the chemical reactions which occurred,
displacing the system from near-equilibrium. The positions of the filament beads are then
updated in a mechanical equilibration cycle by minimizing the total mechanical energy
function U . This is accomplished using the conjugate-gradient minimization algorithm
[312]. The minimization procedure ends when the maximum net force remaining in the
network is below a user-specified force tolerance FT , as result of which the system returns
to near mechanical equilibrium.
A.4 Mechanochemical coupling
An important facet of the dynamics of actomyosin networks is that the chemical reac-
tion rates of the associated proteins depend on the forces they sustain: at high tension
149
the myosin minifilaments will walk and unbind more slowly (stalling and catch-bond be-
havior) whereas the passive cross-linkers are modeled as unbinding more quickly under
tension (slip-bond behavior) [87, 267]. These force-sensitive behaviors thus play the role
of nonlinearly coupling the mechanical state of the actomyosin network to its stochastic
chemical dynamics.
The myosin motors used in MEDYAN are modeled after non-muscle myosin IIA
(NMIIA), which exists in the cell as a minifilament consisting of tens of individual myosin
heads. The chemical dynamics of the myosin minifilaments are based on the Parallel
Cluster Model of Erdmann et al. [90, 93]. In this model, a myosin minifilament contains
a number Ntotal of individual myosin heads and has a binding rate to the actin filament
pair equal to
kfil,bind = Ntotalkhead,bind, (A.6)
where khead, bind is the individual myosin head binding rate. In MEDYAN, Ntotal is
uniformly randomly selected between a minimum and maximum number of heads each
time a minifilament binds. The bound myosin minifilament has a number of bound heads
N0bound under zero tension equal to the duty ratio ? times the total number of heads:
N0bound = ?Ntotal. (A.7)
The duty ratio is determined by the individual head unbinding rate:
khead,bind
? = , (A.8)
k0head,unbind + khead,bind
where k0head,unbind is the head unbinding rate under zero tension. Under tension Fext the
bound myosin minifilament has altered walking and unbinding rates as well as an altered
number of bound heads. The number of bound heads under tension is given by
{ }
Fext
N 0bound(Fext) = min Ntotal, Nbound + ? , (A.9)Ntotal
150
where the parameter ? = 2.0 is chosen to fit experimental data. The myosin minifilament
walking rate under zero tension is
0 1? ?kfil,walk = s khead,bind, (A.10)?
where s is called the stepping fraction, defined as the ratio of the user-specified real
distance between binding sites on the filament dstep to the distance between binding
sites on the computational cylinder representing the filament segment dd : s = steptotal d .total
Equation A.10 is based on the PCM and is explained Refs. [42, 93]. Under tension, the
myosin minifilament walking rate is altered according to a formula of the Hill type:
{ }
F ? F
kfil,walk = max 0
stall ext
0.0, kfil,walk , (A.11)Fstall + Fext/?
where the stall force Fstall is the maximum tension a minifilament can sustain before it
stops walking, and where ? = 0.2 is another parameter chosen to fit to experimental
data. The myosin minifilament will unbind from the pair of actin filaments under zero
tension with a rate
0 ( ( khead,bindNtotalkfil,unbind = ) ) . (A.12)
k0 +khead,bind
exp log head,unbind0 Ntotal ? 1khead,unbind
This non-obvious expression is the inverse of the mean residence time of the minifilament
as determined using the PCM. Under tension, the myosin minifilament unbinding is
modeled with Kramers-type catch-bond behavior:
{ ( )}
0 ?Fextkfil,unbind(Fext) = kfil,unbind max 0.1, exp , (A.13)Nbound(Fext)F0,head
where F0,head is the characteristic force a single myosin head catch-bond, and the mini-
mum unbinding factor 0.1 is a parameter to chosen to ensure the possibility to unbind
under arbitrarily large tension. We assume for myosin minifilaments that the stretching
151
constant is given by
Kbound,str = Khead,strNbound, (A.14)
where Khead,str is the stretching constant for an individual head; this equation assumes
the bound heads share the load in parallel.
The unbinding of passive cross-linkers (e.g. ?-actinin) are modeled as Kramers-type
slip-bond: ( )
Fext
k 0linker,unbind(Fext) = klinker,unbind exp , (A.15)F0,linker
where F0,linker is the characteristic force of the cross-linker slip-bond.
Finally, the actin filament will polymerize with a rate that exponentially decreases
with the component of the sustained force along the polymerizing tip, Fext. This depen-
dence is based on the Brownian ratchet model of Peskin et al. [101]:
( )
0 ? Fextkpoly(Fext) = kpoly exp , (A.16)F0,poly
where F 00,poly is the characteristic force of the Brownian ratchet model, and kpoly is the
zero-force polymerization rate.
Any of the above characteristic forces F0 may be converted to a corresponding char-
acteristic distance x0 via
F0 = kBT/x0, (A.17)
where kBT is the thermal energy, casting expressions of the form Fext/F0 to the form
Fextx0/kBT .
152
Appendix B
Supporting information for Chapter
2
B.1 Details of BC model
The BC model consists of the following set of 11 coupled ODE?s:
dGT
= kT Toff, +T+ + k
T T D T
off, - T? + knexG ? krenexG (B.1)dt
?GTN(kTon, + + kTon, -)
dGD
= kD TD + kD TD + kPi TPi + kPi TPi + k GToff, + + off, - ? off, + + off, - ? renex (B.2)dt
?k GD ?GDnex N(kD Don, + + kon, -)
dF T
= GTN(kTon, + + k
T T T T T T
dt on, -
)? koff, +T+ ? koff, - T? ? khydF (B.3)
+k FPirehyd
dFPi
= k ThydF ? k PirehydF ? kPi TPioff, + + ? kPi P i P idt off, -
T? ? kphosF (B.4)
+k DrephosF
dFD
= GDN(kD D Pi D D Don, + + kon, -) + kphosF ? krephosF ? kdt off, +
T+ (B.5)
?kD Doff, -T?
153
dT T? = kT GT (TPi + TD) + (kPi TPi + kD D T Ton, ? ? ? off, ? ? off, ?T? )?? ? khydT? (B.6)dt
+k TPirehyd ? ? kT T Pi D D D Toff, ?T? (?? + ?? )? kon, ?G T?
dTPi? = k T Pi T T D D Pi P ihydT? ? krehydT? + (koff, ?T? + koff, ?T? )?? ? kphosT? (B.7)dt
+krephosT
D ? kPi P i? off, ?T? (?T? + ?D? )? TPi T T D D? (kon, ?G + kon, ?G )
dTD? = kD GDon, ? (T
T
? + T
Pi
? ) + (k
T
off, ?T
T
? + k
Pi
off, ?T
Pi D
? )?? + k
Pi
phosT
dt ?
(B.8)
?k D D D T Pi T T DrephosT? ? koff, ?T? (?? + ?? )? kon, ?G T?
where
T ( Pi)
?T
T
= ?
T
1? ?? (B.9)N N
D T
D
? = ?? (B.10)N
?Pi? = 1? ?T? ? ?D? . (B.11)
In Table B.1 we list the meaning and values of the rate constants used in our implemen-
tation of the BC model.
154
Label Reaction Value
kT Polymerization of GT to barbed end 11.6 ?M?1 s?1on, +
kTon, - Polymerization of GT to pointed end 1.3 ?M
?1 s?1
kToff, + Depolymerization of F
T from barbed end 1.4 s?1
kToff, - Depolymerization of F
T from pointed end 0.8 s?1
kDon, + Polymerization of GD to barbed end 2.9 ?M
?1 s?1
kDon, - Polymerization of GD to pointed end 0.13 ?M
?1 s?1
kD Doff, + Depolymerization of F from barbed end 5.4 s
?1
kDoff, - Depolymerization of F
D from pointed end 0.25 s?1
kPioff, + Depolymerization of F
Pi from barbed end 1.4 s?1
kPioff, - Depolymerization of F
Pi from pointed end 0.8 s?1
khyd ATP hydrolysis converting F T to FPi 0.3 s?1
krehyd ATP condensation converting FPi to F T  0.3 s?1
knex Nucleotide exchange converting GD to GT 0.01 s?1
krenex Nucleotide exchange converting GT to GD  0.01 s?1
kphos Inorganic phosphate release converting FPi to 0.002 s?1
FD
kphos Inorganic phosphate capture converting FD to  0.002 s?1
FPi
Table B.1: Rate constants in the BC model. The prefix ?re? indi-
cates the reverse of a kinetically dominant forward reaction (i.e. nearly
irreversible reactions). The value of rate constants for these reverse re-
actions is taken to be equal to the corresponding forward reaction rate
multiplied by a small parameter . We typically take  = 0.01. Some
of these reactions, such as the nucleotide exchange reaction, are second
order reactions. For example the proper rate of reaction for conversion
of GD to GT is k? Dnex[G ][ATP ]. We treat such cases as pseudo-first or-
der reactions by assuming that the concentration of the species which we
don?t track is constant and that its concentration is contained in the rate
constant used. Thus k = k?nex nex[ATP ] in our model. This assumption of
constant concentrations of free ATP, ADP, and Pi is reasonable in cellular
environments. All values are taken from Refs. 12, 183.
We note that the original equations in the BC model did not include reversible
reactions as shown here. This amounts to setting krenex, krehyd, and krephos to 0 Equations
B.1-B.8. The interpretation of ?ij is the probability that the subunit adjacent to the j tip
is in the i hydrolysis state. The equations of motion of these variables in principle depend
on the hydrolysis state of the subunit next to them toward the center of the filament,
and Equations B.1-B.8 represent the truncation of this set of recursion equations. This
is accomplished by assuming that ?ij depends only on the tip subunit hydrolysis state
through Equations B.6-B.8. These equations were arrived at by inspecting the time
155
Figure B.1: Time course of the concentrations of the various species
following the addition of 3 ?M of GT and 3 ?M of GD actin to a bath
with a number concentration N = 0.017 ?M of seed filaments. The
mean-field BC (smooth curve) model accurately predicts the shape of the
time-courses resulting from the stochastic simulation (noisy curve).
course of the ?i ij and the Tj variables and discerning the equations which approximately
related the two. The system of Equations B.1-B.8 does not admit and analytical solution
but can be numerically integrated with appropriate initial conditions specified [183].
We check the accuracy of the truncation assumption of the BC model by comparing a
predicted time course to the results of a simulation using the software package MEDYAN.
MEDYAN was developed to perform coarse-grained simulations of active networks and
combines stochastic chemical algorithms with detailed mechanics as well as coupling
between reaction rates of force-sensitive chemical reactions and the mechanical state of
the species involved [42]. In Figures B.1 and B.2 we show the simulated time courses as
well as the mean-field prediction of the BC model.
The steady state vector xss satisfying f(xss) = 0 can be found by numerically solving
the root of the right hand sides of Equations B.1-B.8. Equations B.1-B.5 sum to zero,
reflecting the conservation of total actin M encoded in these reactions. Also each of
the two sets of Equations B.6-B.8 sum to zero, reflecting the conservation of plus end
156
Figure B.2: Time course of the concentrations of the various tip species
at the plus (A) and minus (B) ends, following the addition of 3 ?M of GT
and 3 ?M of GD actin to a bath with a number concentration N = 0.017
?M of seed filaments. The noisiness of the stochastic trajectories results
from the small copy number of filaments. Note how quickly the these
concentrations attain their steady state values with this parameterization.
This implies that not tracking these variables explicitly, as in the QSSA
and CT models is valid for most of the trajectory.
157
tips and minus end tips. Thus the system x? = f(x) represented by Equations B.1-B.8
is linearly dependent, and no solution to exists f(xss) = 0 unless we specify additional
equations. These additional equations are
GT +GD + F T + FPi + FD = M (B.12)
T T? + T
Pi + TD? ? = N?. (B.13)
For unbranched filaments considered here, the number of plus end tips is equal to the
number of minus end tips: N+ = N? = N . Solving f(xss) = 0 gives one solution for
which all variables are non-negative, so there is a unique realistic steady state solution
for a given set of parametersM? and N . The eigenvalues of the BC Jacobian evaluated at
the equilibrium point J? = ?f ??x ss indicate the stability of the steady state. 3 of the 11x=x
eigenvalues are zero, corresponding to the 3 linear conservation laws in our system. This
implies that the dynamics of the BC model lie on an 8-dimensional submanifold of the full
11-dimensional variable space, and this submanifold is determined by the parameters M
and N . The remaining eigenvalues are negative, indicating that the unique non-negative
vector xss is attracting and stable. We note that this equilibrium point of the dynamics
actually corresponds to a non-equilibrium steady state of the chemical system, since this
state corresponds to actin treadmilling, fueled by ATP hydrolysis. This chemical driving
is manifested in the values of certain pseudo-first order reaction rate constants such as
knex.
B.2 Steady state concentrations in the BC model
We show here that the independence of the steady state concentrations of some species on
M is also present in the BC model. We do this by consideration of the system of algebraic
equations f(xss) = 0. The Jacobian matrix of system at the steady state (see Equation
2.24) is visualized in Figure B.3. Now that we can grasp which species are coupled to
which other species, we observe that no species depend on F T , FPi, or FD, except those
species themselves. Therefore we could ignore the equations of motions of these variables
158
Figure B.3: Visualization of J? to understand the couplings in the BC
model. If ?xi?x =6 0, then J
?
ij will be nonzero. If this coupling causes xij
to increase, the entry in the matrix is colored blue, and if it causes xi
to decrease, the entry is red. The saturation of the color indicates the
magnitude of the coupling. An ?R? is placed in the plot if that element is
nonzero only due to the inclusion of slow reversible reactions, i.e. when
 > 0. The corresponding plot for the CT model is identical to the top
left 5 by 5 matrix in this plot.
159
and the resulting subsystem of equations would be closed. Now that subsystem will be
linearly dependent, but we can supplement it with the following equations to make it
independent:
T T + TPi? ? + T
D
? = N. (B.14)
The subsystem of equations could now be solved, giving the steady state concentrations
of all species except for F T , FPi, and FD. M , the total amount of actin, does not
enter into any of the equations of the subsystem, and so the solution of that system does
not depend on M . Therefore, the steady state concentration of each species except F T ,
FPi, and FD does not depend on the initial conditions, however they do depend on the
parameter N .
Now we consider the case where  = 0. Referring to Figure B.3, we see that this would
mean that no species depends on FD. By reasoning similar to the above, this implies that
the steady state concentration of each species except FD could be determined without
specifying M . Thus, the steady state concentration of only FD depends on the initial
conditions in this case.
160
Appendix C
Supporting information for Chapter
3
C.1 Discrete variables and the relation to the Gibbs-Duhem
equation
The Gibbs-Duhem Equation from classical thermodynamics states
?
Nid?i = ?SdT + V dp = 0 (C.1)
i?M
whereM represents all chemical species in the system including the solvent, and the last
equality holds at constant temperature and pre?ssure [25]. Applying the product rule1 to
the expression for the Gibbs free energy, G = i?M Ni?i, we get
? ? ?
dG = ?idNi + Nid?i = ?idNi. (C.2)
? i?M i?M i?M
Evaluating i?M ?idNi using stoichiometric coefficients in place of dNi can be shown
to lead to the expression ?G(3).
If the copy numbers are small, we are better served using the discrete difference
operator ? and not the differential difference operator d. The product rule for the
1The product rule can be derived as d(fg) = (f + df)(g + dg)? fg = fdg + gdf + dfdg = fdg + gdf
since the term dfdg is assumed to be negligible. We give this reminder to emphasize the presence of the
cross term, which we do not neglect in the discrete case.
161
discrete difference operator applied to G gives
? ? ?
?G = ?i?Ni + Ni??i + ??i?Ni. (C.3)
i?M i?M i?M
Here we do not neglect the cross-term as we do in the differential product rule, Equation
C.2. We evaluate this expression term by term:
?
?i?Ni = ?G(3) (C.4)
i?M
? ?( )N ? ? Ni ?( ) ( )i i Ni + ? Njj N N
Ni??i = kBT log (C.5)
Ni N N + ?
i?M i?R
? ?( )
j? jP ( ) ( )
N ? ? ??i ?i i Ni + ? ?j N ?j
?Ni??i = kBT log (C.6)
Ni Nj N + ?
i?M i?R j?P
Comparing these to the expressions for the accuracy of the various approximations
given in Appendix B, we have
? ?
Ni??i + ?Ni??i = ?G(0) ??G(3) (C.7)
i?M i?M
and thus, combining all the terms in Equation C.3, we recover our exact expression
?G(0), and this approach can be seen as an alternate derivation of the main result. To
summarize, for small, discrete copy numbers, we obtain corrections to the expression for
the change in Gibbs free energy accompanying chemical reactions that are not captured
by the constraints imposed by the Gibbs-Duhem Equation, which assumes that the copy
numbers are continuous quantities.
162
C.2 Accuracy of the approximations
We calculate the accuracy of the approximations ?G(1),?G(2), and ?G(3) by taking the
difference of these quantities with ?G(0). We have
( )
N + ? N+?
(?G(1) ??G(0))/kBT = log ? ? (C.8)N
( )
N + ? N+? ?( )N N ( )i ? N Nj? i j(?G(2) ?G(0))/kBT = log + logN Ni ? ?i N? j + ?j( ) (i R ) j?P( )(C.9)
? N + ?
N+? ? N Ni??i ? N Nj+?ji j
(?G(3) ?G(0))/kBT = log +logN N
? i
? ?i N + ?
i R j? j jP
(C.10)
The accuracy of ?G(1) depends only on N for a given reaction, whereas the remaining
approximations depend also on the values of Ni and Nj . These observations reflect
the fact that, in order to arrive at the expression for ?G(1), we leveraged the size of
N compared to ?, and to arrive at the expressions for ?G(2) and ?G(3) we further
successively leveraged the sizes of Ni, Nj compared to ?i, ?j .
C.3 ?G of solvent fluctuations
Here we first calculate an approximate expression for the change in Gibbs free energy of
the system accompanying a fluctuation of n solvent molecules from compartment A to
compartment B. We have
? ( ) ( )
initial Ni,AG = N ??0 + k T log +N ??0,?
Ns,A
i,A (i B ) s,A s + kBT log?? NA NAi L Ni,B
+ N 0i,B ??i + kBT log NB
i?L ( )
+N ??0,?
Ns,B
s,B s + kBT log , (C.11)
NB
163
where L is the set of solute species. After the transfer of n solvent molecules from A to
B we have
? ( ) ( )
final 0 Ni,A 0,? Ns,A ? nG = Ni,A ??(i + kBT log +)Ns,A ??s + kBT logNA ? n NA ? ni?L? Ni,B
+ N 0(i,B ??i + kBT log N + ni? BL )
0,? Ns,B + n+Ns,B ??s + kBT log . (C.12)
NB + n
Taking the difference and simplifying, we arrive at the exact expression
NNAA (Ns,A ? n)Ns,A?n N
NB
B (N
Ns,B+n
s,B + n)
?G = kBT log
(N N ?
.
n N
A ? n) A N s,A (NA + n)NA+n NN s,Bs,A s,B
(C.13)
To understand the magnitude of ?G for typical values of n, Ns,A, and Ns,B compared
to NA and NB, we first make the assumption that the two compartments initially have
the same number of solvent molecules, i.e. that Ns,A = Ns,B ? Ns. Next we assume that
the solvent molecules dominate the proportion of total molecules, allowing us to write
NA ? NB ? N . These approximations will hold in the limit that the number of solute
molecules is much less than the number of solvent molecules for each compartment. We
next introduce the small parameters
?
i?LNi,AA = , (C.14)
N
and ?
i?LNi,BB = , (C.15)
N
which capture the dilutions of the two compartments, and
n
? = , (C.16)
N
which represents the relative size of the fluctuation. For N = 109, we typically have
? ? 10?4.5 (since n ? N1/2) and  ? 10?6. Substituting these parameters in Equation
164
C.13, we have
(
N2N
?G = kBT log ?
(N(1? ?))N(1??)(N(1 + ?))N(1+?) )
(N(1?  ? ?))N(1?A??)(N(1?  + ?))N(1?B+?)A B
(N(1?  ))N(1?
. (C.17)

A A
)(N(1?  N(1? )B)) B
We expand this expression to first order in A and B, and then to second order in
?. The result is
1
?G ? (A ? B)N?kBT + (A + B)N?2kBT
2
1 n2
= (A ? B)nkBT + (A + B) kBT. (C.18)
2 N
The observed numerical agreement between Equations C.13 and Equations C.18 is
close for realistic values of the parameters: for N = 109A , NB = 5 ? 108,  ?6A = 10 ,
 = 3? 10?6B , and 1/2n = NA , the prediction of Equation C.13 is ?0.0632401 kBT , and
the prediction of Equation C.18 is ?0.0632436 kBT . The first term in Equation C.18
will dominate if the solute dilutions in the two compartments are very different from
each other. In this case, we may compare the size of this change in Gibbs free energy
to that accompanying the diffusion of a solute from compartment B to compartment A.
This latter change in Gibbs free energy will be approximately k T log AB  . If we nowB
set A = aB, where a is of order 1 (typically it will fall in the range [1/10, 10]), then
?G for the solvent fluctuation will be kBT (a ? 1)Bn and ?G for the solute diffusion
will be kBT log a. The product Bn will typically on the order of ? 10?1.5, so one can
see that for typical values of the parameters the change in Gibbs free energy from a
solute diffusion event will be significantly greater in magnitude than that from a solvent
fluctuation. If the dilutions are very similar, then A ? B ? 0, and the second term in
2
Equation C.18 dominates. This term is on the order of (A+B)k nBT since N ? 1. These
changes in Gibbs free energy will typically be much smaller than those accompanying a
chemical reaction or inter-compartment diffusion of the solute. Thus we may neglect the
activity of the solvent in tracking the Gibbs free energy of the system.
165
Appendix D
Supporting information for Chapter
4
D.1 Parameterization
Parameterization of the CT model for the purpose of tracking free energy changes during
simulation trajectories consists of choosing values of the rate constants (kinetic parame-
ters) and of ?G0 (thermodynamic parameters) for all reactions in the model. Wherever
possible, values from the literature are used. Experimental measurements have deter-
mined rate constants for every reaction, however for some reactions the value of ?G0
hasn?t been reliably measured, to the best of our knowledge. Below, we describe a
technique to solve for these unknown values.
For reversible reactions, where the forward and reverse rate constants k+ and k? are
known, such as (de)polymerization and (un)binding of cross-linkers, ?G0 can be found
from
k?
?G0 = kBT logKeq = kBT log (D.1)
k+
Literature values for the equilibrium constants or of ?G0 are used for irreversible reac-
tions, for which k? is often too small to determine from direct measurement. Irreversible
reactions in this system include all reactions except for (de)polymerization reactions. For
reactions for which literature values of ?G0 are unavailable, it is possible to solve for
?G0 values based on a self-consistency condition [98, 180]: the sum of the ?G0 values
166
around a closed loop of reactions in which the number of molecules has not experienced
a net change must be zero since the free energy is a state function (equivalently, by
Equation D.1, the product of equilibrium constants around any such loop must be equal
to one). Writing several such closed loops of reactions leads to a system of equations that
can be solved for the unknown variables. Not all possible loops result in independent
equations, but we were able to determine the values of two unknown parameters using
the loops illustrated in Figure D.1.
The loops in Figure D.1 imply the following independent system of equations:
KTKhydKrelK
Pi
nex = K KATP (D.2)
KTKhydKphosKnex = K
DKATP (D.3)
where Khyd represents the hydrolysis of ATP by F T , Kphos represents the release of Pi
by FPi, Knex represents nucleotide exchange converting GD to GT , KATP represents the
hydrolysis of ATP in solution producing ADP and Pi, and Krel represents the release of
phosphate by GPi.
Values from the literature [12, 89, 91] can be used to determine 6 of the 8 variables
in Equations D.2 and D.3, which thus represent two equations in two unknowns: Krel
and Knex. The resulting parameters are listed in Table D.1.
Note that it is possible to draw loops such as those in Figure D.1 that would imply
that the equilibrium constants for polymerization and depolymerization of, for example
GT , should be the same at the plus and minus ends of the filaments. This condition
is not borne out by the experimental values of these equilibrium constants, and this
discrepancy is a recognized outstanding problem [12]. Here, we use the literature values
for these equilibrium constants and employ the reaction loop method only to determine
the parameters Krel and Knex.
For reversible binding of myosin filaments, results from the PCM are used to describe
binding and unbinding rates, and therefore Keq [90, 93]. The filament binding rate is
167
A) T
... K+ + . . . +
KATP Khyd
... + + + . . . +
KPiKnex K
... rel+ + + ... + +
B) T
... K+ + . . . +
KATP Khyd
... + + + . . . +
K
K phosnex KD
... + + + . . . + +
ATP Actin ATP 
ADP-Pi Actin Pi
ADP Actin ADP
Figure D.1: Diagrams representing sequences of reactions, with the
involved species drawn between each reaction, resulting in independent
relations between the equilibrium constants. The meaning of these equi-
librium constants is provided in the main text. Loops are assumed to
proceed in the clockwise direction, and arrows that point opposite to this
direction indicate that that reaction is occurring ?backwards?, i.e. from
products to reactants. Polymers are shown as connected chains of sub-
units, while the ?+? sign indicates different solvated species.
168
given as
kfil, bind = Ntkhead, bind, (D.4)
and the unbinding rate is
[( )N ]k t ?1head, bind + khead, unbind
kfil,unbind = Ntkhead, bind ? 1 , (D.5)
khead, unbind
where khead, bind and khead, unbind describe the binding kinetics of a single myosin head,
and Nt is the number of heads in the filament (c.f. Appendix A) [93]. The resulting
expression for ?G0 is
[( )N ]
0 k
t
fil, unbind k? head, bind
+ khead, unbind
?G = kT = kT log ? 1 . (D.6)
kfil, bind khead, unbind
We assume that we have chemostatted concentrations of ATP, ADP, and Pi, which
we account for implicitly via the effect of these concentrations on the kinetic and thermo-
dynamic parameters of certain reactions. Thus these species are not explicitly tracked.
The concentrations of these species affect the change in Gibbs free energy associated
with the following reactions:
? myosin filament walking
? nucleotide exchange (GD ? GT )
? phosphate release by F and G-actin (FPi ? FD, GPi ? GD)
The effect of the concentrations of ATP, ADP, and Pi for the these reactions is to simply
change the reaction quotient Q which changes ?G via ?G = ?G0 + kBT logQ. For
instance the nucleotide exchange reaction can be written explicitly as
ATP +GD ? ADP +GT (D.7)
and the change in Gibbs free energy is
( )
CADPC T
?G = ?G0 + kBT log
G . (D.8)
CATPCGD
169
To treat the concentrations of ATP and ADP implicitly, we rewrite the reaction as
GD ? GT (D.9)
for which the change in Gibbs free energy is
( )
? C T
?G = ?G0 + k GBT log , (D.10)
CGD
where ( )
0? 0 CADP?G = ?G + kBT log . (D.11)
CATP
A similar approach is taken for the other reactions mentioned above.
The concentration of just ATP (since neither ADP or Pi appear implicitly as reactants
in any of the reactions of the CT model) affects the kinetics of the following reactions:
? myosin filament walking
? nucleotide exchange
To understand the effect of CATP on the myosin filament walking rate, we employ the
five state cross-bridge model of a single myosin head described in Ref. 90. In that model
unbinding of a head from the filament substrate occurs via two pathways: a slip path,
with rate k35, and a catch path, with an effective rate k345. The catch path is a two-step
reaction: the release of ADP with rate k34, followed by the unbinding of the filament
head and binding of ATP with rate k45 = kTCATP . The effective rate constant of the
approximate one-step representation of this reaction is
k34kTCATP
k345 = . (D.12)
k34 + kTCATP
Unless CATP is very low, this reaction rate is limited by k34. Because the head can
unbind by the catch or slip pathway, the rate for unbinding is
k34kTCATP
khead, unbind = k35 + (D.13)
k34 + kTCATP
170
Figure D.2: A discrete model of the sequence of chemical reactions
constituting the myosin head power-stroke cycle. In a minifilament, tens
of such heads work in tandem to processively walk along the actin sub-
strate. Image reprinted from Ref. [90] with permission from the American
Physical Society.
The slip path should also be dependent on ATP concentration, since in state 5 the head
is ATP-bound, however we follow the authors of [90] in neglecting this dependence since
the slip path only becomes active under large load.
The rate of the nucleotide exchange reaction also depends on CATP and also occurs
in two steps. The full reaction, with ATP and ADP explicitly included, is
GD +ATP ? G? +ATP +ADP ? GT +ADP (D.14)
where G? represents actin with no bound nucleotide. Following [183], this reaction is
approximated as a one-step irreversible reaction with ATP and ADP included explicitly,
Equation D.9. We can write the rate knex of this approximate reaction as:
kD??k??TCATP
knex = ? kD?? (D.15)
kD?? + k??TCATP
where kD?? is the dissociation rate of ADP, k??T is the second order rate constant
of ATP association, and the approximation holds except at low concentrations of ATP
171
when ADP dissociation is no longer the rate-limiting step. The values of k??T and kD??
are presented and discussed in Refs. 13, 15, 313.
Reaction Rate Constant ?G0 (kBT )
GT poly at plus end 11.6 (?Ms)?1 -2.12 a [183]
GT depoly at plus end 1.4 s?1 2.12 a [183]
GT poly at minus end 1.3 (?Ms)?1 -0.51 a [183]
GT depoly at minus end 0.8 s?1 0.51 a [183]
GPi poly at plus end 3.4 (?Ms)?1 -2.81 a [12]
GPi depoly at plus end 0.2 s?1 2.81 a [12]
GPi poly at minus end 0.11 (?Ms)?1 -1.75 a [12]
GPi depoly at minus end 0.02 s?1 1.75 a [12]
GD poly at plus end 2.9 (?Ms)?1 0.59 a [183]
GD depoly at plus end 5.4 s?1 -0.59 a [183]
GD poly at minus end 0.09 (?Ms)?1 1.03 a [183]
GD depoly at minus end 0.25 s?1 -1.03 a [183]
Pi release by F-actin 0.002 s?1 -7.31 b,e [12, 183]
ATP hydrolysis by F-actin 0.3 s?1 -10.0 [91, 183]
Pi release by G-actin 5 s?1 -10.77 c,e -
Nucleotide exchange 0.01 s?1 e -6.76 c,e [183]
Cross-linker binding 0.7 (?Ms)?1 -0.85 a [42]
Cross-linker unbinding 0.3 s?1 0.85 a [42]
Myosin head binding 0.2 s?1 d - d [42]
Myosin head unbinding 1.708 s?1 d - d [42]
Myosin filament walking - f -14.5 e [209]
Table D.1: Kinetics and thermodynamic parameters describing re-
actions in the CT model as well as cross-linker and myosin filament
(un)binding and myosin filament walking.
a - Values of ?G0 determined via Equation D.1.
b - ?G0 determined from Keq given in Ref. 12.
c - ?G0 determined using constraints as described above.
d - Parameters describing the myosin filament obtained via Equations D.4, D.5, D.6.
e - Depends implicitly on CATP , CADP , and CPi; given value applies to standard
state.
f - Calculated in simulation using results of PCM, see [42].
In all the studies in this thesis, implicit nucleotide concentrations are taken to be
CATP = 8 mM , CADP = 7 ?M , and CPi = 1 mM , corresponding roughly to the
amounts found in human muscle after exercise [209]. Other parameters of the system,
172
including mechanical constants, diffusion rates, screening lengths, and boundary cutoffs
are given below.
Parameter Description Value
General Simulation Parameters
kBT Thermal energy 4.1 pN ? nm
Lcomp Cubic compartment side length 500 nm
Nx, Ny, Nz Number of compartments in each dimension 2, 2, 2
Lcyl Filament cylinder equilibrium length 27 nm
?t Length of chemical evolution step 0.05 s
FT Force tolerance of mechanical minimization 1 pN
Mechanical Parameters
Kfil,str Actin filament stretching constant 100 pN/nm [42]
bend Actin filament bending energy 2690 pN ? nm [42,
314]
Kvol Cylinder excluded volume constant 105 pN/nm4 [42]
Khead,str NMIIA head stretching constant 2.5 pN/nm [315]
K?,str ?-actinin stretching constant 8 pN/nm [316]
boundary Boundary repulsion energy 41 pN ? nm a
? Boundary repulsion screening length 2.7 nm b
Mechanochemical Parameters
NNMIIA,bind Binding sites per cylinder for myosin motors 8 c
N?,bind Binding sites per cylinder for ?-actinin 4 d
dstep NMIIA minifilament step size 6.0 nm [315]
Nmin, Nmax Range of number of NMIIA heads per minifila- 15, 25 e [317]
ment
F Stall force of NMIIA minifilament 50 pN fstall
F0,head Characteristic force of NMIIA catch-bond 12.6 pN [93]
F0,? Characteristic force of ?-actinin slip-bond 17.2 pN [9]
F0,poly Characteristic force of actin Brownian ratchet 1.5 pN [10]
lM Equilibrium length of NMIIA minfilament 175? 225 nm [42]
l? Equilibrium length of ?-actinin 30? 40 nm [42]
Diffusion Constants
kactin,diff Diffusion constant of actin monomer 20 ?Ms?1 [42]
k Diffusion constant of ?-actinin 2 ?Ms?1?,diff [42, 226]
kmotor,diff Diffusion constant of NMIIA minifilament 0.2 ?Ms?1 [42]
Table D.2: All other parameters used in the simulations reported in
this chapter.
a - Chosen for the energy scale to be 10 kBT .
b - Chosen as the the length of a G-actin monomer.
c - Chosen to allow the spacing between binding sites to be roughly equal to its
physiological value near 6 nm [315].
173
d - Chosen to allow the spacing between binding sites to be roughly equal to its
physiological value near 30 nm [17].
e - Chosen to given an average Ntotal = 20 in approximate agreement with literature
values [317].
f - A wide range of values are found in the literature for the stall force of the minifil-
ament. We take an order of magnitude estimate for this parameter based on the stall
force of a single head (taken here as 5 pN , estimated as dstepKhead,str [42]) times the
number of bound heads in the minifilament (taken here as 10). This parameter choice
is empirically valid as it yields observable network contraction.
Cylinder equilibrium lengths Lcyl are chosen as 27 nm with 4 binding sites per cylin-
der for myosin filaments and 1 binding site per cylinder for cross-linkers, giving approxi-
mately physiological values for stepping distances of myosin motor filaments and spacing
along actin filaments of bound ?-actinin. We note that the form of the mechanochemical
models has been changed from those used in Ref. 42; the modeling used here is current
as of MEDYAN v3.2, and we refer readers Appendix A for further details.
D.2 Mean-field model of treadmilling dissipation
To validate the methods for quantifying dissipation using MEDYAN against a simpler
representation of actin filament treadmilling, we developed a mean-field description of
the dissipation resulting from chemical reactions in the CT model (see Chapter 2). Mean-
field models of the trajectory of the vector of concentrations of species, C(t), have been
formulated previously as an 11-dimensional system of ODEs ([183]), and in the CT model
as a 5-dimensional system of ODEs ([99]). These models describe the polymerization of a
concentration Nfil of actin filaments in a pool of actin subunits of total concentrationM .
The subunit species tracked by these models are distinguished by their polymerization
state and by the hydrolysis states of the nucleotide to which they are bound. The meaning
of ?mean-field? in this context is the assumption that reacting species are well-mixed
over the entire system volume, therefore obeying mass-action kinetics and deterministic
dynamics which can be represented by ODEs. For an instantaneous value of C, we
define a function D?(C) representing the instantaneous rate of dissipation due to a set
174
of reactions ?. Thus a solution of a mean-field C(t) = N(t)/?, allows us to construct the
trajectory of the dissipation rate, D?(C(t)). This function cannot capture the dissipation
due to the activity of myosin filaments or cross-linkers, or from relaxation of mechanical
stress, because these aspects are not included in the these mean-field models describing
filament treadmilling. The benefit of such a mean-field model is that one can perform
systematic variation of parameters with limited computational demands, and we use it
here to study the effect of the parameters Nfil and M on the dissipation due to filament
treadmilling.
The function D?(C(t)) can be written as a sum over the reactions ? ? ? of the
instantaneous rate of change of the solution?s Gibbs free energy due to that reaction:
?
D?(C(t)) = ?G?(C(t))r?(C(t)). (D.16)
???
The expressions for ?G?(C(t)) for different reactions are described above. The instan-
taneous rate of reaction ?, r?(C(t)), is written as usual for mass-action kinetics as
?
r (C(t)) = ?k C?i? ? i (D.17)
i?R
where R is the set of reactant species for reaction ?, and where we have included the
conversion factor ? to convert r ?1?(C(t)) to units of s .
To facilitate the study of dissipation due to filament treadmilling, we define a set of
reactions ? in the CT model which constitutes the dominant cycle a subunit undergoes
in the treadmilling process: a) polymerization of GT to the plus end, b) hydrolysis of
ATP by F T , c) release of Pi by FPi, d) depolymerization of GD from the minus end, and
e) nucleotide exchange converting GD to GT . We refer to this set of 5 reactions as the
main treadmilling pathway (MTP). Alternative sequences of reactions whose net effect
is similarly the conversion of one molecule of ATP to ADP and Pi are considered as of
secondary importance and not included in this analysis.
This mean-field description of entropy production can be compared to the descrip-
tion of entropy production rates for chemically reactive systems that emerged from the
175
Brussels school of thermodynamics [318, 319]. The results of that school include the min-
imum entropy-production principle applicable in the linear regime [198], and the general
evolution criterion applicable even in the the nonlinear regime [320]. Their formalism
typically considers the total entropy production rate as a volume integral over the local
entropy production rate density, which is itself written as a sum over fluxes multiplied
by their corresponding thermodynamics forces defined at each point of the system. This
sum is decomposed into terms representing diffusion and terms representing chemical
reactions, and the terms representing the chemical reactions are written such that the
fluxes reflect the net reaction rate at that point in the system. In contrast, our mean-
field description neglects concentration gradients and resulting diffusion fluxes, as we
assume a homogeneous distribution of the chemical species. Equation D.16 then repre-
sents only the chemical contribution to the entropy production, as a sum over the rates
and affinities of the reactions in the system, implicitly integrating over the homogeneous
system volume. We treat the forward and reverse direction for some chemical reaction as
separate terms in Equation D.16, so these rates cannot be considered fluxes which would
include the reverse rate as well. This allows for more general sets of reactions which
might include effectively irreversible processes for which the reverse rate is negligible.
However the set of reactions ? could be chosen to include a reverse reaction for each
forward reaction, with the result that these pairs of terms represent fluxes along the
reversible reaction pathway. The parsimonious reaction set MTP is chosen not to fully
describe the rate of entropy production in the system, but to allow easy visualization of
the main contributions to the entropy production. We do not pursue the connection of
our treatment to the formalism of the Brussels school further here, but we lastly note
that our results are compatible with their minimum entropy production principle, as
shown in Figure D.3.
We first verified that the mean-field model of MTP dissipation agreed with results
from MEDYAN simulations, to illustrate consistency between these approaches. In Fig-
ure D.3 we display the close match between the trajectory of MTP dissipation over a
2000 s run from these two approaches. Note that, to allow direct comparison, only the
176
changes of Gibbs free energy resulting from reactions in the MTP set are visualized for
both approaches here, i.e. the contribution from diffusion and other non-MTP reac-
tions in the MEDYAN simulation are not visualized. We also turned off in MEDYAN
the force-sensitive decrease in polymerization rate when the filament tips push against
the simulation hard-wall boundaries, since this effect is not captured in the mean-field
modeling [101]. We used a simulation volume of 1 ?m3 and initial conditions of equal
amounts (10 ?M each) of GT and GD actin in a 0.08 ?M pool of seed filaments con-
taining F T . The dissipation rate decreases nearly monotonically, attaining a minimal
steady state value after tens of seconds. In Figure D.3 we also display the individual
contributions to the sum in Equation D.16. Initial polymerization of GT to the plus end
constitutes the majority of the initial dissipation. As this polymerization process slows
after about 1 second, the hydrolysis of ATP by the now relatively abundant F T becomes
the dominant contribution. As hydrolysis then slows after about 10 seconds, the total
dissipation rate reaches a steady state value of roughly 80 kBT/s. In Figure D.3 we also
plot the mean-field prediction of the trajectory of the reacting species? concentrations.
We next simultaneously varied the total concentration of actinM and the concentra-
tion of actin filaments Nfil, and determined the total dissipation integrated along each
trajectory as well as the steady state dissipation rate. As M was varied, we held the
initial concentration of each species proportionally the same: 49 % GT , 49 % GD, and 2
% F T . As shown in Figure D.4, the integrated dissipation over 2000 s was observed to
increase monotonically with both M and Nfil. Quantitatively, the integrated dissipation
depends on the choice of initial proportions, however we found that the shape of the de-
pendence on Nfil and M is largely independent of initial proportions (data not shown).
Total dissipation increases with M simply because more actin is available to hydrolyze
ATP during the approach to steady state. For large amounts of actin, increasing Nfil, the
number concentration of filaments, allows increased polymerization of GT , which consti-
tutes a large contribution to total dissipation during the early stages of the trajectory.
177
Figure D.3: Results of the mean-field modeling of MTP dissipation.
Top: Comparison of DMTP(C(t)) calculated using the mean-field model
(with time increments of 0.01 s), with ?Gchem rates measured during
MEDYAN simulation (with time increments of 0.1 s). Middle: Plot of the
contributions of each reaction in the MTP to the total dissipation. The
items in the legend represent reactions in the MTP, which are described
in the main text. Bottom: Plot of the trajectories of the reacting species
concentrations. The notation for each species is described in the main
text.
178
Figure D.4: Left : The total dissipation integrated over 2000 s trajec-
tories as M and Nfil are varied. Right : The steady state dissipation rate
over the same range of M and Nfil.
Increasing Nfil also shifts the steady state concentration of GT downward (Figure D.5),
implying that more GT has been polymerized during the approach to steady state. A
loose analogy can be made of a one lane road compared to a multi-lane highway during
heavy traffic to describe this situation. As Nfil is increased with M fixed, the steady
state dissipation rate increases concavely, as shown in Figure D.4. The steady state con-
centration of F T also increases concavely (Figure D.5), representing higher rates of ATP
hydrolysis. The contributions of each reaction to the total dissipation rate at steady
state as Nfil is varied is illustrated in Figure D.6. The lack of dependence of the steady
state dissipation rate on M can be explained by the fact that increasing M increases the
steady state concentration of only FD, not of any other species [99]. In other words, all
extra actin accumulates in the form FD as M is increased. This species is essentially in-
ert, since the depolymerization rate of FD is controlled by the concentration of filaments
Nfil. Thus the steady state dissipation has no dependence on the total amount of actin.
Furthermore, it has no dependence on the initial concentrations, since it is known that
the steady state vector of concentrations does not depend on initial conditions [99].
179
Figure D.5: The concentrations at steady state of the various actin
subunit species as the concentration of filaments Nfil is varied. These
curves have no dependence on initial conditions, except FD which will
increase linearly with the total concentration of actin subunits M ; any
additional actin subunits in the system will accumulate in the form FD
at steady state.
180
Figure D.6: Contributions of each reaction in the main treadmilling
pathway to the total dissipation rate at steady-state, as the concentration
of filaments Nfil is varied. These curves have no dependence on the initial
concentrations of the different subunit species.
181
Appendix E
Supporting information for Chapter
6
E.1 Weibull plots
To assess whether stretched exponential functions describe the empirical distributions of
|?U?| and ?U+, we first define Q(u) = ln (? ln (P (|?U?| ? u))). The degree to which
the plots ofQ(u) against ln(u) appear to be linear serves as a check of the appropriateness
of modeling P (|?U?|) as a stretched exponential, or Weibull, distribution [273]. On
the basis of these Figures E.1 and E.2 we conclude that the Weibull distribution is a
satisfactory choice for all values of V . In the Chapter 6, the Weibull parameters k and
? were determined by fitting the stretched exponent ke?(x/?) to the observed CCDF
P (|?U?|) on a log-scale, that is, by fitting ?(x/?)k to ln (P (|?U?|)) using standard
nonlinear fitting routines. Treating these functions on a log-scale ensured a better fit to
the distribution tails which are of most interest in the present case.
E.2 Computing filament displacements
The area between the two filaments x and y is triangulated using the beads comprising
the filaments ({x }nx?1 and { }ny?1i i=0 yj j=0 ) as vertices, where nx is the number of beads in x
and similarly for ny. To compute the displacement of filament x during the time interval
?t, we set y to the new configuration of x at the end of the interval. The triangles come
182
V ( m3)
0.125
2 1.0
3.375
8.0
0 15.625
2
4
6
8
10
6 4 2 0 2 4 6 8
log(u)
Figure E.1: Plots of the function Q(u) = ln (? ln (P (|?U?| ? u))) for
different volumes V along with a fitted line, where P (|?U?|) is the ob-
served CCDF obtained from five runs of each volume.
Q(u)
183
V ( m3)
2 0.1251.0
3.375
8.0
0 15.625
2
4
6
8
10
8 6 4 2 0 2 4 6 8
log(u)
Figure E.2: Plots of the function Q(u) = ln (? ln (P (?U+ ? u))) for
different volumes V along with a fitted line, where P (?U+) is the ob-
served CCDF obtained from five runs of each volume.
Q(u)
184
in pairs for most of the filament lengths, as shown using the dark and light colors of green
of Figure E.3. If nx and ny are unequal (say nx < ny), extra triangles are added using
the last bead in x, xnx?1, as the only vertex in filament x. The sum of these triangle
areas Atot is divided by the average of the two filament contour lengths Lx and Ly to
give the measure of distance d = 2AtotL +L .x y
E.3 Parameterization
The following table lists the parameters chosen for the simulations presented in this
Chapters 5 and 6. However, in 5, the motor stall force is varied and does not correspond
to values given below, and the cylinder discretization lengths are chosen differently. As
described in that chapter, the cylinder lengths are either 27 nm (with bend = 2690 pN ?
nm) or 108 nm (with bend = 672 pN ? nm). Additionally, in finite size scaling studies
the number of components Nx, Ny, Nz are varied.
185
Figure E.3: Illustration of how the area between two filaments x and
y is triangulated to allow calculation of the distance between them. The
beads comprising the filaments are labeled xi, yj , and areas between
triplets of beads are labeled Ai,j where the lowest indices of the beads xi
and yj in the triplet are used.
186
Parameter Description Value
General Simulation Parameters
kBT Thermal energy 4.1 pN ? nm
Lcomp Cubic compartment side length 500 nm
Nx, Ny, Nz Number of compartments in each dimension 2, 2, 2
Lcyl Filament cylinder equilibrium length 54 nm
?t Length of chemical evolution step 0.05 s
FT Force tolerance of mechanical minimization 1 pN
Mechanical Parameters
Kfil,str Actin filament stretching constant 100 pN/nm [42]
bend Actin filament bending energy 1344 pN ? nm [42,
314]
Kvol Cylinder excluded volume constant 105 pN/nm4 [42]
Khead,str NMIIA head stretching constant 2.5 pN/nm [315]
K?,str ?-actinin stretching constant 8 pN/nm [316]
boundary Boundary repulsion energy 41 pN ? nm a
? Boundary repulsion screening length 2.7 nm b
Mechanochemical Parameters
NNMIIA,bind Binding sites per cylinder for myosin motors 8 c
N?,bind Binding sites per cylinder for ?-actinin 4 d
dstep NMIIA minifilament step size 6.0 nm [315]
Nmin, Nmax Range of number of NMIIA heads per minifila- 15, 25 e [317]
ment
Fstall Stall force of NMIIA minifilament 100 pN f
F0,head Characteristic force of NMIIA catch-bond 12.6 pN [93]
F0,? Characteristic force of ?-actinin slip-bond 17.2 pN [9]
F0,poly Characteristic force of actin Brownian ratchet 1.5 pN [10]
lM Equilibrium length of NMIIA minfilament 175? 225 nm [42]
l? Equilibrium length of ?-actinin 30? 40 nm [42]
Chemical Parameters
kactin,diff Diffusion constant of actin monomer 20 ?Ms?1 [42]
k?,diff Diffusion constant of ?-actinin 2 ?Ms?1 [42, 226]
k Diffusion constant of NMIIA minifilament 0.2 ?Ms?1motor,diff [42]
kactin,poly,+ Actin plus-end polymerization 11.6 ?Ms?1 [12]
kactin,poly,- Actin minus-end polymerization 1.3 ?Ms?1 [12]
k ?1actin,depoly,+ Actin plus-end depolymerization 1.4 s [12]
k ?1actin,depoly,- Actin minus-end depolymerization 0.8 s [12]
k ?1head,bind NMIIA head binding 0.2 s [88]
k0head,unbind NMIIA head unbinding under zero tension 1.7 s
?1 [42, 88]
k?,bind ?-actinin binding 0.7 ?Ms?1 [72]
k0?,unbind ?-actinin unbinding under zero tension 0.3 s
?1 [72]
Table E.1: All parameters used in the simulations reported in Chapter
6 and 7.
a - Chosen for the energy scale to be 10 kBT .
187
b - Chosen as the the length of a G-actin monomer.
c - Chosen to allow the spacing between binding sites to be roughly equal to its
physiological value near 6 nm [315].
d - Chosen to allow the spacing between binding sites to be roughly equal to its
physiological value near 30 nm [17].
e - Chosen to given an average Ntotal = 20 in approximate agreement with literature
values [317].
f - A wide range of values are found in the literature for the stall force of the minifil-
ament. We take an order of magnitude estimate for this parameter based on the stall
force of a single head (taken here as 10 pN , estimated as dstepKhead,str [42]) times the
number of bound heads in the minifilament (taken here as 10). This parameter choice
is empirically valid as it yields observable network contraction.
E.4 Shannon entropy of tension distribution
The simulation volume of 1 ?m3 is discretized into 103 cubic voxels, each 0.1 ?m in linear
dimension. Let i, j, k = 1, ..., 10 index these voxels, which are an analysis tool and not
related to the reaction-diffusion compartments used in MEDYAN. After each simulation
cycle, the mechanical components of the cytoskeletal network (i.e. the filament cylinders,
the myosin motors, and the passive cross-linkers) are each under some compressive or
tensile force Tn, where n indexes the mechanical component. There are other mechanical
potentials involving these components, but we focus here only on the tensions Tn. Each
mechanical component has a center of mass rn, and we introduce the indicator function
?ijk(rn) which is equal to 1 if rn is inside voxel i, j, k and 0 otherwise. The total tension
magnitude inside voxel i, j, k is
?
|T |ijk = |Tn|?ijk(rn). (E.1)
n
The discrete non-negative scalar field |T |ijk is converted to a probability distribution
Pijk by normalization:
|T |ijk
Pijk = ? . (E.2)
ijk|T |ijk
188
Finally, we introduce the discrete Shannon entropy of this distribution at time t as
?
H(t) = ? Pijk(t) lnPijk(t). (E.3)
ijk
The units of H are nats, and large values indicate a homogeneous spatial distribution
of tension magnitudes throughout the network. Reported trends using this metric are
essentially independent of the discretization length.
E.5 Machine learning implementation details
To forecast the occurrence of cytoquakes, we resorted to using a high-dimensional ma-
chine learning (ML) model (3 layer feed-forward neural network) after it was found that
several simple features in the eigenspectrum which we believed might reflect mechanical
stability (for instance the value of the smallest positive eigenvalue) did not by themselves
significantly correlate with cytoquake occurrence. We pose the forecasting of cytoquakes
as a binary classification problem. A trajectory ?U(t) = U(t+ ?t)?U(t) at QSS (after
1,000 s) is converted to a binary sequence such that each t for which ?U(t) ? ?UT ,
as well as the tW = 0.15 previous seconds (i.e. 3 previous time points) are classified
as cytoquakes, and the rest are not. This tW window is chosen to help overcome the
stochasticity inherent in the chemical dynamics which, along with the instantaneous me-
chanical stability we are using as a predictor, controls cytoquake occurrence. We focus
here on the five runs of conditions C3,3. ?UT = ?100 kBT is chosen to lie well in the tail
of the distribution of |?U?| for this condition and therefore distinguishes rare events, as
shown in Figure 6.2. With these choices, ? 10% of samples across all runs are labeled
as events in the classification problem.
The predictors of the model capture information about the network?s mechanical
stability. The ordered sets of eigenvalues {?k}3Nk=1 at each time t is padded by adding
zero eigenvalues between the unstable (?k < 0) and stable (?k ? 0) parts of the spectrum
to maintain a fixed input dimension across all time points and runs. We then collect these
eigenvalues into a tuple M(t) such that the first element of M(t) is the largest negative
189
A B
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0 10 20 30
Number of Components
C D
1.0 0.80
0.78
0.8
0.76
0.6 0.74
0.72
0.4
0.70
0.2 0.68
AUC: 0.810 0.66
0.00.0 0.2 0.4 0.6 0.8 1.0 oft tiff , I U , IFPR s s I +, U , U
,
Figure E.4: A: Cumulative explained variance from PCA of the ?
1, 600 eigenvalues {? (t)}3Nk k=1. B: Schematic depiction of the feed-forward
neural network architecture. C: ROC curves for the model using only
{? }3Nk k=1 as input and trained on a single run of condition C3,3, with five
realizations of the stochastic batch network training and their average
shown. The ROC curve of a random model is plotted as the red dotted
line. D: Bar plot indicating the AUC of ROC curves using different
combinations of inputs for the model trained on data collected from all
runs of condition C3,3. From left to right, the labels indicate that the
model inputs are: {? }3Nk k=1; {?k|0 ? ? 3Nk < ?T }; {?k|?T ? ?k}; {?k}k=1
and {rk}3Nk=1; U , using a logistic regression model; {?k}3Nk=1, {r 3Nk}k=1,
and U ; {? }3N 3Nk k=1, {rk}k=1, and U with forecasting done for large positive
increments ?U > 100 kBT . Error bars indicate uncertainty from five
realizations of stochastic batch training.
TPR Cumulative Explained Variance
AUC
190
?k at time t and the last element is the largest positive ?k at time t. We optionally
include the the inverse participation ratios {r }3Nk k=1 in this vector by first adding zeros in
the places of the set {r }3Nk k=1 corresponding to where zeros were added in the set {?k}3Nk=1,
and then interleaving the ?k and rk in the now doubly sized tuple M(t), so that now for
example the first two elements of M(t) correspond to the largest negative ?k and the
associated rk at time t. The tuples M(t) are then linearly rescaled, so for each element
Mi(t) the average over all times of a run is 0 and the variance is 1. These rescaled tuples
are labeled M?(t).
When only the ?k are included then M?(t) has ? 1,600 dimensions, and with the
rk are also included it has ? 3,200 dimensions. To avoid overfitting the model, we first
reduce the dimensionality of M?(t) via principal component analysis (PCA) using all QSS
time points in a run. We choose 30 dimensions as the size of the reduced tuple m(t)
because this allows for more than 95% of the variance of M?(t) to be explained when just
the ?k are included as shown in Figure E.4.A. Model performance appreciably decreases
when fewer than 30 dimension are used and improves only marginally if more are used. A
row of ones is added as a 31st dimension to m(t) as a bias for the neural network. As an
additional indicator of the network?s mechanical stability we also consider its mechanical
energy U at time t. U(t) is linearly rescaled to give U?(t) so that it has zero mean and
unit variance. We then optionally augment with input tuple m(t) with the U?(t) as a
32nd dimension.
We can treat the data from all five runs of condition C3,3 separately or combine all
data together to train a larger model. Model performance is generally found to be better
when trained on data from a single run, however by combining data from all runs we
probe more general underlying trends that are not specific to the network organization
of one run. When describing trends from varying model inputs, as in Figure E.4.D, we
focus on results obtained by combining all runs due to their greater generality.
For a single run there are ? 20,000 samples, giving 100,000 samples when combining
all runs. When combining runs, we first rescale and perform PCA on the predictors using
only the data within a single run, and then concatenate the resulting m(t) with their
191
associated labels into a larger data set. This way the relative variation of the predictors
compared to their typical values for a particular organization of the actomyosin network
is retained, and the typical values of particular network organizations themselves affect
the model inputs to a lesser degree.
We used the Python modules scikit-learn and Keras with a Tensorflow back end to
train a deep feed-forward neural network and a logistic regression model for the binary
classification problem [321, 322]. The 31 or 32-dimensional (depending on if U?(t) is
included as a predictor) input tuple m is fed into three fully connected hidden layers
Li, i = 1, 2, 3, each with either 30 or 100 nodes depending on if the data consists of
a single run (20,000 samples) or of all five runs (100,000 samples). Each node in the
hidden layers uses a rectified linear unit activation function. The output of the network
is two nodes using a softmax activation function whose values are p and 1 ? p, where
p is the predicted probability of a cytoquake event at that time t. This architecture
is schematically illustrated in Figure E.4.B. The network is trained for either 400 or
200 epochs using a categorical cross-entropy loss function with Adam optimization in
stochastically chosen batches of either 1,000 or 10,000 samples, depending on the whether
the single or multiple run data sets, respectively, are used. The cytoquake samples are
given a higher weight (?3) than the non-cytoquake samples during training. A L2
penalty of 0.05 is used to curb overfitting. When using only U?(t) as a predictor, a
logistic regression model is fit using the same sample weights.
Of all the data samples, we use 2/3 to train the model with and validate the model on
the remaining 1/3. We repeat these random training/testing set splits to gather statis-
tics on model performance. The binary classification procedure involves the probability
threshold pT (such that p > pT means the model predicts a cytoquake). Model perfor-
mance is measured by varying pT from 0 to 1 and measuring the true positive rate (TPR,
the proportion of actual cytoquakes correctly predicted as such) and false positive rate
(FPR, the proportion of actual non-cytoquakes incorrectly predicted as cytoquakes) on
the test data; the locus of these points forms the receiver operator characteristic (ROC)
curve. A random model would have FPR = TPR, so an area under the curve (AUC)
192
of the ROC curve greater than 0.5 indicates a good model, and a perfect model would
have an AUC of 1. One can also consider precision-recall (PR) curves, which contain
points in the space of model precision (the proportion of predicted cytoquakes which
were actual cytoquakes) and recall (the same as TPR). A random model would have the
same precision, equal to the proportion of actual cytoquakes in the testing data, for all
values of recall as pT is varied, giving an AUC equal to that proportion.
When the test data is unbalanced, i.e. when there are many more non-cytoquake
events than cytoquake events, it has been shown that the AUC of the PR curve is a more
faithful metric for model performance (since a model may score a high AUC of the ROC
curve by overestimating that events are not cytoquakes) [323, 324]. To overcome this
limitation of ROC curves, which we believe has a more intuitive interpretation that PR
curves, we balance the testing data, keeping all cytoquake events and randomly keeping
an equal number of non-cytoquake events. We confirmed that trends observed in the
AUC of the ROC curves as the model is varied also hold when considering the AUC of
PR curves on the full test set.
In Figure E.5 we show examples of these PR and ROC curves on the training and
testing data for a model trained on a single run. The very high AUC of the PR and ROC
curves evaluated on the training data indicates that the model has nearly perfected its
prediction on those samples and may indicate overfitting, however this high performance
generalizes nicely to the unseen testing data. Note that the AUC of the ROC evaluated
on the testing data is significantly higher than shown in Figure E.4.D reflecting the
generally higher performance of models trained on data from a single run compared to
models trained on data from all runs.
Finally, as a sanity check, we confirmed that randomly shuffling the labels on the
training set decreases performance on the training set and causes the performance on
the test set to decrease to that of a random model, as shown in Figure E.6.
Applying the model using the Hessian eigenspectrum as the input, we obtained an
AUC of 0.81 when using data from a single run of condition C3,3 (Figure E.4.C) and
of 0.70 when using data from five runs, i.e. from five different network realizations. In
193
Figure E.4.D, we display the effects of varying the machine learning model inputs on
prediction performance, reflecting the degree to which cytoquake occurrence depends on
the various inputs. We point out that these trends from varying the model inputs are not
particularly strong, contributing only marginal changes (though greater the measured
uncertainty) to the model performance. These differences are less than the difference
resulting from combing all five runs in a data set rather than using one run. We report
them here mainly out of completeness, rather than in support of some strong conclusion.
Uncertainty in AUC from five repetitions of stochastic batch training is roughly 0.01
for all reported values. Keeping only the eigenvalues of the soft modes does not harm
performance (AUC 0.71), while keeping only the stiff modes does harm performance
(AUC 0.68). Performance is not harmed (AUC 0.72) upon augmenting the input with the
inverse participation ratios {r (t)}3Nk k=1. Interestingly, we found that a logistic regression
model using only the mechanical energy U(t) as an input feature performs well (AUC
0.74, with a smaller uncertainty around 0.002 for this simpler model), reminiscent of
the debate concerning one neuron vs. deep learning models of earthquake aftershock
prediction [286, 287]. This logistic regression model has learned an optimal cutoff for U
that indicates instability and likely cytoquake occurrence. We may seemingly conclude
that the machine learning model using the Hessian eigenspectrum as an input has merely
learned what the mechanical energy is, however we find that by far the best performance
results from combining {?k(t)}3Nk=1, {rk(t)}3Nk=1, and U(t) in the ML model, reaching an
AUC of 0.79 when using data from all five runs. This suggests that the learned features of
the Hessian eigenspectrum are not redundant given U , i.e. that their mutual information
is low. Finally, we found that prediction of large positive increments (?U > 100 kBT )
is also possible, with an AUC of 0.74 when combining all inputs.
194
A B
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
AUC*: 0.808 AUC: 0.993
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
C Recall D FPR
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
AUC*: 0.363 AUC: 0.855
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Recall FPR
Figure E.5: A: PR curve evaluated for a model using {?k}3Nk=1, {r }3Nk k=1,
and U as inputs trained on data from a single run at QSS of condition
C3,3 and evaluated on the training data. The red line indicates the per-
formance of a random model on the data set. The asterisk on the AUC
indicates that the fraction of cytoquake samples in the data set (for this
run ? 0.06) has been subtracted from the actual AUC, to give the area
between the black and red curves. B: ROC curve for the same model
evaluated on the training data. C: PR curve for the same model eval-
uated on the balanced testing data. D: ROC curve for the same model
evaluated on the balanced testing data.
Precision Precision
TPR TPR
195
A B
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
AUC*: 0.383 AUC: 0.918
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
C Recall D FPR
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
AUC*: -0.009 AUC: 0.462
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Recall FPR
Figure E.6: A: PR curve evaluated for a model using {? }3Nk k=1, {rk}3Nk=1,
and U as inputs trained on data from a single run at QSS of condition
C3,3 and evaluated on the training data, when the training data labels
have been randomly shuffled. The red line indicates the performance
of a random model on the data set. B: ROC curve for the same model
evaluated on the training data. C: PR curve for the same model evaluated
on the balanced testing data. D: ROC curve for the same model evaluated
on the balanced testing data.
Precision Precision
TPR TPR
196
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