Table Of ContentTracking nonequilibrium in living matter and
self-propelled systems
Etienne Fodor
To cite this version:
Etienne Fodor. Tracking nonequilibrium in living matter and self-propelled systems. Physics [physics].
Université Paris Diderot, 2016. English. tel-01345451v2
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Université Paris Diderot (Paris 7) - Sorbonne Paris Cité
École Doctorale “Physique en Île-de-France”
Thèse de doctorat
Discipline : Physique théorique
présentée par
Étienne Fodor
Signatures hors de l’équilibre
dans les systèmes vivants et actifs
Tracking nonequilibrium in living matter
and self-propelled systems
dirigée par Paolo Visco et Frédéric van Wijland
Soutenue le 4 juillet 2016 devant le jury composé de
Eric Bertin examinateur
Chase P. Broedersz rapporteur
David S. Dean rapporteur
Jean-François Joanny président
Paolo Visco co-directeur de thèse
Frédéric van Wijland directeur de thèse
Acknowledgments
I would like to warmly thank my Ph.D. advisors, Paolo Visco and Frédéric van
Wijland, for their support, guidance and encouragement all along the three years
of my Ph.D. thesis. I have highly benefited from all the enthusiasm, clarity and
rigor that characterize their supervision. I am indebted to Nir S. Gov for the
recurrent discussions which have largely contributed to shape my conception of
active fluctuations in living matter. I am very grateful to Julien Tailleur for sharing
his vision of active matter which has motivated the second part of my thesis, and
I sincerely thank Julien for his overall support. I am also thankful to Cesare
Nardini and Michael E. Cates for their collaboration in the study of self-propelled
particles as an active matter system. Besides, I would like to thank the members
of my thesis committee: Eric Bertin, Chase P. Broedersz, David S. Dean and
Jean-François Joanny, for accepting to evaluate my Ph.D. work. In particular,
I am grateful to the referees, Chase P. Broedersz and David S. Dean, for their
comments which enabled me to improve this manuscript.
Moreover, I would like to express my sincere gratitude to the three different
experimental groups with whom I had the opportunity to collaborate. I am grateful
to David A. Weitz and Ming Guo for sharing their results on tracer dynamics
inside living melanoma cells, as a first insight into the captivating issues raised
by the intracellular dynamics. I thank Timo Betz and Wylie W. Ahmed for the
fruitful collaboration that we had all along my Ph.D. thesis. I really enjoyed
our collaborative effort to analyze the dynamics within living mouse oocytes as a
fascinating nonequilibrium material. I am thankful to Marie-Hélène Verlhac and
Maria Almonacid who provided us with the oocytes. My sincere thanks also go
to Daniel Riveline, Vishu Mehandia, Jordi Comelles and Raghavan Thiagarajan.
Our collaboration has been a stunning opportunity to complement our study of
active fluctuations in living matter with epithelial tissues.
Furthermore, I am obliged to Hisao Hayakawa for providing me with the oppor-
tunity to visit the Yukawa Institute of Theoretical Physics at Kyoto University.
The two visits during the second and third year of my Ph.D. were exciting ex-
periences to discover the academic research conducted at the YITP and in the
Physics Department of Kyoto University. I have largely benefited from Hisao’s
experience to gain a deeper view into the intriguing phenomena which feature
non-Gaussian fluctuations. I thank the Ph.D. students of the Advanced Statis-
tical Mechanics group for the many interesting discussions that we had, notably
Kiyoshi Kanazawa and Tomohiko Sano. Besides, I am very grateful to Shin-
ichi Sasa, Takahiro Nemoto and Andreas Dechant for sharing their insight into
nonequilibrium statistical mechanics.
Eventually, I would like to thank the various members of the laboratory Matière
et Systèmes Complexes for the very nice working atmosphere in which I enjoyed
studying during the past three years. In particular, I am grateful to François
Graner, François Gallet, Atef Asnacios, Sylive Hénon, Fabien Montel, Jean-François
Berret and Jean-Baptiste Fournier for insightful discussions. Besides, I also thank
the many Ph.D. students who made the everyday life in the laboratory so ani-
mated: Thomas, Gwen, Tanguy, Simon, Mourtaza, Agnese, Alex, Sham, David,
François, Iris and others.
Contents
Introduction 1
1 Modeling the fluctuations of passive tracers 7
1.1 Tracer in a thermal bath . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Nonequilibrium dynamics driven by active fluctuations . . . . . . . 12
2 Living matter: a paradigm of nonequilibrium systems 17
2.1 Structure and dynamics of the intracellular environment . . . . . . 18
2.2 Measuring fluctuations and response . . . . . . . . . . . . . . . . . 20
2.2.1 Statistics of tracer displacement . . . . . . . . . . . . . . . . 21
2.2.2 Mechanical properties at the subcellular scale . . . . . . . . 24
2.3 Models for the intracellular mechanics and dynamics . . . . . . . . 27
3 Active cage model of fluctuations in living cells 31
3.1 Caging dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Model predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 Statistics of tracer displacement . . . . . . . . . . . . . . . . 36
3.2.2 Energetics of active fluctuations . . . . . . . . . . . . . . . . 39
Paper A: Energetics of active fluctuations in living cells . . . . . . . 43
4 Colloidal tracers in living melanoma cells 55
Paper B: Activity-driven fluctuations in living cells . . . . . . . . . . . . 59
5 Vesicle dynamics in living mouse oocytes 67
Paper C: Active mechanics reveal molecular-scale force kinetics in living
oocyte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Paper D: Nonequilibrium dissipation in living oocytes . . . . . . . . . . . 107
6 Vertex fluctuations in epithelial layers 119
Paper E: Active fluctuations are controlled by molecular motor regula-
tions in cell monolayer . . . . . . . . . . . . . . . . . . . . . . . . . 123
7 Nonequilibrium properties of persistent self-propelled particles 137
7.1 Self-propelled particles as an active matter system . . . . . . . . . . 138
7.2 Interacting particles under persistent fluctuations . . . . . . . . . . 140
7.3 Effective equilibrium regime . . . . . . . . . . . . . . . . . . . . . . 146
Paper F: How far from equilibrium is active matter? . . . . . . . . . 149
7.4 Collective modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Main results and outlook 163
A Persistent self-propelled particles: approximate dynamics 167
B Persistent self-propelled particles: Dean-Kawasaki equation 173
Paper G: Generalized Langevin equation with hydrodynamic back-
flow: Equilibrium properties 177
Paper H: Active cell mechanics: Measurement and theory 183
Paper I: Modeling the dynamics of a tracer particle in an elastic
active gel 195
Paper J: Active cage model of glassy dynamics 204
Bibliography 212
Papers 231
Introduction
The aim of statistical mechanics is to describe the properties of macroscopic sys-
tems from the sole knowledge of their microscopic constituents and of their inter-
actions. The predictions of statistical mechanics offer the opportunity to compare
a wide range of systems by using a reduced number of macroscopic quantities.
Equilibrium systems are characterized by very specific properties both at the dy-
namical and stationary levels. An important feature of equilibrium systems is the
time reversibility of their dynamics. It constrains the relaxation after a (small)
perturbation to be fully encoded in the spontaneous stationary fluctuations. More-
over, fluctuations in equilibrium are entirely controlled by only two parameters: the
temperature and the friction coefficient with the surrounding thermostat, which
endow these fluctuations with a strong sense of universality. No further details
from the thermostat are at play.
Waiving the constraints of equilibrium opens the door to a wide variety of
nonequilibrium dynamics. The first nonequilibium systems that come to mind are
systems caught during their relaxation towards equilibrium. Others can be main-
tained out-of-equilibrium by applying an external field enforcing a steady flux, such
as a particle or charge current, or an energy flux. Yet another class of nonequi-
librium systems comprises systems in which energy is injected and dissipated at
the microscopic level of their individual components [1]. These are called active
systems, and they are the main focus of the present study. The energy stored in
the environment, most often in a chemical form, is converted into mechanical work
to produce directed forces and thus directed motion.
By contrast to equilibrium settings, the breakdown of equilibrium laws in ac-
tive systems can be used to extract quantitative information about the microscopic
active processes making up the energy reservoir. One can access the kinetic de-
tails of the fluctuations, to be characterized in terms of time, length and energy
scales. In equilibrium, the reversibility of the dynamics is enforced by the detailed
balance principle: the forward and reverse transitions between microstates are
equally probable in a steady state. The existence of an arrow of time only emerges
2 Introduction
at the macroscopic level as a result of a coarse graining of the dynamics [2]. By
contrast, the arrow of time is already defined at the microscopic level in active
systems because of the irreversibility of the dynamics. Recent methods of stochas-
tic thermodynamics have been proposed to extend thermodynamic concepts when
fluctuations are of paramount importance [3]. They provide a powerful framework
to relate the breakdown of time reversal to the microscopic energy conversion at
the basis of the nonequilibrium dynamics. We will discuss some of the most fruitful
ideas of this developping field in Chapter 1.
From individual tracers in living systems ...
A paradigm of active systems are living systems. In living cells, it is the contin-
uous injection of energy provided by adenosine triphosphate (ATP) which initially
triggers the activity of intracelullar nonequilibrium processes. For instance, the
molecular motors can convert the chemical energy provided by ATP hydrolysis
into a mechanical work to exert forces within the cell. The ensuing fluctuations
are referred to as active fluctuations, distinct from the thermal fluctuations already
present in the absence of nonequilibrium activity. The effect of these fluctuations
is apparent in a large variety of living systems, from individual crawling cells to
cell aggregates and epithelial tissues. They drive the dynamics of intracellular
components, such as proteins, organelles and cytoskeletal filaments [4, 5]. Tracer
particles are injected in living cells to probe these fluctuations. Alternatively,
the intracellular environment is reconstituted in vitro as minimal model systems.
Recent progress in tracking methods allow one to gather a large amount of statis-
tics to analyze the tracer displacement. Moreover, the tracer can be manipulated
to measure the response of the system: it reveals the viscoelastic properties of
the intracellular medium [6–8]. These techniques, known as microrheology, are
presented in our overview in Chapter 2. Combining measurements of the sponta-
neous fluctuations and of the response, the departure from equilibrium is generally
quantified by a frequency dependent effective temperature [9]. Yet, its physical
interpretation is at best limited. Not only does it lack any microscopic interpre-
tation, unlike the kinetic energy of thermal agitation, but it also depends on the
measured observables.
To go beyond this characterization of the intracellular nonequilibrium proper-
ties, one has to rely on a modeling of the tracer dynamics. Based on experimental
observations, we propose a phenomenological picture for the constant remodeling
of the intracellular environment in terms of an active cage subjected to random
hops. Our minimal model, which we present in Chapter 3, reproduces qualitatively
and quantitatively the fluctuations and the departure from equilibrium reported
experimentally. It provides a useful framework to analyze fluctuations and re-
sponse in actual living systems, allowing one to extract information about the
Introduction 3
intracellular activity. These applied quantitative aspects come in the subsequent
Chapters 4, 5 and 6.
To test our predictions with experimental data, we consider three different liv-
ing systems. First, we treat the dynamics of tracers injected in living melanoma
cells under three conditions: motor inhibited cells, ATP depleted cells, and un-
treated cells as a control. We demonstrate that our predictions are consistent with
a series of measurements, supporting the validity of our phenomenological picture.
We provide a quantitative characterization of active fluctuations in terms of time,
length and energy scales (Chapter 4). Second, we propose a detailed analysis of
fluctuations in living oocytes as probed by intracellular vesicles. We estimate the
energy fluxes between the active processes, the tracer and the thermostat. In par-
ticular, we reveal that the efficiency of energy transduction from the cytoskeleton
remodeling to the tracer motion is very low (Chapter 5). Eventually, we investigate
the dynamics of epithelial tissues through the fluctuations of tricellular junctions,
named as vertices. The analysis of vertex fluctuations provides a synthetic read-
out of the effect of inhibitors acting on the molecular pathway regulating motor
activity (Chapter 6).
... to interacting self-propelled particles
Another canonical example of active systems are the ones made of interact-
ing self-propelled constituents. The first experimental studies of such systems
were concerned with biological systems in which the emerging phenomenology
results from various complex ingredients. As an example, the interplay of the
self-propulsion, of the alignment interaction, and of the hydrodynamics interac-
tion drive the dynamics in dense swarms of bacteria [10]. To create minimal
biomimetic systems, motile colloids with well-controlled properties have been syn-
thesized in the past decades. As an example, Janus particles have two different
sides with distinct physical and/or chemical properties [11]. Such a symmetry
breaking induces a local gradient in the surrounding environment, of either ther-
mal, electric or chemical origin, which results in a self-propelled motion. Inspired
by such experimental systems, recent theoretical works have focused on simple
models of interacting active particles. These have shed light on the mechanism of
the transition to collective motion in the presence of aligning interactions [12], and
shown the possibility of a motility-induced phase separation (MIPS) even when
the pair interaction between particles is purely repulsive [13, 14].
Despite the nonequilibrium nature of active particles, it is often difficult to
precisely pinpoint the truly nonequilibrium signature in their emerging properties.
For instance, MIPS is not associated with the emergence of steady-state mass cur-
rents. Even for systems with steady currents, the connection to equilibrium physics
can sometime be maintained – the transition to collective motion amounts in some
