Thesis (pdf) - Espci

Studies of dynamical phenomena
in soft-matter and physical
biology
—————————————-
Ph.D. thesis
September 2008
—————————————-
Jan Brugués Ferré
Ph.D. advisors:
Jaume Casademunt Viader
Pierre Sens
Programa de doctorat de Física Avançada
Bienni 2002-2004
Departament d’Estructura i Constituents de la Matèria
Universitat de Barcelona
La Physique de la particule au solide
Université Pierre et Marie Curie

Studies of dynamical phenomena
in soft-matter and physical biology
Ph.D. thesis, Jan Brugués Ferré
Ph.D. advisors:
Jaume Casademunt Viader
Pierre Sens
Programa de doctorat de Física Avançada
Bienni 2002-2004
Departament d’Estructura i Constituents de la Matèria
Universitat de Barcelona
La Physique de la particule au solide
Université Pierre et Marie Curie
September 2008

Acknowledgements
This thesis has been something really hard, and has lasted probably too much.
During all this time I met quite a lot of people which has somehow helped me
finnish the thesis.
I would first like to thank my two PhD advisors, Pierre Sens and Jaume
Casademunt, who made this thesis possible.
Jaume, moltes gràcies per la teva confiança i per haver-me donat l’oportunitat
de fer la transició de física d’altes energies a biofísica. Gràcies per tot el que
m’has ensenyat, per la teva paciència, per tots aquests bons moments, per
preocupar-te per mi sempre i sobretot per la teva amistat.
Pierre, it is difficult to express in words how grateful I am. To begin with,
thanks for accepting me as a student, I am honored to be your first PhD student.
I have learned a lot from you, both scientifically and personally. Thanks
for your sense of humor and criticism. Thanks for pushing me when I needed,
and being so patient with me during the writing of this thesis. I’ll never forget
how much of your and your daughter time you have spent with me correcting,
even if it was late in the night. Above all, thank you for being a friend, I will
really miss all that time with you at ESPCI.
M’agradaria també agraïr a en Quim, que va ser el meu director de tesis
durant els primers anys en què vaig treballar sobre teoria de cordes. Gràcies
per ser sempre sincer amb mi i pels teus bons consells, i per ser comprensiu
malgrat la meva immaduresa.
Thank you very much Jean-François, Guillaume, Benoit and François for
the great scientific discussions and collaborations we had in Paris.
Una de les persones responsables que hagi acabat fent biofísica és l’Otger,
gràcies per presentar-me a en Pierre. Gràcies per la teva amistat, les converses
sobre biofísica i noies, i els nostres kahales. Gràcies per estar al meu costat
en els moments difícils de París. Durant tots aquests anys, moltes persones

ii Acknowledgements
m’han ajudat i estat al meu costat, d’entre elles hi ha alguns dels meus millors
amics i companys de feina. Gràcies Pau per ser com ets, les nostres crispis i
aguantar-me en els meus moments ”pedra a la sabata”. Al Tonir, també per
ser un bon amic i per tot el que em passat junts durant l’època de cordes.
Gràcies també al Guillermo, al membranelo, al Carlos, al Tonim, al Minervo
i tants d’altres que m’oblido però que han estat part de tots aquests anys.
M’agradaria agraïr especialment a la Maria, que sempre ha estat al meu
costat recolzant-me i ajudant-me en els pitjors moments, i per què la teva
amistat és una de les coses més preuades i importants que tinc. Gràcies també
a la Maria Sureda per ser una bona amiga i pels nostres cafés on ambdós ens
hem desfugat a gust.
Vull agraïr a la Julia tota la paciència infinita que ha tingut durant tots els
darrers mesos de la tesi, en els quals no només m’ha ajudat i ha sacrificat tant
del seu temps, sino que ha compartit amb mi part de la seva vida. Gràcies
també a la Gandi.
Moltes gràcies Ana Maria, per ser molt més que la meva professora de
piano, pels teus bons consells, les nostres converses i per tot aquest temps
en el que m’has ensenyat tantíssimes coses, entre les quals, a multiplicar el
temps.
Per acabar, vull agraïr molt profundament la meva família per suportarme
i ajudar-me en els moments més complicats, i per, en general, ser-hi quan
els he necessitat. Pels bons consells de la mimi (no n’hi ha prou en ser...) i la
mami, i per estar sempre pendent que tot m’anés bé. Si he aconseguit el que
he aconseguit és gràcies a vosaltres. Gràcies al nen i la nena i al meu pare per
estar sempre accessibles. Gràcies al iaiu i al padrí, que per malgrat no ser-hi
sempre m’han servit d’exemple a seguir. Aquesta tesi us la dedico a vosaltres.

Contents
1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Probing elastic anisotropy from defect dynamics in Langmuir
Monolayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 The monolayer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Brewster angle microscopy . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Theoretical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.1 Elastic free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.2 Point defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.3 Defect motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.4 Enhanced negative defect dissipation . . . . . . . . . . . . . . . 16
2.4.5 Ratio of defect velocities . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Self-organization and cooperativity of weakly coupled
molecular motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 Molecular motor modelization. . . . . . . . . . . . . . . . . . . . . 24
3.1.2 The two state model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Self-organization and cooperativity of weakly coupled
molecular motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 Mean-field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

iv Contents
3.2.3 Hard-core repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.4 Long range interaction - transition to mean field . . . . . . 39
3.2.5 Attractive interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1 Force transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.2 Motor efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Membrane-cortex interactions: Micropipette experiments
and blebbing cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Modeling cell mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Cellular membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Energy cost of membrane deformation . . . . . . . . . . . . . . 54
Membrane tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.2 Cytoskeletal cortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Maxwell model for actin cortex viscoelasticity . . . . . . . 59
Contractile stress in the cortex. . . . . . . . . . . . . . . . . . . . . 62
Cytoskeletal cortex thickness . . . . . . . . . . . . . . . . . . . . . . 63
Mechanical equilibrium of the cell . . . . . . . . . . . . . . . . . 65
4.3 Membrane-cortex adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.1 Kinetic model for membrane-cortex adhesion . . . . . . . . 67
4.3.2 Micropipette experiments as membrane-cortex
adhesion probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Influence of the link concentration . . . . . . . . . . . . . . . . . 73
Effect of mysoin II activity . . . . . . . . . . . . . . . . . . . . . . . 74
Adhesion energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.3 Energetic description of a bleb . . . . . . . . . . . . . . . . . . . . 80
4.4 Dynamical response of a cell to a controlled mechanical
perturbation using a micropipette set up . . . . . . . . . . . . . . . . . . . 86
4.4.1 Micropipette experiments as membrane tension and
adhesion probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.2 Micropipette static suction pressure . . . . . . . . . . . . . . . . 91
4.4.3 Effect of the loading rate . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4.4 Can actin stop a moving membrane? . . . . . . . . . . . . . . . 93
4.5 Experimental observation of the different regimes . . . . . . . . . . 98
4.5.1 Non equivalence of initial aspiration and retraction . . . 99
4.5.2 Micropipette suction regimes . . . . . . . . . . . . . . . . . . . . . . 100
Saltatory and no cortex regime . . . . . . . . . . . . . . . . . . . . 101
Cell oscillations in micropipette . . . . . . . . . . . . . . . . . . . 103

Contents v
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.A Membrane flow dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.A.1 Dissipation during membrane-cortex detachment . . . . . 110
4.A.2 Dissipation during membrane flow inside a micropipette110
5 General conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
A Resum en català . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.1 Auto-organització i cooperativitat de motors moleculars
interactuant feblement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.2 Mesura de l’anisotropia elàstica a través de la dinàmica de
defectes en monocapes de Langmuir . . . . . . . . . . . . . . . . . . . . . 120
A.3 Interaccions de membrana i còrtex: Experiments amb
micropipeta i cèl . lules amb blebs . . . . . . . . . . . . . . . . . . . . . . . . . 123
B Résumé en Français . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.1 Auto-organisation et coopérativité des moteurs moléculaires
interagissant faiblement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.2 Mesure de anisotropie élastique a travers de la dynamique
de défauts en monocapes de Langmuir . . . . . . . . . . . . . . . . . . . . 130
B.3 Interactions de membrane et cortex : Expériences avec
micropipette et cellules avec blebs . . . . . . . . . . . . . . . . . . . . . . . 133
B.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

1
General introduction
In recent years, research in biology has evolved towards a more quantitative
analysis. The use of physical approaches have improved the understanding
of many biological processes including membrane and cytoskeleton interactions,
cell division, mechanotransduction and cell movement. In vitro approaches
which try to identify the minimum amount of components needed
to reproduce particular phenomena in living systems, are also an important
tool for biological research.
The study of biology is so vast that spans several orders of magnitude in
both force and length. Typical length scales range from a few nanometers for
a molecular motor, to a few meters corresponding to the size of a mammalian.
Forces range from a few piconewtons 1 (stall force of a molecular motor), to
several newtons (earth attraction on a human being is of about ∼ 500 N).
We will focus on the cellular and sub-cellular scale. A cell has a typical
diameter of 10 µm, and contains many organelles and proteins, including
molecular motors, which are in the range of a few nanometers, capable of generating
forces of piconewtons, and they are the smallest entities that we will
consider. Our unit scale of energy is thus ∼ pN × nm, which is of the order
of the thermal energy kBT ∼ 4 pN nm (the energy released by hydrolization
of ATP is of about 8 kBT ). As a consequence we expect that thermal fluctuations
play an important role at these microscopic scales. Moreover, living
organisms continuously consume energy (ATP), which maintains them in an
out of equilibrium steady state. Thermal equilibrium state corresponding to
the death of the organism. Any physical approach at this scale must explicitly
take into account these fluctuations and can not be described in terms of ener-
1 1 pN = 10 −12 N

2 1 General introduction
gies. In this context, out of equilibrium statistical physics is a good candidate
in trying to model processes at this microscopic scale.
At a more coarse grained level, the cell can be characterized by its shape
and mechanical properties. Traditionally, the physics of the state of the matter
(condensed matter), dealt with the macroscopic states of the matter that
arise from the physical properties of the microscopic constituents (atoms or
molecules) namely, solid and liquid. However, the state of cells and other
materials such as polymers, colloids, foams or gels, do not correspond either
to liquid or solid. They have a large degree of ordering and are definitely
not simple liquids. Furthermore, they are fairly soft materials that can be deformed
by thermal fluctuations. The ”softness” of these materials lead to a
recent field in physics named soft matter. The typical energy scale for cell
membrane deformation is of κ ∼ 10 − 100kBT , which is slightly larger than
thermal fluctuations. Soft-matter is a good candidate to study coarse grained
macroscopic properties of the cell where thermal fluctuations are averaged
although play a role in determining shape and mechanical properties.
At cellular scales, inertial forces are negligible when compared to viscous
forces (the Reynolds number Re ≪ 1), and Aristotelian physics, where velocity
and force are linearly related, govern generically cellular and sub-cellular
phenomena (molecular motor motion, intracellular transport, cell deformation).
In recent years, many qualitative advances in experimental techniques,
ranging from microscopy, micromanipulation to gene transfection, have led
to an enormous quantity of new data and observations at the cellular level
which require fundamentally new concepts to help their understanding. In
this context, physicists and biologists live an exciting period where physical
concepts can help to describe and understand new general cellular properties,
which at the same time can be directly accessed experimentally. Soft-matter
and statistical physics are the obvious candidates to bridge between biology
and physics, although new physical and biological tools will probably emerge
as new phenomena is observed. My personal point of view is that the present
state in biophysical research is qualitative similar to that in the early years of
particle physics, where the first particle accelerators led to the discovering of
new fundamental particles.
The large amount of interesting and fundamental problems to be understood,
the ease of experimental comparison, the beauty to relate simple concepts
such as geometry, topology, or elasticity to biological phenomena, and
the personal and scientific enrichment of the interdisciplinary work involved,

1 General introduction 3
led me, irremediably, to devote this thesis and my future research to biophysics.
This thesis do not respond to a pre-established scheme, although there is a
logic underlying the pattern followed, which starts from a string theory PhD
student during the first two years of my doctorate, and ends as a biophysicist.
The transition between such a perpendicular fields has resulted in the following
thesis, where aspects of purely soft matter concerning topological defects
appear first, followed by a study of some cooperativity of molecular motors,
and ends with a purely biophysical approach of membrane-cytoskeleton interactions
which explicitly connects with biological experiments. Yet from
the beginning I tried to work on problems that allowed me to interact with
experiments and researchers from other fields.
This thesis is composed of three chapters which are devoted to soft-matter
and physical biology. The first chapter belongs to the field of topological defects.
The study of topological defects is of general relevance in many areas of
physics and biology, because their main universal properties can help understand
systems that presumably exhibit topological defects, such as the mitotic
spindle during cell division, using simple condensed matter experiments, like
in liquid crystals. We consider the motion of defects and their interaction,
which is not fully understood. In this regard, we study the dynamics of annihilation
of two defects in Langmuir monolayers using a liquid crystal model.
We are able to extract the dependence of the elastic anisotropy of the material
on the surface pressure measuring only the ratio of velocities at which the
defects approach each other.
In the second chapter of the thesis, we consider some aspects of selforganization
and cooperativity of molecular motors. Molecular motors are
proteins that convert chemical energy into mechanical work at a molecular
scale, and are responsible of many biological phenomena, ranging from muscle
contraction to cellular transport and cell motility. Molecular motors usually
cooperate to generate large forces, as in the case of muscle contraction.
Although there are several theoretical works considering the coupling between
molecular motors, there is not yet a full understanding of cooperativity
and force distribution among clusters of molecular motors. We consider several
representative weakly interactions between motors, namely, short range
and long range repulsion, and weak attraction, and we study their collective
behavior, force distribution and efficiency when a force is applied in the foremost
motor. Our approach is mainly based on Langevin simulations using a
two state model for a molecular motor.

4 1 General introduction
Finally, in the third chapter of the thesis we consider the interaction between
the cellular membrane and the cytoskeletal cortex. The plasma membrane
and the cytoskeletal cortex are amongst the most important parts of the
cell. They participate in providing cell shape, separate the extracellular media
from the cytosol, act as mechanotransducers, and are responsible for cell motion.
The adhesion between cortex and membrane is thus fundamental for the
proper functionality of a cell. Failure or weakening of this adhesion induces
a phenomenon named blebbing, which consist of highly dynamical spherical
protrusions that nucleate as a consequence of membrane unbinding and
retract with the appearance of a new cortex under the bare membrane. The
properties of bleb formation and membrane-cortex adhesion can be experimentally
addressed using mainly micropipette aspiration. We propose both
a kinetic and energetic model for cortex and membrane adhesion which we
use to understand the dependence of the adhesion strength on the active state
of the cell and external perturbations such as osmotic shocks or micropipette
aspirations. Once addressed the concept of adhesion energy, we consider the
dynamical response of the membrane and cortex to a mechanical perturbation.
In particular, we focus on the treadmilling properties of the cortex and
its viscoelastic response using a Maxwell like description for the short time
elastic and long time viscous behavior of the cortex. Using the micropipette
aspiration set up as a model system, we theoretically identify all the possible
aspiration regimes and we compare them with both existing experimental
measurements, and experiments we have conducted on the Entamoeba histolytica
system in collaboration with B. Mauguis and F. Amblart.

2
Probing elastic anisotropy from defect dynamics in
Langmuir Monolayers
2.1 Introduction
The study of topological defects is of general relevance in many areas of
physics and biology, ranging from cosmology to superfluid helium or cell
division. An important motivation arises from the universality of the defects,
which occur in any system with order parameter. Their main properties are
independent of the underlying physics, determined only by symmetries, the
dimension of the order parameter and the defect charge. This universality has
been lately exploited by condensed matter systems such as liquid crystals
which allow to perform relatively easy experiments yielding knowledge in
completely different physical areas.
Of particular interest is the study of the motion of a defect, and the interaction
and annihilation dynamics of defect pairs, which have been extensively
studied by several groups (Ryskin and Kremenetsky, 1991; Svenˇsek
and ˇZumer, 2002; Blanc et al., 2005). A remarkable feature of defect pairs annihilation,
both observed experimentally and numerically, although not fully
understood, is the enhanced mobility of the positively charged defect with respect
to its negative counterpart. Elastic anisotropy (inequality of the elastic
constants) and hydrodynamic effects arising from defect motion (backflow)
have been shown to generically contribute to explain this asymmetry. In fact,
the latter is dominant in the context of bulk (3-d) liquid crystals (Tóth et al.,
2002; Svenˇsek and ˇZumer, 2002; Oswald and Ignés-Mullol, 2005; Blanc
et al., 2005), thus hindering the possibility to develop a simple method to
quantitatively relate material elasticity to defect dynamics, which would be
an interesting alternative to traditional methods to determine the elastic constants
(de Gennes and Prost, 1995; Oswald and Ignés-Mullol, 2005).

6 2 Probing elastic anisotropy from defect dynamics in Langmuir Monolayers
We will study the opposite scenario by studying defect dynamics in Langmuir
monolayers spread at the air/water interface (Hatta and Fischer, 2003),
where the hydrodynamic effects can be neglected. Contrarily to bulk (3-d)
liquid crystals, where rotational and translational viscosities are of the same
order of magnitude, local molecular rotation without hydrodynamic effects is
possible in Langmuir monolayers (2-d). Then, defect dynamics can be simply
explained by the effect of material elasticity. As a result, measurements
of differential defect mobilities enable us to probe the elastic anisotropy of
the two-dimensional material.
In this chapter, we will report on quantitative studies of defect dynamics in
Langmuir monolayers with a polar nematic arrangement. The observed asymmetry
of defect mobility will be explained only by topological properties of
the defects, and a quantitative analysis in terms of a simple theoretical model
that rules out backflow effects will be presented. The model is exploited to
extract the dependence of the elastic anisotropy on surface pressure and to estimate
the compressibility-dependent elastic constants of the 2-d system. This
indirect procedure is demonstrated to be an advantageous alternative over direct
measurements, unpractical in these quasi-two-dimensional systems.
The results reported in this chapter correspond to the reference (Brugues
et al., 2008a).
2.2 Experimental setup
Experiments are performed in a shallow thermostated Teflon cuvette where
a monolayer is spread over an area delimited by two mobile barriers, which
allows to control the surface pressure, Fig. 2.1.
2.2.1 The monolayer
The monolayer is composed of the photosensitive azobenzene amphiphile
8Az3COOH. All classes of azobenzenes are composed of a common core
of two phenyl rings linked by a N = N double bond 1 that can adopt two
different geometrical configurations: the trans configuration, where the two
phenyl rings are oriented in opposing directions, Fig. 2.2a (left), and the cis
configuration, where the two phenyl rings are oriented in the same direction,
Fig. 2.2a (right). The trans isomer exhibits an orientational order parameter
due to its geometrical and elecrical properties.
1 Therefore, the molecule cannot rotate in the double bond axis.

(a) (b)
Π
CCD CCD
Brewster Angle
Microscopy
Π
Chloroform
solution
Π
2.2 Experimental setup 7
Fig. 2.1. (a) Schematics of the experimental setup. Spread monolayers of the azobenzene amphiphile
8Az3COOH are prepared by depositing drops of a chloroform solution on a surface
of pure water contained in a Teflon cuvette. A lateral pressure Π is applied by adjusting the
surface area. Orientational order inside the trans droplets is measured by a Brewster angle
microscope (BAM) (Ignés-Mullol et al., 2004). (b) Images obtained using the experimental
setup courtesy of Jordi Ignés-Mullol.
Azobenzenes exhibit two main properties that can be exploited for our experimental
purposes. On one hand, transitions between trans and cis isoforms
can be induced by exposure to appropriate wavelengths of light (photoisomerization),
Fig. 2.2a, and by thermal relaxation towards the trans configuration
which has a favorable energetic configuration, Fig. 2.2b. Due to the dramatically
different geometrical properties of the trans and cis conformations, mixtures
of both isomers organize differently at the air/water interface. On the
other hand, mesophases of the trans isomer, appear at relatively low area densities.
As a consequence, monolayers composed of these azobenzenes exhibit
long-range orientational order from the trans isoform, but positional disorder
(Tabe and Yokoyama, 1994), which allow topological defects to easily
move.
(a) (b)
N
N
hν
Energy
N
hν kBT
N
′ , N
N
N N
Configuration
Fig. 2.2. Azobenzene photoisomerization. Transition between trans form (left) and cis form
(right) can be induced using appropriate wavelengths of light (a). Alternatively, the cis configuration
will thermally relax to the energetically stable trans configuration (b).

8 2 Probing elastic anisotropy from defect dynamics in Langmuir Monolayers
Nonirradiated 8Az3COOH trans isomer monolayers exhibit complicated
stripe like patterns (Crusats et al., 2004), Fig. 2.3a. If the azobenzene solution
is exposed to room light, a mixture of cis and trans isomers is obtained which,
after spreading a monolayer, is organized in circular droplets of a birefringent
trans phase with a central topological defect embedded in an isotropic cis
phase (Ignés-Mullol et al., 2005), Fig. 2.3b. We will use spread monolayers
of trans and cis isomers to study annihilation of topological defects that arise
from the fusion of a pair of droplets (see below).
(a) (b)
Fig. 2.3. (a) Nonirradiated textures of 8Az3COOH trans isomer monolayers, image modified
from (Crusats et al., 2004). (b) Circular droplet of a trans isomer embedded in a cis phase.
Both images are obtained using Brewster angle microscopy (BAM).
Imaging of thin films at a gas/liquid interface can be achieved by several
techniques (for instance fluorescence microscopy and AFM). Many of
these techniques perturb the system, either by probe compounds (such as fluorescent
dye) or by direct interaction (AFM). A non invasive technique that
allows to study such systems without perturbing its original state is based on
the Brewster angle, Fig. 2.4.
2.2.2 Brewster angle microscopy
For a beam of a p-polarized light (parallel to the plane of incidence), there
is an angle of incidence θ (Brewster angle) at which no reflection occurs,
Fig. 2.4 (left). When a thin film is placed between the two phases, the optical
properties of the system change, and a small amount of light is reflected, Fig.
2.4 (right). In general, the polarization of the reflected light differs from the
incident light.
A Brewster angle microscope (BAM), is built using a source of p-polarized
light (a laser and a polarizer) and a receptor (a CCD camera and a polarizer),
Fig. 2.1a. We use a calibration of the gray-scale distribution in the BAM im-

θ θ
air
thin film
subphase
Fig. 2.4. Scheme of the Brewster angle.
2.3 Results 9
ages and the orientational parameter as reported in (Ignés-Mullol et al., 2004),
Fig. 2.5.
Grey level
φ(deg)
Fig. 2.5. Gray level as a fucntion of the polar angle for a droplet of trans isomer in a matrix of
cis isomer, image modified from (Crusats et al., 2004).
2.3 Results
For a wide range of experimental conditions, the organization of the amphiphilic
molecules inside the circular domains is characterized by a uniform
tilt (around 45 ◦ (Crusats et al., 2004)) with respect to the air-water interface
normal, Fig. 2.6. Molecular ordering can thus be described by a twodimensional
vector field n. Constant-angle anchoring at the droplet boundary
(Crusats et al., 2004; Reigada et al., 2004) results in the inclusion of inner
point defects of total charge +1 (Mermin, 1979; Tabe et al., 1999). Small
enough droplets feature a single s =+1 defect near the center (in the core

10 2 Probing elastic anisotropy from defect dynamics in Langmuir Monolayers
region of these singularities, the molecular tilt becomes normal to the interface
(Tabe et al., 1999)), and a pure bend texture around the core (see Fig.
2.8). Since the system is not chiral, equal amount of droplets with clockwise
and counter-clockwise bending textures are present in the monolayer.
(a) (b)
θ
x
z
φ n
y
Fig. 2.6. (a) Constant tilt of the amphiphilic molecules with respect to the air-water interface
normal (z). The orientational order of the domains is described by the director field n, defined
as the projection of the molecule on the monolayer plane (xy). (b) Director field on the
monolayer plane (red) of a droplet with a positive +1 defect.
Coalescence of droplets with the same chirality is hindered and seldom
observed. On the other hand, droplets with opposed chirality fuse abruptly
(Fig. 2.8), creating a pair of new point defects upon intersection. Fusion of
droplets with the same chirality is energetically unfavorable since involves the
creation of a line defect consisting of antiparallel arrangements of molecules
in the contact line between droplets. The energy barrier to overcome scales
then with the size of the droplet, the proportionality constant being the elastic
splay constant of the material. For droplets with opposite chirality, the contact
line between droplets have parallel director fields, Fig. 2.7.
(a) (b)
Fig. 2.7. Coalescence of two droplets of (a) opposite chirality and (b) same chirality.

2.3 Results 11
Upon droplet fusion, a pair of defects appear at the surface of the newly
formed droplet. Since the total charge must be preserved inside the closed domain,
we consistently assign a −1/2 charge to each boundary defect. These
semi-integer defects are necessarily confined to the droplet boundary since
they are not allowed inside polar nematic domains. In the simplest route to
collapse, typical in the fusion of droplets of different size, one of the originally
central s =+1 defects is annihilated in a two-step mechanism, Fig.
2.8A. First it migrates to merge at the boundary with one of the newly created
singularities, resulting in a +1/2 boundary defect. The latter, and the
remaining −1/2 boundary defect approach following an asymmetric dynamics
along the contour of the fused droplet, with +1/2 defects always moving
faster (see Fig. 2.8B for a representation of defect trajectories). Relaxation of
the droplet shape following coalescence is much faster than defect dynamics.
As a result, droplet circularity does not change measurably during the final
stages of defect collapse, where a roughly constant linear velocity along the
curved boundary is observed. The ratio of linear velocities in this regime measured
for different surface pressures is used below to quantify the asymmetry
in defect mobilities.
-30 -20 -10 0
time (s)
! (C)
!
!
" !
!
Fig. 2.8. (A) Experimental coalescence process of a clockwise and an anti-clockwise " bend
domain at T = 32◦C and Π = 0.3mN m−1 leading to a droplet with a single s =+1 defect.
BAM analyzer is set at 60◦ counterclockwise from the plane of incidence, which includes the
vertical axis in the images. A sketch of the molecular field is shown below each experimental
image. Merging of one of the inner s =+1 defects and one of the two −1/2 boundary defects
created upon fusion (b) results in a ±1/2 pair of boundary defects that attract and annihilate
(d) following the arclengths towards their meeting point shown in (B). The straight lines yield
the average linear velocity in the pseudo-constant velocity regime prior to collapse. Elapsed
times from panel (b) are 90 s (c) and 124.6 s (d). The ruler is 100µm long. Video image
digitization and processing (Rasband, 2006) is used to set the quasi-circular droplet contour
as fixed in space.
L ,L
+ -(µm)
20
0
-20
-40
-60
-80
(B)

12 2 Probing elastic anisotropy from defect dynamics in Langmuir Monolayers
2.4 Theoretical description
The mutual elastic attraction between defects of opposite charge is the responsible
of the annihilation of the boundary ±1/2 defects experimentally observed.
The velocity at which the defects annihilate depends on the amount of
energy dissipated as they approach each other. The defect movement involves
a translational flow of the polar molecules along the direction of motion and
reorientation of the director field through rotation of the polar molecules. The
dissipation has then two main contributions that arise from backflow and the
dissipation during molecule reorientation.
Rotational viscosity in our system is of about 10 −10 kg s −1 (Feder et al.,
1997). The effective translational viscosity results from the coupling between
the monolayer and the subphase (Stone, 1995). The subphase has a viscosity
of the order of 10 −3 kg m −1 s −1 and the dissipation occurs in a length scale
of the order of the system, i.e., the droplet radius (∼ 50 µm), leading to an
effective translational viscosity two orders of magnitude larger than the rotational
viscosity. Recent numerical analysis (Svenˇsek and ˇZumer, 2002) has
shown that the effect of elastic anisotropy and backflow cannot be decoupled
in the study of asymmetric defect mobilities in bulk liquid crystals. Contrary
to bulk liquid crystals, where rotational and translational viscosities are of
the same order of magnitude, our system exhibits a larger effective translational
viscosity with respect to the associated to molecular reorientations. As
a consequence, the motion of the defect involves only reorientations of the
director field, which allows to study the effects of the elastic properties of the
material on the defect motion alone. Actually, it has recently been shown that
complex reorientations of the molecular field can take place in this system
without noticeable hydrodynamic effects (Burriel et al., 2006).
2.4.1 Elastic free energy
We have seen in the experimental section that for a wide range of experimental
conditions, the organization of the amphiphilic molecules inside the
circular confined domains is characterized by a uniform tilt with respect to the
air-water interface normal (see Fig.2.6). Consequently, we describe our system
with a two-dimensional director field n, and a generic elastic free energy
that contains the lowest order in the director field distortions,
F = d 2 x
KS
2 (∇ · n)2 + KB
(∇ × n)2
2
, (2.1)
where KS and KB are splay and bend constants (de Gennes and Prost, 1995;
Oswald and Ignés-Mullol, 2005). Earlier studies on this system suggest that

2.4 Theoretical description 13
the effect of anisotropic boundary conditions can be neglected with respect to
inner elasticity in the regimes where pure bend textures are obtained (Ignés-
Mullol et al., 2005). More generally, Eq. 2.1 should include contributions
from a density order parameter so as to account for the finite compressibility
of this system (Hinshaw et al., 1988; Tabe et al., 1999). Nevertheless, at a
fixed lateral pressure, explicit density contributions can be renormalized into
effective elastic constants and Eq. 2.1 is of general applicability, keeping in
mind that the value of the elastic constants will depend on the applied pressure.
a b
v−
Fig. 2.9. The motion of a negative defect along the curved boundary (a) can be mapped into
that of a defect in rectilinear motion (b). This mapping is justified because the director field
around ±1/2 boundary defects rotates when defects move along the curved boundary. In (a),
we see a sketch of a −1/2 defect with a virtual −1/2 part outside the boundary (in dashed
lines) moving at a velocity v− and its director field is rotated accordingly to its motion (the
director normal to the boundary is preserved as the defect moves as indicated with the lines
pointing to the center of the droplet). In addition, the effect of the curvature of the droplet on
the velocities is negligible in our experimental conditions (droplet radii considered in the range
30-60 µm ). In (b), a −1 defect is decomposed in a −1/2 defect and a virtual −1/2 defect in
dashed lines. This corresponds to having a −1/2 defect in the boundary and consequently, the
results of rectilinear ±1 defect motion can be extrapolated to that of semi-integer boundary
defects.
In the experimental section, we observed topological defects of charges
±1 in the bulk and ±1/2 on the boundary, Fig. 2.8. Fractional defects in a
nematic order parameter can not exist on the bulk by topological constraints
and must be confined on the boundary. Fig. 2.8A shows that the director field
around ±1/2 defects rotates when defects move along the curved boundary,
that is, the angle between the ±1/2 director field and the tangent to the
boundary is kept constant as the defect moves, Fig. 2.9a. Therefore, the director
field around a defect moving along the boundary can be mapped into that
of a defect in rectilinear motion. On the other hand, each boundary defect can
be regarded as a ±1 with half its spatial extend being virtual. This allows to

14 2 Probing elastic anisotropy from defect dynamics in Langmuir Monolayers
extrapolate the results of rectilinear ±1 defect motion to that of semi-integer
boundary defects, Fig. 2.9b.
2.4.2 Point defects
Let us consider an isolated point defect in the director field of charge s and
rotational symmetry. We use polar coordinates (r,φ) and express the director
field using the angle ψ with respect to the radial direction, Fig. 2.10, nr =
cosψ and nφ = sinψ. Minimization of the free energy (Eq. 2.1) gives,
(1 + α cos2ψ) ∂ 2ψ = α sin2ψ
∂φ2 ∂ψ
∂φ
2
− 1 , (2.2)
where α ≡ (KS −KB)/(KS +KB) is a measure of the elastic anisotropy. α < 0
(resp. > 0) favors splay (bend) textures, and α = 0 is the isotropic case.
y
r
n ψ
Ψ
Fig. 2.10. Definition of the director field angular coordinates.
(a) (b) (c) ψ
Fig. 2.11. Solutions for s =+1 defects corresponding to the energetically favorable solution
to equation 2.2 for (a), α < 0, which corresponds to a pure splay configuration, in this case
ψ = 0. (b), α > 0, which corresponds to a pure bend configuration, ψ = π/2. And (c), α = 0,
where the energetic cost of splay and bend configurations weight the same. The solution is
a spiral, ψ = ψ0, where ψ0 ∈ (0,2π), that combines splay and bend contributions. In (c) the
angle ψ between the director and the polar direction is sketched.
φ
x

2.4 Theoretical description 15
For s =+1, the energetically favorable solution to Eq. 2.2 for α > 0 is
ψ+ = ±π/2 (pure bend configuration), and ψ+ = 0,π for α < 0 (pure splay),
Fig. 2.11. For s = −1, the director field around the defect is given in implicit
form by (Landau et al., 1995)
φ[ψ−,k−] ≡ k−
ψ−
0
1 + α cos2x
1 + αk2 1/2 dx, (2.3)
− cos2x
where the integration constant k− is determined by φ[−4π,k−]=2π, which
is the topological condition for the director field of a −1 defect in polar coordinates,
ψ[φ + 2π] =ψ[φ] − 4π, for φ = 0. Since a negative defect is a
mixture of splay and bend configurations, for α = 0, the defect is deformed
with respect the corresponding isotropic one, minimizing the splay or bend
domain depending on the sign of the anisotropy, Fig. 2.12.
(a) (b) 0
(c)
ψ
-1
-2
-3
-4
-5
-6
0 0.5 1 1.5 2 2.5 3
Fig. 2.12. Deformation of a −1 defect. (a) Negative defect for isotropic elastic constants. (b)
The director angle ψ as a function of the polar angle φ, for isotropic elastic constants (dashed
line) and for an anisotropy α = 0.9 (solid line). (c) Negative defect for an anisotropy α = 0.9.
2.4.3 Defect motion
During defect motion, the director field is distorted. We have shown that back
flows are negligible in our system and consequently, in a quasistatic approximation,
we can assume that Eq. 2.3 holds during motion and describes the
instantaneous director field during rectilinear defect motion at velocity v.
The dissipation rate per unit length associate to rotations of the director field
is (Kleman and Laverntovich, 2003)
Σ = γ
dS
φ
2 ∂Ψ
, (2.4)
∂t

16 2 Probing elastic anisotropy from defect dynamics in Langmuir Monolayers
where Ψ is the angle between the director field and the polar axis, Ψ = ψ +φ
(see Fig. 2.10), and γ is the rotational viscosity of the medium. Because of
the symmetry in the solutions of Eq. 2.3, the dissipation can be decomposed
in the product of a radial and angular contribution, being this last part the
responsible for differences in dissipation for s = ±1 defects,
Σ = γv 2
R
log
rc
2π
0
dφ sin 2 (φ − φ0)
2 ∂Ψ
, (2.5)
∂φ
where φ0 is the direction of movement, rc is the core radius of the defect and
R is the system size, which in this case is the droplet radius. Regardless of the
accuracy of the logarithmic dependence in the drag force expression (Ryskin
and Kremenetsky, 1991), what is relevant for our analysis of the relative velocities
(see below) is the fact that R and rc can be assumed to be roughly the
same for the two defects of opposite charges 2 .
Using the trigonometric equality sin 2 (φ − φ0) =1/2(1 − cos2(φ − φ0))
and the defect ±1 symmetry, we have that (∂Ψ/∂φ) 2 [φ +π/2]=(∂Ψ/∂φ) 2 [φ]
and consequently, the dissipation is independent of the direction of motion,
and takes the simple form
Σ = γv 2
R
log
rc
1 2π
dφ
2 0
2.4.4 Enhanced negative defect dissipation
2 ∂Ψ
. (2.6)
∂φ
The dissipation is proportional to the sum of the rotation squared of each polar
molecule over the spatial extension of the defect. As a consequence, for
any global fixed rotation of a defect, the least dissipative way to rotate each
polar molecule would be, if possible, a uniform rotation. A uniform rotation
of each polar molecule means a totally symmetric defect, which in the case
of ±1 defects with elastic anisotropy, corresponds to the +1 defect. Since
the negative defect is a mixture of bend and splay configurations, the elastic
anisotropy introduces an inhomogeneous distortion on the director field and
as a consequence the defect is squashed in the splay or bend domain depending
on the sign of α (see Fig. 2.12). A global fixed rotation of the negative
defect carries a local nonuniform rotation of the director field in the regions
where the splay or bend has been increased or decreased. As a consequence,
the dissipation will generically be larger for negative defects.
2 More accurate estimations of those parameters result in a residual sublogarithmic dependence
on the elastic anisotropy which can be neglected.

2.4 Theoretical description 17
To prove this statement more rigorously, we consider the angular part
of the dissipation, 2π
0 dφ (∂Ψ/∂φ)2 . Recalling the definition of the defect
charge,
2π
0
dφ ∂Ψ
∂φ
= Ψ(2π) −Ψ(0) ≡ 2πs, (2.7)
we show that the overall absolute value of the rotation of the director field
Ψ in both positive and negative defects is the same. In the case of a positive
defect, the derivative of the director field with respect to the angular direction
is equal to 1 independently of the anisotropy, whereas in the negative case,
it is a function of the angular direction and the anisotropy (Eq. 2.3). As a
consequence,
2π
0
dφ
2 ∂Ψ+
=
∂φ
1
2π
dφ
2π 0
∂Ψ±
2 2π
≤ dφ
∂φ 0
2 ∂Ψ−
, (2.8)
∂φ
where in the last step we have used a general inequality of functional analysis
(the continuous version of the discrete triangular inequality). The immediate
implication of this result is that, due to elastic anisotropy, isolated positive
defects are always faster than negative ones (independently of the anisotropy
sign).
2.4.5 Ratio of defect velocities
The power supplied per unit length needed to maintain the defect moving at
a fixed velocity v is P = fv, where f is the drag force, and must be equal to
the dissipation rate Σ. This leads to the expression
f− 1
2 γv−
⎡
R 2π
log dφ ⎣1+ 1
1+αk2 ⎤2
− cos2ψ− ⎦ ,
rc
0
k−
rc
1+α cos2ψ−
R
f+ πγv+ log , (2.9)
for the negative and positive defect respectively. Although the attractive force
between defects is of elastic origin and can be in general a complicated function
of the position, the anisotropy α, and the viscosity of the material can
vary as a function of the lateral pressure, to compare the ratio between velocities
we only need to know that both the defects feel the same force f
in opposite directions (Newton’s third law), and the same material viscosity.
Using the expression for the drag force in Eq. 2.9 for the +1 and −1 defects,

18 2 Probing elastic anisotropy from defect dynamics in Langmuir Monolayers
v+
=
v−
1
⎡
2π
dφ ⎣1 +
2π 0
1
1 + αk
k−
2 ⎤2
− cos2ψ− ⎦ .
1 + α cos2ψ−
(2.10)
As seen in Fig. 2.13, there is a wide range of anisotropy α for which the
ratio of velocities is roughly constant, and the asymmetry is only significant
for |α| tending to 1. The ratio v+/v− is invariant if KB and KS are interchanged
(Svenˇsek and ˇZumer, 2002), i.e., if α us replaced by −α, since the
change α →−α results only in a rotation of π/2 of the solution of Eq. 2.2.
Thus the overall rotation of n due to defect motion is the same, leaving invariant
the integrand in Eq. 2.10.
2.5 Discussion
v−/v+
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
|α|
Fig. 2.13. Plot of the solution of Eq. 2.10.
We can now use Eq. 2.10 to determine the ratio KS/KB by measuring defect
velocities under different monolayer conditions.
To this end we have conducted experiments varying the surface pressure,
which is kept below 4 mN m −1 . Above this value, the range of motion of
the slow defect cannot be resolved in our system. Similarly, temperature is
maintained at 32 ◦ C, a compromise between the hindered mobility at lower
temperatures and the fast dynamics above this value. Droplet radii were considered
in the range of 30 - 60 µm, with no significant effect in the results.
Experimental ratios of defect velocities (defined as the linear velocity
along the curved boundary) as a function of lateral pressure are shown in Fig.
2.14 in terms of the mean and standard deviation of several experiments. The

2.5 Discussion 19
ratio v−/v+ can be transformed, using Eq. 2.10, to yield α vs. Π and, therefore,
the ratio of the elastic constants as a function of Π (Fig. 2.15). Our measurements
can be combined with the results of Feder et al. (Feder et al., 1997)
whose analysis of orientation fluctuations in this system state that the value
for the geometric mean of KS and KB is of about K ≡ √ KSKB ∼ (40±25)kBT ,
and insensitive to pressure variation. This way, we can estimate KS and KB as
a function of Π, (Fig. 2.15), using
− K2
α = K2 S
. (2.11)
+ K2
K 2 S
In the zero-Π limit (lowest anisotropy) KS KB 10 −19 J. As Π increases,
there is a rapid increase (resp. decrease) of KS (resp. KB) until Π 1 mN/m,
where further variation of these parameters cannot be resolved. These values
are consistent with estimations found in the literature (Tabe et al., 1999) and
values extrapolated for thin suspended liquid crystal films (Rosenblatt et al.,
1979).
v−/v+
1
0.8
0.6
0.4
0.2
0
0 1 2 3 4
Π (mN m −1 )
Fig. 2.14. Experimental measurements of the relative velocities of boundary defects moving
towards their annihilation as a function of the lateral surface pressure.
We can qualitatively relate these results to the known tendency of azobenzene
derivatives to form supramolecular aggregates (H-aggregates), whose
presence can be revealed here by spectroscopy methods (Pedrosa et al., 2002).
In short, aggregates form in such a way that the azobenzene planes of neighboring
molecules are parallel, stacking perpendicularly to the molecular tilting
direction. Because of this, a splay arrangement of the molecular field
inside the axi-symmetric droplets would be energetically more demanding
than its bend counterpart, since the former would impose a certain curvature

20 2 Probing elastic anisotropy from defect dynamics in Langmuir Monolayers
α
!
1.00
0.95
0.90
0.85
0.80
0
K S, K B (J)
1
10 -17
10 -18
10 -19
10 -20
10 -21
0
2
"(mM m -1 )
Π (mN m −1 )
1 2 3
"(mM m -1 Π (mN m ) −1 )
Fig. 2.15. Anisotropy parameter, α, as a function of Π, as obtained from the data in Fig. 2.14.
The inset shows the value of the elastic constants KS (), and KB (◦) estimated as described
in the text.
on the H-aggregates. The known increase of the extension of aggregates with
Π (Ignés-Mullol et al., 2005) can therefore justify the observation that splay
distortions are increasingly less favorable. In fact, molecules cease to behave
as individual monomers at Π 2 mN/m, which may justify the levelling off
of the elastic constants at moderate pressures. Nevertheless, this qualitative
argument cannot explain why the behavior is so dramatically different even
with modest extension of the aggregation. Molecular dynamics simulations
could be employed to address these issues (Allen, 1996).
2.6 Conclusions
We have studied Langmuir monolayers where defect dynamics can be analyzed
in the absence of backflow. As a consequence, differential defect mobilities
can be traced back to elastic anisotropy. We have shown that this simplified
scenario can be exploited with the aid of a simple model to use dynamical
measurements to gain quantitative knowledge of the dependence of
rather elusive material parameters, here the elastic constants, on a basic thermodynamic
control parameter of the monolayer such as the surface pressure.
3
4
4

Acknowledgments
2.6 Conclusions 21
The work in this chapter has been done in close collaboration with J. Ignés-
Mullol and F. Sagués, in particular, the experimental measurements performed
on the 8Az3COOH monolayers.

3
Self-organization and cooperativity of weakly coupled
molecular motors
3.1 Introduction
The collective behavior of molecular motors plays an important role in many
biological phenomena, including muscle contraction, cellular transport, cell
motility and cell division.
In recent years, the appearance of new in vitro experiments allowed the
observation of single motor proteins. These experiments included gliding essays,
optical tweezers and micromechanical force measurements. Their results
revealed new insights into the basic principles of single molecular motors,
and inspired new models for cooperative motors. In that context, molecular
motors attached to actin (myosin), microtubule (kynesin or dynein) filaments
or rigid cargoes, were modeled as rigidly coupled motors (i.e., sharing
equally any external force applied) (Leibler and Huse, 1993; Juelicher
and Prost, 1995; Julicher et al., 1997; Badoual et al., 2002; Klumpp and
Lipowsky, 2005; Klumpp et al., 2006). These models resulted in non linear
force-velocity relationships and dynamic transitions with spontaneous bidirectional
movement in motors lacking intrinsic directionality (as observed
experimentally in the case of NK11, a mutant of NCD (Endow and Higuchi,
2000)). However, the assumption of rigid coupling may result inappropriate
in certain conditions such as in gliding essays for instance, when the distance
between motors is large enough. Relaxing the backbone rigidity and letting
the motors interact elastically, leads to a cross-over behavior of the motors
from non linear to linear force-velocity curves (Vilfan et al., 1998).
More recent experiments have emphasized that for intracellular traffic,
the cooperative action of groups of processive motors is required when they
are attached to fluid-like cargos, such as vesicles or membrane tubes (Koster
et al., 2003; Leduc et al., 2004). In this context, motors are not rigidly cou-

24 3 Self-organization and cooperativity of weakly coupled molecular motors
pled, and although they may interact, they are free to move independently
from each other. This new scenario of motor cooperativity inspired new theoretical
models based on lattice formulation, which consider both attractive
and repulsive interactions (Campas et al., 2006). These models reproduce
and explain qualitatively the ability of groups of motors to pull on membrane
tubes, but are unable to predict the force distribution among the motors since
they are based on discrete hoping rates and not on mechanistic models.
Inspired on the case of motors pulling a liquid cargo, we will consider
the situation of weakly coupled molecular motors that cooperate to overcome
an external force that is applied to only one or a few motors in the front.
Our approach will be based on the simplest possible two state model for
a motor (Julicher et al., 1997). At the price of missing a detailed description
of specific cases, our mechanistic approach (in contrast to the lattice approach)
allows a quantitative analysis of cluster efficiency and force transmission
among the motors. We aim to gain new insights into generic phenomena
concerning force transmission between motors and the connection between
motor interactions, velocity-force curves and the collective performance of
motor clusters.
3.1.1 Molecular motor modelization
Following (Julicher et al., 1997), we will briefly review the theoretical models
of translationary molecular motors. Molecular motors are basically proteins
that convert, at a molecular scale, chemical energy into mechanical work.
Translationary molecular motors move unidirectionally along polar filaments
made of periodic structures (actin filaments and microtubules). Many models
for molecular motors are inspired on ”thermal ratchets” by Feynman (Feynman
et al., 1966) which explain unidirectionality out of periodic distribution
of temperatures with appropriate asymmetry (Büttiker, 1987; Landauer,
1988). This approach, although interesting, it is not appropriate for molecular
motors which are isothermal machines.
The general mechanism behind the non vanishing displacement of a brownian
particle in an asymmetric periodic structure without any thermal gradient
or macroscopic force, consists in driving the system out of the thermodynamic
equilibrium, violating detailed balance, in which case energy is
constantly added to the system (ATP consumption in the case of molecular
motors), and the time symmetry is broken. There are three different mechanisms
that maintain the system out of thermodynamic equilibrium in a periodic
isothermal ratchet (Julicher et al., 1997),

3.1 Introduction 25
Fluctuating forces. The particle is submitted to a periodic potential and fluctuating
forces that, although having a zero averaged value, have non trivial
temporal correlations that do not satisfy a fluctuation-dissipation theorem,
and encode the energy source.
Fluctuating potentials. The particle is submitted to a periodic potential which
value depends on time. In this case, the fluctuating forces are Gaussian white
noise that obey a fluctuation-dissipation theorem. In this example the energy
source is implicitly manifested in the time dependence of the periodic potential.
Fluctuating state transitions. The particle has different states which transition
rates do not satisfy detailed balance (the ratio of the two state rates it is
not given by the energy difference between the two states). The dynamics of
transitions between states is added independently and may reflect chemical
reactions (ATP consumption as example for molecular motors). The fluctuating
forces satisfy again a fluctuation-dissipation theorem.
We will focus on a simple description of the third mechanism with only
two states which basically represent an attached state and a weakly bound
state, aiming to extract generic features of the motion of a motor which will
allow a straight forward extension to the case of many motors.
3.1.2 The two state model
We consider the simplest fluctuating state transitions model, which is a
molecular motor with only two states, corresponding to the bound state
(power stroke) and a weakly bound state (detached state). The equation of
motion for the position of the motor, x, is a Langevin equation,
dx
λi
dt = −∂xUi(x)+ fi(t), (3.1)
where λi is a friction coefficient that in general may depend on the state of
the motor, and fi(t) is a fluctuating force that satisfy a fluctuation-dissipation
theorem,
〈 fi(t)〉 = 0, 〈 fi(t) f j(t ′ )〉 = 2λikBT δ(t −t ′ )δij. (3.2)
The potential Ui has the filament symmetries and reflects the interaction between
the motor and the filament. As a consequence the potential must be
both periodic and asymmetric. The conformational change of the protein
which leads to the motor movement is represented by the sliding over the

26 3 Self-organization and cooperativity of weakly coupled molecular motors
asymmetric potential, Fig. 3.1. In this model we assume only a conformational
change which leads to the power stroke of the motor. Any other conformational
change is considered instantaneous when compared to chemical
reaction times of the order of milliseconds.
(a)
(b)
ATP
ATP ADP P
P ADP
ADP
Unbinding Diffussion Binding Power stroke
(c) α1 β2
α2
Fig. 3.1. Equivalence between a molecular motor cycle (a) and the two state model (b-c)
explained in the text.
In Fig. 3.1 we schematically show the equivalence between a motor cycle
and the two state model. Initially, the motor is attached to the filament in
what is called ”rigor state” until a molecule of ATP binds to the motor, which
detaches from the filament and hydrolyzes the ATP molecule,
β1
ATP ⇋ ADP + P, (3.3)
this reaction represents the transition from the bound state (1) to the unbound
state (2). Associated to this reaction there is a chemical potential difference
∆µ ≡ µAT P − µADP − µP which measures the free-energy change per ATP
consumed. At the unbound state, when a phosphate molecule P is released
the motor attaches to the filament and performs the power stroke after the
molecule of ADP is released. The transition from the unbound state (2) to the
bound state (1) is thermal (passive),
M − ADP − P ⇋ M + ADP + P. (3.4)

Transition rates
3.1 Introduction 27
There are two transition rates associated to each of the two different reactions
(Eq 3.3 and 3.4), Fig. 3.1c. α1,2 correspond to the hydrolysis of ATP,
and their ratio depends on the difference of energy between the two states,
∆U ≡ U1 −U2, and the difference of chemical energy ∆µ. β1,2 correspond to
the thermally induced release of phosphate and ADP, and their ratio satisfies
detailed balance,
α1(x) −∆U(x)+∆µ
= exp
(3.5)
α2(x) kBT
β1(x) −∆U(x)
= exp
, (3.6)
β2(x) kBT
The global transition rates between the two states w1,2, include both α and β
rates: wi ≡ αi + βi, which in terms of α2 ≡ α, and β2 ≡ β,
−∆U
w1(x) = (α exp∆µ + β)exp
(3.7)
kBT
w2(x)=α + β, (3.8)
which do not satisfy detailed balance unless ∆µ = 0. When a cell has excess
of ATP (∆µ > 0), the motors are driven out of thermodynamic equilibrium
and perform a directional movement along its associated polar filaments. In
physiological conditions, ∆µ ∼ 10kBT , the system is far from equilibrium,
and α exp∆µ ≫ β. If in addition ∆U ≫ kBT , which is equivalent to neglect
thermally excited transitions with respect to conformational changes,
w1(x)=α exp∆µ (3.9)
w2(x)=α + β. (3.10)
For simplicity we can assume U2 to be a flat potential, and de-exitations (corresponding
to the rate w2) to be spatially delocalized. Moreover, it is reasonable
to assume that the transition from the bound to the unbound state, which
depends on ATP binding, is localized at the minimum of U1, corresponding
to the fact that ATP is not likely to bind to a motor during the ”power stroke”.
In Fig. 3.2, we schematically represent the potentials and the transition rates
between the two states.

28 3 Self-organization and cooperativity of weakly coupled molecular motors
Ui
wi
δ
ℓ
Fig. 3.2. Schematics of the molecular motor as a ratchet. The bound state corresponds to an
asymmetric periodic potential U1, with a periodicity ℓ. The bound state corresponds to a flat
potential U2. The transition rate from U1 to U2 state, w1 is localized at the minimum of the
bound state potential (with a width δ). De-excitations are delocalized, w2 =constant.
3.2 Self-organization and cooperativity of weakly coupled
molecular motors
We are interested in the cooperativity of N infinitely processive motors moving
along a one-dimensional filament. An external force F opposing the motion
is applied only on the foremost motor. The force transmitted to the rest of
the motors is given by motor-motor interactions. We consider the interaction
between motors to be a short-ranged (non strongly-binding) potential including
a hard-core repulsive part. By non bindging potential we mean that the
mean distance between motors diverges at the steady state, in contrast to the
case of strongly coupled motors, Fig. 3.3.
We use a two state model as described in section 3.1.2, which is characterized
by a bound and unbound potential and their respective transition rates.
We represent the ratchet potential as Ui(xi,ki), where the index i ∈ (1,N) is a
label for the i-th motor, xi is its position, and ki ∈ (1,2) is a discrete stochastic
variable representing the state of the motor. Finally, the interaction between
motors is represented by the potential W(xi − xk). The equation of motion for
the motors is described by a set of Langevin equations equivalent to Eq. 3.1,
λ ˙xi = −U ′
i (xi,ki) −∑ W
k=i
′ (xi − xk) − Fδ1i + ζi(t), (3.11)
where λ is a friction coefficient, which is taken to be the same for both bound
and unbound states. ζi(t) is the thermal noise that satisfy 〈ζi(t)ζ j(t ′ )〉 =
w1
U2
U1
x
w2
x

3.2 Self-organization and cooperativity of weakly coupled molecular motors 29
W (r)
2
1.5
1
0.5
0
-0.5
0.1 0.2 0.3 0.4 0.5
Fig. 3.3. Several examples of interacting potentials between motors. In black, hard-core potential;
in blue, a weak-attractive (see text for a definition of weak-attractive) potential; in green,
a long-range potential, and in dotted red, an example of strongly binding potential.
2kBT λδijδ(t −t ′ ) (Gaussian process), and its strength is defined as the diffusion
coefficient, D = kBT /λ.
We restrict our present analysis to motors that move on the same filament,
and as a consequence, they share the same ratchet potential, Ui(xi,ki) ≡
U(xi,ki), that represent the interaction between filament and motor. Each motor
switches states independently, that is, transition rates of motor i depend
only on xi (Julicher et al., 1997), in contrast to previous studies where motors
switch at the same time (Derényi and Ajdari, 1996).
For simplicity we define U(x,1) ≡U1(x) as a completely asymmetric sawtooth
potential with period ℓ and height U, and a sliding velocity v = U/(λℓ),
which represents the ”power stroke” of the motor. U(x,2) ≡ U2 is modeled as
a flat potential, Fig. 3.4.
Hereinafter we define U1 and U2 as the bound and unbound states respectively.
The lifetime of U2 is τ and de-exitations are delocalized. Transitions
from U1 to U2 are localized at the minimum of U1(x) (see section 3.1.2) and
are much faster than any other process during a motor cycle (τ and the ”power
stroke”), and are considered instantaneous, which corresponds to a large excess
of ATP (∆µ ≫ 0).
3.2.1 Simulations
We will numerically solve the set of Langevin equations using a standard
Euler method for the integration of differential equations. The discretized
version of the set of equations, Eq. 3.11, reads,
r

30 3 Self-organization and cooperativity of weakly coupled molecular motors
Ui
U
ℓ
v
√ Dτ
Fig. 3.4. Two state potential of a molecular motor. The state U1 is a totally asymmetric potential
with period ℓ with localized transitions at each minima. The unbound state U2 is modeled
as a flat potential with delocalized transitions after a deterministic time τ.
xi( j + 1)=xi( j)+
U2
U1
x
δki1v −∑ W
k=i
′ (xi( j) − xk( j)) − Fδi1
∆t (3.12)
+(−2Dlog(χ)∆t) 1/2 cos(2πχ), (3.13)
where the last term is a discretization of the thermal noise ξi(t), using a uniform
distributed stochastic variable, χ ∈ (0,1) (Sancho et al., 1982). For the
transitions between states U1 and U2, we define the following rules: From U1
to U2,
If(ℓ + δ >| mod(xi,ℓ) |>ℓ − δ), ki = 2, (3.14)
which corresponds to a transition localized in a small interval δ from the
minimum of the potential (see Fig.3.2). For the transition from U2 to U1, the
motor remains at U2 for a deterministic interval of time τ, and then changes
to U1.
Simulation parameters
The fixed simulation parameters are
• The time step ∆t is fixed to 5 × 10 −5 s.
• The periodicity of the potential U1 is fixed to 8 nm which is roughly the
length of a heterodimer of α-β tubulin (a microtubule monomer).
• The interval around the potential minimum for motor transition is fixed to
δ = 0.002ℓ.
• The de-excitation time of the state U2, τ = 1/w2 is fixed to 50 ms.

3.2 Self-organization and cooperativity of weakly coupled molecular motors 31
• The value of the maximum of the U1 potential is chosen U/kBT = 20,
which is roughly the energy released for ATP hydrolysis.
• The motor friction coefficient is fixed to λ = 4×10 −4 kg s −1 , which leads
to a sliding velocity over the potential U1 of about 0.2µm s −1 , resulting in
a motor velocity consistent with KIF1A velocities (Okada and Hirokawa,
1999; Nishinari et al., 2005; Endow and Higuchi, 2000).
In the following sections, we will consider the cooperativity and selforganization
of molecular motors for different interaction potentials, namely,
hard-core repulsion, weak-attraction and long-range repulsion. We will compare
their velocity-force curves to a mean field calculation (that we will define
in the following). We will see that depending on the interaction between motors,
clusters of motors of sizes different from N self-organize and move at
different velocities. Finally, will consider the distribution of forces transmitted
among the motors and the cluster efficiency.
3.2.2 Mean-field
The degree of cooperativity between motors depends on the correlations between
positional and internal degrees of freedom of different motors. Consider
the case of two motors that are connected by a rigid rod, Fig. 3.5. Their
collective motion depends critically on the state of each motor. There are four
instantaneous velocities that can be defined depending on the state of each
(see Fig. 3.5). This case corresponds to the maximum cooperativity between
motors and the internal degrees of freedom of the motors (state 1 or 2) are
correlated with the positional degree of freedom. Their collective behavior
and extension to N motors have been extensively studied in (Juelicher and
Prost, 1995; Julicher et al., 1997; Badoual et al., 2002) and reveal nonlinear
deviations of the force-velocity relationship.
Consider now the opposite case: two motors that interact with a soft (nonbinding)
potential and an opposing force 1 F acting on the first motor. If the
resulting interacting force varies slowly with respect to the period of the periodic
potential ℓ: ℓW ′′ (r)/W ′ (r) ≪ 1, the interaction force remains insensible
to a motor cycle and its spatial fluctuations, so the motors feel the same
interacting force 〈−W ′ (r)〉 −W ′ (〈r〉) =F/2 irrespective to the particular
motor state, and move at an average velocity v that corresponds to the velocity
of a single motor with an applied force F/2, Fig. 3.6. The intuitive picture
corresponds to represent the motion of a motor as a vehicle (which is over
1 We need to apply a force otherwise the mean distance between motors diverges and they
do not interact with each other.

32 3 Self-organization and cooperativity of weakly coupled molecular motors
(a) (b) (c)
Fig. 3.5. Schematics of two motors connected by a rigid rod. When both motors are in the
diffusive or bound state (a) and (b), they behave as single a motor (with half the total force).
(b) In the crossed configurations, the advancing probability of the motor in the diffusive state
do not rely on noise since the bound motor is pushing (pulling), and the motor advances
deterministically.
damped), with its engine that has an asymmetric combustion cycle (the cycle
of the motor) but moves at a constant velocity. The two motor case would correspond
to two cars that are pushing together against a force, sharing exactly
the same force, and hence moving at the velocity they would move if they
were alone pushing against half of the applied force (Fif. 3.6b). This softly
enough interacting potential results in the intuitive result that a force applied
to a cluster of N motors is equally shared and their velocity VN is related to
the velocity of a single motor, VN(F)=V1(F/N).
(a) (b)
F
F/2 F/2
〈 ˙x〉 = v v
Fig. 3.6. Schematics of the interaction of two motors satisfying mean field. (a) the spring
connecting the two motors represents a soft connection that is insensible over a period ℓ. In
(b), the car analogy, where an over damped car is moving at a constant velocity although its
internal motor has asymmetric combustion cycles.
More rigorously, we define the mean field approximation for a cluster of
N motors as satisfying the relation:
〈−W ′ (r)〉 −W ′ (〈r〉), (3.15)
so that correlations between positional and internal degrees of freedom of
different motors are neglected. The steady state solution of Eq. 3.11 for
the mean field approximation implies that the force between two motors is
F

3.2 Self-organization and cooperativity of weakly coupled molecular motors 33
constant, and, consequently, each motor behaves as an isolated motor subject
to an equal share F/N of the external force. The collective dynamics
is then reduced to the single-motor problem with VN(F) =V1(F/N) and
FN(V )=NF1(V ). The physical picture corresponds to N ballistic motors defined
solely by V1(F).
It is illustrative to obtain the exact velocity-force curve for N motors in
mean field approximation, or equivalently, for a single motor at a force F/N.
We use these values later as a reference to compare with interacting potentials
that not satisfy the conditions for mean field approximation.
The mean velocity over a time t corresponding to m cycles is given by
〈 ˙x〉 = ∆x
t ≡ ∑m i=0 δxi
∑ m i=0 δti
= 〈δx〉
, (3.16)
〈δt〉
where ∆x is the total distance traveled by the motor during the time t. This
distance can be decomposed in steps of distances δxi, which correspond to
the advanced distance per cycle. Similarly, the time t is the sum of intervals
δti corresponding to the time lapse of the i-th cycle. Finally, the mean velocity
is the ratio of the mean displacement and the mean time lapse per cycle. We
compute the mean velocity in two steps: first, the mean displacement and
second the corresponding mean time per cycle,
Mean displacement per cycle
We consider the initial position of the motor as an arbitrary minimum of the
asymmetric potential U1, Fig. 3.7. The motor instantaneously changes its state
to U2 and starts diffusing for a time τ. During this diffusive state, the motor is
drifted by the external force at a velocity F/λ. After a time τ the motor deexitate
to the state U2 at a position x with a gaussian probability distribution
centered at x0 ≡−Fτ/λ,
P(x)= 1
(x + Fτ/λ)2
√ exp −
2πa 2a2
, (3.17)
where a2 ≡ Dτ. At the state U2, the motor diffuses and slides over the ramp of
the potential towards the minimum. The probability of going forward towards
the minimum of the potential (and not going backwards) is given by (Parmeggiani
et al., 1999)
(F −U1/ℓ)z
P→ 1 − exp − . (3.18)
kBT

34 3 Self-organization and cooperativity of weakly coupled molecular motors
When the external force approaches the force due to the power stroke, U1/ℓ,
the advancing probability is considerably reduced. For the sake of simplicity,
we will neglect here the noise in the U1 state, and the motor always move
forward towards the minimum, advancing a multiple of the potential period,
nℓ, each cycle.
Expressing the transition position from the state U2 to U1, x ≡ (n−1)ℓ+z
(see Fig. 3.7), the mean displacement per cycle is 〈δx〉 = ℓ∑ ∞ n=−∞ np(n), with
p(n) the probability of falling at any position between n(ℓ − 1) and nℓ,
〈δx〉 = ℓ
∞
∑
n=−∞
ℓ
n
0
1 (ℓ(n − 1)+z + Fτ/λ)2
dz√
exp − . (3.19)
2πDτ 2Dτ
For an external force F = λℓ/(2τ), the motor at state U1 after the transition
time τ is positioned at half the period of the potential from the initial position.
In this case, the probability of advancing forward or backwards (+nℓ or
−nℓ) is exactly the same and the mean displacement is δx = 0. This force
corresponds to the stall force of the motor if λℓ/(2τ) < U/ℓ, otherwise, the
stall force is U/ℓ. Defining the two dimensionless parameters α = vτ/ℓ, the
ratio of the lifetime of the state U2 and the sliding time on the state U1, and
β ≡ ℓ/(4Dτ) 1/2 , the ratio of the potential period and the diffusive displacement
in U2, we can express the stall force as,
F 0
s = λv min
1, 1
2α
where the super-index 0 stands for mean field.
Mean time lapse per cycle
, (3.20)
The time lapse corresponding to advancing a distance nℓ is the sum of the U2
life-time τ and the sliding time on the U1 state, δt = τ +ts, with ts =(ℓ−z)/v,
see Fig. 3.7. The mean time lapse is then 〈δt〉 = ∑ ∞ n=−∞ δt(n)p(n), with p(n)
the probability of falling at any position between n(ℓ − 1) and nℓ,
〈δt〉 =
∞
∑
n=−∞
ℓ
0
dz τ +
ℓ − z
v
1 (ℓ(n − 1)+z + Fτ/λ)2
√ exp −
2πDτ 2Dτ
(3.21)
Finally the mean velocity V 0 ≡〈˙x〉 is the ratio of Eq. 3.19 and Eq. 3.21,
which in terms of the dimensionless parameters α, β and the dimensionless
force f ≡ F/(λv), can be expressed as

3.2 Self-organization and cooperativity of weakly coupled molecular motors 35
F τ/λ
(i)
t =0
(ii)
z
(iii)
0 nℓ − 1 x nℓ
Fig. 3.7. Schematics of the meanfield calculation for the force-velocity curve of N motors,
which corresponds to a single motor with a rescaled force. (i) transition from the U1 to U2
state and drift due to the external force. (ii) the motor de-excitate to the state U1 at a distance x
from the origin with an associated probability (see text). (iii) the motor slides over the potential
U1 until a new minimum at nℓ is reached. As explained in the text, backwards transitions are
neglected in the limit of vanishing noise.
where
φ( f ) ≡ 1
2
erfc(αβ f )+
V 0 φ( f )
( f ) = (1 − f ) , (3.22)
α + φ( f )
∞
∑
n=1
(erfc(β(n + α f )) − erfc(β(n − α f )))
,
(3.23)
and erfc(x) ≡ x
0 dxexp(−x2 ) is the complementary error function. The veloc-
ity of a free motor ( f = 0) is given by the simple expression V 0 (0)= v
1+2α .
In this calculation we have neglected the noise contribution during the sliding
in the U1 state. This contribution might be important, specially near the
stall force if Fs ∼ U/ℓ, in which case the probability of a backward step tends
to unity (see Eq. 3.18). In general, our approximation should be valid for
U/ℓ ≫ λℓ/(2τ) or a small noise intensity such that the length diffused in a
typical sliding time ∼ ℓ/v is much smaller than the period of the potential,
D ≪ vℓ. The parameters we are considering correspond to this last condition.
In Fig. 3.8, we plot velocity-force curves for different noise intensities
and de-excitation τ, showing that the disagreement with our approximation is
fairly small for small noise. Hereinafter, we will use the values F 0
s and V 0 (0)
calculated to compare with the non mean field cases.

36 3 Self-organization and cooperativity of weakly coupled molecular motors
V1/V1(0)
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
F/Fs
Fig. 3.8. Velocity-force curves for the mean field calculation (solid line) (Eq. 3.22), the corresponding
simulation with noise turned off in the U1 state (solid symbols), and simulation
with noise in both states (empty symbols). The parameters used are: τ = 0.05 and D = 0.25
(back and circles); τ = 0.05 and D = 0.025 (red and squares); τ = 0.175 and D = 0.25 (blue
and diamonds); τ = 0.175 and D = 0.025 (green and triangles). Both velocities and forces are
normalized to the analytical zero force velocity and the stall force (see text).
3.2.3 Hard-core repulsion
Significant deviations from mean field are expected generally for shortranged
repulsive interactions (Campas et al., 2006) and for small or moderate
noise strength in intermediate force range. In our model, we expect correlations
between motors to enhance the motor performance in general as in the
rigidly bound motors (see Fig. 3.5). However, not all the motor configurations
in short-range repulsive interactions contribute to increase the velocity with
respect to mean field, Fig. 3.9.
We will study the case of a hard-core potential between motors, modeled
by a truncated Lennard-Jones potential (see Fig. 3.3),
W(r)= 4ε
σ 6 σ 12
−
r6 r12
r ≤ 2 1/6 σ
(3.24)
ε r > 2 1/6 σ.
We analyze the dependence of the deviation from mean field with respect to
(i) β, the dimensionless parameter which is the ratio of the potential period
and the diffusive displacement in the state U2. Since the period of the potential
is fixed (ℓ ∼ 8 nm), varying β stands for varying noise intensity, Fig. 3.10. (ii)
the hard-core parameter σ, which represents effectively the size of the motor

3.2 Self-organization and cooperativity of weakly coupled molecular motors 37
Fig. 3.11. We generally obtain enhanced motor performance with respect to
mean field, VN(F) > V1(F/N), for most of the parameter values. However,
there is a small region of parameters for which the deviation with respect to
mean field vanishes or even become negative.
Deviation from mean field
The deviation from mean field can be explained by considering the 2 motor
case with each of the possible motor-motor state combinations, Fig. 3.9.
There are two configurations of the motors that correspond to a mean field
contribution: the two motors in the bound state U1, Fig. 3.9(b) and the two
motors in the state unbound U2. These two configurations satisfy that the
mean force is equally shared by the two motors and the velocity corresponds
to the velocity a single motor would have in half the force applied. The
crossed configurations are the responsible for mean field deviation and illustrate
the effect of the correlations between motors. In Fig. 3.9a, the foremost
motor is in the unbound state and the second motor is sliding in the bound
state. A single motor in U2 would drift at a velocity −F/λ. In the case of two
motors, the second transmits the power stroke to the foremost, which finally
moves at a velocity v−F/(2λ) (> 0 if F is smaller than the stall force) which
deterministically enhances the advancing probability of the motor which do
not rely on noise, leading to an effect that persists for small noise, O(D 0 ),
while V1(F) falls as O( √ Dexp(−1/ √ D)). As a consequence, this motor configuration
always contribute towards gain in motor performance. For the last
configuration, corresponding to the first motor in the U1 state and the second
in the U2 state, the first motor is moving alone since the second motor is diffusing
and in mean, not moving. As a consequence, the first motor moves at a
velocity v − F/λ, which is smaller than the mean field velocity v − F/(2λ),
and the deviation from mean field is negative.
The relative weights between configurations U1-U2 and U2-U1 determine
the amount and sign of deviation from mean field. A large positive deviation
from mean field will occur when the configuration U1-U2 dominates,
this is, when the time during which the second motor pushes the first one
is maximized. This time depends strongly on the particle size, or, σ. For
mod(σ,ℓ) ∼ 1, this effect disappears completely since the pushing distance
tends to 0: if both motors are in contact, they coincide in the U1 minimum
and consequently they change to the U2 state together. On the contrary, configuration
U2-U1 is in general present, so we expect a negative deviation from
the mean field for σ close to the period length ℓ. For intermediate values
of σ, we expect a gain with respect to mean field, because the second mo-

38 3 Self-organization and cooperativity of weakly coupled molecular motors
tor will generically push against the first motor. In particular, due to symmetry,
the maximum performance will correspond to mod(σ,ℓ) ∼ 1/2, Fig.
3.11. The amount of deviation from mean-mean field, depends on the correlations
between motors. If entropic repulsion between motors dominates,
D
(V1(0)−V1(F))ℓ
≫ 1, correlations between motors are fully suppressed, and we
recover the mean field approximation results. For very weak noise, the increased
motor performance becomes dramatic, Fig. 3.10.
(a) (b) (c) (d)
v − F/(2λ) −F/(2λ)
v − F/(2λ)
v − F/λ
Fig. 3.9. Schematics of cooperativity in the case of two motors with hard-core repulsion. (a)
the second motor pushes the foremost motor which is at the diffusive state. This configuration
represents a net gain with respect to mean field, since the advancing mechanism for the foremost
motor do not rely on thermal fluctuations as in the mean field case. (b) the two motors
slide over the U1 potential, sharing the external load as in mean field. (c) the foremost motor
takes over all the load when sliding over the U1 potential whilst the second motor diffuses.
This configuration corresponds to a loss with respect to mean field since a larger force is applied
on the foremost motor. (d) the two motors diffuse on the U2 state sharing the external
load as in mean field approximation.
N motors
For large N, we find that VN( f ) converges to a limiting curve VN → V∞( f ),
that corresponds to a force per motor f that tends to a constant. One would
be tempted to state that the force per motor tends the corresponding mean
field approximation, that is, the force per motor is equally distributed and
f = F/N. As we will see in section 3.3.1, the force distribution is not homogeneous
throughout the motors, although its rescaled value saturates to a
constant. In this sense, we recover the extensive scaling as in mean field, but
with a V∞( f ) = V1( f ), Fig. 3.12. This is an important difference with respect
to the prediction of the lattice model in (Campas et al., 2006), where VN(F)
(the non rescaled velocity-force curve) was found to saturate with N to a limiting
curve VN(F) → ˜V (F), i.e. they obtained a non-extensive scaling of force,
except near stall. Consequently even with an unlimited supply of motors, the
cluster could not advance against an arbitrarily large force, contrarily to our

3.2 Self-organization and cooperativity of weakly coupled molecular motors 39
VN/V 0
1 (0)
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
F/(NF 0 s )
Fig. 3.10. Velocity-force relationship for N = 1, 2 motors, α = 0.15 and hard-core repulsion
(repulsive part of a Lennard-Jones potential truncated at r = 2 1/6 σ, with ε = 1 and σ = 0.2ℓ).
Circles correspond to β = 10; empty circles, N = 1 and N = 2 with mean field; solid circles, N
= 2. Squares correspond to β = 44.7; empty squares, N = 1 and N = 2 with mean field; solid
squares, N = 2. Axes are normalized by D = 0 values of the 1-motor stall force and the 1-motor
free velocity (see text).
V2(2F0)/V1(F0)
1.8
1.6
1.4
1.2
1
0.8
0.6
0 0.5 1 1.5 2 2.5 3
Fig. 3.11. Dependence on the hard-core size σ (in units of ℓ) of the gain with respect to mean
fied for an external force F0 = 1 3 F0
s .
result which states that any arbitrary force can be overcome if enough motors
are supplied.
3.2.4 Long range interaction - transition to mean field
Early in section 3.2.2, we described the criteria for mean field approximation.
In the last section, we showed that increasing noise leads to mean field
σ

40 3 Self-organization and cooperativity of weakly coupled molecular motors
VN/V 0
1 (0)
1
0.8
0.6
0.4
0.2
0.5
0 5 10 15
N
20 25 30
0
0 0.2 0.4 0.6 0.8 1
F/(NF 0 s )
Fig. 3.12. Convergence with the number of motors N, of velocity vs force per motor, for
α = 0.15 and β = 14.1. Diamond, N = 1; inverted triangle, N = 2; triangle, N = 3; empty
square, N = 5; empty circle, N = 10; solid square, N = 15; solid circle, N = 30. In the inset,
the convergence for a fixed force F0 = 1/3F 0
s
(see Fig. 3.10). Another route to mean field is to impose a soft interacting potential
such that the variation of the force over a ratchet period is negligible,
ℓW ′′ (r)/W ′ (r), which ensures the decoupling of internal and spatial degrees
of freedom between motors. This condition is easily achieved for a (longranged)
interaction of the form W(r) ∼ exp(−r/λ), with λ ≫ ℓ. Adding to
this potential a hard-core part to prevent motor-motor crossing, we end up
with a long range interaction (see Fig. 3.3),
σ 6 σ 12
4ε −
W(r)= r6 r12
+ Aλ exp − r
r ≤ 2
λ
1/6 σ
Aλ exp − r
r > 2
λ
1/6 (3.25)
σ,
where the parameter A, sets the intensity force of the long-range potential.
We expect a crossover force between hardcore repulsion and long-range interaction,
if the force provided by the long-range part is not enough to overcome
the force per motor: −W ′ (2 1/6 σ) < F/N. In Fig. 3.13, we show several
plots of the 2-motor velocity-force curves for different long-range intensities
intensity. The transition region from mean field to cooperative behavior
corresponds qualitatively to a cross-over force F ∗ 2A/λ exp(−2 1/6 σ/λ).
Above the cross-over force, the velocity-force curve approaches the hardcore
curve (region in between the top and bottom curve in Fig. 3.13). The gain
in velocity from respect to mean field in this region grows linear with the force
until it reaches the hard-core region, inset of Fig. 3.13, with a slope that is
VN (NF0)/V1(F0)
2.5
2
1.5
1

3.2 Self-organization and cooperativity of weakly coupled molecular motors 41
independent of the long-range intensity. This linear growth can be attributed
to the cooperativity of the motors which slowly grow (since λ ≫ σ) until
it saturates to the maximum value given by the close packing of the motors
(hard-core part). The fact that the cooperativity increases linearly with the
force in the long-range interaction may have influence in the distribution of
forces in larger clusters (N > 2), since an unevenly distribution of forces in a
motor cluster could lead to the same motor velocity due to the cooperativity
increase in the more loaded regions (see section 3.3.1).
V2/V 0
1 (0)
1
0.8
0.6
0.4
0.2
∆V2/V 0
1 (0)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0 0.1 0.2 0.3 0.4
∆F/(2F 0 s )
0
0 0.2 0.4 0.6 0.8 1
F/(2F 0 s )
Fig. 3.13. Velocity force curves for long range interaction for A = 0.5,1,1.5,2 and 10. In the
inset, velocity gain with respect to mean field for the transition region between the mean field
curve (inferior curve) and hard-core curve (superior curve) as a function of the increase of
force from the cross-over, for each value A. In all cases, λ = 10ℓ.
3.2.5 Attractive interaction
We have seen in section 3.2.3 that the enhanced cooperativity from mean field
arises from the interplay between the gain from the crossed configurations
U1-U2, where the trailing motor pushes the diffusing leading motor (see Fig.
3.9a), and the loss from the crossed configurations U2-U1, where the leading
motor moves alone and therefore slower than mean field (see Fig. 3.9c). In
the case of rigidly coupled motors, the crossed configuration U2-U1 gives
also a net gain because the leading motor pulls the diffusive trailing motor
increasing the advancing probability dramatically (see Fig. 3.5b).
If a weak (non-binding) attractive interaction is added to the hard-core repulsion
(a Lennard-Jones potential in our case), a new scenario emerges in

42 3 Self-organization and cooperativity of weakly coupled molecular motors
which the collective behavior differs qualitatively from both short-ranged interactions
and rigidly coupled motors. Consider first the case of 2 motors. For
vanishing external force, the attractive interaction is not binding (the mean
distance between motors diverge with time), and the velocity is equal to the
free motor. As we increase the force in the leading motor, the effective attractive
potential between motors increases, Fig. 3.14a and the push-and-pull
mechanism is increased. As a consequence, we find that when increasing the
force, the mean velocity of the pair is increased with respect to the free motor
velocity, Fig. 3.14b.Upon further increase of the load, the velocity V2 must
eventually decrease below V1(0), Fig. 3.14b.
(a) (b)
W (r)+Fr
4
3
2
1
0
Binding
Non binding
0 0.5 1 1.5 2
r
V2(F )
2.6
2.5
2.4
2.3
2.2
2.1
2
0 0.2 0.4 0.6 0.8 1
Fig. 3.14. (a) Effective attractive potential resulting of the sum of a non binding attractive
interaction between motors and a non vanishing external force (red). Non binding attractive
potential in the absence of external force (black). (b) Non monotonous velocity-force curve
for 2 attractive motors.
This behavior has strong implications for the self-organization in the case
where many motors are available.
Cluster dynamics
In the case of N motors, we expect a non-trivial self-organization of motors
as a consequence of the non-monotonous velocity-force curve for 2 motors
(see Fig. 3.14b), that exhibits velocities that are larger than the free motor
velocity. In fact, for forces such that V2(F) > V1(0), a cluster formed by the
two first motors escape from the rest moving at faster velocity. When F is
increased beyond the value for which V2(F)=V1(0), the motor following the
cluster is captured and the new cluster of 3 motors will start increasing its
velocity with the force again as in the 2 motor case. As a consequence, if
N motors are available, the curve VN(F) exhibits N − 2 ”rebounds” at V1(0)
F

3.2 Self-organization and cooperativity of weakly coupled molecular motors 43
before decreasing below that value (when no more motors are available), each
one consisting in the capture of an additional motor by the cluster, Fig. 3.15.
VN/V1(0)
1.15
1.1
1.05
1
0.95
0 0.5 1 1.5 2 2.5
Fig. 3.15. Velocity-force curve for the case of attractive forces with α = 0.15, β = 14.1, ε = 1
and σ = 0.2ℓ.
The remarkable behavior depicted in Fig. 3.15 (for which the depth of the
attractive well is only 10% of U) shows a genuinely self-organized collective
behavior that has no counterpart in single-motor dynamics or in rigidly
coupled motors.
Transient bimodality and hysteresis
In the neighborhood of force values Fi, where VN(Fi) =V1(0), we expect
transient stable clusters of i + 1 and i + 2 motors, where i is the i − th rebound
corresponding to the force Fi. If we consider the case of 3 motors, we
expect to observe clusters of both 3 and 2 motors near the force for which
V3(F)=V1(0). For smaller forces, although the stationary cluster is formed
by 2 motors, a finite long lived cluster of 3 motors will appear transiently if
the motors start close enough. If we plot an histogram of velocities as a function
of time, we indeed observe initially a peak at a velocity V3 corresponding
to the cluster of 3 motors, that eventually disappears while another peak of
velocity V2 corresponding to the cluster of 2 motors, remains stationary for
large times, Fig. 3.16b. This transient bimodality implies the appearance of
metastability and hysteresis as we move along the force coordinate since the
capture of a motor (when F surpasses Fi from below) or the release of one
motor (when F decreases below Fi from above) takes a finite time (see Fig.
3.16a)
F

44 3 Self-organization and cooperativity of weakly coupled molecular motors
(a) (b)
V3
2.5
2.4
2.3
2.2
2.1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
F
Fig. 3.16. (a) Velocity-force curve for 3 motors. Arrows indicate hysteric transitions from the
metastable extensions for N = 2 and N = 3. (b) Transient bimodality for the decay of the
3-motor metastable cluster to the 2-motor (downward arrow in (a)). In all cases, α = 0.15,
β = 14.1 and σ = 0.2ℓ.
3.3 Discussion
3.3.1 Force transmission
An important aspect for motor kinetics is how the external force is transmitted
through the cluster of motors, since force affects dramatically the motor
detachment rate. If each motor in the cluster shares the same amount of force,
the cluster will be more stable and able to sustain larger forces than in the
case of unevenly shared force, where the motors sustaining more load will
tend to detach faster. Depending on the motor reservoir, this effect could lead
to a catastrophe event in which all the motors in the cluster would detach, decreasing
dramatically the stall force of the cluster. An extension of the present
work dealing with cluster kinetics including detachment dependent on force
is work in progress (Brugues and Casademunt, 2008). In this section, we are
interested in the force distribution in the cluster depending on the interaction
potential. In Fig. 3.17a, we show the force per motor due only to interaction 2
as a function of the position, N = 1 being the first motor, without adding the
power stroke part. Surprisingly, the force distribution is not in general uniform.
For hard-core interaction, there exist a plateau of maximal sustained
force near the front of the cluster that exceeds the force corresponding to
mean field approximation, and a zone of about 3 motors that sustain smaller
2 We do not add the force due to the ratchet potential since the total force per motor must be
the same in order for the cluster to move at the same velocity.
t
V2
V3
V

3.3 Discussion 45
forces than mean field. Using the long-range interaction, we have seen that
the cooperativity gain increases with the interaction force between motors
until it reaches a saturation value for motors which are very close to each
other. This effect is enough to explain the non uniform distribution of forces
between motors. In Fig.3.17b, we show the force distribution for the case of 7
motors. We can see that for a large long-range intensity (Eq. 3.25), we recover
mean field, this is, the force is uniformly distributed among the motors. As
soon as we decrease the intensity, the external force will reach the cross-over
force and some motors will begin to cooperate (in particular, the motors in the
front), being able to sustain larger forces than the non cooperating motors, for
the same cluster velocity (see Fig.3.13). The increase of cooperativity is linear
for a range of forces, until it saturates, in which case the force distribution
corresponds to the hard-core case (see Fig. 3.17b). For the attractive case, we
observe that the force is nearly uniformly distributed.
We can reduce the effect of the cooperativity to 2 motors: For the case of
hard-core interaction, the deviation from mean field corresponds to the rear
motor ”pushing” against the first motor. As a consequence, provided that any
motor in the cluster has a motor behind it, it would perform better. The last
motor of the cluster though, it is not pushed by any other motor, and as a
consequence, it will perform worse, and will also push less efficiently. Since
the whole cluster move at the same velocity, the rear motors must sustain
lower forces to compensate its lower performance. Thus, the non uniform
distribution arises naturally from the motor cooperativity.
For the case of attractive interaction, the deviation from mean field corresponds
to both motors pushing and pulling (respectively) to each other, and
as a consequence, the cooperativity is a symmetric effect (not only a rear motor
pushing the foremost one), and as a consequence, all motors in the cluster
perform nearly equal. The last one being slightly less efficient because it lacks
the pushing effect.
In general, one could experimentally imagine to be able to test the kind
of motor interaction by testing not only the force-velocity curves, but also
the stability of the cluster which depends dramatically on the motor force
distribution (Brugues and Casademunt, 2008).
3.3.2 Motor efficiency
The efficiency of a motor is defined as the ratio of mechanical work performed
to chemical energy consumed,

46 3 Self-organization and cooperativity of weakly coupled molecular motors
0.4 (a) (b)
(fi − fmf )/fmf
0.3
0.2
0.1
0
-0.1
0 2 4 6 8 10 12 14 16 18 20
(fi − fmf )/fmf
0.6
0.4
0.2
0
-0.2
-0.4
1 2 3 4 5 6 7
Cluster position Cluster position
Fig. 3.17. Relative averaged force distribution with respect to mean field among the motors
forming a cluster with an applied external force F = N ( fmf = 1). (a) Distribution for hardcore
interaction for 4, 7, 8, 19 and 20 motors (circles), and for attractive interaction for 10
motors (squares). (b) Distribution of forces for 7 motors for attractive interaction (squares),
hardcore interaction (empty circles) and long-range interaction with intensity A = 0.5 (solid
blue circles), A = 3 (solid red circles) and A = 10 (solid black circles).
η(F)= FV(F)
, (3.26)
r(F)∆µ
where r(F) is the chemical reaction rate, that in our case is the number of
excitations per unit time (Julicher et al., 1997; Parmeggiani et al., 1999),
which corresponds to transitions from the U1 to U2 state. In a mean field
approximation, N motors consume N times as molecules of ATP per unit
time as a single motor, rN(F) =Nr1(F/N), and they sustain N times the
force of a motor for a given velocity, FN(V )=NF1(V ). The efficiency of a
motor cluster within mean field approximation reads ηN(F) =η1(F/N): a
motor cluster has proportionally more motive power, but it is as an efficient
machine as a single motor.
In the case of short-ranged repulsion and weak-attraction, the efficiency
of the motors is generically increased, more strongly for the attractive case,
Fig 3.18a. The enhanced efficiency, is a result of both the increase of power
with respect to mean field VN(F)F ≥ V1(F/N)F, and the decrease of the rate
of ATP consumption rN(F) ≤ Nr1(F/N), Fig. 3.18b.
The counter-intuitive decrease in ATP consumption when the motors
move faster 3 can be qualitatively explained by the same cooperativity mechanism
that increases the velocity with respect to mean field (see section 3.2.3).
Consider the case of two motors. In mean field, the two motors are independent,
move at the same velocity and we only need to consider the consump-
3 One would expect that if a motor moves faster, completes more cycles in less time, and
therefore, should in principle consume more ATP.

10 (a) (b)
ηN /η max
1
8
6
4
2
0
0 2 4 6 8 10 12 14
F
r(F )/r(0)
2
1.5
1
0.5
3.3 Discussion 47
0
0 2 4 6 8 10 12 14
Fig. 3.18. (a) Efficiency normalized to the maximum value for N = 1. Empty circles, N = 1 and
N = 3,5 for mean field; squares, N = 3, triangles N = 5 (solid symbol, attractive case, empty
symbol, repulsive case). (b) Rate of ATP consumption normalized to the rate of consumption
at vanishing force, corresponding to 5 motors. Empty circles for mean field, empty triangles
for hard-core repulsion, and solid triangles for attraction. The parameters used are α = 0.15,
β = 14.1 and ε = 1.
tion of one of them. If we focus on the process of advancing one period of
the potential, Fig.3.19a, after the transition from the U1 to U2 state, the motor
diffuses with a negative drift due to the external force and the probability to
de-exitate to the previous period is higher than advancing to the next potential
period. For small forces and small noise, when the motor de-excites to
the previous period, the distance to the minimum U1, z ∼ Fτ/λ is smaller
than the period length z ≪ ℓ and hence, the time for re-excitation to state
U2 is much shorter than the sliding time corresponding to advance a period.
As a consequence, there are many fast transitions between state U1 and U2
before the motor advances one period at which many ATP molecules are consumed
(one for each transition). At increasing forces this effect is increased
because the probability of fast transitions is increased, until a threshold force
at which the consumption exhibits a maximum and increasing the force has
the effect of reducing the sliding time in the previous fast transitions until the
force is equal to the ratchet force and the motor do not slide anymore (in our
parameters, the stall force is given by the siding force).
In the case of motors cooperating with each other (attractive and hardcore),
the scenario is completely different. The probability of advancing a
period is increased by the cooperation between motors, Fig. 3.19b, and the
motor performs nearly a single slow cycle per ATP consumed. This effect is
enhanced with the force since the motors are kept in contact and it is even
more important in the attractive case (as discussed in earlier sections).
F

48 3 Self-organization and cooperativity of weakly coupled molecular motors
(a)
(b)
Many fast cycles
+
one slow cycle ∼ one slow cycle
Fig. 3.19. Qualitative explanation of the differences in ATP consumption between mean field
and cooperative case. In (a), a single motor representing mean field performs many fast cycles
between states U1 and U2 corresponding to de-excitations to the previous period due to the
force driven drift, before advances a period performing a slow cycle. As a consequence, most
of the ATP is consumed corresponds to fast cycles, increasing the rate of consumption. In
(b), cooperativity between motors enhances the probability to advance one period, and as a
consequence, most of the ATP consumed correspond to slow cycles, decreasing the rate of
consumption.
For large clusters, we have seen that the distribution of forces (the average
force of a single motor in the cluster) is not uniform. The foremost motors
sustain larger forces than the rear ones, but move at the same velocity. As a
result, motors in the front have an enhanced cooperativity which is translated
in a higher motor efficiency (Brugues and Casademunt, 2008).
3.4 Conclusions
We have considered the collective behavior of molecular motors for hard-core
repulsion, weak attractive interaction and long-range repulsive interaction.
Their collective performance is generally optimized with respect to mean
field. For the case of attractive motors, new cluster dynamics phenomena
arise which involves hysteresis and transient bimodality. If a large enough
reservoir is available, the cluster moves at a velocity greater than the free
motor, and the cluster size is dynamically readjusted depending on the force
applied. Finally, we have considered the cluster efficiency and force distribution
which is a crucial assumption in some recent works concerning collective
behavior (Campàs et al., 2008).
It remains an open question whether biological systems might take advantage
of these effects, possibly combining them with the regulation of processivity.
Nevertheless, our results may be relevant in the broad context of
nonequilibrium physics, and could be used for artificial design of motors. It
is worth remarking that some inputs of our model, such as the interactions
and the friction coefficient (defining the noise strength), involve not only the

3.4 Conclusions 49
motor itself but also the cargoes. Consequently, it should be feasible to experimentally
tune parameters in broad ranges, and in particular find regimes
where the present predictions are observable.
A limitation of the present analysis for biological or biomimetic applications
is that finite processivity and force-dependent kinetics have been
neglected. This could be easily incorporated in the model (Brugues et al.,
2008a).

4
Membrane-cortex interactions: Micropipette
experiments and blebbing cells

52 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
4.1 Introduction
The plasma membrane is essential for the cell. It encloses the cell, defines
its boundaries, mediates the exchange of materials between the cytosol and
the extracellular media, and contains proteins that act as sensors of external
signals which allow the cell to react and adapt its behavior in response to environmental
changes. Underneath the membrane lies the cytoskeletal cortex, a
highly dynamic network of actin filaments that reorganizes continuously. The
cortex works in conjunction with the membrane, and is primary responsible
for cell shape, cytokinesis, movement and mechanotransduction. Interactions
between cortex and membrane are fundamental for all these functions. The
plasma membrane and cortex are held together by both molecular interaction
between both lipids and membrane proteins and cytoskeletal proteins. The
loss of adhesion induces the formation of blebs, which are highly dynamical
spherical deformations of the cell surface lacking cytoskeleton. If the cell is
to survive, those patches of unbound membrane must eventually heal with
the appearance of a new cortex that retracts the protrusion. Although in most
cases blebs are an indication of cell death, there are some examples where
cells take advantage of blebs to move.
In this chapter, we first investigate the mechanism of cortex and membrane
adhesion. Our approach is based on two completely different descriptions.
On one hand, we properly consider the kinetics of the membrane and cortex
linkers, at the price of losing spatial correlations between links. On the other
hand, we use a coarse grained description for the adhesion in terms of an
effective macroscopic density of adhesion energy. While the first description
allows a better understanding of the dynamical properties of the adhesion,
and in particular, the dependence of the adhesive strength in terms of the
active state of the cell, it fails to predict any scale of bleb nucleation. The
second description do not treat properly the link dynamics, but it is able to
qualitatively describe bleb nucleation and bleb statistics.
The second part of the chapter is devoted to the study of the dynamical
response of the membrane and cortex to mechanical perturbations, that
eventually lead to an unbinding of the membrane from the cortex. In particular,
we consider how the viscoelasticity (elastic behavior at short time and
viscous behavior at longer time) and the treadmilling (polymerization and
depolymerization) of the cortex affects the membrane-cortex instability. We
propose a simple Maxwell like model for the cortex which allows us to determine
all possible dynamical regimes under a controlled perturbation using
a micropipette aspiration set up. We compare our theoretical predictions with

4.1 Introduction 53
previous works found in the literature and our experiments specially conducted
to observe the different dynamical regimes.

54 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
4.2 Modeling cell mechanics
4.2.1 Cellular membrane
The cellular membrane is a self-assembled bilayer containing lipids and many
other proteins (Alberts et al., 2002). Membrane lipids have a polar head and a
hydrophobic tail which is sandwiched between the hydrophilic head groups.
Within a vesicle, the lateral diffusion of a lipid is of about 1µm 2 s −1 (Alberts
et al., 2002), which means that an average lipid molecule would diffuse the
typical cell length (∼ µm) within a few seconds. In biological membranes,
this diffusion is reduced by a factor of 3-10, due to complex structures in the
membrane such as lipid rafts (Dietrich et al., 2001) and other obstacles like
membrane proteins and cysoskeleton.
In this section we briefly describe the mechanical properties of the membrane:
energy cost of membrane deformation and membrane tension.
Energy cost of membrane deformation
Any deformation of the membrane can be decomposed in a longitudinal deformation
(which includes stretching and shearing) and an out-of-plane deformation.
Each mode of deformation has its energetic cost, and contributes
to the total energy of a membrane deformation.
Shearing corresponds to a deformation on the plane that conserves area,
which microscopically speaking corresponds to a rearrangement of the lipids
in the leaflet. Since the membrane behaves as a two dimensional fluid, the
energetic cost of shearing the membrane is negligible in comparison to the
other two modes of deformation.
Stretching corresponds to an in plane deformation that changes the area
of an element of the surface, which corresponds to varying the lipid density
by direct expansion or contraction of area per molecule. Defining the relative
change in area α ≡ ∆A/A (areal strain), its associated energy per unit surface
is
es = 1
2 Kaα 2 , (4.1)
where Ka is the elastic moduli for direct area expansion with values for typical
lipid vesicles of about 10 −1 Nm −1 (Evans and Rawicz, 1990).
Bending, corresponds to the energy cost of out of plane deformations.
Microscopically, the cost to bend a membrane is related with the different
amounts of direct stretching of molecules in each leaflet, the density of

4.2 Modeling cell mechanics 55
molecules being changed locally depending on the curvature of the deformation.
The bending energy density of a two dimensional membrane can
be expressed in terms of two invariants involving the two principal curvatures
(c1 ≡ 1/R1 and c2 ≡ 1/R2) of a two dimensional surface (Helfrich and
Servuss, 1984),
eb = 1
2 κ(c1 + c2 − c0) 2 + ¯κc1c2. (4.2)
The spontaneous curvature c0 is in general nonzero whenever the two sides of
the membrane are not identical. Microscopically, this corresponds to a different
lipid composition or density, or it is associated to a different composition
of the aqueous media facing the two leaflets of the membrane. The sum of the
principal curvatures, or mean curvature, is associated with an elastic modulus
κ, and the product of principal curvatures or Gaussian curvature with
¯κ. The Gauss-Bonnet theorem shows that the Gaussian curvature is a topological
invariant, and consequently, we will neglect its contribution since we
will restrict ourselves to surfaces with spherical topology and constant bending
modulus ¯κ. Typical measured values for the bending modulus are κ ∼10
kBT (Evans and Rawicz, 1990), and consequently, thermal fluctuations induce
large undulations.
Membrane tension
As opposed to the bending rigidity and the elastic moduli for direct area expansion,
the membrane tension, which by definition is the derivative of the
membrane energy with respect to the area, it is not a well-defined microscopic
quantity. The total microscopic area, the area per lipid times the total number
of lipids, is not experimentally measurable. The reason for that are the thermal
induced fluctuations in the membrane which are expected to be important
since the the bending modulus is of the order the thermal energy. Some
fluctuation modes will be experimentally accessible whereas others will not
be optically resolved (modes of wavelength smaller ∼ µm). The apparent
membrane area then will not correspond to the microscopic total area, Fig.
4.1. Imposing tension in the membrane increases the apparent area by removing
fluctuation modes, each of them carrying the same amount of energy
(Equipartition theorem), and therefore, there is an energetic cost associated
to the increase of apparent area even though the total amount of microscopic
area is conserved. We can define then an effective tension that will be a superposition
of an area increase due to reduction of membrane undulations plus a

56 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
Fig. 4.1. Schematic representation of a fluctuating membrane at different length scales. In
solid line, the real membrane area. In dashed line, the apparent membrane area.
direct expansion in area per molecule. The effective membrane tension is related
to the areal strain by (Helfrich and Servuss, 1984; Fournier et al., 2001;
Evans and Rawicz, 1990),
α − α0 = kBT
8πκ log
κπ2 /A + γ
κπ 2 /A0 + γ0
+ γ − γ0
Ka
, (4.3)
where A and A0 are the actual and a reference apparent area respectively, γ0 is
the initial tension corresponding to A0, and α − α0 is the difference between
the actual and initial relative excess area between the apparent and real area.
The tension initially grows exponentially with the relative increase of area
(entropic regime) and its followed by a linear increase that is related with the
direct increase of area per molecule (elastic regime). The crossover tension
between the two regimes is of the order γ ∼ 510 −4 N/m, or equivalently,
α ∼ 2-5% (Evans and Rawicz, 1990), which is not very far from the tension
needed to disrupt the membrane.
Cells have their own mechanisms to regulate membrane tension to new environmental
conditions in order to prevent, for instance, burst under a sudden
decrease of external pressure. There are mainly two regulatory mechanisms:
membrane and bulk exchange through the membrane, Fig. 4.2. The former
involving both vesicular transport between the interior of the cell (Golgi apparatus)
and the cell membrane (endocytosis and exocytosis), and sequestered
membrane in form of membrane invaginations, like in the case of caveolae
(Alberts et al., 2002). The second one utilizes several types of channels
embedded in the membrane that by either active (involving energy consumption)
and/or passive (tension or carrier mediated gates) transport, induce a
bulk flux between the interior and exterior of the cell.

(a) (b)
(c)
4.2 Modeling cell mechanics 57
Fig. 4.2. Different mechanisms of membrane tension regulation. In (a), several examples of the
presence of both active and passive channels in the membrane, which regulate the membrane
tension. (b) Electron micrograph of a fibroblast showing the presence of caveolae in the cellular
membrane. As tension is applied on the membrane, the caveolae may unfold maintaining
the membrane tension constant. (c) Electron micrographs of the formation of clathrin-coated
vesicles from the plasma membrane during endocytosis. Membrane recycling regulates membrane
tension by changing the amount of lipids in the bilayer. (Images from (Alberts et al.,
2002)).
4.2.2 Cytoskeletal cortex
Actin cortex is involved in many mechanical processes of the cell, ranging
from large scale phenomena such as cell locomotion and cell shape remodeling,
to small scale phenomena including endocytosis. Actin forms a dynamical
cross-linked gel that is attached to the membrane, where predominantly
polymerizes 4.3. Actin is cross-linked by several proteins, the most common
being filamin and α-actinin, which concentration affects dramatically the cortex
elastic properties. The average polarization of the cortex points away from
the membrane.
There are other actin associated proteins of the ERM family such ezrin,
which take part on the anchoring of the cortex to the membrane. Finally, actin
filaments can associate to myosin motors forming the so called actomyosin
complexes or microfilaments. Myosin II self-assemble into bipolar filaments
that act on actin generating stress in the cortex by sliding each filament with
respect to each other, Fig. 4.3c. Actomyosin complexes can form very highly
organized structures capable of generating large amount of stress, as in the
case of myofibrils in muscle contraction. In the cortex, actomyosin complexes
organize randomly as part of the highly crosslinked network of actin filaments
with a mesh size of about 10-100 nm.
Actin polymerization is regulated by several cortex proteins such as
formin, profilin or the multiprotein complex Arp2/3 which promote actin
polymerization, and by certain lipids composing the plasma membrane such

58 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
as PIP2 (Raucher et al., 2000; Sheetz et al., 2006). Although the precise mechanism
remains unclear, it is widely accepted that actin in the cortex polymerizes
mainly under the membrane and depolymerizes along the cortex thickness.
As a consequence, the cortex treadmills from the membrane at a velocity
which is roughly the polymerization velocity, Fig. 4.3.
(a) (b)
(c)
h ∼ 500 nm ξ ∼ 100 nm
+
−
−
− − Contractile stress
−
Fig. 4.3. Schematics of the cytoskeletal cortex. (a) The cortex is a thin layer of cross-linked
actin filaments (red) formed under the membrane. (b) The actin network forming the cortex
is characterized by a constant treadmill of material from the membrane. (c) Molecular motors
(Myosin) slide actin filaments with respect to each other creating contractile stress in the
cortex.
Viscoelastic properties of the cysotkeleton
Several rheological techniques may be used to characterize the mechanical
properties of the cortex and the cytoskeleton of the cell in general. These
methods include laser tweezers, AFM and magnetic bead rheology, among
others (Pullarkat et al., 2007). The complex structure and heterogeneity of
the cytoskeleton makes quantitative rheological measurements particularly
difficult. Moreover, elastic properties may be expected to depend on the active
state of the cell. However, the viscoelastic behavior under an external
deformation consisting of an elastic response at short time and a viscous flow
at long time is a general property of the cytoskeleton. Unfortunately, linear
response characterized by a single relaxation time scale (see below) is not sufficient
to account for the cytoskeleton viscoelasticity at all time scales, and
a crossover from a linear viscoelastic behavior to a power law stress stiffening
regime is experimentally observed (Fernández et al., 2006; Trepat et al.,
2007), which corresponds to a continuum of relaxation time scales.
+
+
+
vp
vd
−

Theoretical modeling of the cytoskeleton
4.2 Modeling cell mechanics 59
The modeling on cell mechanical properties can be divided into two main
approaches, which consider passive, viscoelastic properties of the cell and
active mechanisms of cell mechanics, respectively. The split of the mechanics
of the cell into an active and passive regimes is justified by the different
response times of the cell in rheological experiments, seconds to minutes for
a passive response and several minutes for an active response (Thoumine and
Ott, 1997).
The purely passive elastic response has been modeled considering basically
two different frameworks. Tensegrity, which stands for structures consisting
of elements under tension (actin cables) and elements under compression
(microtubules) (Ingber, 1993), and soft glass rheology, where the competition
between entropy in crosslinked actin network and bending of individual
filaments can account for the crossover from linear elasticity to non-linear
stiffening (Wilhelm and Frey, 2003; Head et al., 2005; Head et al., 2003a;
Head et al., 2003b; Head et al., 2003c)
The model of active mechanisms consists on generalized hydrodynamic
theories of polar maxwell viscoelastic gels formed by polar filaments (actin
or microtubule filaments) which are maintained out of equilibrium by constant
consumption of energy (ATP) (Kruse et al., 2004; Juelicher et al., 2007;
Joanny et al., 2007).
Here, we are interested in the actin cortex, which is a highly dynamic
thin layer, Fig. 4.3. We aim to use the simplest model that accounts for viscoelasticity
and treadmilling, and which includes an active stress driven by
actomyosin complexes. We consider that polymerization occurs mainly in the
cortex, and we assume that the interaction between motors and actin leads to
a contractile effective stress which depends only on myosin concentration.
Maxwell model for actin cortex viscoelasticity
We will work assuming a linear viscoelastic regime for the cytoskeleton,
which is at least observed experimentally for certain conditions (Thoumine
and Ott, 1997). The simplest viscoelastic model that incorporates short time
scale elastic behavior and large time scale fluid behavior is the so called
Maxwell model. For short time scales, the cortex behaves as a solid, and
the internal stress σij obeys σij = 2Euij (Landau et al., 1995), where uij ≡
1/2(∂iu j + ∂ jui) is the strain tensor, ui being the deformation field,and E the
Young modulus of the cortex. In the opposite limit, the cortex behaves as
a fluid, and the stress is given by σij = 2ηuij ˙ (Landau and Lifshitz, 1987),

60 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
where η is the viscosity of the cortex. Defining τ as the crossover time scale
between the two regimes, the viscosity and Young modulus are related by
η ∼ τE, which follows from equating both elastic and viscous stress at the
crossover time τ. The linear interpolatory equation that describe both limiting
cases is given by,
1 d
+ σij = 2E
τ dt
duij
. (4.4)
dt
In order to illustrate that this equation represent both solid and fluid behavior,
consider a periodic deformation where σij and uij are proportional to e −iωt .
Eq. 4.4 reads,
σij = 2Euij
. (4.5)
1 + i/ωτ
For large frequencies ω ≫ 1/τ, the stress is σ 2Euij and the deformation
corresponds to an elastic solid. For small frequencies ω ≪ 1/τ, the stress is
σ −2iωτEuij = 2τEuij, ˙ which is the usual expression for a viscous fluid
with viscosity Eτ. The microscopic origin of the relaxation time τ is the kinetics
of the cross-linking proteins, which bind and unbind to actin filaments,
creating and breaking crosslinks continuously, relaxing any tension stored
between filaments. Typical orders of magnitude for the cortex parameters are
E = 103 Pa, τ = 10 s, and η = 104 Pa s (Wottawah et al., 2005).
Treadmilling in the cortex induces an inwards convective flow of material
from the membrane with a velocity related to the polymerization velocity, Fig.
4.3. In order for Eq. 4.4 to account for treadmilling, we need to transform the
time derivative d/dt in a convective time derivative, d/dt ≡ ∂/∂t + vi∂i.
The presence of myosin induce a contractile stress in the cortex. At
equilibrium, the myosin stress σm and the cortex elastic stress balance, and
∇σ + ∇σm = 0, where σm > 0 since myosin stress is compressive, see Fig.
4.4.
Considering a gel growing in a cylinder, the force balance in the angular
direction is automatically satisfied, and the radial component reads
∂
∂r r (σrr + σm) − (σθθ + σm)=0, (4.6)
where we are assuming the myosin contraction to be isotropic. The radial
stress applied on the membrane is σrr(R)+σm(R). Myosin needs a characteristic
time ∼ 1/kon to attach to actin, and since the gel grows from the
membrane, this corresponds to a characteristic length from the membrane

(a) (b)
θ
r
h
σmdS3
4.2 Modeling cell mechanics 61
σmdS4
σmdS2
σmdS1
Fig. 4.4. Sketch of the coordinates in the cortex (a), and stress applied by the surrounding
myosin on a piece of cytoskeletal cortex (b).
∼ vp/kon, where myosin has not been able to attach to the membrane. As a
consequence, the density of myosin at the membrane is zero, so the stress applied
on the membrane is only an elastic stress, which depends on the myosin
concentration throughout the gel,
σrr(R)= 1
R
dr(σθθ + σm) ≡
R R−h
γc + γm
, (4.7)
R
where γc and γm are the lateral and myosin tension in the cortex. Similarly,
for spherical geometry the normal stress at the membrane is
σrr(R)= 2
R2 R
rdr (σθθ + σm). (4.8)
R−h
A gel growing away from a curved surface needs to either concentrate or
dilate in order to accommodate the curvature, which induce stress in the gel
that depend on its thickness (Noireaux et al., 2000; Sekimoto et al., 2004).
The contribution of this effect to the normal stress of order O (h/R) 2 .
The cortex thickness is of about 500 nm, which is much smaller than
the radius of the cell, R ∼ 20µ. Using a thin layer approximation, we can
treat the cortex as a two dimensional surface, averaging all quantities along
the radial direction. We can also neglect the geometric contribution to the
stress in comparison to the active stress generated by myosin, which is of
the order O (h/R) (see next section). Therefore, we write first an effective
maxwell equation for a flat surface, and then for small deformations of the
cortex, we relate the normal and lateral stress by force balance. The tangential
component of the stress tensor is

62 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
1 ∂
+
τ ∂t + vα∂
β + vp∂z σαβ = 2E ˙u αβ, (4.9)
where the greek indices stand for longitudinal directions along the surface,
and z is the normal direction in which the material convects at a veloc-
ity vp. Averaging this
last equation with respect to the slab thickness h
, in a thin film approximation, we obtain
Ā ≡ 1/h h
0 A(z)dz
1 d
+ ¯σ
τ∗ αβ = 2E ˙ū
dt
αβ, (4.10)
where we have assumed that actin polymerizes at the membrane surface with
no lateral stress, σ αβ(z = h) =0, and we have used that in the thin shell
approximation σ αβ(z = 0) ∼ ¯σ αβ. The total time derivative now stands for
the convective derivative over the longitudinal directions, d/dt ≡ ∂t + vα∂ β ,
and τ ∗ ≡ τ + h/vp is an effective relaxation time that includes the effect of
the treadmilling along the normal direction.
As already mentioned in this section, we consider the actomyosin complexes
to create an effective stress which depends only on the density of motors.
Its particular value is a function of the relative orientation of the actin
filaments and myosin concentration in the cortex (Juelicher et al., 2007). The
stress profile of the actomyosin complexes depends both on the treadmilling,
which limits the amount of myosin present in the cortex, and on its bulk density.
In the following section we explicitly calculate the active stress in the
cortex using linear kinetics for myosin.
Contractile stress in the cortex.
We can use linear kinetics to write the local mass conservation for the density
ρ of actomyosin, which is related with the total amount of active stress in the
cortex,
dρ
dt = kon(ρs − ρ) − ko f f ρ, (4.11)
where kon and ko f f are rates of attachment and detachment, and ρs is the maximum
density of actomyosin in the gel, which is related to the density of actin
filaments in the gel, and the total number of bulk myosins. The time derivative,
d/dt ≡ ∂/∂t + v·∇ is a convective derivative that takes into account the
treadmilling of material towards the interior of the cell at a velocity vp. The
steady state profile of myosin is given by

ρ(z)= ˜ρs
1 − exp
4.2 Modeling cell mechanics 63
− z
λ
, (4.12)
where z is the coordinate normal to the membrane, ˜ρs ≡ ρskon/(kon + ko f f )
is the mean density of myosin in the absence of actin turnover, and λ ≡
vp/(kon + ko f f ) is a typical length scale below which convection dominates
myosin binding.. When actin turnover is slow, the attachment length becomes
small and the whole cortex is decorated with myosin. On the contrary, if actin
turnover is very fast compared to attachment kinetics, the attachment length
can be of the order of the cortex thickness The parameter that controls the
amount of density in the cortex is then h/λ, h being the cortex thickness.
Assuming a linear relationship between the actin stress and the myosin
density, and defining ε as the amount of stress per actomyosin, the total lateral
tension due to myosins is,
h
γm ≡ ε
0
ρ(z)=σmsλ
h
λ +
exp − h
− 1 , (4.13)
λ
where σms ≡ ε ˜ρs is the stress that the mean density of myosin would perform
on the gel in the absence of actin turnover. If the gel thickness is very small
compared to the typical length needed for a myosin to attach to the gel (λ ∼
kon/vp), myosin density is strongly limited by the treadmilling process and
the lateral tension is
γm ∼ σms(h 2 /λ), (4.14)
In the opposite limit, myosin kinetics is fast and the maximum stress is
reached almost on the entire gel thickness,
γm ∼ σmsh. (4.15)
Actin turnover and gel thickness affects dramatically the amount of stress
stored in the cortex. The cortex thickness is in turn regulated by several mechanisms,
including the stress stored in the cortex. In next section, we briefly
review the mechanisms that may fix the cytoskeleton thickness.
Cytoskeletal cortex thickness
Actin filaments grow preferentially from the membrane due to the presence
of actin polymerization promoters in the membrane (PIP2 as an example
(Raucher et al., 2000; Sheetz, 2001)). After a certain length, actin filaments
are cross-linked by proteins present in the cortex (mostly filamin and

64 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
α-actinin (Alberts et al., 2002)) becoming a meshwork with gel like properties
that treadmills at a velocity given by the polymerization velocity vp. The
equilibrium between polymerization and depolymerization fixes the steady
state thickness of the cortex. Considering that the main contribution from
actin depolymerization takes place at the gel surface,
˙h = vp − vd(h), (4.16)
where vp and vd are the polymerization and depolymerization velocities respectively.The
gel will reach its steady state when the depolymerization velocity
at the gel surface equals the polymerization velocity, vp = vd(h). Three
mechanisms may control the thickness of the gel: biochemical regulation,
monomer diffusion, and stress induced depolymerization.
The amount of actin monomer concentration determines the polymerization
and depolymerization velocity of actin. Moreover, several proteins including
Cofilin regulate the turnover rate of actin by adjusting actin monomer
concentration (Carlier et al., 1997), providing the cell with a mechanism to
regulate its cortex thickness.
Since the cortex polymerizes mainly from the surface, actin monomers
must diffuse along the cortex thickness, and hence, the polymerization velocity
can be limited by this process (Noireaux et al., 2000).
All symmetry-breaking models (Sekimoto et al., 2004; van Oudenaarden
and Theriot, 1999; Mogilner and Oster, 2003) consider a coupling between
tensile stress and gel dissociation. The microscopic origin of the dissociation
can be the unbinding of cross-links in the cortex, which disrupts the gel-like
structure, or direct depolymerization of actin filaments. Both depolymerization
and disruption of the gel structure can be effectively described with a
depolymerization velocity which increases with the tensile stress. Kramers
rate theory (Kramers, 1940; Plastino et al., 2004) suggests an exponential
dependence of the depolymerization velocity with the stress (Noireaux et al.,
2000; Plastino et al., 2004; Gerbal et al., 1999; Juelicher et al., 2007),
vd = v 0 d exp
σ
, (4.17)
where v 0 d is the depolymerization velocity at vanishing stress, and σ0 is
a reference stress related to actin and cross-linker kinetics. We will assume
that the polymerization velocity at the membrane is constant and that the gel
is growing without lateral stress. Notice that the a finite thickness solution
is ensured by the increase of active contractile tension due to myosin as a
function of the cortex thickness (see Eq. 4.13) (Juelicher et al., 2007).
σ0

Mechanical equilibrium of the cell
4.2 Modeling cell mechanics 65
In section 4.2.2, we have considered the amount of stress generated by myosin
in the cortex (see Eq. 4.13). Myosin stress is transmitted to the membrane via
deformation of the actin cortex and via proteins that link the actin cortex
to the membrane. The normal stress transmitted to the membrane is given
by 2(γc + γm)/R, where γc is the lateral tension in the cortex and γm is the
myosin tension (see Eq. 4.7), and where we are neglecting corrections of
order O (h/R) 2 .
At steady state, the shape of the cell is determined by the balance of the
outward forces from the osmotic pressure and the inward forces from the
membrane and cortex, Fig. 4.5. This static force balance (membrane not moving),
leads to the Laplace law including the tension in the cortex,
∆P = 2 γ + γc + γm
. (4.18)
R
At static equilibrium, the strain rate vanishes ( ˙R = 0), and as a consequence,
the lateral stress in the cortex is σθθ = 0 1 (see Eq. 4.10). Then, the only
tension contribution at equilibrium from the cortex corresponds to myosin
tension, γm. A cell may thus be pressurized by increasing myosin activity or
density.
Laplace law stated above (Eq. 4.18), is only valid if the cortex stress is
transmitted to the membrane. The cortex is adhered to the membrane by actin
associated proteins which are responsible for force transmission. Failure of
these links leads to separation of the membrane from the cytoskeleton. The
next chapter is devoted to the stability of the membrane and cortex adhesion
and its consequences to the cell behavior.
1 Here we are neglecting the stress created by the curvature which is of the order O (h/R) 2 .

66 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
(a) (b)
Pc
P0
Fig. 4.5. Schematics of the forces involved in the cell at steady state. In (a), the internal, Pc,
and external, P0, pressures exert a normal force on the membrane and cortex. The normal projection
of the myosin tension in the cortex (b) is transmitted to the membrane through proteins
that link the cortex and the membrane, and balances the resulting force at the membrane.

4.3 Membrane-cortex adhesion
4.3 Membrane-cortex adhesion 67
In this section we describe the membrane-cortex adhesion using two different
approaches:
(i) The first approach uses a kinetic model for its linkers, where we will
consider the cortex as a flat and rigid surface, and we will leave for forthcoming
sections the effect of the cytoskeleton visco-elasticity. We will find the
adhesion stability conditions as a function of kinetic parameters of the linkers
and the total force applied on the membrane, which may result from pressure,
cytoskeleton tension or any external perturbation (such as an osmotic
shock or the action of a micropipette aspiration). As a result, we will obtain
that cooperativity between bonds determines the stability of membranecortex
adhesion: below a critical force links can to the load and the adhesion
of membrane and cytoskeleton is stable. Above a critical force, there is a
global link failure and an abrupt transition from a stable adhesive membrane
to an unstable membrane detached from the cytoskeleton. Finally, we will
show how adhesion depends critically on the pre-stressed state of the cortex
and the density of linkers, and we will compare our model with quantitative
experiments found in the literature (Merkel et al., 2000).
(ii) The second approach is an energetic description of the detachment
process, which considers the cost of creating a protrusion detached from the
cortex. Balancing the energy cost of membrane-cytoskeleton detachment and
the membrane tension with the energy gain from the pressure, we find an
abrupt transition for membrane detachment.
4.3.1 Kinetic model for membrane-cortex adhesion
The sketch of the system is depicted in Fig. 4.6. The density of links ρb between
membrane and cortex is represented by springs that attach and detach
continuously at rates kon and ko f f respectively. A force per unit area f is
shared by the bonds, that are stretched by a small amount x, which is a molecular
length scale. When the link and membrane forces are not balanced, the
membrane flows and the stretching increases at a rate given by viscous dissipation.
In appendix 4.A, we show that the dominant term in the dissipation
is due to the flow of cytosol through the cortex meshwork, which in terms
of dissipation per unit area is ˙E ∼ ˙x 2 ηch/ξ 2 , where ηc is the cytosol viscosity
(∼ 3.10 −3 − 2.10 −1 Pa s (Charras et al., 2008)), h is the thickness of the
cortex (∼ 500 nm), and ξ is the cortex mesh size (∼ 10 − 100 nm).
Defining the effective viscosity per unit length, η ≡ ηch/ξ 2 , a free membrane
under a pressure ∆P would move at a velocity ˙x ∼ ∆P/η, and would

68 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
(a) (b)
x
ξ
kxρb
f
η ˙x
δ
koff
Fig. 4.6. (a) Schematics of the links between membrane and cortex. (b) Linkers have an association
and dissociation rate that depend on the energy profile of the link (black curve). When
a force is applied on the bond, its energy landscape and associated rates are modified (red
curve).
need a typical time t ξ ∼ ξη/∆P to inflate to a distance ξ , which is the mean
distance between links (see Fig. 4.6a). For typical pressure values in micropipette
experiments, ∆P ∼ 100 − 1000 Pa (Merkel et al., 2000; Rentsch
and Keller, 2000), t ξ is of about 10 −6 − 10 −3 s. If we compare this time with
a dissociation attempt frequency starting at about ∼ 10 9 − 10 10 s −1 (Evans,
2001), we realize that the inflation time needed for spatial correlation between
neighboring links (that is, the time needed to inflate to a distance ξ ) is several
orders of magnitude larger than the typical link kinetics time scale 2 . As a
consequence, we can assume that links are not spatially correlated although
they cooperate to overcome the force applied on the membrane. Basically,
there are a lot of attachment and detachment events while the membrane is
moving. This image breaks down if the detached patch of membrane is sufficiently
large so that the links in the center have no chance to rebind as the
patch of membrane inflates.
We will use the mean density of bonds when considering the force balance
at the membrane, which is justified by the fast kinetics of attachment and
detachment,
kon
η dx
dt = f − kxρb, (4.19)
where k is the stiffness of the links, which we take of about 10 −2 pN nm. The
linear kinetics equation for the number of attached links reads,
2 The threshold pressure for spatial correlation is calculated considering the pressure needed
to inflate the membrane a length ∼ ξ in a time ∼ 1/kon, and is of about 10 7 Pa. We will see
that the links typically become unstable for much smaller pressures.

4.3 Membrane-cortex adhesion 69
dρb
dt = kon(ρ0 − ρb) − ko f f ρb, (4.20)
where ρ0 is the saturation density for bound links (ρ0 ∼ 1/ξ 2 ). The kinetic
rates kon and ko f f are determined by the energy landscape of the link, Fig.
4.6b. The energy barrier to overcome in order to dissociate a link changes as
an external force is applied (or equivalently, an external potential −xf), Fig.
4.6b, and the off rate increases exponentially with the applied force (Kramers,
1940; Kampen, 2007),
ko f f = k 0 o f f exp[kxδ/kBT ], (4.21)
where δ ∼nm is a molecular scale of the link which is related to the maximum
of the energetic barrier (see Fig. 4.6b), and kx is the stretching force on a
link. The rate of rebinding, kon depends in general on the substrate and link
stiffness (Evans, 2001), but for the sake of simplicity, we assume it here to
be constant 3 .
Before considering the dynamical system given by equations Eq. (4.19)
and (4.20), it is convenient to rescale the variables with the relevant scales to
identify the dimensionless parameters that govern the stability of the system.
Time is rescaled by the attachment time scale tb ≡ 1/kon, and τ ≡ t/tb. The
stretching length scale is related with the typical force per link and the stiffness
ls ≡ f /(kρ0), and z ≡ x/ls. Finally, we rescale the link density by the
saturation density ¯ρb ≡ ρb/ρ0, and the set of equations Eq. (4.19) and (4.20)
become,
ζ dz
dτ = 1 − z ¯ρb, (4.22)
d ¯ρb
dτ = −(1 + χ expαz) ¯ρb + 1, (4.23)
where ζ ≡ η/(ktb) is the ratio between hydrodynamic relaxation and link
kinetics time scales (ζ ≫ 1), χ ≡ k 0 o f f /kon compares the on and off rates, and
α =
f /ρ0
, (4.24)
kBT /δ
is the ratio of the typical force per link and a typical force from thermal noise.
We investigate the linear stability of the stationary points of these two nonlinear
equations, ˙z = ˙¯ρb = 0, given by
3 We have considered several dependences of kon on both geometry and force and the qualitative
behavior of our analysis remain unchanged.

70 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
¯ρbs = 1
, (4.25)
zs
zs − 1 = χ expαzs. (4.26)
A stationary stretching that is finite and linearly stable, corresponds to a stable
adhesion between membrane and cortex. The complete unbinding of the
membrane from the cortex corresponds to a vanishing density of links (or
equivalently, a diverging stretching (Eq. 4.25)). ). Equations Eq. (4.25) and
(4.26) do not have solutions in general for every value of the parameters χ
and α. The parameter space that defines the region of stationary solutions is
given by the condition α < α ∗ (Fig. 4.7a), with
χ =
1
α∗ exp(1 + α∗ . (4.27)
)
The space of stationary solutions ( Eq. (4.27)), that corresponds to a mem-
(a) 3
(b) 30
χ
2.5
2
1.5
1
0.5
SOLUTION
SPACE
NO SOLUTION
0
0 0.2 0.4 0.6 0.8 1
α
z
25
20
15
10
5
0
NO
SOLUTION
0.1 0.15 0.2 0.25 0.3 0.35 0.4
Fig. 4.7. (a) Parameter space for the existence of stationary solutions. Notice that the only relevant
parameters are the ratio of on and off rates χ = k 0 o f f /kon and the rescaled force per bond
α = f /(kBT /δ). (b) Stationary normalized stretching z as a function of the normalized force
α for χ = 0.9. Unstable solution (dotted red), stable solution (solid green) and the condition
for solution stability, z < (1 + α)/α (solid black).
brane bound to the cortex, may contain regions of stable or unstable solutions
depending on the value of the different parameters. The use of linear stability
analysis allows to understand the behavior of the stationary solutions under
small perturbations δρb ≡ ρb − ρbs and δz ≡ z − zs. The evolution of a linear
perturbation around the stationary solution of Eq. (4.22) and (4.23) is given
by δ ˙z
δ ˙ρb
= 1
− 1/ζ
zs α(1 − zs)
− z 2 s /ζ
− z 2
δz δz
≡ Λ , (4.28)
s δρb δρb
α

4.3 Membrane-cortex adhesion 71
The stability is obtained from the eigenvalues of Λ, or equivalently, its determinant
and trace . Since the trace Tr(Λ), is always negative, the stationary
solution is stable if the determinant is positive, det(Λ) > 0, or equivalently, if
1 + α(1 − zs) > 0. Using Eq. 4.26 we find that the condition that defines the
stable parameter space is exactly the same condition for solution existence
defined in Eq. 4.27 . Moreover, since there are always a pair of stationry
solutions, (z 1 s ,z 2 s ), the condition for positive determinant implies that within
this pair of stationary points, the smaller corresponds to the stable solution
whereas the larger to the unstable one, see Fig. 4.7b. In other words, the solution
that corresponds to the links being closer to the cortex and equivalently,
larger link density, is the stable solution, and ceases to exist for α > α ∗ (see
Eq. 4.27).
It is striking that the linear stability of the solution does not involve the
friction η, or in terms of rescaled quantities, ζ . Intuitively one should expect
that links are stable if within the time of a detachment and rebinding event,
the membrane barely move, otherwise, the stretching of the link becomes too
large to be sustained. In other words, one would expect that the stability of the
bonds is strongly dependent on the ratio between hydrodynamic relaxation
(how fast the membrane moves) and bond kinetics, which is the definition
of ζ . The reason for ζ not playing any role in the stability of the stationary
solutions, comes from the fact that at linear level, the stability is dictated by
the sign of the slope in the perturbations close to the stationary point (whether
the tendency is to go towards or away the stationary point), but not by its value
(in this case, given by ζ ). When the perturbation is strong enough, we need
to include non-linear terms in order to know the evolution towards the stable
or unstable branch. The non-linear analysis would depend on the values of
ζ since it determines whether the bond kinetics is fast enough to drive the
system back to equilibrium or, whether, if the membrane has moved too far
for the bonds to drive it back. The nonlinear effects results in a regime near
the unbinding transition that is nonlinearly unstable.
Our discussion reveals that membrane-cortex adhesion is stable for a
limited range of forces that depend only on the link kinetic parameters
(χ = k 0 o f f /kon), Fig. 4.7, and becomes unstable for a critical force per link,
that as a function of the critical rescaled force α ∗ , is
f ∗ b = kBT
δ α∗ , (4.29)
where α ∗ is the critical rescaled force given by Eq. 4.27. The origin of the
force f on the membrane is in general the pressure applied on the membrane,
which origin include external perturbations (the action of micropipette aspi-

72 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
ration or osmotic shocks) and active terms (myosin activity in the cytoskeleton).
From section 4.2.2, we know that the force per unit area that acts on the
bonds at equilibrium, ∆P − 2γ/R is proportional to the tension generated by
the presence of myosin in the cortex (Eq. 4.18), feq = 2γm/R, and the rescaled
force is
α = ∆P − 2 γm
R
δ
ρ0kBT
, (4.30)
where R is the radius of curvature. High internal pressure, a low membrane
tension or a high myosin activity can all contribute to membrane unbinding.
4.3.2 Micropipette experiments as membrane-cortex adhesion probes
Micropipette experiments allow to quantitative probe membrane-cortex adhesion.
The size of the detachment area is fixed by the radius of the pipette, and
the pressure in the micropipette is controlled. We use the model discussed in
the previous section, which treats properly the link kinetics, to interpret some
experimental measurements (Merkel et al., 2000).
During a micropipette experiment, a drop of pressure is applied on a region
of the cell (∼ R 2 p) and the force per unit area with respect to the equilibrium
state,
feq = 2 γm
R
γ
= ∆P − 2 , (4.31)
R
is increased by an amount Pe − Pp ≡ ∆P ′ , where Pe is the pressure of the external
cellular media and Pp is the pressure inside the micropipette. Assuming
the perturbation to be instantaneous 4 , and the cytoskeleton to be a rigid (non
deformable) surface, the total force per unit area acting on the links after the
micropipette suction is,
f = ∆P ′ + 2 γm
, (4.32)
R
where the second term in the right hand side is the pre-stressed state of the
cell, or equivalently, the equilibrium force per unit area.
Experimental results are often expressed in terms of the percentage of
cells for which the plasma membrane is detached from the cytoskeleton at
a given pressure (Merkel et al., 2000). We expect the membrane to abruptly
detach from the cortex above a critical pressure. These results vary significantly
with both Talin (a protein that links actin and membrane) and myosin
concentration, see Fig. 4.8.
4 In section 4.4 we will treat carefully the response of the cortex.

Influence of the link concentration
4.3 Membrane-cortex adhesion 73
The critical rescaled force per link α ∗ depends only on χ, the ratio between
on and off rates, which is an intrinsic property of the links (Eq. 4.27). For a
given cell type, we can assume that χ is fixed, and consequently α ∗ . In the
previous section we have seen that there is detachment for a critical force
per link. Recalling the relation between the critical rescaled force α ∗ and the
critical force per unit area f ∗ (Eq. 4.24),
f ∗ ∗ kBT
= ρ0α , (4.33)
δ
we realize that the attachment can be weakened either by decrease of the
available density of links (in the mentioned experiments (Merkel et al., 2000),
Talin), or by decreasing the cortex density. In both cases, the critical force for
unbinding should decrease linearly.
decreasing the available density of links, which can be obtained by decreasing
the total amount of actin associated proteins (in the mentioned experiments,
Talin (Merkel et al., 2000)), or decreasing the linking sites available
in the actin cortex (by weakening the cortex using latrunculin A or cytochalasin
D; or decreasing filamin concentration which increases the mesh size),
should decrease linearly the force needed to induce the instability.
In wild type amoeba (Merkel et al., 2000), Fig. 4.8a, there is a clear shift
of about ∼ 150−200 Pa between the pressure needed to unbind membrane at
the front and at the rear part of the cell, which corresponds to the more active
and the more stable part of the cell respectively. This shift can be attributed to
a different concentration of Talin between these two regions.This is supported
by experiments performed in a mutant lacking Talin, for which the threshold
pressure is the same in the rear and the front, and also exactly equal to the
front of the wild type cell, Fig. 4.8b.
In the case of a mutant lacking Myosin II, the values for the micropipette
critical pressure may differ from the wild type cell because the total force
per unit area is a combination of both pressure and myosin driven tension
(see paragraph below), but the shift in pressure between a myosin null cell
with Talin and without Talin should be of the same order as in the wild type
cell, according to our predicted linear relation between force and link density
(Eq. 4.33). In Fig. 4.8c-d, those results are reported obtaining a difference
between the front and the rear part of the cell of about 150 − 500 Pa, which
is of the same order as in the wild type cell. On the other hand, the absolute
value in critical micropipette pressures in the wild type and the null-myosin
cell differs by a factor of five. This increase in the stability of membranecortex
adhesion in the absence of Myosin can also be accounted with our

74 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
(a)
aspirated cells (%) aspirated cells (%)
suction pressure (Pa)
−Talin
(c)
(b) (d)
suction pressure (Pa)
aspirated cells (%)
aspirated cells (%)
−MII
suction pressure (Pa)
−MII
−Talin
suction pressure (Pa)
Fig. 4.8. Percentage of cells for which the membrane is unbound from the cytoskeleton in
the micropipette for wild-type (a), Talin null (b), Myosin null (c), and Myosin and Talin null
cells (d). Figure modified from (Merkel et al., 2000). The circles correspond to the cell front,
and triangles to the cell rear. Knockout of Talin leads to the same threshold pressure for the
cell front, but changes the cell rear to the same value as the cell front, indicating that Talin
concentrates on the cell rear. In the absence of myosin the pressure for unbinding is much
larger.
model considering the explicit relation between the force per unit area and
the micropipette pressure plus myosin driven cortical tension, as we explicitly
discuss in the following paragraph.
Effect of mysoin II activity
The effect of myosin activity in micropipette experiments leads to intriguing
results. As reported in (Merkel et al., 2000), the absolute value in critical micropipette
pressures in a wild type and a myosin null cell differs by a factor of
five. Consequently, the stability of membrane-cortex adhesion is enhanced by
reducing myosin activity, an effect that in (Merkel et al., 2000) is attributed
to the myosin induced softening of the cortex.
This effect can be explained very naturally by our model: given an available
density of links, ρ0, the critical force per unit area, f ∗ , is fixed by Eq.
4.33. Since the force is a combination of myosin driven cortical tension and

4.3 Membrane-cortex adhesion 75
micropipette pressure, (Eq. 4.32), we expect that a decrease of myosin activity
(which reduces the cortical tension) should lead to an increase of the
critical micropipette pressure,
∆P ′ ∗ kBT
= ρ0α
δ
− 2γm . (4.34)
R
Because of Myosin activity, in a wild-type cell, the cortical tension in the
cortex is different from zero, and the links are pre-stressed. As a consequence,
a weaker pressure suction is needed to unbind the membrane from the cortex,
which occurs for moderate pressures of about 200 − 300 Pa. Contrarily, in a
myosin-null cell, the cortical tension in the cortex vanishes and the links that
are not initially pre-stressed, leading to a tighter adhesion between membrane
and cortex. A larger micropipette pressure difference is needed in this case in
order to unbind the membrane from the cortex, see Fig. 4.9.
Comparing the critical pressures in both wild type and myosin-null cells
(Fig. 4.8 a-c or b-d), we can estimate the amount of myosin driven tension in
a cell,
γm = R
2 (∆P−MII − ∆P), (4.35)
where R stands for a radius between the cell and micropipette radius. The typical
amoeba radius in the reported experiments is of about 10 µm (Merkel
et al., 2000), resulting in a myosin tension of about γm ∼ 5×10 −3 N/m, which
is at least two orders of magnitude of the typical membrane tension of a vesicle,
contributing to the 80% of the force needed to unbind the membrane from
the cortex in non pre-stressed links, or in terms of aspiration pressure, ∼ 800
Pa from the ∼ 1000 Pa needed to detach the membrane. Myosin driven tension
is thus the main force opposing the outward pressure from the cell and
giving shape to the cells, the membrane acting only to transmit the internal
pressure to the cortex.
A slightly increase of myosin activity, in particular a 25% increase in
myosin driven contraction, which in terms of aspiration pressure represents
∼ 200 Pa, leads to instability (∼ 1000 Pa, see Fig. 4.8). As a consequence,
amoebas are close to the instability threshold in normal conditions.
The advantage of being close to the instability threshold using myosin
contraction, that accounts for the 80% of the critical force, could be related
with the ability of the cell to fine tune its response under external stimuli
(food or undesired environmental conditions) initiating cell movement by a
localized influx of calcium through cell membrane. Asymmetric distribution
of Talin is another possible way to drive direct motility of cells.

76 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
Using the myosin-null experiments, Fig. 4.8c, in which case the micropipette
pressure is directly related with the available density of links (Eq.
4.34) we can estimate the relative concentration of Talin ρt with respect to
the total linker concentration ρ0,
ρt
ρ0
= ∆P−MII − ∆P −MII,T
∆P −MII , (4.36)
which is of about 10 − 30% of the saturation density ρ0. Assuming the saturation
density to be limited by the mesh size, ρ0 ∼ 1/ξ 2 ∼ 100 links/µm 2 ,
Talin density should be of about 10 links/µm 2 . This small density of Talin
links seems to be enough to drive direct motion in amoebas by its asymmetric
distribution. It remains to be unveiled how those proteins are distributed
unevenly within the membrane.
We have not estimated yet the values for α ∗ or χ. Using a value of 100
links/µm 2 for the saturation density of links and a link length of the order of
nanometers, we find that
α ∗ = ∆P−MII δ
, (4.37)
kBT ρ0
is of about 4. In other words, the critical force per link (Eq. 4.29) is four times
the thermal force of the link, and is of the order of 16 pN. From the estimates
of α ∗ , we can use Eq. 4.27 to evaluate the ratio of on and off rates, χ ∼ 10 −3 .
We have seen that understanding membrane-cortex adhesion require a
proper account of myosin activity and link density.How this microscopic kinetic
description links with a continuum coarse grained description by means
of an adhesion energy is addressed in the following paragraph. Many experiments
claim to measure adhesion energy as if it was a constitutive parameter
of the cell. We have shown that this is not the case, and adhesion depends on
the state of the cell, in particular on myosin activity. We will give an explicit
dependence in terms of myosin tension and link density, in the framework of
a rigid cortex 5 .
Adhesion energy
Small interlink distances (∼ 100 nm) compared to a typical micropipette radius
of few µm, suggests a continuum energetic description for the membranecortex
adhesion (Sheetz, 2001; Merkel et al., 2000; Brochard-Wyart et al.,
5 Considering the cortex as a rigid surface leads to an over estimation of the adhesion energy
but qualitatively predicts the same behavior.

∆Pp(Pa)
1800
1600
1400
1200
1000
800
600
400
200
15 Pa µm 2
4.3 Membrane-cortex adhesion 77
0
0 20 40 60 80
ρt =0
100 120
ρ0(1/µm 2 )
1/ξ 2
∆P −MII
p
∆P −MII,T
p
∆Pp
∆P −T
p
Fig. 4.9. Theoretical predictions for the critical aspiration pressure in a micropipette experiment
as a function of the available density of links ρ0 according to Eq. 4.34. In solid black
and red, a cell containing and lacking myosin II respectively. The slope is the same in both
cases and equal to 15 Pa µm 2 (see Eq. 4.34). The red dashed line shows the case for Talin null
mutants (as reported in (Merkel et al., 2000), whereas black dashed line corresponds to a cell
with an available link density of order one linker per mesh size.
2006). The value of this adhesion energy is considered as a parameter accessible
through micropipette or tether extrusion experiments. However, the
value of the adhesion energy should in general be a function of the density of
links, which in turn depend on the state of the cell. As a consequence, the relationship
between pressure applied (in the case of a micropipette aspiration)
and the adhesion energy is far from being straight forward.
In our kinetic model, the work needed to detach the membrane is the external
force applied on the membrane times the displacement before unbinding,
which is given by the difference of the critical stretching x ∗ and the initial
equilibrium stretching xeq, see Fig. 4.10,
w = f ∗ (x ∗ − xeq), (4.38)
where we assume that x ∗ −xeq is very small. In other words, we need to bring
the membrane to a distance from the cortex that corresponds to the critical
stretching (see Fig. 4.7b). As in the case of the critical aspiration pressure
in a micropipette experiment, the adhesion energy will depend on the level
of stress in the cytoskeleton, which determine the mean number of attached
links to be broken, and consequently, it will depend on the available density

78 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
of links ρ0 and on the myosin driven cortical tension γm. In what follows
we will derive the expression of the adhesion energy and we will obtain a
non linear relation with the link density and the myosin tension, contrarily
to the intuitive linear dependence on the available density of links. As an
example, we will consider the adhesion energies that would correspond to the
experiments done by (Merkel et al., 2000) in wild type and mutant amoebas
lacking myosin II or Talin.
f
δx
xeq
Fig. 4.10. Schematics of the work needed to unbind the membrane. Initially, the cell is at
equilibrium and the stretching is given by xeq. In order to unbind the membrane, we need to
apply a force that drives the links towards the critical stretching x ∗ .
In terms of rescaled quantities (z = xkρ0/ f , ¯ρ0 = ρ0ξ 2 and α = f δ/(kBT ρ0)),
we can express the adhesion per unit area (Eq. 4.38) as
w = w0 ¯ρ0 1 − zeqαeq
z∗α ∗
, (4.39)
where w0 ≡ (kBT /δ) 2 α ∗ (1 + α ∗ )/(kξ 2 ) is the maximum adhesion energy
that corresponds to non pre-stressed links in the cortex (no myosin activity),
and, zeq and αeq are the equilibrium rescaled stretching and force respectively.
At equilibrium, the rescaled force depends on the myosin driven cortical tension
which sets the equilibrium of the cell and the force per link by Eq. (4.32),
feq = 2γm/(Rρ0),
x ∗
αeq = 2γm δ
, (4.40)
ρ0R kBT
which using the values obtained in the previous paragraph for myosin tension
γm ∼ 5.10 −3 Nm −1 , leads to a rescaled force of about 3, where we have
assumed the available density of links to be ∼ 1/ξ 2 . In the case of the mutant
lacking Talin, the link density is reduced by a 10 − 30%, and αeq ∼ 4.

4.3 Membrane-cortex adhesion 79
The equilibrium rescaled stretching is related with the equilibrium rescaled
force by Eq. 4.26. Finally, the critical rescaled force and stretching are given
respectively by Eq. (4.26) and (4.27).
2 w
ρ0ξ
w0
1
0.8
0.6
0.4
0.2
-Talin
0
0 1 2 3 4
αeq
-MII
-T, -MII
Wild type
Fig. 4.11. Characterization of the effective binding energy. Reducing the link density (-T) has
two effects: decrease the global factor in the adhesion energy that is proportional to the link
density, but also increase the pre-stressed state of the cell. Myosin perturbations (-MII) set the
value of the binding energy in the curve.
The expression for the adhesion energy per unit area, Eq. (4.39), contains
the intuitive linear dependence with the saturation number of links, w ∼ w0 ¯ρ0,
which only depends on intrinsic properties of the links, namely the on and off
rates, and the stretching constant of the links; and a correction that takes into
account the pre-stressed state of the cell. As the link density is reduced, not
only the adhesion energy decreases linearly, but also the level of pre-stress on
each link increases, thus reducing significantly the adhesion energy, Fig. 4.11.
The linear dependence in the link density of the adhesion energy w0 ¯ρ0 should
only be observed in the absence of myosin II. As a consequence, it does not
make sense to infer from a particular experiment an absolute adhesion energy
for a certain cell, since the particular measured value will depend in general
on the state of the cell which is in most cases unknown. In the experiments
done in wild type and mutant amoebas by (Merkel et al., 2000), we expect
four different values for the adhesion energy for the wild type and the three
different mutants. Using for the link stiffness k ∼ 10 −2 pN nm −1 (Evans,
-T

80 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
2001), the maximum (bare) adhesion energy w0 is about 3 mJ m −2 , and this
is what would be measured in a mutant lacking myosin II, i.e., an equilibrium
state which is not pre-stressed. For a mutant lacking Talin and myosin II
(αeq = 0), the adhesion energy is reduced by a 10 − 30%. For a wild type cell
and a mutant lacking Talin (αeq ∼ 3 and 4 respectively) the adhesion energies
are reduced by a 60% and 80% respectively, see Fig. 4.11 and Table. 4.1. The
dramatic increase in the adhesion energy for a cell lacking myosin activity
but with the same density of links available, which can be of the order of
the 250%, illustrates the importance of the equilibrium pre-stressed state of a
cell in the experimental measurements of adhesion energies and detachment
pressures.
Cell type w (mJ m −2 )
Wild type 1.2
Talin-null 0.6
Myosin II-null 3
Myosin II and Talin-null 2.4
Table 4.1. Table for the predicted adhesion energies w for wild type and mutants lacking
myosin or Talin, corresponding to the experiments by (Merkel et al., 2000).
So far we have considered a description for the membrane-cortex adhesion
that describes properly the dynamics of the linkers. For the sake of simplicity
we have considered the cortex as a flat rigid surface, which yields to an
overestimation of the adhesion energy and the threshold for membrane unbinding.
The effect of the cortex viscoelasticity will be explicitly considered
in forthcoming sections. Additionally, membrane unbinding usually involves
a characteristic nucleation length scale, as in the case of blebbing cells (Charras
et al., 2006; Charras et al., 2008), which our present mean field description
considering no spatial information is unable to describe. Adding lateral correlation
in the model is straightforward conceptually (Garrivier et al., 2002) but
involves extra nonlinearities which do not allow analytical treatment. Instead,
in the next section we propose a coarse grained description based on energy
considerations (Sheetz et al., 2006; Charras et al., 2008).
4.3.3 Energetic description of a bleb
The previous description of membrane-cortex adhesion assumes no lateral
correlation between the links, which corresponds to small deformations of
the membrane. When a large enough patch of membrane detaches (Sb ≫ ξ 2 ),

4.3 Membrane-cortex adhesion 81
see Fig. 4.12, there is lateral correlation between links, and deformation satisfies
a force balance that involves pressure, membrane bending and membrane
tension must be considered. In this case, a macroscopic energetic approach
that takes in account the bleb shape is justified for nucleation sizes larger
than the mesh size ξ . The notions of link density and on and off rates are replaced
by an effective adhesion energy density w which is the energetic cost
per unit area of detaching membrane from the cytoskeleton 6 .
(a) (b) (c)
Vb
u
Sb
Fig. 4.12. Nucleation of a bleb. (a) The detachment of a large enough area Sb can lead to the
formation of a bleb (b) or an eventually resealing of the membrane.
Assuming for now that the pressure and the membrane tension are constant,
and taking into account that the curvature contribution to the energy is
negligible compared to the tension contribution (κ/(γRb) ∼ 10 −8 ), the cost
of energy of forming a bleb is simply
∆E = −∆PVb + γ∆S + wSb, (4.41)
where ∆S is the increase of membrane area when the bleb is formed. In the
linear regime (u ≪ Rb), (see Fig. 4.12b) , the volume and the excess area
are Vb ∼ S 2 b /R and ∆S ∼ S2 b /R2 respetivelly. Minimization of the energy, Eq.
4.41, with respect to the radius of the bleb leads to Laplace law, Rb = 2γ/∆P,
and the energy in terms of the bleb detached area becomes
∆E = wSb − ∆P2
4γ S2 b. (4.42)
For small detached area, the energy increases linearly with the area, whereas
for large bleb detached area, the quadratic term dominates and the energy
decreases monotonously, Fig. 4.13. Consequently, there is an energy barrier
∆E ∗ corresponding to the nucleation of a bleb of detached area S∗ b given by
6 The adhesion energy w has to be thought of depending of the pressure and cell activity, as
we have seen in the previous section. However here

82 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
∆E
∼ wSb
S ∗ b = 2 wγ
,
∆P2 (4.43)
∆E ∗ = w2γ .
∆P2 (4.44)
∆E ∗ = w2 γ
∆P 2
S ∗ b =2 wγ
∆P 2
2 ∆P
∼−
4γ S2 b
Fig. 4.13. Energy profile of a bleb in terms of detached area Sb. For small bleb detached area
(Sb ≪ γw/∆P 2 ), the energy grows linearly with the adhesion energy. As the bleb surface becomes
larger, the gain of energy driven by the pressure dominates over the adhesion energy and
the energy decreases monotonously. The maximum of energy, ∆E ∗ ∼ w 2 γ/∆P 2 , corresponds
to an energy barrier to nucleate a bleb and the detached area, S ∗ b = 2wγ/∆P2 , corresponds to
the critical area above which a bleb will nucleate.
Above the critical detached surface S∗ b , forming a bleb is energetically
favorable, whereas for smaller surfaces a bleb will not form. Associated
to the energetic barrier ∆E ∗ , there is a characteristic time for bleb nucleation
τn. The waiting time for a fluctuation to overcome a barrier grows exponentially
with the energy according to Kramers theory (Kramers, 1940),
τn ∼ exp(∆E ∗ /kBT ), with a pre-factor that depends on the unbinding properties
of the links ,
τn = τ 0
w2γ n exp
∆P2 1
ET
Sb
, (4.45)
where ET is an energy scale that can be related with local pressure, thermal,
or active fluctuations. The nucleation is an indication of the likelihood of
bleb formation. When the argument of the exponential in Eq. 4.45 is smaller
than one, a bleb is likely to occur, whereas for larger values the waiting time

4.3 Membrane-cortex adhesion 83
diverges and a bleb event occurs very unlikely. The nucleation of a bleb is
in principle a very complicated process, and we do not aim to give an exact
derivation, but an order of magnitude for the critical pressure,
∆P ∗
w2γ ∼
ET
1/2
. (4.46)
According to Eq. 4.42, once the bleb is nucleated, it grows indefinitely.
This is because we have considered a bleb as a small perturbation of the
cell decoupling both cell and bleb energies and considering the former as a
constant. Stabilization of the bleb can result from an increase of membrane
tension or a decrease of internal pressure as the bleb grows.
Mean number of blebs in a cell
Using the concept of bleb nucleation time, we can write a qualitatively coarse
grained description for the mean number of blebs in a cell. Blebs nucleate at
an average rate given by the inverse of the nucleation time τn. At the same
time, blebs have a life time, which includes a fast inflation and slower retraction
τb, Fig. 4.14. The equation governing the mean number of blebs is given
by
˙
Nb = Nt − Nb
τn
− Nb
, (4.47)
τb
where Nt represents the maximum number of blebs in the cell, corresponding
to the number of blebs that would occupy all the cell area.
Fig. 4.14. Although blebs are highly dynamical, one can define an average number of blebs as
a balance between bleb nucleation and bleb retraction.

84 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
At steady state, the mean number of blebs is constant and is given by
Nb = τb
Nt. (4.48)
τn + τb
In general, the life time of a bleb τb, depends on the volume of the mature
bleb. We consider the life time of the bleb to be dominated by the retraction
process which is controlled by the growth and contraction of a new actomyosin
cortex. The contraction time is limited by myosin activity, and we
consider it to be constant and of the order of minutes(Charras et al., 2008).
As we have shown previously, the nucleation time τn depends exponentially
on the energy cost to create a bleb, τn = τ0 n exp w2γ/(∆P 2ET ) . The presence
of blebs should affect the mechanical state of the cell, both pressure
and its tension, ∆P(Nb), γ(Nb). In the former case, a bleb reduces the drop
of pressure when the cytosol and small vesicles flow through the cell cortex
towards the bleb. Big organelles and vesicles remain in the interior of the cell
an the osmotic pressure in the bleb decreases. In the case of the membrane
tension, the increase of area of the cell, which is linear with the number of
blebs, reduces membrane fluctuations and eventually stretching of the membrane,
which causes an increase of the membrane tension (see section 4.2.1).
Linearly we can express both effects as
′
∆P = ∆P0 1 − β Nb , (4.49)
γ = γ0(1 + β ′′ Nb), (4.50)
where ∆P0 and γ0 are the pressure and tension the cell would have in the
absence of blebs .Both the drop of pressure and increase of membrane tension
tend to reduce the probability of forming more blebs,
τn ∼ τ 0
w2γ0 n exp
∆P2 0
1 + βNb
ET
, (4.51)
where β = 2β ′ + β ′′ . Plugging this equation to Eq. 4.48, we obtain a selfconsistent
expression for the number of blebs, or equivalently, the pressure or
tension in the cell,
Nb = Nt
1 + τ0
n w2γ0 exp
τb ∆P2 0
1 + βNb
ET
−1
. (4.52)
This equation relates the number of blebs with the value of the pressure
and tension that the cell would have in the absence of blebs, which can be
seen as a control parameter that can be varied by external (such as osmotic
shock) and internal (such as myosin activity) stress.

4.3 Membrane-cortex adhesion 85
Below we arbitrary focus on the pressure variation and keep the membrane
tension constant. Considering the variation of the membrane tension could
be easily applied and leads to the same qualitatively results. We can solve
Eq. 4.52 as a function of the normalized pressure in the absence of blebs,
∆P0/ w2γ0/ET ) 1/2 , which we use as a control parameter, Fig. 4.15. For
small values of the control parameter, blebs are not formed. The transition
from a non-blebbing cell to a blebbing cell is given by Nb ∼ 1. After blebs
start to form, the pressure remains nearly constant because blebs buffer the
cell internal pressure, even if the pressure the cell would have in the absence
of blebs is increased (see Fig. 4.15),
∆Pmax ∼
0.8
0.6
0.4
0.2
w 2 γ0
ET
1
log(τ0 1/2
. (4.53)
n /τb)
(a) (b) (c)
∆P/ w 2 γ0/kBT 1/2
0.0
0.0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
∆P0/ w 2 γ0/kBT 1/2
Fig. 4.15. Cell pressure and number of blebs as a function of the normalized pressure in the
absence of blebs (b). Initially, no blebs are present in the cell and the cell pressure is sensitive
to external perturbations, ∆P ∼ ∆P0. Above a pressure threshold (dotted line), blebs start to
form and the pressure saturates to a constant value as long as new blebs can be formed. In
(a), an Entamoeba histolytica with a single bleb corresponding to the region near the pressure
threshold. In (c), M2 cells from (Charras et al., 2006), corresponding to the region of many
blebs.
Blebs can be seen in non healthy cells, possibly because of weak cytoskeleton
adhesion, but can also serve physiological purposes such as motility
(Charras and Paluch, 2008; Coudrier et al., 2005). In this case, blebbing
produces a diffusion like motion (Coudrier et al., 2005).
In this chapter, we have considered both kinetic and energetic properties of
the membrane-cortex adhesion. Any visco-elastic property of the cortex has
been neglected for simplicity. In the following chapter, we use a micropipette
set up both experimentally and theoretically to study the reaction of a cell to
a mechanical perturbation.
0.8
0.6
0.4
0.2
Nb/Nt

86 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
4.4 Dynamical response of a cell to a controlled mechanical
perturbation using a micropipette set up
In section 4.3 we analyzed the stability of the membrane-cortex adhesion using
both a simple kinetic model for the protein linkers and a coarse grained
energetic description. We have deliberately neglected the viscoelastic properties
of the cortex in order to simplify the analysis and focus only on the
instability. In this section we study the reactivity of the cytoskeletal cortex
to dynamical perturbations, which depends both on cortex viscoelasticity and
actin polymerization.
Generically, the cytoskeletal cortex behaves as a viscoelastic material,
with an elastic response at short time , and a fluid response at longer time.
Under micropipette aspiration or any other pressure perturbation such as an
osmotic shock, pressure builds up a transient elastic stress on the cortex that
eventually relaxes after a time τ ∗ (see Eq. 4.5). During the elastic deformation,
proteins that link the membrane to the cortex are put under tension. The
elastic stress built in the cortex may eventually reach the critical force per link
at which the cortex-membrane adhesion becomes unstable (see section 4.3).
The section is organized as follows: First, we consider the case of an instantaneously
applied pressure in the micropipette. This case illustrates the
effect of the elastic deformability of the cortex on the force experienced by the
linkers. Second, we consider a pressure that is applied at a constant rate. The
comparison between pressure loading rate and cortex relaxation rate fixes a
maximum transient force experienced by the linkers, which strongly depends
on the loading rate. As a consequence, there is not an absolute value of micropipette
pressure at which the membrane-cortex adhesion become unstable
but a continuos range of pressures depending on the time scale to build up
pressure in the micropipette.
4.4.1 Micropipette experiments as membrane tension and adhesion
probes
There are several techniques that may serve as a direct measure of membrane
tension, membrane tether extraction, which consists of pulling a membrane
tether out of the cell membrane using optical tweezers (Hochmuth et al.,
1996), micropipette aspiration (Evans and Rawicz, 1990), and analysis of
fluctuation spectrum (Pécréaux et al., 2004), to cite some examples.
Micropipette aspiration is able to both perturb and measure the response
to a perturbation, while using a membrane tether allows only to measure
the membrane tension. For practical reasons, micropipette aspiration (Fig.

4.4 Dynamical response of a cell to a controlled mechanical perturbation using a micropipette set up 87
4.16) is more convenient to measure the membrane tension area dependence
and to be able to observe the cross-over between the entropic and elastic
regime (Evans and Rawicz, 1990) (see Eq. 4.3), because the area drown in
the micropipette is a sizable fraction of the total area of the cell (typical micropipette
radii are of about 5-10 µm). We first review some of the results of
micropipette aspiration experiments in vesicles and in cells.
Vesicles
Experiments on vesicles have been extensibly done for many years (Evans
and Rawicz, 1990; Merkel et al., 2000; Rentsch and Keller, 2000; Rentsch
and Keller, 2000; Sit et al., 1997). Initially the vesicle is at equilibrium and
force balance (Laplace law) is satisfied,
Pc − P0 = 2 γ
, (4.54)
R
where Pc and P0 are the internal and external pressures respectively (see Fig.
4.16). When a drop of pressure is applied locally inside the micropipette,
the vesicle deforms, its radius of curvature increases, and eventually a new
equilibrium is reached, Fig. 4.16,
1
∆P = 2γ
Rp
− 1
, (4.55)
R(L)
where Rp and R are the micropipette and vesicle radii, L is the length of the
tongue inside the micropipette, and ∆P ≡ P0 − Pp is the difference between
the external pressure and the pressure inside the micropipette (see Fig. 4.16).
Notice that equation (4.55) only has a stable solution if the membrane tension
γ increases with the tongue length L (or area), since as the radius of the vesicle
R decreases by volume conservation as the vesicle is being sucked, so does
the force opposing force suction, which is the constant micropipette curvature
(1/Rp) minus the vesicle curvature (1/R).
The membrane tension does increase with the vesicle area (Eq. 4.3), and
an equilibrium shape for a given tongue length can always be obtained until
the radius of the cell reaches the radius of the pipette for which the vesicle is
completely swallowed. Using equation (4.55), which determines the tension
needed for the new equilibrium, and the relationship between the increase of
tongue length and the areal strain α (Evans and Rawicz, 1990),
α ∼ 1
2
Rp 2
3
Rp ∆L
−
, (4.56)
R R
Rp

88 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
R0
Pc
R(L)
P0
Rp
L
Pp
Fig. 4.16. Schematics of a micropipette experiment on a vesicle.
we can measure experimentally the relationship between the tension and the
areal strain (Eq. 4.3). Micropipette experiments have been used to show the
cross-over between a regime where the tension is dominated by thermal fluctuations
(entropic) and a regime where the tension is dominated by direct
stretching of lipids (elastic) (Evans and Rawicz, 1990) (see Eq. 4.3).
In vesicles, the new equilibrium is reached by the balance of the micropipette
pressure and the area dependent tension (Eq. 4.55). In a cell, the
cytoskeletal cortex also contributes to the force balance of the cell (see section
4.2.2 and Eq. 4.18), and any elastic deformation contributes to balancing
the pressure. The stationary force balance equivalent to the case of the vesicle,
Eq. 4.55, is
1
∆P = 2(γ(L)+γm) −
Rp
1
. (4.57)
R(L)
We are interested in the way the cell reaches this new equilibrium. There are
two possible scenarios taking into account the cortex-membrane stability:
(i) The membrane-cortex complex deforms continuously until the new equilibrium
is reached, Fig. 4.17a→b.
(ii) The membrane-cortex deforms until the critical force on the linkers is
reached and the membrane unbinds from the cortex. At this stage, the
bare membrane reaches a different equilibrium with a larger tongue length
resulting either from the increase of membrane tension, Fig. 4.17a→c
or from the formation of a new cortex which stops the membrane, Fig.
4.17a→e. In the former case, after a characteristic polymerization time, a
new cortex is finally formed under the bare membrane. In the presence of
myosin, contractile tension is built in the cortex which presumably brings
the membrane-cortex to the same final equilibrium than in the previous
process, Fig. 4.17c→d and 4.17f→b.

4.4 Dynamical response of a cell to a controlled mechanical perturbation using a micropipette set up 89
(c)
(d)
(a)
(b)
Fig. 4.17. In a micropipette experiment, the cell reaches the new equilibrium through three
different scenarios. The initial state of the cell corresponds to a the membrane attached to the
micropipette under moderate micropipette suction. When the suction increases, the cell deforms
and a membrane-cortex tongue advances towards the interior of the micropipette. If the
transient elastic tension in the cortex due to the deformation is smaller than the critical tension
for membrane-cortex instability, the cell reaches the new equilibrium deforming continuously
from (a) to (b). If the tension in the cortex exceeds a critical value, the membrane detaches
from the cortex. A new transient equilibrium is reached either by increase of membrane tension
(c) or polymerization of a new cortex (e). In the former case, a new cortex eventually
repolymerizes underneath the bare membrane, and myosin driven contraction drives the cell
to the final equilibrium, (c) to (d), which corresponds to the final configuration (b). In the second
case, as the new cortex is polymerized, myosin contraction drives the cell to the final state
(f) which coincides with the final state (b).
These scenarios differ on the transient elastic tension built on the cortex,
which results from the elongation of the tongue inside the micropipette, and
the way the membrane is transiently stopped. In order to quantitatively describe
these regimes we need to both consider the maximum elastic stress
stored in the cortex during the aspiration and compare the force per link
with the force threshold (Eq. 4.29) obtained in membrane-cortex adhesion
analysis, section 4.3, and consider the conditions for which the membrane is
stopped either by a new cortex or by the increase of membrane.
In our analysis, we will consider that myosin driven tension in the cortex
is constant 7 , and thus can be obtained by the initial equilibrium before
aspiration, Fig 4.17a,
L
7 Keeping the myosin tension constant assumes that the thickness of the gel is roughly constant
during the aspiration process, and second, myosin adapts fast to the new cytoskeletal
Rp
R0
R
(e)
(f)

90 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
1
∆P(0)=2(γm + γ(0)) −
Rp
1
,
R0
(4.58)
where ∆P(0) is the pressure of a reference state, where the membrane is
attached to the micropipette.
When the micropipette pressure ∆P is decreased, a tongue starts to advance
at a velocity that is dictated by viscous dissipation. The dominant term
in the dissipation (see Appendix 4.A) should results from membrane flow
through links to the cortex, and leads to an effective viscosity that increases
linearly with the length of the tongue, η(L) ∼ µbRp(Rp + L)/ξ 2 , where µb
is a two dimensional viscosity for the bilayer of about 3 × 10−6 Pa m s
(Hochmuth et al., 1982), and ξ is the mean distance between links which is
roughly the mesh size of the cortex. The dynamical force balance including
the dissipative force is given by
µb
Rpξ 2 (L + Rp)˙L
1
= ∆P − 2(γ(L)+γc + γm)
Rp
− 1
. (4.59)
R(L)
The radius of the cell is related with the increase of tongue length assuming
volume conservation, R(L) ∼ R0(1 − R2 p∆L/(4R3 0 )). For typical experimental
conditions, R0 ∼ 20µm, Rp ∼ 5µm, and the increase of tongue
length is a few Rp. As a consequence, the cell curvature can be considered as
constant R ≈ R0. The membrane tension depends generically on the tongue
length through the areal strain (see section 4.4.1), α ∼ 1/2∆L/Rp((Rp/R) 2 −
(Rp/R) 3 ). For our experimental values the areal strain is of about 0.02∆L/Rp (Brugues
et al., 2008b). We know from experiments (Evans and Rawicz, 1990) that
the crossover from the entropic to the elastic regime in vesicles occurs for
γ ∼ 5 × 10−4 N/m, which corresponds to α ∼ 2-5%, so for an increase of the
membrane tongue of about two micropipette radii, we would start to stretch
directly the lipids of the membrane in a vesicle and the tension would vary
linearly with the areal strain (see Eq. 4.3 in section 4.2.1). In a cell, we do not
know the amount of available membrane, and the dependence of the tension
in the areal strain does not need to coincide with the case of a vesicle. However,
we will assume linear response, with a stretching modulus ¯Ka which is
expected to be smaller than the actual lipid stretching modulus,
γ ∼ γ0 + ¯Ka
Rp 2
3
Rp ∆L
−
≡ γ0 + ζ∆L, (4.60)
2 R R Rp
area and its density is not affected by the elastic stress. These restrictions could be easily
relaxed, but the main results remain qualitatively the same.

4.4 Dynamical response of a cell to a controlled mechanical perturbation using a micropipette set up 91
where ζ is the increase of tension per unit length of about 450 N m−2 . For
pressures of about 1000 Pa, the membrane tension balances the pressure for
lengths between two and three times the radius of the micropipette (Merkel
et al., 2000; Rentsch and Keller, 2000; Brugues et al., 2008b), and ζ should be
of the order of 450 N m−2 , which roughly corresponds to direct stretching of
the lipids. We consider the dissipation to be independent of the micropipette
length, and write Eq. 4.59 as
µb ˙
1
∆L = ∆Pp − 2(ζ∆L + γc)
ξ 2
Rp
− 1
, (4.61)
R
where now ∆Pp ≡ Pp − Pp(0) is the difference of pressure between the reference
pressure and the final applied pressure, and we have used Eq. (4.58)
for the initial force balance. Assuming for simplicity that the cortex do not
propagate inside the cell, since that would involve shear stress, the strain rate
in the micropipette is simply ˙L/L. The time evolution of the cortical tension
is given by the maxwell like equation
1 d
+
τ∗ dt
γc = Eh ˙L
. (4.62)
L
The equations 4.60 and 4.62 will be solved numerically below. The most important
feature is that the stress reaches a maximum connected to the viscous
relaxation at large time. We will give analytical approximations for this maximum
in the elastic and viscous limits.
4.4.2 Micropipette static suction pressure
For an instantaneous applied pressure, the maximum stress occurs in the elastic
regime and is relaxed after a time τ∗ . The solution of Eq. 4.62 in the elastic
regime gives
L
γc ∼ Ehlog , (4.63)
L0
where L0 ≡ L(0) ∼ Rp. We have seen that for tongue lengths of about two
or three times the radius of the micropipette, the membrane tension is large
enough to balance micropipette pressures of about 1000 Pa, which is the typical
order of magnitude of pressure needed to induce membrane-cortex detachment.
In this approximate treatment, we linearize the expression of the
cortical tension, γc ∼ Eh∆L/Rp, and the equation of motion for the increase
of tongue Eq. 4.61, reads

92 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
˜η ∆L ˙ = ∆Pp − 2 ζ + Eh
∆L
,
Rp ˜Rp
(4.64)
where 1/ ˜Rp ≡ (1/Rp − 1/R), and ˜η ≡ µb/ξ 2 . The solution for the increase
of tongue length is
∆L(t)=
∆PRp ˜Rp
1 − exp
2(ζ Rp + Eh)
− 2(Rpζ + Eh)
˜ηRp ˜Rp
t
. (4.65)
The maximum stress stored in the cortex is then given by γc(τ ∗ )=Ehlog(1+
∆L(τ ∗ )/Rp). Although the pressure in the micropipette is set instantaneously,
the flow of membrane and cortex is not instantaneous, but fixed by dissipation.
As a consequence, the rate of increase of cortex tension, and its maximum
value, are limited by this dissipation.
4.4.3 Effect of the loading rate
Experimentally it is never possible to set an instantaneous pressure. Instead,
a characteristic time scale to build the final value is required. This time scale
may affect dramatically the response of the cortex to the pressure, and consequently,
the force on the links and the probability of membrane unbinding.
Applying the same final pressure with two different rates may result in a
membrane-cortex that is stable if the loading rate is small and unstable if the
loading rate is larger.
We model the pressure ramp as an exponential relaxation to the final value
which interpolates a linear ramp plus a saturation towards the final pressure,
∆Pp(t)=∆P∞ 1 − e −βt
, (4.66)
where β is the characteristic pressure rate. Depending on the value of β, we
expect the cortex to behave as a solid or as a fluid. For βτ ∗ ≫ 1, the final
pressure is reached instantaneously compared to the relaxation of the cortex,
and we can use the results obtained above, Eq. 4.65. For βτ ∗ ≪ 1, the cortex
reacts as a viscous fluid,
γc ∼ τ ∗ Eh˙L/L, (4.67)
and the equation of motion for the tongue length in the micropipette reads,
˜η ∆L ˙ = ∆P∞ 1 − e −βt
− 2 ζ∆L + τ ∗ Eh ˙L
1
. (4.68)
L ˜Rp

4.4 Dynamical response of a cell to a controlled mechanical perturbation using a micropipette set up 93
Linearizing the expression of the viscous tension γc, and defining an effective
viscosity ηv ≡ ˜η + 2τ ∗ Eh/(Rp ˜Rp) ∼ 10 8 Pa s m −1 as the sum of the
membrane and cortex viscosities, we can solve the increase of tongue length
in the micropipette, Eq. 4.68
∆L(t)= ˜Rp∆P∞
2ζ
1
1 − β/β ζ
1 − e −βt + β
e
βζ −β
ζ t
− 1
, (4.69)
where β ζ ≡ 2ζ /( ˜Rpηv) ∼ 10 s −1 . Since τ ∗ ∼ 10 s, and we are in the regime
βτ ∗ ≪ 1, we have that β ≪ β ζ and the tongue length is roughly
∆L(t) ∼ ˜Rp∆P∞
2ζ
βt + β
e
βζ −β
ζ t
− 1
. (4.70)
The maximum stress is obtained imposing ˙γc = 0 in Eq. 4.67. The time at
which the stress reaches its maximum is given by,
with c ≡ 1 +
tmax = ˜Rpηv
2ζτ∗ −c
−c − ProductLog(−e ) , (4.71)
4Rpζ 2
β ˜R 2 pηv∆P∞ . The maximum stress is given by γc(tmax), Fig. 4.18.
In order to obtain the detachment pressure on the micropipette as a function
of the loading rate, we equate the maximal critical force per link ξ 2 (γm +
γc(max))/Rp, to the value obtained in section 4.3, 16 pN. In Fig. 4.19, we
plot the critical pressure as a function of the loading rate.
If the critical force per link is reached, the membrane detaches from the
cortex. After a transient membrane flow, the tongue eventually stops. Two
possible mechanisms may contribute to stop the membrane tongue, increase
of tension (see Fig. 4.17c) and repolymerization of new cortex (see Fig. 4.17e.
In the following section we discuss both contributions.
4.4.4 Can actin stop a moving membrane?
There are mainly two mechanisms that may contribute to stopping the membrane
tongue. First, as the tip grows the membrane tension increases with
the total apparent membrane area in the pipette (see Eq. 4.60 and section
4.2.1), and a force balance is reached that only involves membrane tension.
Second, actin polymerization underneath the membrane is promoted by nucleators
(Charras et al., 2006), and eventually, a new cytoskeletal cortex may
be formed under the moving membrane, which stress contributes to balancing

94 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
γmax/γ ∞ max
1
0.8
0.6
0.4
0.2
γc (KPa µm)
0.3
0.25
0.2
0.15
0.1
0.05
0
0 10 20 30 40 50 60
t (sec)
0
0 20 40 60 80 100
βτ ∗
Fig. 4.18. Maximum tension stored in the cortex normalized to the tension under infinite loading
rate, as a function of the loading rate times the relaxation time, for a final pressure of 1 KPa
and for τ∗ = 15 sec (blue), 50 sec (red) and 100 sec (green). The maximum tension saturates
as a result of membrane dissipation, ∆L ˙ ∼ ∆Pf / ˜η. In the inset, the evolution of the tension in
the cortex as a function of time for different loading rates, β = 1, 0.25, 0.1, 0.05 and 0.025
sec−1 .
P (KPa)
0.8
0.6
0.4
0.2
UNBINDING
0
0 0.2 0.4 0.6 0.8 1
β (s −1 )
Fig. 4.19. Critical micropipette pressure for membrane unbinding as a function of the loading
rate β for a typical degradation time of 30 seconds. The critical force per bond used is 16 pN,
as obtained in section 4.3.

4.4 Dynamical response of a cell to a controlled mechanical perturbation using a micropipette set up 95
the pressure. In what follows, we will consider the conditions for cortex repolymerization
under a moving membrane, and we will compare the relative
effect of re-polymerizing a new cortex and the increase of membrane tension
in stopping a moving tongue.
In section 4.2.2, we discussed the mechanisms that control the cortex
thickness. A simple dynamical evolution for the thickness is given by Eq.
4.16, ˙h = vp − vd(h). Its stationary solution (˙h = 0), results from the balance
between polymerization and depolymerization velocities, which in general
depend on stress and monomer concentration.
Actin filaments nucleate at the membrane and grow towards the interior
of the cell (Alberts et al., 2002; Charras et al., 2008). When they reach a
characteristic size, which is roughly the mesh size of the gel ξ , filaments
overlap and start crosslinking, leading to an elastic gel structure. In order to
have a gel that is able to sustain and exert stress, the thickness must reach
a critical value which is at least the crosslinking length ξ . This condition
defines a threshold for the depolymerization velocity above which a stable
layer of new cytoskeleton can not be formed: the polymerization velocity is
equal to the depolymerization velocity at a thickness equal to the mesh size,
vp ∼ vd(ξ ).
In general, the balance between polymerization and depolymerization velocities
depends on the stress stored in the cortex and the actin monomer
concentration and diffusion. Since a cortex is always formed once the membrane
has stopped (Merkel et al., 2000; Charras et al., 2008), and experiments
using photobleaching recovery suggest a large diffusion constant for g-actin
of D ∼ 10 2 µm 2 /s(Lanni and Ware, 1984) 8 , we will not consider any effect of
the monomer concentration and difusion here.
In our case, the gel grows on a membrane that is moving. Assuming the
gel grows under no stress under the membrane, it becomes stressed as it ages
away from the membrane, Fig. 4.20. The total amount of stress stored in the
cortex depends on how fast the gel is recycled, or equivalently, the time during
which a layer of gel is stretched before it is depolymerized, t ∼ h/vp. The
depolymerization velocity vd increases exponentially with the stored stress
(see Eq. 4.17), so a gel is expected to be thinner in a moving membrane.
The effective stress in the gel is in general given by a viscoelastic maxwell
like equation, Eq. 4.10, with a relaxation time scale τ ∗ which includes both
the effect of cytoskeleton viscoelasticity and treadmilling (polymerization
and depolymerization). Here we consider very thin gels, so the relaxation
8 Which corresponds to mean displacement of 10µm after one second, much larger than the
typical advancing velocity ˙L < µm/s

96 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
(a) (b)
L
˙L
vm =0 vm = ˙ vp
L
l
ξ
vd
l + δ
Fig. 4.20. Schematics of a gel growing on a moving membrane. (a) The membrane tongue
in the micropipette moves as a consequence of the suction pressure. (b) Actin filaments grow
under the membrane and cross-link at a distance of about the mesh size of the gel. The resulting
cortex is stretched due to the membrane displacement.
time τ ∗ is dominated by treadmilling, in which case τ ∗ ∼ h/vp, and we
expect it to be smaller than the loading time scale due to the membrane
flow tη ∼ ηRp/∆P. As a consequence, the gel behaves as a viscous fluid
(σ ∼ Eτ ∗ ∇ ˙u), with a viscoelastic stress at the outer layer of the gel,
σ ∼ E h
vp
˙L
L + σm(h). (4.72)
Intuitively, the amount of deformation u stored at a distance h from the
membrane is given by the increase of tongue length during a time equal to
the relaxation time τ ∗ , u ∼ τ ∗ ˙L. Since we are considering very thin gels we
can assume that h is smaller than λ, where λ is the characteristic binding
length of myosin (see section 4.2.2), myosin tension is strongly limited by
the treadmilling process and σm(h) ∼ σmsh/(2λ), where σms is the stress corresponding
to the maximum density of myosin attached to the cortex (see
section 4.2.2).
The condition that allows the cortex to grow thicker than the typical mesh
size ξ becomes,
vp ≥ vd(ξ )=v 0 d exp
E ξ
σ0 L
˙L
vp
+ σmsξ
, (4.73)
σ02λ
where σ0 is a reference stress related to actin and cross-linker kinetics, and v 0 d
is the stress free depolymerization velocity. From this last relation we obtain
the condition the tongue velocity ˙L, and its length L, must satisfy in order for a
new crosslinked gel to be able to polymerize under the membrane. Neglecting
any variation of the pressure or the tension, while the tongue advances in the
micropipette, the membrane velocity is given by η ˙L ∼ ∆P − γ/Rp 9 . Since
9 There is no elastic term arising from the cytoskeleton because it is yet to be formed.

4.4 Dynamical response of a cell to a controlled mechanical perturbation using a micropipette set up 97
the tongue velocity is bounded, the stress (∼ ˙L/L) will decrease in time as the
tongue length increases. As a consequence, there is a tongue length scale L σ c
above which the visco-elastic stress is small enough to allow the appearance
of a new cytoskeletal cortex,
∆L σ c = Rp∆P − γ
ηRp
ξ
vp σ0 log vp/v0 d
E
. (4.74)
− σmsξ /2λ
In absence of membrane motion, myosin stress is in general an important
contribution to regulate cortex thickness (see section 4.2.2). Typical gel thicknesses
are of about 500 nm. Since we aim to find the conditions for gel generation
(ξ ≪ 500 nm), the elastic stress resulting from membrane motion must
be larger than myosin stress and we can typically neglect the myosin contribution
contribution in Eq. 4.74.
Once the condition for polymerizing a gel is fulfilled, the new cortex slows
down the membrane, which reduces the tongue velocity and in turn, favors
gel growth. As the thickness increase, myosin is recruited, which increases
the stress in the cortex (see Eq. 4.72). After the initial slowing down of the
membrane tongue due to visco-elastic stress, the cortex is able to finally stop
the membrane if the tension created by myosin10 balances the difference of
pressure, Rp∆P = γ + γm, which gives a lower bound for the gel thickness h,
necessary to stop the membrane: γm(h) ≥ Rp∆P − γ,
h ≥ λ
1 + Rp∆Pp − γ
+ ProductLog
σms
−exp
−1 − Rp∆P − γ
σms
≡ hs,
(4.75)
where we have used Eq. 4.13 for the myosin tension. For small suction
pressures, the gel thickness needed is very small h ≪ λ, and the critical
thickness grows as the squared root of the micropipette pressure, hs ∼
( λ
σms Rp∆P − γ) 1/2 . For larger suction pressures, a thicker cortex is required,
and the myosin kinetics is fast enough to reach the maximum stress almost in
the entire gel thickness (h ≫ λ), and the critical thickness is linear with the
pressure, hs ∼ (Rp∆P − γ)/σms.
The critical length scale for cortex repolymerization must be compared to
the length at which the membrane is stopped by the increase of membrane
tension. If the length for a gel to grow is smaller, the gel is able to stop the
membrane by itself, otherwise, the increase of membrane tension stops the
membrane before a gel has grown.
10 Note that if the membrane stops, the elastic stress induced on the cortex is only myosin
stress after the relaxation time.

98 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
The evolution of the tongue in absence of cortex stress is η ˙L ∼ ∆P−(γ0 +
ζ∆L)/Rp, where ζ is the rate at which the tension increases with the increase
of tongue length L (see Eq. 4.60). This leads to the characteristic length L γ c,
at which the membrane is stopped by the increase of membrane tension,
The ratio of the two length scales is
Lσ c
L γ c
∆L γ c = Rp∆P − γ0
. (4.76)
ζ
∼ Eξζ
log
ηRpσ0vp
v 0 d
vp
. (4.77)
Taking the reference stress σ0 as of the same order of magnitude than the
Young modulus, the ratio of lengths is of about 1 − 10, so in general, the
gel is able to grow before the increase of membrane tension stops the membrane.
However, both lengths can be of the same order of magnitude, and the
different phenomena may not be distinguishable.
4.5 Experimental observation of the different regimes
We have theoretically considered the unbinding dynamics of the membranecortex
complex focussing on micropipette experiments, although the results
are of general applicability for any dynamical perturbation in the cell. As
discussed in section 4.3, there is a critical force per bond at which the
membrane-cortex attachments become unstable (Eq. 4.29 in section 4.3).
This force was translated to a static critical suction pressure by the relation
f ∗ =(∆P + 2γm/R)ξ 2 . Comparison with experiments (Merkel et al., 2000)
allowed to determine the energy adhesion and a quantitative understanding
of the discontinuous transition between stable and unstable attachments, as a
function of linker concentration and myosin activity.
Earlier in this chapter, we have seen that those concepts are not static
and that the stability criteria depends on the dynamics of the perturbation
(see Fig. 4.19). As an indication of this phenomenon, (Merkel et al., 2000)
reports an apparent counter intuitive observation: after the initial instability
in which the membrane unbinds from the cytoskeletal cortex, the growth of
a new cortex is able to retract the tongue back to a position that was initially
unstable, although the pressure in the micropipette is kept constant, Fig. 4.21.
In the original work, (Merkel et al., 2000) propose as an explanation that the
density of linkers is increased in order to sustain larger pressures.

4.5 Experimental observation of the different regimes 99
4.5.1 Non equivalence of initial aspiration and retraction
Our theoretical analysis propose an alternative explanation that does not involve
any active response of the cell, but relies on the dynamical origin of the
perturbation and viscoelastic behavior of the cortex, Fig. 4.21.
Initially, the micropipette aspirates the cell at a certain rate (supposedly
instantaneous), and the cortex is elastically deformed. As already explained,
the membrane detaches as a consequence of the force applied on the links,
which is the sum of the elastic deformation due to the suction process and the
myosin tension. Consequently, the final pressure and the loading rate must
lie on the unstable region described in Fig. 4.19, which we represent as the
process (i) circle in Fig. 4.21b. When the membrane reaches its new equilibrium
length (by increase of membrane tension or by cortex repolymerization),
a new cortex grow under the membrane, which after recruiting myosin,
it retracts the tongue towards the cell. In this case, the loading rate at which
the force is applied is given by the actin sliding velocity driven by myosin.
Molecular motor velocities are in the range of several nanometers per second
(see chapter 3), which is at least one order of magnitude smaller than the initial
velocity of the tongue dictated by viscous dissipation. As a consequence,
during the retraction process, the contribution from the elastic deformation in
the cortex is smaller than in the aspiration case (see Fig. 4.18). In terms of the
phase diagram (Fig. 4.19), the retraction process corresponds to the same final
pressure but to a smaller loading rate. Eventually, the rate of retraction (or
null rate) could correspond to a stable attachment (process (ii) of Fig. 4.21b),
in which case, once the new equilibrium is reached (the retracted tongue), the
membrane would not detach again.
In contrast, cellular blebbing is mainly caused by myosin contraction,
which implies that the loading rate is the same than the retraction rate. In
other words, the myosin tension alone must overcome the critical value for
membrane unbinding, which corresponds to the criteria derived in section 4.3.
The periodicity arises from the fact that during both detachment and retraction,
the force per link remains the same, contrarily to the micropipette case,
where the initial suction process induces an extra transient force in the links,
which is not present during retraction.
From the previous analysis, we expect other possible regimes, which summarizing
include: saltatory increase of the tongue during the suction process
(Rentsch and Keller, 2000), corresponding to several repolymerization
of cortical layers which become unstable; oscillatory behavior near the equilibrium
tongue length, corresponding to a cortical layer reaching the unbinding
transition while the tongue is retracting; and a regime where no cortex

100 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
(a)
L
P
γm
Cytoskeleton
Saturation from contraction
membrane tension
Unbinding
t
t
P (KPa)
(b)
0.8
0.6
0.4
0.2
(ii)
0
0 0.2 0.4 0.6 0.8 1
β (s −1 )
Fig. 4.21. Non equivalence between the suction process and the cortex retraction. (a) Evolution
of the membrane tongue, micropipette pressure and myosin tension as a function of
time. Initially, the membrane unbinds from the cortex and the tongue grows until it reaches a
saturation value due to increase of membrane tension or cortex repolymerization. During the
retraction process, the membrane may not detach from the cortex although the pressure in the
micropipette is the same than during the extraction. (b) Same phenomenon explained using the
phase diagram for critical pressure (see Fig. 4.19). Initially (i), the pressure is applied at a high
rate, and this triggers membrane unbinding (region in red). After saturation, the final pressure
is kept constant but the loading rate is dictated by myosin activity, and is much smaller, so the
new point in the phase diagram corresponds to a stable attachment (ii).
can be formed, which corresponds to the case where the membrane velocity
is above the polymerization threshold and the cell is finally engulfed inside
the micropipette.
To this end, we have collaborated with Benoit Mauguis and François Amblart
(Brugues et al., 2008b), in order to conduct several experiments on the
Entamoeba histolytica system (Coudrier et al., 2005), to experimentally observe
several of the possible predicted regimes.
4.5.2 Micropipette suction regimes
We can distinguish three phases during a micropipette experiment, which
are the initial growth of the tongue, saturation, and retraction processes (see
Fig 4.21).
The first phase may occur while the membrane is still bound to the cytoskeleton,
or after its unbinding.As already discussed the second phase may
be due to an increase of membrane tension or to the contractile activity of the
cytoskeleton. Since we have already discussed the role of membrane tension
(i)

4.5 Experimental observation of the different regimes 101
earlier in this chapter, we will neglect the effect of the increase in membrane
tension in the following discussion.
Saltatory and no cortex regime
We have seen in section 4.4.4, that a new cortex can repolymerize under a
moving membrane above a threshold tongue length, which is determined by
the suction pressure, or equivalently, the velocity at which the tongue moves
(Eq. 4.74). If the condition is fulfilled, a new cortex is polymerized and a
force opposing the motion slows down the membrane. For a critical thickness
hs (Eq. 4.75), the cortex is able to stop the membrane, in which case we reach
the phase two of retraction. However, we have seen that myosin contraction
can lead to membrane unbinding 11 , if the critical force per link is surpassed,
or equivalently, if a threshold cortex thickness is reached,
hb Rp
σmsξ 2 f ∗ , (4.78)
where we have used the limit where the myosin kinetics is faster than the cortex
recycling (h > λ, see Eq. 4.13). If the unbinding thickness hb is smaller
than the stopping thickness, the new cortex is able to slow down the membrane
transiently until the detachment increases the tongue velocity again,
leading to a saltatory like movement (Fig. 4.25). This regime is experimentally
observed in Fig. 4.22a→e. The theoretical prediction is depicted in Fig.
4.22e, and is based in the force balance between the pipette pressure and the
myosin tension, Eq. 4.61, and the stress dependent dynamical evolution of
the gel thickness, Eq. 4.16.
Interestingly, the micropipette suction force increases as the tongue length
increases, because this corresponds to a decreasing cell radius, and a correspondingly
increasing Laplace pressure that drives the cell towards the micropipette.
If the increase of membrane tension is not sufficient to stop the
membrane, the amoeba will be irremediably swallowed by the micropipette,
with an increasing membrane velocity as more amoeba is sucked. The maximum
velocity is roughly ∆ ˜P/ ˜η with ∆ ˜P is the Laplace pressure 12 , in which
case, the threshold length for cortex polymerization is (Eq. 4.74)
∆L σ c ∼ ∆ ˜Pξ
˜ηvp
E
σ0 log(vp/v 0 d ).
(4.79)
11 Decreasing the micropipette radius increases the force on the bonds for the same cortex
stress.
12 ∆ ˜P 1Rp
1
≡ ∆P − γ − .
Rc(L)

102 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
(a)
(c)
(f)
10µm
10µm
(b)
(d)
10µm
10µm
(e)
∆L(µm)
30
25
20
15
10
(a)
(c)
(b)
(d)
5
0 5 10
t (sec)
15 20
5µm 5µm 5µm
Fig. 4.22. Saltatory regime (a-e), and transition to no cortex regime (f). In (a-d) sequences
of cortex repolymerization (a) and (c), which stops the advancing tongue, followed by detachment
processes (b) and (d) with an abrupt acceleration of the tongue motion. We define
this regime as saltatory since although the membrane nearly stops, it does not retract. In (e) a
qualitative theoretical prediction of the saltatory process in which the critical cortex thickness
needed to stop the tongue hs corresponds to the critical thickness for bond unbinding hb, using
the parameters, vp = 50 nm s −1 , hc = 200 nm and ∆P = 400 Pa . (f) In the same experiment,
we observe a transition from the saltatory regime to the no cortex regime in which several attempts
of repolymerizing a gel (white arrows) are frustrated and the tongue advances until the
whole amoeba is engulfed into the micropipette. In these series of snapshots, the micropipette
has been moved
If the threshold length is similar to the length of the amoeba inside the micropipette
(∼ 40µm, ∆P ∼ 1000 Pa) for the maximal velocity, we do not
expect to observe cortex repolymerization. In particular, a transition from a
saltatory regime (cortex polymerization) to a no cortex regime could be observable
since the threshold length increases as the amoeba is swallowed.
In Fig. 4.22f, we report, during the same suction experiment as in the Fig.
4.22a→d, a transition to a non cortex regime, in which several partial cortex
repolymerization are observed (white arrows), before the amoeba was finally
swallowed.
Another aspiration regime is observed when the cortex is able to repolymerize,
with a thickness below the unbinding transition, but unable to stop
the membrane. In this case, one should observe an initial slope of the membrane
trajectory, corresponding to the motion of the free membrane, followed
by a change in the slope due to the opposing force from the new cortex layer
(Fig. 4.25). Experimentally, this case is difficult to observe unless filamentous

4.5 Experimental observation of the different regimes 103
actin is fluorecently labeled, since it is not possible to distinguish by confocal
microscopy alone if the membrane decreases its velocity by increase of
membrane tension or due to the formation of a new cortex.
Cell oscillations in micropipette
Cortex unbinding may also occur after retraction has been initiated (hs < hb <
h), in which case, we expect to observe an oscillatory regime, similar to the
saltatory regime, which approaches the saturation length by a series of retractions
and expansions (Fig. 4.25). This regime is experimentally observed, in
Fig. 4.23 we show two snapshots of the oscillatory behavior with the corresponding
tracking. The theoretical prediction is depicted and plotted with the
experimental measurements, showing a qualitatively good agreement. Notice
that this regime corresponds with the aspiration phase, since the mean position
of the tongue increases and eventually saturates (Fig. 4.23).
38
(a) (b)
10µm
10µm
L(µm)
36
34
32
30
28
26
0 20 40 60 80 100 120 140 160
t (sec)
Fig. 4.23. Tracking of the amoeba oscillations inside a micropipette of 10 µm using a suction
pressure of about 400 Pa. (a) two snap shots corresponding to a maximum and a minimum of
a period (top and bottom respectively). (b) plot of the tracked oscillations (tracking of images
using ImageJ (Rasband, 2006)) and theoretical prediction (solid line), with the mean length
evolution (dotted line). The parameters used are ∆P ∼ 400 Pa, vp ∼ 50nm s −1 , hc ∼ 600 nm.
The period of the measured oscillations is well defined, Fig. 4.24, and is
given by the time needed for the gel to reach the critical thickness at which
the threshold force is reached and the gel detaches. It is roughly given by
τ ∼ hb/vp, or
τ ∼ Rp
vp
f ∗ b . (4.80)
σmsξ 2

104 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
Remarkably, for a given cell, the critical unbinding force, the myosin density,
cross-linkers density and polymerization velocity are fixed, so the period of
oscillation depends only on the radius of the micropipette, but not on the micropipette
pressure. Using the value for σms and the critical force obtained in
section 4.3 (σms = γm/h ∼ 10 4 Pa, f ∗ ∼ 16 pN), and typical values for polymerization
velocity vp ∼ 50nm s −1 and crosslinker distance ξ ∼ 100 nm, we
obtain an oscillation period of about 15 seconds for the micropipette radius
used in the experiments (Rp ∼ 5µm), which is in good agreement with the
period measured of about 13 ± 9 seconds (Fig. 4.24).
Number
35
30
25
20
15
10
5
0
0 5 10 15 20 25 30 35
Period (s)
Fig. 4.24. Histogram of the oscillation period obtained for 4 different amoeba with a pipette
radius Rp =5µm. The average period τ = 13 sec would correspond to a gel thickness of about
250µm, which is within the range of typical cortex thickness.
More exhaustive experiments varying the radius of the micropipette or
using drugs affecting polymerization velocity or crosslinker density would
allow to quantitatively test our prediction for the oscillation period, specially
the fact that it does not depend on the micropipette pressure. An indication
of this phenomena is given by the sharp period measured in different cells in
which the micopipette pressure was not the same, but varied in order to reproduce
the oscillations (Fig. 4.24). Unfortunately, changing the radius of the
micropipette Rp is difficult, because small Rp requires large suction pressure
in order to see the effects, and large Rp allows the cell to crawl by itself inside
the micropipette, even without applying any pressure.
Finally, during the retraction phase, two regimes can be observed. On one
hand, the stable retraction of the tongue occurs when the final thickness of the
gel is below the critical unbinding thickness. In this case, the tongue reaches
an equilibrium length given by the force balance Rp∆P − γ(L)=γm. On the

4.6 Conclusions 105
other hand, the oscillations reported in the aspiration phase will continue once
the saturation phase has been reached since the final thickness will still be
larger than the critical thickness, but with a vanishing mean displacement.
hb
h
Retraction
Oscillatory
hb >hs
Saltatory
h
Steady
flow
hs >hb
Fig. 4.25. Phase diagram of the different cortex regimes depending on the thickness at which
the links remain stable hb and the thickness needed to stop the membrane hs. The diagonal
divides the space in a region where hb > hs, upper region, and a region where hs > hb, lower
region. In this diagram, we choose an arbitrary value of h. Depending on the value of hs and
hb we observe four different regimes. The retraction regime, corresponding to hb > h > hs.
The oscillatory regime, corresponding to h > hb > hs. The saltatory regime, which differs
from the oscillatory regime in the fact that the former do not stop the membrane at any time,
hs > h > hb. The steady flow regime, where h < (hs,hb), and the tongue is stopped by the
increase of membrane tension.
In Fig. 4.25, we show a phase diagram for all possible regimes observed
in a micropipette experiment.
4.6 Conclusions
We have exhaustively considered the phenomenon of membrane-cortex adhesion
and its stability under pressure perturbations. We have followed two
main steps:
First, we have used two different approaches based on both a model for
the membrane-cortex linkers, which takes properly into account the kinetics
of attachment and detachment, and an energetic description of the detachment
process, to describe the adhesion between the membrane and cortex.
We have shown that within both approaches, there is a global link failure and
an abrupt transition from a stable adhesive membrane to an unstable membrane
detached from the cytoskeleton for a critical force on the links.
hs

106 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
The strength of the adhesion depends strongly on both myosin concentration,
which sets a force on the links that increase the likelihood of membrane
detachment, and link concentration, which determines the amount of force
per link. Our results show that the adhesion energy is not an absolute quantity
experimentally accessible, but depends on the active state of the cell, which
can account for differences in the measured adhesion energy of up to 250%.
We give a comprehensive explanation of previous works found in the literature
(Merkel et al., 2000), where myosin and link perturbations lead to a priori
surprising results concerning the values of the critical unbinding pressure and
adhesion energy.
We have described the mean number of blebs in a cell using a coarse
grained kinetic model. We are able to distinguish two qualitative different
regimes: a diffusive state where the cell uses blebbing to move, characterized
by a mean number of blebs close to one, and a state where the cell blebs
extensively, in which case blebs serve as a pressure buffer keeping the internal
pressure constant.
Second, we have considered the dynamical response of a cell to mechanical
perturbations, and in particular, how the viscoelasticity and the treadmilling
(polymerization and depolymerization) of the cortex affects the transmission
of force to the links and the membrane-cortex instability. Based on
experimental evidence, we have used the simplest Maxwell like model that
describes the short time elastic behavior and long time viscous behavior of
the cytoskeletal cortex, and takes explicitly into account both the treadmilling
properties of the cortex and the active stresses generated by myosin concentration
in the cortex. We found that the unbinding instability is highly sensitive
to the rate at which a pressure perturbation is applied in comparison with
the relaxation time scale of the cytoskeletal cortex.
Focussing on micropipette experiments, which allow to both measure mechanical
properties of the cell and control the suction pressure, we have identified
several possible regimes during a micropipette extraction: A stable deformation
where cortex and membrane deform while remaining adhered. A
deformation which involves an initial detachment followed by a repolymerization
of new cortex with stable adhesion. In this last regime, the cortex may
eventually become unstable as the elastic stress in the cortex increases due
to myosin activity, leading to either a saltatory or oscillatory regime. If the
adhesion of the new cortex remains stable, the tongue inside the micropipette
retracts towards the cell. In order to understand both the effect of the repolymerization
of a new cortex and the increase of membrane tension in the process
of stopping the detached membrane, we have modeled the growth of the

4.6 Conclusions 107
cortex thickness under a moving membrane. We found that there is a critical
membrane velocity above which a stable gel can not be formed.
We have tested our theoretical predictions on experiments found in the
literature (Merkel et al., 2000), and on experiments we have conducted on the
Entamoeba histolytica. We have experimentally identified several predicted
regimes and we have found good agreement with the theoretical prediction.
Our results represent a full characterization of a mechanical perturbation
on the membrane cortex interface and provides a better understanding of the
adhesion between membrane and cortex, ruling out the concept of an absolute
value for the adhesion energy during experimental measurements.
Aknowledgements
The work in this chapter has been done in close collaboration with B. Mauguis
and F. Amblart, in particular, the experimental measurements conducted
in the Entamoeba histolytica system.

108 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
4.A Membrane flow dissipation
In this appendix, we will consider the energy dissipation of a membrane that
flows away from the cortex. The membrane velocity depends on this dissipation
and it is involved in several sections of the present chapter, and it is
particularly important in determining the loading rate on the cortex which
determines the elastic stresses stored in the cortex, and consequently, its response
to external perturbations such as osmotic shocks and micropipette experiments.
The appendix is organized in two parts that correspond to the two
relevant geometries that we encounter in our analysis: the dissipation during
the initial membrane-cortex detachment, which can be extrapolated to the
dissipation during bleb inflation, and the dissipation during membrane flow
inside a micropipette (see Fig 4.27).
Membrane motion during bleb inflation or inside a micropipette involve
flow of both cytosol and membrane. The increase of volume in both blebs and
tongues in micropipette aspiration, needs a continuous income of membrane
and cytosol from the rest of the cell. In general, the energy dissipation of a
liquid flow depends on its velocity gradients and viscosity,
˙E = µ dV (∇ · v) 2 , (4.81)
where µ is a viscosity, v the velocity of the flow, and the integral runs over
the liquid volume. In the case of interest, we need to add the contributions
from both the membrane and cytosol.
Membrane dissipation
The membrane behaves as a two dimensional incompressible fluid. As a
consequence, the velocity profile through the neck defined by the area of
membrane-cortex detachment, is given by lipid conservation within the membrane
area,
2πr˙r = ˙A, (4.82)
where r is the distance from the the center of the detached area (see Fig.
4.26a), and A is the area of the bleb or micropipette tongue. Since the membrane
coming from the rest of the cell is attached to the cortex, it must flow
through protein linkers, where the velocity must vanish, Fig. 4.26b. As a result,
there is a velocity gradient between the protein linkers of about ˙r/ξ ,
being ξ the mean distance between links. The total dissipation due to membrane
motion is thus given by the sum of a term that comes from the global

4.A Membrane flow dissipation 109
gradient of the flow (∼ ˙A/r 2 ) and a term from the local gradient of the flow
at the protein linkers (∼ ˙A/(ξ r)),
1 Acell
˙Em ∼ µb log +
ξ 2 S
1
˙A
S
2 , (4.83)
where S is the area of the detached patch, µb ∼ 3 × 10 −6 Pa s m (Hochmuth
et al., 1982) is the two dimensional viscosity of the membrane (µb ≡ eηb,
where e and ηb are respectively the thickness of the bilayer and the viscosity
of the membrane). The first term in the membrane dissipation Eq. (4.83)
corresponds to the local dissipation at the binders and the second term corresponds
to the global dissipation. The former dominates for patches that are
larger than the mean distance between linkers (S > ξ 2 ), and it is always the
case for blebs (S ∼ µm 2 ) and micropipette experiments (Rp ∼ µm).
(a) (b)
˙r
Fig. 4.26. Sketch of the flow of lipid membrane from the cell body to the patch of detach
area (dotted line). In (a), the velocity profile depicted in lines. (b) The flow must vanish at the
protein linkers, and as a result, there is a gradient of velocity between protein linkers.
Cytosol dissipation
Assuming the cytosol behaves as an incompressible fluid, the velocity profile
is given by mass conservation, ˙r ∼ ˙V /r 2 , where V is the volume of the bleb
or tongue in the micropipette. If the cortex remains stable under the patch of
detached membrane, which is the case for the initial detachment, the cytosol
must flow through the network of actin filaments that form the cortex, which
induces velocity gradients of about ∼ ˙r/ξ (see Fig. 4.27), as in the case of the
membrane flowing through linkers, assuming that the mean distance between
linkers and the cortex mesh size are roughly the same. The total dissipation
contains a global gradient of the flow (∼ ˙V /r 3 ) and a global contribution
(∼ ˙V /(r 2 ξ )), restricted to the thickness of the cortex, h,
r
ξ

110 4 Membrane-cortex interactions: Micropipette experiments and blebbing cells
h 1
˙Ec ∼ ηc +
Sξ 2 S3/2
˙V 2 , (4.84)
where ηc ∼ 3 × 10 −3 -2 × 10 −1 Pa s is the cytosol viscosity (Charras et al.,
2008). As in the case of the membrane, the dissipation resulting from the
cytosol flow through the cortex dominates.
4.A.1 Dissipation during membrane-cortex detachment
The dissipation from the membrane deformation that follows the detachment
from the cortex, see Fig. 4.27, has a contribution from the membrane flow
through linkers and a contribution from cytosol flow through the cortex (Eq.
4.83 and 4.84),
˙E ∼ µb
ξ 2 ˙A 2 + ηch
Sξ 2 ˙V 2 . (4.85)
For small deformation of the membrane, ˙V ∼ S ˙x and ˙A ∼ x ˙x, where x is the
normal coordinate to the cortex. For small enough distances (x ∼ 10 nm),
the relevant contribution to the dissipation is the flow of cytosol through the
cortex. For larger distances between the cortex and the membrane, the dissipation
is mainly due to the membrane drag through the linkers. For initial
detachment, which is relevant in the kinetic stability analysis in section 4.3,
the dissipation is then
and the effective viscosity η = Sηch/ξ 2 .
˙E ∼ Sηch
ξ 2 ˙x 2 , (4.86)
4.A.2 Dissipation during membrane flow inside a micropipette
During micropipette aspiration experiments the cortex and membrane may
form a tongue inside the pipette that advances at a certain velocity, Fig. 4.27
(see section 4.4). In this geometry, the main contributions to the dissipation
are the membrane flow through the binders, and the cytosol flow towards the
micropipette (in this case there is no flow through the cortex as it is moving
with the membrane). The former main contribution comes from the cancelation
of the membrane velocity at each binder of the cytoskeletal network, and
it is related with the rate of area increase in the micropipette, Eq. 4.83. For the
micropipette geometry, ˙A = Rp ˙L, where L is the tongue of membrane inside

(a) (b)
4.A Membrane flow dissipation 111
Fig. 4.27. Schematics of the membrane (green) and cytosol (black) flows in the initial detachment
(a) and micropipette experiment (b) geometry. During the initial detachment (a),
membrane must be dragged from the cell (green arrows) and cytosol must flow (black arrows)
through the cortex (in red). In the micropipette experiment (b), membrane is dragged from the
cell and cytosol flows towards the micropipette. In this case, the cortex is deformed from the
micropipette-cell contact, and follows the motion of the membrane.
the micropipette, and the membrane flow in the cell ˙r at a distance r from
the micropipette contact is ˙r ∼ Rp ˙L/r. At the interior of the micropipette, the
cortex is deformed as the membrane moves and the gradient of velocity at
the cortex is ˙L/L. The incompressibility of the membrane leads to a constant
velocity inside the micropipette ˙L. The relative velocity between the membrane
and cortex which is the responsible of the viscous dissipation inside
the micropipette is then ˙L − z˙L/L, where z is the coordinate along the micropipette.
The resulting viscous dissipation is then the sum of the membrane
drag from the cell and inside the micropipette and the flow of cytosol towards
the micropipette (second term in Eq. (4.84)),
˙E ∼ Rpµb
ξ 2
Rp log
Rcell
Rp
+ L
˙L
3
2 + Rpηc ˙L 2 . (4.87)
The cytosol term is typically four orders of magnitude smaller than the contribution
from the membrane flow, and the effective viscosity η(L) is then
η(L) ∼ µbRp
ξ 2 (Rp + L). (4.88)

5
General conclusions
In this work we have studied three problems from soft matter to physical
biology.
In the first part, we have studied defect dynamics in Langmuir monolayers.
An intrinsic asymmetry between velocities of positive and negative
defects is observed during the annihilation process. We have shown that this
asymmetry results from the different topology of the negative and positive
defects, and the asymmetry of the elastic constants of the material. Using a
simple quantitative model based on liquid crystals, we have been able to relate
dynamical measurements (in this case, the defect velocity), to thermodynamic
properties of the material (in this case, the elastic constants), as a function of
the surface pressure. This procedure opens new ways to measure material
properties which are otherwise very difficult to obtain. A possible continuation
of the present work is to consider similar analysis of the wave generation
phenomena induced by light. As explained in the chapter, these materials are
photosensitive, and irradiation using the appropriate wave length induces a
transition between a polar to a non polar molecule. This effect is interesting
because it generates dynamical wave patterns (Crusats et al., 2005; Burriel
et al., 2006) that depend on the topological charge and their separation. A
similar approach as the used in this chapter could lead to new strategies to
measure the elastic properties of the material using irradiation of light. More
interestingly, the conformational change induced by light, resembles the conformational
change used by molecular motors to produce work (Browne and
Feringa, 2006). In the spirit of biomimetic experiments, the study of artificial
molecular motors build from azobenzene molecules would be of special
interest in understanding cooperative phenomena or self organization of real
molecular motors in a system which is can be both much more controlled and
tuned.

114 5 General conclusions
In the second part, we have considered the collective behavior of molecular
motors for several weak interactions, namely, hard-core repulsion, weak
attractive interaction and long-range repulsive interaction. We have shown
that their collective performance is generally optimized with respect to the
corresponding meanfield scenario, which underestimates the cooperativity of
molecular motors in terms of velocity and efficiency. The deviation from a
mean field picture can be understood within a two state model. We obtain a
nearly uniform distribution of forces along the motor cluster. Force distribution
along the cluster is important since dynamics of motor attachment and
detachment depend critically on the applied force. In this chapter, we only
wanted to elucidate some generic cooperativity phenomena, so we neglected
the finite processivity and force dependent kinetics of molecular motors. A
more accurate model accounting for real molecular motors should be considered,
able to deal with dimeric motors. A future direction of the present work
will carefully examine the force distribution on the cluster of motors and its
implications on self-organization. A non uniform distribution of forces (as
in the case of long-range interaction) will presumably lead to a dynamically
unstable cluster that will break towards a smaller stable cluster with a larger
(hard-core), but uniformly distributed forces. It would be interesting also to
determine how general are these cooperative effects and whether is possible
to recover the same effects using simple cluster rules concerning the non trivial
additivity of velocities (a n = 2 cluster goes faster than a n = 1 cluster
with half the applied force), which could lead to shock waves and other interesting
phenomena. The main flaw of this chapter is the completely lack
of direct experimental comparison. In this regard, experiments using a GFP
labelled monomeric molecular motor KIF1A, attached to different types of
cargoes, for instance magnetic beads, allowing to fine tune the motor-motor
interaction, will be of special interest to test our predictions.
In the third part, we have considered the adhesion between plasma membrane
and cytoskeletal cortex and its stability under pressure perturbations.
We have used two different approaches based on both a kinetic model for
membrane-cortex linkers, and an energetic description of the detachment process.
We have shown that within both approaches, there is a global link failure
and an abrupt transition from a stable adhesive membrane to an unstable detached
membrane for a critical force on the links. We have shown that the stability
of the adhesion depends strongly on the state of the cell, and in particular,
on both myosin concentration and link concentration. Our results suggest
that the adhesion energy is not an absolute quantity which is experimentally
accessible, but depends on the state of the cell. We have described the mean

5 General conclusions 115
number of blebs in a cell using a coarse grained kinetic model. We are able to
distinguish two qualitatively different regimes: a state where cells use blebbing
to move, corresponding to a state where there are rarely more than one
bleb at a given time, and a state where the cell blebs extensively, in which case
blebs serve as a mechanical buffer keeping the internal pressure or membrane
tension constant. Finally, we have considered the dynamical response of a cell
to mechanical perturbations, and in particular, how the viscoelasticity and the
treadmilling process of the cortex affects the membrane-cortex stability. We
have shown that the instability of the membrane-cortex depends on the rate
at which a pressure perturbation is applied in comparison with the relaxation
time scale of the cytoskeletal cortex. Focussing on the micropipette aspiration
set up, we have identified several possible regimes during a micropipette
extraction: A stable deformation where cortex and membrane deform in conjunction.
A deformation which involves an initial detachment followed by
a repolymerization of new cortex with stable adhesion. In this last regime,
the cortex may eventually become unstable as the elastic stress in the cortex
increases due to myosin activity, leading to both a saltatory and oscillatory
regimes. If the adhesion of the new cortex remains stable, the cell may retract
all the way to the initial state although the perturbation remains applied. We
have modeled the growth of the cortex thickness under a moving membrane
and we have found that there is a critical membrane velocity above which a
stable gel can not be formed. We have tested our theoretical predictions on
experiments found in the literature and on experiments we have conducted
on the Entamoeba histolytica in close collaboration with B. Mauguis and F.
Amblart.
This chapter is specially relevant because it covers many different physical
possibilities regarding adhesion and micropipette aspiration, which can
thoroughly be compared to experiments. However, more experiments need to
be done, specially regarding the dependence of adhesion on myosin and link
density. Labeling of actin in the micropipette experiments will provide evidence
of the several reported regimes, specially for the threshold for cortex
repolymerization. The difficulty of using several radii of micropipettes, do not
allow to clearly demonstrate the dependence of the oscillations in the radius
of curvature alone. It will be also interesting to experimentally address the
threshold pressure for membrane detachment as a function of the loading rate
using the same micropipette set up used with the Entamoeba histolytica. We
are currently working towards this direction in collaboration with B. Mauguis
and F. Amblart.

116 5 General conclusions
As general remarks, this thesis served more as a path or transition towards
a purely biophysical research, than a closed and definite project. The knowledge
and tools learned will hopefully help me to address new biological problems.
I personally believe the future directions in physical biology will tend
to point towards the relation between forces and functionality, ranging from
the coupling between external stimuli and gene network regulation, to the relation
between development and cell morphology and stress. One example
of future direction with involve concepts considered in this thesis and also a
large amount of available data and observations, as well as important medical
applications, is the study of the stability of the mitotic spindle both theoretically
and experimentally. There, concepts from soft-matter (liquid crystals)
and statistical physics (molecular motors) could help to elucidate how such a
dynamical structure auto-organizes and is able to sustain and generate forces
in the range of nano newtons. In addition, the variety of available techniques
allows to compare any possible model prediction with in vivo and reconstituted
systems.

A
Resum en català
El present treball versa sobre aspectes dinàmics de fenòmens relacionats
amb la biofísica i la matèria condensada. L’estructura de la tesi està dividida
en tres parts principals: en primer lloc estudiem la dinàmica de defectes
topològics en monocapes de Langmuir, amb l’intenció de relacionar les propietats
dinàmiques (velocitats) amb els paràmetres materials del sistema (constants
elàstiques). La segona part, està dedicada a la cooperativitat de motors
mol . leculars. Estudiem el mecanisme físic responsable de l’increment
en l’eficiència de motors cooperatius en funció del tipus d’interacció entre
motors. Finalment, estudiem la interacció del cortex i la membrana cel . lular,
donant ènfasi a la relació entre la dinàmica dels lligams entre la membrana i
còrtex, i la viscoelasticitat d’aquest darrer.
A.1 Auto-organització i cooperativitat de motors moleculars
interactuant feblement.
Els motors moleculars són proteïnes capaces de convertir l’energia provinent
de la hidròlisi de l’ATP en treball mecànic, que és utilitzat per generar forces
que permeten una bona part dels moviments cel . luars. El seu comportament
col . lectiu juga un paper important en molts processos biològics, incloent contracció
muscular, transport, motil . litat i divisió cel . lular.
En el darrers anys, el desenvolupament de nous experiments ”in vitro”, ha
permés d’observar motors mol . leculars aïllats. Aquest experiments incloen
assaigs lliscants, pinces òptiques i mesures micromecàniques de força. Els
resultats d’aquests experiments han revelat noves idees sobre els principis
bàsics del motors mol . leculars i han inspirat nous models pel seu comportament
col . lectiu. En aquest context, existeixen força models considerant els
motors enganxats a filaments rígids, i per tant, rígidament acoblats. Aquest

118 A Resum en català
models prediuen un comportament no lineal de la relació entre força aplicada
i velocitat dels motors, que ha estat mesurada experimentalment.
L’assumpció de motors rígidament lligats, no és sempre vàlida. En experiments
recents centrats en tràfic intercel . lular, s’ha demostrat que l’acció
conjunta de grups de motors mol . leculars és necessària quan aquests estan enganxats
a càrregues de tipus fluid (com ara vesícules o tubs de membrana).
En aquest context, però, els motors no estan rígidament lligats (malgrat poder
interactuar entre ells), i poden moure’s de manera indepèndent respecte els
altres motors. Aquest nou escenari de cooperativitat ha inspirat nous models
de cooperació entre motors basats en models discrets, capaços d’explicar
qualitativament l’habilitat de grups de motors d’estirar tubs de membrana
conjuntament. Entre les limitacions d’aquests models, rau la impossibilitat
d’obtenir la distribució de forces de cada motor.
Motivats pel cas de motors estirant una càrrega líquida, en el present
treball estudiarem la situació corresponent a motors interactuant feblement,
sotmesos a una força aplicada només al primer motor, que seria l’equivalent
al front del tub de membrana. La nostre aproximació al problema es basarà
en l’ús del model més simplificat possible de dos estats per un motor. El
nostre model mecanístic ens permetrà obtenir informació quantitativa sobre
l’eficiència i transmissió de força de grups de motors. El nostre objectiu en
aquest treball és aclarir fonomens genèrics de transmissió de força entre motors,
la seva connexió amb el tipus d’interacció entre motors, corves de forçavelocitat
i rendiment col . lectiu de grups de motors.
El model de dos estats per un motor mol . lecular, es basa en el fet que hi ha
dues principals configuracions durant el moviment del motor, corresponents
a un estat lligat, responsable de fer avançar el motor, i un estat deslligat,
on el motor difon. Durant l’estat lligat, el motor està sotmés a un potencial
periòdic que representa la periodicitat del filament on el motor es mou. El
potencial corresponent a l’estat deslligat és pla. La transició de l’estat lligat
a l’estat deslligat es produeix al voltant del mínim de potencial, i correspon
a l’absorció d’una molècula d’ATP. La transició de l’estat deslligat al lligat
està deslocalitzada i té una vida mitja fixada, representant l’alliberació d’un
fosfat un cop l’ATP ha estat hidrolitzat.
En aquest model, el moviment dels motors mol . leculars és descrit amb una
equació de Langevin, que integrarem númericament per tres tipus d’interacció:
repulsiva de curt abast, feblement attractiva i repulsiva de llarg abast.
El grau de cooperativitat entre motors, depèn de la correl . lació entre graus
de llibertat posicionals i ineterns. En el cas de dos motors fortament lligats,
la cooperació es fa evident, ja que quan els estats dels motors són creuats (un

A.1 Auto-organització i cooperativitat de motors moleculars interactuant feblement. 119
en estat lligat i l’altre en estat deslligat), el motor que esta deslligat, enlloc
de difondre, és empés o estirat per l’altre motor en l’estat lligat. Per tant,
el motor en l’estat deslligat avança un periode de manera determinista, i no
degut a fluctuacions com en el cas d’un motor aïllat. En la situació oposada,
on dos motors interactuen amb un potencial molt suau, petites variacions en
la posició dels motors de l’ordre de la periodicitat del filament, es descorrellacionen
amb els estats del motor i perdem la cooperativitat. Aquest cas, el
definirem com a camp mitjà, ja que els motors en l’estat estacionari senten
el mateix potencial d’interacció i la seva velocitat és equivalent a la velocitat
d’un motor amb una força externa N vegades més petita. En el límit de soroll
petit, es pot calcular la corva força-velocitat de manera exacta pel cas de
camp mitjà, que utilitzarem com a referència per comparar amb els altres
casos d’interacció.
En primer lloc, considerem una interacció repulsiva de volum exclós, que
modelem amb un potencial de Lennard-Jones truncat. Genèricament obtenim
un guany respecte el cas de camp mitjà, que depèn del tamany dels motors
moleculars respecte a la periodicitat del filament. A mida que augmentem el
nombre de motors, el guany relatiu va disminuint fins que satura a un valor
constant.
En segon lloc, considerem un potencial d’interacció repulsiu de llarg
abast, que modelem amb el potencial de volum exclós, més un potencial que
decau exponencialment amb un paràmetre que mesura la suavitat. Quan la
força externa supera un cert valor, el potencial suau no és suficient per compensar
la força i els motors comencen a veure el potencial de volum exclós.
Per tant, per un rang de forces, observem que els motors es comporten com
a camp mitjà, però a partir d’un cert valor, els motors comencen a cooperar i
observem una transició cap al comportament descrit pel potencial de volum
exclós.
Finalment, considerem una interacció feblement attractiva. En aquest cas,
observem un fenomen en principi poc intuitiu.
Conclusions
Hem considerat el comportament col . lectiu de motors moleculars per interaccions
de volum exclós, attracció feble i interacció repulsiva de llarg abast.
El redniment col . lectiu és genéricament superior a la del cas de camp mitjà.
Pel cas de motors atractius, observem un nou fenomen de dinàmica de grup
de motors que involucra histèresi i bimodalitat transitòria. Si la reserva de
motors es prou gran, el tamany del grup de motors es reajusta dinàmicament
depènent de la força aplicada. Finalment, hem considerat l’eficiència i la distribució
de força per cada motor del grup. Una limitació del present treball pel

120 A Resum en català
que fa a l’aplicació biològica o biomimètica, és que hem negligit la processivitat
i la depèndència en la força de les transisions entre estats dels motors.
Aquests aspectes els poden incorporar fàcilment en el model i representen la
futura direcció del present treball.
A.2 Mesura de l’anisotropia elàstica a través de la dinàmica de
defectes en monocapes de Langmuir
L’estudi de defectes topològics és genèricament rellevant en força àrees tant
de física com en biologia, comprenent desde cosmologia a heli superfluid
o divisió cel . lular. Una part important de la motivació pel seu estudi ve del
seu grau d’universalitat, com passa amb qualsevol sistema amb paràmetre
d’ordre. Les seves propietats principals són indepèndents de la underlying
física, i estan només determinades per simetries, la dimensió del paràmetre
d’ordre, el defecte i el sistema. Aquest grau d’universalitat ha estat utilitzat
per sistemes pertanyents a la matèra condensada, com ara els cristalls
líquids, que permeten, a través d’experiments relativament senzills, obtenir
informació sobre àrees de la física comletament diferents.
L’estudi del moviment de defectes, la seva interacció i anihilació és particularment
interessant i ha estat estudiat extensivament per nombrosos grups
d’investigació. Un aspecte sorprenent del fenomen d’anihilació, que malgrat
no estar completament entès, és observat tan experimentalment com
mitjançant càlcul numèric, és la sistemàtica diferència de velocitats entre
defectes negatius i positius, sent el darrer sempre més ràpid. S’ha demostrat
que efectes hidrodinàmics i l’anisotropia de les constants elàstiques
(diferència entre les constants elàstiques) contribueixen genèricament a explicar
aquesta assimetria en les velocitats. De fet, en el context de cristalls liquids
3-dimensionals, l’efecte hidrodinàmic domina per sobre de l’anisotropia
elàstica, impedint qualsevol intent de relacionar quantitativament l’elasticitat
del material amb la dinàmica dels defectes, cosa que seria una alternativa
força interessant als mètodes tradicionals per determinar constants elàstiques.
El nostre estudi considerarà la situació oposada, on els efectes hidrodinàmics
són negligibles. Per aquest fí, hem considerat la dinàmica de defectes en
monocapes de Langmuir exteses a l’interface aire/aigua. Contràriament als
cristalls líquids 3-dimensionals, on la viscositat rotacional i tranlacional són
del mateix ordre de magnitud, en monocapes de Langmuir (2-dimensionals),
les molècules poden rotar sense generar efectes hidrodinàmics. En aquest
cas, la dinàmica de defectes, i més concretament, l’assimetria de velocitats
es pot explicar només a partir de les propietats elàstiques del material. Com

A.2 Mesura de l’anisotropia elàstica a través de la dinàmica de defectes en monocapes de Langmuir 121
a conseqüencia, mesures de la diferència en mobilitat dels defectes negatius
i positius, ens permetrà d’obtenir informació sobre l’anisotropia elàstica del
material.
En el present treball està dividit en dos parts. En primer lloc, explicarem
el dispositiu experimental utilitzat per mesurar la dinàmica de defectes. A
continuació, demostrarem que l’observada assimetria en la mobilitat dels defectes
pot ser explicada mitjançant únicament propietats topològiques dels
defectes. Presentarem també un model quantitatiu que permetrà extreure la
depèndència de l’anisotropia elàstica de la pressió superficial, i estimar les
constants elàstiques del sistema.
Dispositiu experimental
La part experimental del treball ha estat feta mitjançant un recipient de tefló
on la monocapa s’exten sobre una àrea limitada per dos barreres mòvils, que
hem utilitzat per controlar la pressió superficial.
La monocapa és composa del fotosensible azobenzé amfifílic 8Az3COOH.
Totes les classes d’azobenzé estan constituides d’un nucli format per dos
anells de fenil lligats a un doble enllaç de nitrogen, que poden adoptar dos
configuracions geomètriques diferents: la configuració trans i la configuració
cis. Degut a les propetats geomètriques i elèctriques, l’isomer trans mostra un
paràmetre d’ordre orientacional.
Els azobenzens presenten dos propietats que usarem pels nostres propòsits
experimentals. Per una banda, es poden induir transisions entre les configuracions
cis-trans irradiant les molècules amb llum d’una longitud d’ona
apropiada. Les molécules cis relaxen espontàniament a la configuració trans
per fluctuacions tèrmiques. Degut a les propietats geomètriques completament
diferents, les barreges de isòmers cis i trans s’organitzen de manera
diferent a l’interfície d’aire/aigua. Per altra banda, les mesofases de l’isomer
trans, tenen una densitat superfícial relativament petita. Consqüentment, les
monocapes compostes d’aquest isomer presenten un ordre orientacional de
llarg abast, però un desordre posicional, cosa que permet que els defectes es
moguin amb facilitat.
Barrejes de tran-cis isomers on la concentració de cis és dominant, resulta
en la formació de gotes d’isomer trans, que presenten un defecte de càrrega
+1 al centre. Utilitzarem aquestes gotes per estudiar l’anihilació de defectes
resultant de la fusió d’un parell de gotes.
El métode utilitzat per la visualització del paràmetre d’ordre, està basat
en el concepte d’angle de Brewster, que és l’angle d’incidència en el qual un
raig de llum polaritzat es reflecta completament. Si dipositem una capa de

122 A Resum en català
material molt fina al pla d’incidència, una petita part de la llum es reflectirà,
presentant en general una polarització diferent a la de la llum incident. Calibrant
apropiadament la senyal rebuda, es pot mesurar unívocament la direcció
del paràmetre d’ordre orientacional.
Els resultats experimentals revel . len les molècules amfifíliques dins de
les gotes, es caracteritzen piter formar un angle constant respecte la normal
de l’interfície aire/aigua. Per tant, podem descriure l’organització molecular
utilitzant un vector 2-dimensional. Al fusionar-se dues gotes, degut a la
conservació de càrrega topològica, apareixen dos defectes de càrrega fraccionaria
−1/2 a la frontera. Típicament, un dels dos defectes fraccionaris es
fuciona amb un dels dos defectes centrals, creant un defecte +1/2 a la frontera.
A continuació, els dos defectes de la frontera s’atrauen degut a la càrrega
oposada i s’acaben fusionant. El defecte positiu sempre es mou més depressa
que el negatiu.
Descripció teòrica
L’atracció elàstica de defectes de càrrega oposada és la responsable de l’anihilació
dels defectes ±1/2 a la frontera. La velocitat a la que els defectes es mouen
depèn de la quantitat d’energia dissipada. En monocapes de Langmuir, la
viscositat rotacional deguda a la reorientació de les molécules, és dominant
respecte a la viscositat translacional de les molècules. Per tant, l’explicació
de l’assimetria de velocitats entre els defectes positius i negatius rau només
en l’anisotropia de les constants elàstiques.
Per tal de descriure els defectes, utilitzem una energia lliure que inclou
els ordres més baixos en la distorsió de la direcció de les molècules.
L’energia conté dos termes caracteritzats per dos constants elàstiques: un
terme d’obertura, anomenat ”splay”, que penalitza les configuracions de
molècules, que partint d’un mateix punt, formen un angle entre elles, i un
terme de curvatura entre molècules, anomenat ”bend”, que penalitza configuracions
curvades de molècules paral . leles.
La forma de les solucions de defectes positius no depèn de l’anisotropia de
les constants, mentres que pel cas de defectes negatius, aquests es deformen
depèndent de l’anisotropia de les constants.
La dissipació que els defectes creen al moure’s, es pot demostrar que és
l’energia que costa moure un defecte al voltant del seu eix. Per el cas en
que l’anisotropia és nul . la, cada molècula, tant del defecte negatiu com positiu,
rota la mateixa quantitat i per tant, els dos defectes dissipen la mateixa
quantitat d’energia. Per contra, quan l’anisotropia és diferent de zero, la deformació
del defecte negatiu implica que hi ha certes molècules es mouen

A.3 Interaccions de membrana i còrtex: Experiments amb micropipeta i cèl . lules amb blebs 123
menys i d’altres, que s’han de moure més respecte al cas isòtrop. Degut al
fet que la dissipació és una funció quadràtica en la rotació del defecte, els
defectes negatius sempre dissipen més, i per tant, es mouen més lentament.
Utilitzant l’expressió de la dissipació dels defectes demostrem que el quocient
de velocitats del defecte negatiu i positiu depèn només de l’anisotropia
de les constants elàstiques. Per tant, mesurant experimentalment les respectives
velocitats dels defectes quan s’anihilen, podem obtenir el valor de
l’anisotropia. Modificant la pressió superficial mitjançant el nostre dispositiu
experimental, obtenim la depèndència de l’anisotropia en funció de la pressió.
Utilitzant el fet que la mitjana geomètrica de les constants elàstiques s’ha demostrat
en la literatura que és indepèndent de la pressió, podem relacionar
directament el valor absolut de les constants amb l’anisotropia que obtenim.
Els nostres resultats mostren que la contribució ”splay” incrementa amb la
pressió, mentres que la contribució ”bend” disminueix. Aquests resultats són
consistents amb la literatura i amb el fonomen anomenat ”aggragats H”, que
consisteix en la preferència de les molècules d’azobenzé de formar agregats
on els plans de les molècules són paral . lels. Per tant, augmentant la pressió,
s’afavoreix aquest fenòmen que penalitza la contribució ”splay”, en contra de
la contribució ”bend”.
Conclusions
Hem estudiat monocapes de Langmuir, on la dinàmica de defectes pot ser explicada
únicament per l’anisotropia de les constants elàstiques. Hem mostrat
que el nostre sistema, juntament amb un model senzill, permet d’utilitzar
mesures dinàmiques per obtenir informació de la depèndència de paràmetres
del material (en aquest cas, les constants elàstiques), generalment difícils de
mesurar directament, en funció de paràmetres termodinàmics bàsics de la
monocapa com la pressió superficial.
Aquest treball ha estat fet en col . laboració amb els doctors J. Ignés-Mullol
i F. Sagués.
A.3 Interaccions de membrana i còrtex: Experiments amb
micropipeta i cèl . lules amb blebs
La membrana cel . lular és essencial per la cèl . lula. Serveix d’envolcall, defineix
les fronteres, fa de mediadora de l’intercanvi de materials entre el cistosol
i el medi extracel . lular i conté proteïnes que actuen com a sensors dels
senyals externs que permeten a la cèl . lula reaccionar i adaptar el seu comportament
en resposta a canvis en l’ambient. Sota la membrana hi ha el còrtex,

124 A Resum en català
format per una xarxa d’actina que es reorganitza constantment. El còrtex es
responsable, entre d’altres coses, de la forma de la cèl . lula, el moviment i el
transport cel . lular intern. La membrana i el còrtex estan adherits mitjançant
interaccions febles de lípids de membrana i proteïnes del còrtex i per interaccions
fortes de proteïnes. La pèrdua d’adhesió té com a conseqüència la
formació de blebs, que són unes protuberàncies de forma esfèrica que es formen
quan un troç de membrana es desenganxa del còrtex. Normalment, la
presencia de blebs és indicació de mort cel . lular, encara que hi ha exemples
on les cèl . lules s’aprofiten d’aquest fenòmen per moure’s. En aquesta part de
la tesi, hem estudiat el mecanisme d’adhesió de la membrana i el còrtex, i la
seva resposta dinàmica a perturbacions mecàniques que eventualment tenen
com a resultat la separació de la membrana del còrtex.
Descripció teòrica
Per modelitzar l’adhesió entre la membrana i el còrtex, hem considerat els
lligams, que normalment són proteïnes, com a molles, i per tant, exercint una
força que es proporcional a la seva extensió. La velocitat a la que la membrana
escapa del còrtex ve donada per el balanç de força entre la pressió aplicada a
la membrana i la força recuperadora dels lligams. Els lligams s’enganxen i es
desenganxen constantment de la membrana a una certa freqüència que depèn
de la força que se’ls hi aplica. Considerant l’equació pel balanç de força en
funció de la densitat de lligams i l’equació de conservació del nombre total de
lligams, que determina la fracció de lligams enganxats, obtenim que existeix
una força crítica a partir de la qual els lligams són incapaços de compensar la
pressió i acabant fallant alliberant la membrana del còrtex.
Si considerem ara una descripció energètica del mateix problema, on comparem
el guany d’energia de desenganxar la membrana degut a la pressió i la
pérdua d’energia degut a la pérdua de lligams i a la tensió de la membrana,
obtenim, com en el cas anterior, una inestabilitat on la membrana es desenganxa
per una pressió crítica. La transició de l’estat enganxat a l’estat desenganxat
ve donada per una barrera energètica que disminueix amb la pressió.
Podem definir un temps característic de pas que depèn exponencialment de la
barrera energètica. El temps de pas ens dóna una estimació de la freqüència a
la que un bleb es crea. Al mateix temps, els blebs es retrauen i desapareixen
després d’un temps característic que depèn del temps que es triga a formar
un nou còrtex sota la membrana desenganxada. El balanç entre els blebs que
es formen i els que desapareixen dóna el número mitjà de blebs en una cèllula,
que depèn de la seva pressió interna. Podem identificar dos règims: per
una pressió moderada, la cèl . lula es manté estable i no apareix cap bleb. Quan

A.3 Interaccions de membrana i còrtex: Experiments amb micropipeta i cèl . lules amb blebs 125
la pressió augmenta suficientment, comencen a aparèixer blebs i la pressió
es manté constant. Aquests dos règims poden correspondre a cèl . lules que
utilitzen pocs (un o dos) blebs per moure’s en el cas de pressió moderada,
ocèl . lules que estan morint, on s’observen blebs per tota la superfície de la
cèl . lula.
Experiments basats en perturbacions mecàniques demostren que les cèllules
es comporten elàsticament per temps petits i una de forma fluida per
temps llargs. Aquest comportament, anomenat viscoelàstic, es pot modelitzar
fàcilment amb un model que interpola la descripció d’un sòlid i un liquid,
degut a Maxwell. Hem utilitzat aquesta descripció per modelar el còrtex de la
cèl . lula. Aplicant una pressió externa que creix a una certa velocitat, que utilitzem
com a paràmetre, podem demostrar que la comparació entre el temps
típic de relaxació del còrtex, que determina la transició entre el comportament
elàstic i fluid de la cèl . lula, i l’escala de temps a la que la pressió augmenta
determina la força màxima transmesa als lligams. Intuitivament, la força aplicada
a un lligam serà la quantitat de deformació del còrtex en el temps durant
el qual el còrtex s’ha comportat elàsticament. Aquesta força ha de ser comparada
amb la força crítica a partir de la qual els lligams es trenquen i la
membrana es desenganxa del còrtex.
Degut a que la força transmesa als lligams depèn tant de la pressió final
que apliquem a la membrana com del ritme al qual incrementem la pressió,
obtenim un diagrama d’estabilitat que ens diu que encara que apliquem pressions
molt fortes, els lligams són capaços de no trencar-se si el ritme al que
apliquem la pressió es prou lent. Igualment, per una pressió moderada, podem
aconseguir trencar els lligams si la velocitat a la qual augmentem la pressió
es prou elevada. Aquest resultat és força rellevant en el cas d’experiments
realitzats amb micropipetes, ja que aplicar una pressió mai és un procés instantani.
Finalment, usant els conceptes anteriors, hem identificat tots els possibles
règims d’aspiració mitjançant una micropipeta: Si el ritme d’aspiració i la
pressió són prou petits, la membrana i el còrtex són deformats fins que arriben
a un nou equilibri sense que la membrana i el còrtex es desenganxin.
Si el ritme d’aspiració es prou ràpid, la membrana es densenganxa del còrtex
i flueix fins que un nou còrtex es format. En auqest cas, hi ha dos possibilitats.
Si quan el nou còrtex es contrau crea una força suficientment gran, els
lligams poden tornar-se a trencar generant un moviment oscil . latori on el trencament
i la formació de còrtex es succeeixen consecutivament. Si la força de
contracció del nou còrtex es suficientment petita, la membrana i còrtex es
retrauen contínuament fins al nou equilibri.

126 A Resum en català
Juntament amb un grup experimental de l’Institut Curie, hem comprobat
experimentalment els règims predits teòricament i em reproduit quantitativament
les oscil . lacions observades en el sistema de l’ameba Entamoeba histolytica.
Conclusions
Hem estudiat exhaustivament el fenomen d’adhesió entre la membrana i el
còrtex i la seva estabilitat, sotmesos a una perturbació en la pressió. Hem
usat una descripció microscòpica dels lligams i una descripció energètica
de l’adhesió. En ambdós casos hem obtingut una inestabilitat global on els
lligams es trenquen per una força crítica. Hem descrit el número mitjà de
blebs en una cèl . lula en funció de la pressió, obtenint dos règims qualitativament
diferents. Per una banda, per pressions moderades, la cèl . lula roman
sense blebs, mentre que per pressions superiors, es creen blebs que mantenen
la pressió interna de la cèl . lula constant.
Hem considerat la resposta dinàmica de la cèl . lula a perturbacions mecàniques.
En particular, em emfatitzat com el caràcter viscoelàstic de la cèl . lula afecta
la transmissió de força als lligams i l’estabilitat d’aquests. Hem obtingut que
la pressió crítica en un experiment depen fortament de la velocitat a la qual
s’aplica la pressió. Com a conseqüència, hem identificat tots els possibles
règims dinàmics durant l’aspiració amb una micropipeta, que inclouen: Una
deformació on la membrana i el còrtex romanen enganxats. Una deformació
on la membrana es desenganxa del còrtex i posteriorment el nou còrtex retrau
fins la nova posició d’equilibri, i finalment, una deformació on la membrana
es desenganxa durant la retracció de forma periòdica.
Utilitzant mesures obtingudes de la literatura i experiments realitzats especialment,
hem pogut comprobar les nostres prediccions teòriques.
Aquest treball ha estat fet en col . laboració amb l’estudiant de tesi B. Mauguis
i el doctor F. Amblart.

B
Résumé en Français
Ce travail traite d’aspects dynamiques de certains phénomènes reliés à la
biophysique et la matière condensée. La thèse est divisée en trois parties
principales. D’abord, nous étudions la dynamique des défauts topologiques
des monocouches de Langmuir, avec l’intention de relier les proprietés dynamiques
(vitesse) avec les paramètres matériels du système (constants élastiques).
La deuxième partie est dédiée à la coopérativité des moteurs moléculaires.
Nous étudions le mécanisme physique responsable du partage de force entre
moteurs coopératifs en fonction du type d’interaction entre moteurs. Finalement,
nous étudions l’interaction entre le cortex et la membrane cellulaire, en
mettant l’accent sur le rapport entre la dynamique des liens entre la membrane
et le cortex, et la viscoélasticité de ce dernier.
B.1 Auto-organisation et coopérativité des moteurs moléculaires
interagissant faiblement
Les moteurs moléculaires sont des protéines capables de transformer l’énergie
provenant de l’hydrolyse de l’ATP en travail mécanique. Les forces ainsi
générées permettent une grand partie des mouvements cellulaires. Les processus
collectifs jouent un rôle important dans beaucoup de processus biologiques,
dont la contraction musculaire, le transport, la motilité et la division
cellulaire.
Ces dernières années, le développement des nouvelles expériences ”in
vitro” à permit d’observer l’activité de moteurs moléculaires isolés. Ces
expériences incluent ”gliding assays”, pinces optiques et mesures micromécaniques
de force. Les résultats de ces expériences ont révélé de nouvelles idées
sur les principes basiques des moteurs moléculaires et ont inspiré de nouveaux
modèles pour comprendre leur comportement collectif. Dans ce con-

128 B Résumé en Français
texte, de nombreux modèles suppose que les moteurs sont rigidement liés
les un aux autres. Ces modèles prédisent un comportement non linéaire
de la relation entre la force appliquée et la vitesse des moteurs, qui a été
mesuré expérimentalement. L’hypothèse de moteurs rigidement liées n’est
pas toujours valable. De récentes expériences concentres sur le traffic intracellulaire
ont démontrées que la mise en mouvement de charges fluides
(comme des vésicules ou des tubes de membrane) nécessitait également
l’action de plusieurs moteurs moléculaires. Dans ce contexte, les moteurs
ne sont pas rigidement lies entre eux et peuvent en principe se déplacer de
manière indépendante. Ce nouveau scénario de coopérativité a inspiré de
nouveaux modèles d’interaction entre moteurs basés en modèles discrets, capables
d’expliquer qualitativement l’habileté de groupes de moteurs d’étirer
tubs de membrane conjointement.
Motivés par le cas de moteurs tirant un chargement liquide, dans cet travail
nous allons étudier la situation qui correspond aux moteurs interagissant
faiblement, soumis a une force appliqué seulement au premier moteur
(par exemple, celui qui se trouve a l’extrémité du tube). Nous supposons un
modèle à deux états pour chaque moteur. Notre modèle mécanique nous permet
obtenir des informations quantitatives sur l’efficacité et la transmission de
force dans un groupe de moteurs. Notre objectif dans ce travail est comprendre
des phénomènes génériques de transmission de force entre moteurs, sa
connexion avec le genre d’interaction entre moteurs, courbes de force-vitesse
et efficacité collectif d’un groupes de moteurs.
Le modèle de deux états pour un moteur moléculaire est basé dans le fait
qu’il y a deux configurations principales pendant le mouvement du moteur,
correspondant à un état lié, responsable de faire avancer le moteur, et un état
délié, dans lequel le moteur diffuse. Dans l’état lié, le moteur est soumis à
un potentiel périodique qui représente la périodicité du filament sur lequel
le moteur marche. Le potentiel qui correspond à l’état délié est plain. La
transition de l’état lié à l’état délié correspond a l’absorption d’une molécule
d’ATP. L’état délié a une durée de vie moyenne fixé, représentant la libération
d’un phosphate, une fois l’ATP hydrolysé.
Dans ce modèle, le mouvement des moteurs moléculaires est décrit avec
une équation de Langevin, que nous intégrons numériquement pour trois genres
d’interaction : répulsive à court portée, faiblement attractive et répulsive à
long portée. Le degré de cooperativité entre moteurs, dépend de la corrélation
entre degrés de liberté positionel et interne. Dans le cas de deux moteurs
fortement lies, la coopération est triviale, puisque un moteur dans un état
délié ne diffuse pas, mais se déplace à la vitesse d’un moteur lié voisin. Par

B.1 Auto-organisation et coopérativité des moteurs moléculaires interagissant faiblement 129
conséquence, le moteur en l’état délié avance de manière déterministe et non
a cause des fluctuations comme dans le cas d’un moteur isolé.
Dans la situation opposé, ou deux moteurs interagissent avec un potentiel
très doux, de petites variations en la position des moteurs de l’ordre de la
périodicité du filament ne sont pas corrélés avec l’état du moteur et perdent
la cooperativité. Cet cas, nous allons le définir comme a champ moyenne,
étant donné que les moteurs en l’état stationnaire soufrent le même potentiel
d’interaction et sa vitesse est équivalent à la vitesse d’un moteur avec la force
externe N fois plus petite. Dans le limite de bruit petit, on peut calculer la
courbe force-vitesse de manière exacte pour le cas de champ moyen, que
on utilisera comme référence pour faire la comparaison avec les autres cas
d’interaction.
D’abord, nous considérons une interaction répulsive de volume exclus
que nous modelons avec un intervalle du potentiel de Lennard-Jones. Nous
obtenons un bénéfice par rapport au cas de champ moyen, qui dépend de la
taille des moteurs moléculaires par rapport a la périodicité du filament. Fur à
mesure que on augmente le numéro de moteurs, le bénéfice relatif diminues
jusque s’arrête a un valeur constant.
En deuxième lieu, nous considérons un potentiel d’interaction répulsive
de long portée que nous modelons avec le potentiel de volume exclus, avec
un potentiel que décline de façon exponentiel avec un paramètre que mesure
la douceur. Quand la force externe surpasse un certaine valeur, le potentiel
doux n’est pas suffisante pour compenser la force et les moteurs commencent
a voir le potentiel de volume exclus. Par conséquence pour un rang de forces,
nous observons que les moteurs se conduisent comme a champ moyen, mais a
partir d’un certaine valeur, les moteurs commencent à coopérer et nous observons
une transition vers au comportement décrit par le potentiel de volume
exclus. Finalement, nous considérons une interaction faiblement attrayante.
Dans ce cas, nous observons un phénomène, en principe, peu intuitif.
Conclusions
Nous avons considéré le comportement collective de moteurs moléculaires
par interactions de volume exclus, attraction faible et interaction répulsive
de long portée. Le rendement collectif est supérieur au du champ moyen.
En le cas de moteurs attrayantes, nous observons un nouveau phénomène de
dynamique de groupe de moteurs que implique hysteresis et bimodalité transitoire.
Si la réserve de moteurs est assez grand, la taille du groupe de moteurs
se réajuste dynamiquement en dépendant de la force appliqué. Finalement,
nous avons considéré l’efficacité et la distribution de force pour chaque

130 B Résumé en Français
moteur du groupe. Une limitation du présent travail pour ce qui concerne a
l’application biologique ou biomemitique, c’est que nous avons négligeait la
processivité et la dépendance en la force des transitions entre les états des
moteurs. Ces aspects on les peut incorporer facilement dans le modèle et
représentent la future direction de ce travail.
B.2 Mesure de anisotropie élastique a travers de la dynamique de
défauts en monocapes de Langmuir
L’étude des défauts topologiques est relevant en beaucoup de zones aussi
de physique comme de biologie,comprennent des la cosmologie a heli superfluide
ou division cellulaire. Une partie important de la motivation pour
son étude vient de son degré d’universalité, comme passe t’il avec quelque
système avec paramètre d’ordre. Ses propretés principaux sont seulement
déterminés per symétries, la dimension du paramètre d’ordre, le défaut et
le système. Cet degré d’universalité aété utilise par systèmes appartenants à
la matière condensée, comme les cristaux liquides, qui permettent, a travers
d’expériences relativement simples, obtenir information sur matières de la
physique différents.
L’étude du mouvement de défauts, sa interaction et annihilation est particulièrement
intéressant et il a était étudié par des nombreux groupes de
recherche. Un aspect surprenant du phénomène d’annihilation, qui , malgré
ne pas être entendu, est observé tant expérimentalement comme grâce a un
calcul numérique, est la systématique différence de vitesses entre défauts
négatifs et positifs, soient le dernière toujours le plus rapide. On a démontré
que les effets hydrodynamiques et l’anisotropie des constants élastiques
(différence entre les constants élastiques) contribuaient à expliquer cette
asymétrie en les vitesses. En fait, dans le contexte de cristaux liquides 3dimensionels,
l’effet hydrodynamique domine au dessus de l’anisotropie
élastique, empêchant tous les essais pour rapporter quantitativement l’élasticité
du matériel avec la dynamique des défauts, que serait une alternative très
intéressant aux méthodes connus pour déterminer constants élastiques. Notre
étude prends en considération la situation opposée, ou les effets hydrodynamiques
sont négligeables. Pour cet but, nous avons considéré la dynamique
des défauts en monocapes de Langmuir étendues à l’interface aire/eau. Contrairement
aux cristaux liquides 3-dimensionnels, ou la viscosité rotacional et
translacional sont du même ordre de magnitude, en monocapes de Langmuir(2dimensionnel),
les molécules peuvent roter sans gérer effets hydrodynamiques.
En ce cas, la dynamique de défauts, et en concret, l’asymétrie de vitesses,

B.2 Mesure de anisotropie élastique a travers de la dynamique de défauts en monocapes de Langmuir 131
peut être expliqué seulement a partir des proprettes élastiques de la matière.
Comme a conséquence, mesures de la différence en mobilité des défauts
négatives et positives, nous permettra d’obtenir information sur l’anisotropie
élastique du matériel. Dans cet travail il est divise en deux parties. D’abord,
nous allons expliquer le dispositif expérimental utilise pour mesurer la dynamique
de défauts. En suite, nous démontrerons que l’asymétrie observée
dans la mobilité des défauts peut être explique seulement grâce aux proprettes
topologiques des défauts. Nous allons aussi présenter un modèle quantitatif
qui permettra extraire la dépendance de la anisotropie élastique de la pression
superficiel, et estimer les constantes élastiques du système.
Dispositif expérimental
La partie expérimental du travail à été fait a travers un récipient de Téflon ou
la monocape s’étends sur une zone limitée par deux barrières mobiles, que
nous avons utilise pour surveiller la pression superficielle. La monocape est
composé du photosensible azobenzene amphiphilique 8Az3COOH. Tour les
classes d’azobenzene sont constitues d’un nucleus formé pour deux anneaux
de fenil lies a un double liaison de nitrogène, que peuvent adopter deux configurations
géométriques et électriques, l’isomère trans montre un paramètre
d’ordre d’orientation.
Les azobenzeness présentent deux propretés que nous utiliserons pour
nôtres finalités expérimentales. D’un coté, on peut induire transitions entre
les configurations cis-trans irradiant les molécules avec une lumière d’une
longueur d’onde convenable. Les molécules cis relaxent spontanément à la
configuration trans par fluctuations thermiques. Dû aux propretés géométriques
complètement différent, les mélanges d’isomères cis et trans, s’organisent
de manière différent à l’interphase d’aire/eau. De l’autre coté, les mesofaces
de l’isomère trans,ont une densité superficielle relativement petite. Par
conséquence, les monocapes composes avec cet isomère, présentent un défaut
de chargement +1 au centre. Nous allons utiliser ces gouttes pour étudier
l’annihilation des défauts résultants de la fusion de deux gouttes. Le métope
utilisé pour la visualisation du paramètre d’ordre, est basé en le concept
d’angle de Brewster, que c’est l’angle d’incidence dans lequel un rayon de
lumière polarisé réflexe complètement. Si on dépose une cape de matérielle
très fine au plain d’incidence, une petite partie de la lumière sera reflectie,
présentant en général une polarisation différent a la de la lumière incident.
En calibrant convenablement le signal reçu, on peut mesurer sans erreur, la
direction du paramètre d’ordre d’orientation.
Les résultats expérimentales révèlent les molécules amphiphiliques audedans
des gouttes, se caractérisent pour être partie d’un angle constant

132 B Résumé en Français
par rapport la normal de l’interphase aire/eau. Par consequence, nous pouvons
decrire l’organisation moleculaire utilitsant un vecteur 2-dimensionnel.
Quand deux gouttes se fusionnent, grâce a la conservation du chargement
topologique, apparaîtrez deux défauts de chargement fractionnaire -1/2 a la
frontière. Typiquement, un des deux défauts fractionnaires se fusionne avec
un des deux défauts centrales, en créant un défaut +1/2 a la frontière. Tout
suite, les deux défauts de la frontière s’attirent a cause de le chargement opposé
et s’y fusionnent. Le défaut positif se remue toujours plus vite que le
négatif.
Description théorique
L’attirance élastique de défauts de chargement opposé est la responsable de
l’annihilation des défauts ±1/2 à la frontière. La vitesse a laquelle les défauts
se meuvent dépend de la quantité de l’énergie dissipé. En monocapes de
Langmuir, la viscosité rotationelle a cause de la réorientation des molécules,
est dominant par rapport à la viscosité translationelle des molécules. Par
conséquence, l’interprétation de l’asymétrie de vitesses entre les défauts positifs
et négatifs est seulement en l’anisotropie des constants élastiques.
A fin de décrire les défauts, nous utilisons une énergie libre que inclure les
ordres plus baisses en la distorsion de la direction de les molécules. L’énergie
contient deux termes caractérisés par deux constants élastiques : un terme
d’oberture, nommé ”splay”, qui pénalise les configurations de molécules, que
partant d’un même point, forment un angle entre elles , et un terme de courbature
entre molécules, nommé ”bend”, qui pénalise configurations courbées
de molécules parallèles.
La forme des solutions de défauts positifs ne dépend pas de l’anisotropie
des constants, tandis que en le cas des défauts négatifs, ces-ci se déforment en
dépendant de l’anisotropie des constants. La dissipation que les défauts créent
en se déplaçant, on peut démontrer que c’est l’énergie que coûte déplacer un
défaut au tour de son axe. Dans le cas en que l’anisotropie est nulle, chaque
molécule, soit d’un effet négatif comme positif, rota la même quantité et, par
conséquence, les deux défauts dissipent la même quantité d’énergie. Par contre,
quand l’anisotropie est différent de zéro, la déformation du défaut négatif
implique qu’il y a certaines molécules qui se déplacent moins et d’autres
qu’on de se déplacer plus au cas isotrope. A cause du fait que la dissipation
est une fonction quadratique en la rotation du défaut, les défauts négatifs se
dissipent toujours plus, et par conséquence, se déplacent plus lentement.
Utilisant l’expression de la dissipation des défauts, nous démontrons que
le quotient de vitesses du défaut négatif et positif dépend seulement de

B.3 Interactions de membrane et cortex : Expériences avec micropipette et cellules avec blebs 133
l’anisotropie des constants élastiques. Par conséquence, mesurant expérimentalement
les respectives vitesses des défauts quand elles s’annihilent, nous pouvons
obtenir le valeur de l’anisotropie. En modifiant la pression superficiel a travers
notre dispositif expérimental, nous obtenons la dépendance de l’anisotropie
en fonction de la pression. En utilisant le fait que la moyenne géométrique
des constantes élastiques on a démontré en la littérature que est indépendant
de la pression, nous pouvons mètre en relation directement le valeur absolu
des constants avec l’anisotropie que nous obtenons. Nos résultats nous montrent
que la contribution ”splay” accroisse avec la pression, tandis que la contribution
”bend” diminue. Ces résultats sont consistent avec la littérature et
avec le phénomène nommé ”agrégées H”, que consistent en la préférence
des molécules d’azobenzene de former agrées ou les plans des molécules
sont parallèles. Par conséquence, augmentant la pression, se favorise cet
phénomène que pénalise la contribution ”splay” par rapport a la contribution
”bend”.
Conclusions
Nous avons étudié des monocapes de Langmuir, ou la dynamique des défauts
peut être expliqué uniquement par l’anisotropie des constants élastiques.
Nous avons montré que notre système, joint avec un modèle simple, permet
d’utiliser mesures dynamiques pour obtenir information de la dépendance
de paramètres du matériel)en ce cas, les constants élastiques), généralement
difficiles de mesurer directement, en fonction des paramètres thermodynamiques
basiques de la monocape comme la pression superficiel. Cet travail
a été fait en collaboration avec les docteurs J.Ignés-Mullol et F. Sagués.
B.3 Interactions de membrane et cortex : Expériences avec
micropipette et cellules avec blebs
La membrane cellulaire est essentiel pour la cellule. Elle sert d’enveloppe,
définie les frontières, elle permet l’échange de matériaux entre le cytosol et
le milieu extracellulaire et elle contient des protéines qui agissent comme
senseurs des signaux externes qui permettront à la cellule de réagir et d’adapter
son comportement en réponse à un changements d’environnement. Sous la
membrane, il y a le cortex, formé par un réseau d’actine que se réorganise
constamment. Le cortex influence la forme de la cellule, son déplacement
et le transport intra-cellulaire. La membrane et le cortex sont liés par des
interactions faibles entre les lipides et les protéines du cortex, et par des interactions
fortes entre protéines. La perte d’adhérence a comme conséquence

134 B Résumé en Français
la formation de blebs, qui sont des protubérances sphériques formées quand
un morceau de membrane se détache du cortex. La présence de blebs est souvent
l’indication de la mort cellulaire, bien que certaines cellules utilisent ce
phénomène pour se déplacer. Dans cette partie de la thèse, nous avons étudié
les aspects statiques et dynamiques du détachement membranaire en réponse
à des perturbations mécaniques.
Description théorique
Pour modéliser l’adhérence entre la membrane et le cortex, nous avons relié
la cinétique d’attachement et de détachement d’une liaison entre membrane
et cortex à la contrainte mécanique exercée par les forces de pression et la
tension du cortex. Dans ce modèle, la membrane plasmique joue un rôle stabilisateur
en compensant une partie des forces de pression. Les différents sites
de liaison agissent de concert pour retenir la membrane, et lorsque l’un des
liens cède, les autres sont soumis à une force plus élevée et ont plus de chance
de céder. Nous avons montré que ce phénomène conduit à une rupture globale
de tous les liens, et au détachement de la membrane, au-delà d’une pression
critique.
Nous avons également étudié les aspects énergétiques de la formation
de bleb. En comparant l’énergie nécessaire pour décrocher la membrane
à l’énergie gagnée lorsqu’un bleb gonfle sous l’effet de la pression, nous
obtenons, comme en le cas antérieur, une instabilité où la membrane se
décroche pour une pression critique. La formation d’un bleb est un processus
de nucléation, caractérisé par une barrière d’énergie qui diminue quand la
pression augmente. On peut définir un temps caractéristique de nucléation qui
dépend de façon exponentielle de la barrière énergétique. Une fois formée, les
blebs se rétractent et disparaissent après un temps caractéristique nécessaire
à la formation d’un nouveau cortex sous la membrane décrochée. L’équilibre
entre formation et rétraction des blebs donne le nombre moyen de bleb à la
surface de la cellule. Ce nombre augmente avec la pression interne, et diminue
avec la tension membranaire et l’intensité de l’adhésion. Lorsque des blebs
sont présent, la pression interne et la tension membranaire sont contrôlés par
l’énergie d’adhésion entre membrane et cortex.
Nous avons ensuite appliqué notre étude à des expériences de micropipette,
lors desquelles une cellule est localement aspirée dans une micropipette. La
cellule peut répondre de plusieurs manières à cette perturbation, y compris
en formant un bleb à l’intérieur de la pipette. Nous avons montré que la
réponse de la cellule dépendait de façon critique de la réponse dynamique
du cytosquelette. Il est expérimentalement établi qu’une cellule soumise à

B.4 Conclusions 135
une perturbation mécanique se comporte de manière élastique à temps court,
et visqueuse à temps long. Nous avons utilisé un modèle très simple de fluide
de Maxwell pour modéliser le comportement viscoélastique du cortex de la
cellule. Soumise à une dépression locale, la cellule se déforme tout d’abord
de manière élastique, puis commence à couler comme un fluide visqueux. La
transition entre ces deux comportements correspond à la contrainte élastique
maximum dans le cortex et sur les liens avec la membrane. Plus la perturbation
est rapide, plus la contrainte est élevée. Cette contrainte doit ensuite
être comparée avec la force critique à partir de laquelle les liaisons se cassent
pour savoir si la membrane va se détacher du cortex. La formation d’un bleb
dépend donc autant de la vitesse avec laquelle une perturbation est appliquée
que de l’intensité de cette perturbation. Une fois un bleb formé, il peut être
stabilisé par une augmentation globale de la tension membranaire. Un nouveau
cortex peut alors se former, et le bleb se rétracte. A ce point, nous avons
déterminé les conditions dans lesquelles cette rétraction peut être presque totale,
et les conditions pour qu’un nouvel épisode de détachement se produise
lors de la rétraction, donnant ainsi lieu à des oscillations périodiques de la cellule
dans la pipette. Nous avons également déterminé dans quelles conditions
un cortex polymérisant sous une membrane en mouvement est en mesure de
freiner l’avancée de la membrane.
Nous avons finalement obtenu un “diagramme de phase” complet du mouvement
d’une cellule sous aspiration locale, qui inclue aspiration complète,
aspiration suivie d’une rétraction complète ou partielle, oscillations, et mouvement
saltatoire.
Conjointement avec un groupe expérimental de l’Institut Curie, nous
avons vérifié expérimentalement les régimes prédits théoriquement et nous
montré que notre modèle reproduit quantitativement les oscillations observés
dans le système de l’amibe Entamoeba histolyca.
B.4 Conclusions
Nous avons exhaustivement étudié le phénomène d’adhésion entre la membrane
et le cortex et la stabilité de l’adhésion lorsque la cellule est soumise à
une perturbation de pression. Nous avons employé une description cinétique
et une description énergétique de l’adhésion. Dans les deux cas, nous avons
obtenu une instabilité globale ou les liaisons se cassent au-delà d’une force
critique. Nous avons décrit le nombre moyen de blebs sur une cellule en fonction
de la pression.

136 B Résumé en Français
Nous avons considéré la réponse dynamique de la cellule à des perturbations
mécaniques. En particulier, nous avons montré que le caractère
viscoélastique de la cellule affecte fortement la stabilité de la membrane.
Nous avons également identifié les différents régimes dynamiques possibles
lors de l’aspiration d’une cellule par une micropipette, que incluent : une
déformation où la membrane et le cortex restent attachés, une déformation
où la membrane se décroche du cortex et est ultérieurement rétractée par
un nouveau cortex, ainsi qu’une déformation où la membrane effectue des
mouvements périodiques. En utilisant des résultats de la littérature et des
expériences réalisées durant cette thèse, nous avons pu vérifier nos prédictions
théoriques. Ce travail était fait en collaboration avec l’étudiant de thèse B.
Mauguis et le docteur F. Amblart.

References
Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., and Walter, P.
(2002). Molecular Biology of the Cell, 4th Edition. Garland Science.
Allen, M. (1996). Molecular dynamics calculation of elastic constants in
gay–berne nematic liquid crystals. The Journal of chemical physics.
Badoual, M., Juelicher, F., and Prost, J. (2002). Bidirectional cooperative
motion of molecular motors. Proc Natl Acad Sci USA, 99(10):6696–
6701.
Blanc, C., Svenˇsek, D., ˇZumer, S., and Nobili, M. (2005). Dynamics of nematic
liquid crystal disclinations: The role of the backflow. Phys Rev
Lett, 95(9):97802.
Brochard-Wyart, F., Borghi, N., Cuvelier, D., and Nassoy, P. (2006). Hydrodynamic
narrowing of tubes extruded from cells. Proceedings of the
National Academy of Sciences, 103(20):7660.
Browne, W. R. and Feringa, B. L. (2006). Making molecular machines work.
Nature nanotechnology, 1(1):25–35.
Brugues, J. and Casademunt, J. (2008). In preparation.
Brugues, J., Ignés-Mullol, J., Casademunt, J., and Sagues, F. (2008a). Probing
elastic anisotropy from defect dynamics in langmuir monolayers. Phys
Rev Lett, 1:037801.
Brugues, J., Mauguis, B., Amblart, F., and Sens, P. (2008b). In preparation.
Burriel, P., Ignés-Mullol, J., Reigada, R., and Sagués, F. (2006). Collective
molecular precession induced by rotating illumination in photosensitive
langmuir monolayers. Langmuir : the ACS journal of surfaces and colloids,
22(1):187–93.
Büttiker, M. (1987). Z. Phyis. B, 68:161.

138 REFERENCES
Campas, O., Kafri, Y., Zeldovich, K. B., Casademunt, J., and Joanny, J.-F.
(2006). Collective dynamics of interacting molecular motors. Phys Rev
Lett, 97(3):038101.
Campàs, O., Leduc, C., Bassereau, P., Casademunt, J., Joanny, J.-F., and
Prost, J. (2008). Coordination of kinesin motors pulling on fluid membranes.
Biophysical Journal, 94(12):5009–17.
Carlier, M. F., Laurent, V., Santolini, J., Melki, R., Didry, D., Xia, G. X.,
Hong, Y., Chua, N. H., and Pantaloni, D. (1997). Actin depolymerizing
factor (adf/cofilin) enhances the rate of filament turnover: implication in
actin-based motility. J Cell Biol, 136(6):1307–22.
Charras, G. and Paluch, E. (2008). Blebs lead the way: how to migrate without
lamellipodia. Nat Rev Mol Cell Biol, page 7.
Charras, G. T., Coughlin, M., Mitchison, T. J., and Mahadevan, L. (2008).
Life and times of a cellular bleb. Biophys J, 94(5):1836–53.
Charras, G. T., Hu, C.-K., Coughlin, M., and Mitchison, T. J. (2006).
Reassembly of contractile actin cortex in cell blebs. J Cell Biol,
175(3):477–490.
Coudrier, E., Amblard, F., Zimmer, C., Roux, P., Olivo-Marin, J.-C., Rigothier,
M.-C., and Guillén, N. (2005). Myosin ii and the gal-galnac lectin
play a crucial role in tissue invasion by entamoeba histolytica. Cell Microbiol,
7(1):19–27.
Crusats, J., Albalat, R., Claret, J., Ignés-Mullol, J., Reigada, R., and Sagués,
F. (2005). Photoinduced molecular reorientation dynamics in confined
domains of langmuir monolayers. The Journal of chemical physics,
122(24):244722.
Crusats, J., Albalat, R., Claret, J., Ignés-Mullol, J., and Sagues, F. (2004).
Influence of temperature and composition on the mesoscopic textures of
azobenzene langmuir monolayers. Langmuir, 20(20):8668–8674.
de Gennes, P. G. and Prost, J. (1995). The Physics of Liquid Crystals 2nd ed.
Oxford University Press.
Derényi, I. and Ajdari, A. (1996). Collective transport of particles in a ”flashing”
periodic potential. Physical review E, Statistical physics, plasmas,
fluids, and related interdisciplinary topics, 54(1):R5–R8.
Dietrich, C., Bagatolli, L., Volovyk, Z., Thompson, N., Levi, M., Jacobson,
K., and Gratton, E. (2001). Lipid rafts reconstituted in model membranes.
Biophysical Journal, 80(3):1417.
Endow, S. and Higuchi, H. (2000). A mutant of the motor protein kinesin that
moves in both directions on microtubules. Nature.

REFERENCES 139
Evans, E. (2001). Probing the relation between force - lifetime - and chemistry
in single molecular bonds. Annual review of biophysics and
biomolecular structure, 30:105–128.
Evans, E. and Rawicz, W. (1990). Entropy-driven tension and bending elasticity
in condensed-fluid membranes. Phys Rev Lett.
Feder, A., Tabe, Y., and Mazur, E. (1997). Orientational fluctuations in a
two-dimensional smectic-c liquid crystal with variable density. Phys
Rev Lett, 79(9):1682–1685.
Fernández, P., Pullarkat, P. A., and Ott, A. (2006). A master relation defines
the nonlinear viscoelasticity of single fibroblasts. Biophysical Journal,
90(10):3796–805.
Feynman, R. P., Leighton, R. B., and Sands, M. (1966). The Feynman Lectures
on Physics. Addison-Wesley, Reading, MA), Vol. I, Chap. 46.
Fournier, J., Ajdari, A., and Peliti, L. (2001). Effective-area elasticity and
tension of micromanipulated membranes. Phys Rev Lett.
Garrivier, D., Decave, E., Brechet, Y., Bruckert, F., and Fourcade, B. (2002).
Peeling model for cell detachment. Eur Phys J E, 8(1):79–97.
Gerbal, F., Noireaux, V., Sykes, C., Jülicher, F., Chaikin, P., Ott, A., Prost, J.,
Golsteyn, R. M., Friederich, E., Louvard, D., Laurent, V., and Carlier,
M. F. (1999). On the ‘listeria’propulsion mechanism. Pramana.
Hatta, E. and Fischer, T. (2003). Splitting of an s= 1 point disclination into
half-integer disclinations upon laser heating of a . . . . J. Phys. Chem. B.
Head, D. A., Levine, A. J., and MacKintosh, F. C. (2003a). Deformation
of cross-linked semiflexible polymer networks. Phys Rev Lett,
91(10):108102.
Head, D. A., Levine, A. J., and MacKintosh, F. C. (2003b). Distinct regimes
of elastic response and deformation modes of cross-linked cytoskeletal
and semiflexible polymer networks. Physical review E, Statistical, nonlinear,
and soft matter physics, 68(6 Pt 1):061907.
Head, D. A., Levine, A. J., and MacKintosh, F. C. (2005). Mechanical response
of semiflexible networks to localized perturbations. Physical review
E, Statistical, nonlinear, and soft matter physics, 72(6 Pt 1):061914.
Head, D. A., MacKintosh, F. C., and Levine, A. J. (2003c). Nonuniversality
of elastic exponents in random bond-bending networks. Physical review
E, Statistical, nonlinear, and soft matter physics, 68(2 Pt 2):025101.
Helfrich, W. and Servuss, R. (1984). Undulations, steric interaction and cohesion
of fluid membranes. Nuovo Cimento D, 3(1):137–151.
Hinshaw, G., Petschek, R., and Pelcovits, R. (1988). Modulated phases in
thin ferroelectric liquid-crystal films. Phys Rev Lett, 60(18):1864–1867.

140 REFERENCES
Hochmuth, F. M., Shao, J. Y., Dai, J., and Sheetz, M. P. (1996). Deformation
and flow of membrane into tethers extracted from neuronal growth
cones. Biophys J, 70(1):358–69.
Hochmuth, R., Wiles, H., Evans, E., and Mccown, J. (1982). Extensional flow
of erythrocyte membrane from cell body to elastic tether. ii. experiment.
Biophysical Journal, 39(1):83.
Ignés-Mullol, J., Claret, J., Albalat, R., Crusats, J., Reigada, R., Romero, M.
T. M., and Sagués, F. (2005). Texture changes inside smectic-c droplets
in azobenzene langmuir monolayers. Langmuir : the ACS journal of
surfaces and colloids, 21(7):2948–55.
Ignés-Mullol, J., Claret, J., and Sagués, F. (2004). Star defects, boojums,
and cardioid droplet shapes in condensed dimyristoylphosphatidylethanolamine
monolayers. J. Phys. Chem. B, 108(2):612–619.
Ingber, D. E. (1993). Cellular tensegrity: defining new rules of biological
design that govern the cytoskeleton. J Cell Sci, 104 ( Pt 3):613–27.
Joanny, J.-F., Juelicher, F., Kruse, K., and Prost, J. (2007). Hydrodynamic
theory for multi-component active polar gels. New J Phys, 9:–.
Juelicher, F., Kruse, K., Prost, J., and Joanny, J.-F. (2007). Active behavior of
the cytoskeleton. Phys Rep, 449:3–28.
Juelicher, F. and Prost, J. (1995). Cooperative molecular motors. Phys Rev
Lett, 75(13):2618–2621.
Julicher, F., Ajdari, A., and Prost, J. (1997). Modeling molecular motors. Rev.
Mod. Phys., 69(4):1269–1281.
Kampen, N. V. (2007). Stochastic processes in physics and chemistry.
books.google.com.
Kleman, M. and Laverntovich, O. D. (2003). Soft Matter Physics: An Introduction.
Springer.
Klumpp, S. and Lipowsky, R. (2005). Cooperative cargo transport by several
molecular motors. Proceedings of the National Academy of Sciences.
Klumpp, S., Muller, M., and Lipowsky, R. (2006). Cooperative transport by
small teams of molecular motors. Arxiv preprint q-bio.SC.
Koster, G., Vanduijn, M., Hofs, B., and Dogterom, M. (2003). From the cover:
Membrane tube formation from giant vesicles by dynamic association
of motor proteins. Proceedings of the National Academy of Sciences,
100(26):15583.
Kramers, H. (1940). Brownian motion in a field of force and the diffusion
model of chemical reactions. Physica.

REFERENCES 141
Kruse, K., Joanny, J.-F., Juelicher, F., Prost, J., and Sekimoto, K. (2004).
Asters, vortices, and rotating spirals in active gels of polar filaments.
Phys Rev Lett, 92:078101.
Landau, L. D. and Lifshitz, E. M. (1987). Fluid mechanics. page 552.
Landau, L. D., Lifshitz, E. M., Kosevich, A. M., and Pitaevskii, L. P. (1995).
Theory of Elasticity 3rd Edition. Butterworth-Heinemann.
Landauer, R. (1988). J. Stat. Phys., 53:233.
Lanni, F. and Ware, B. R. (1984). Detection and characterization of actin
monomers, oligomers, and filaments in solution by measurement of fluorescence
photobleaching recovery. Biophysical Journal, 46(1):97.
Leduc, C., Campas, O., Zeldovich, K., Roux, A., Jolimaitre, P., Bourel-
Bonnet, L., Goud, B., Joanny, J.-F., Bassereau, P., and Prost, J. (2004).
From the cover: Cooperative extraction of membrane nanotubes by
molecular motors. Proceedings of the National Academy of Sciences,
101(49):17096.
Leibler, S. and Huse, D. (1993). Porters versus rowers - a unified stochasticmodel
of motor proteins. J Cell Biol, 121(6):1357–1368.
Merkel, R., Simson, R., Simson, D., Hohenadl, M., Boulbitch, A., Wallraff,
E., and Sackmann, E. (2000). A micromechanic study of cell polarity
and plasma membrane cell body coupling in dictyostelium. Biophys J,
79:707–719.
Mermin, N. D. (1979). Rev. Mod. Phys., 51:591–648.
Mogilner, A. and Oster, G. (2003). Force generation by actin polymerization
ii: the elastic ratchet and tethered filaments. Biophysical Journal,
84(3):1591–605.
Nishinari, Okada, Y., Schadschneider, A., and Chowdhury, D. (2005). Intracellular
transport of single-headed molecular motors kif1a. Phys Rev
Lett.
Noireaux, V., Golsteyn, R. M., Friederich, E., Prost, J., Antony, C., Louvard,
D., and Sykes, C. (2000). Growing an actin gel on spherical surfaces.
Biophysical Journal, 78(3):1643–54.
Okada, Y. and Hirokawa, N. (1999). A processive single-headed motor: Kinesin
superfamily protein kif1a. Science.
Oswald, P. and Ignés-Mullol, J. (2005). Backflow-induced asymmetric collapse
of disclination lines in liquid crystals. Phys Rev Lett, 95(2):27801.
Parmeggiani, A., Juelicher, F., Ajdari, A., and Prost, J. (1999). Energy transduction
of isothermal ratchets: Generic aspects and specific examples
close to and far from equilibrium. Phys Rev E, 60(2):2127–2140.

142 REFERENCES
Pécréaux, J., Döbereiner, H.-G., Prost, J., Joanny, J.-F., and Bassereau, P.
(2004). Refined contour analysis of giant unilamellar vesicles. The European
physical journal E, Soft matter, 13(3):277–90.
Pedrosa, J., Romero, M., and Camacho, L. (2002). Organization of an amphiphilic
azobenzene derivative in monolayers at the air-water interface.
J. Phys. Chem. B.
Plastino, J., Lelidis, I., Prost, J., and Sykes, C. (2004). The effect of diffusion,
depolymerization and nucleation promoting factors on actin gel growth.
European Biophysics Journal.
Pullarkat, P., Fernandez, P., and Ott, A. (2007). Rheological properties of the
eukaryotic cell cytoskeleton. Physics Reports, 449(1-3):29–53.
Rasband, W. S. (1997-2006). Imagej. U.S. N.I.H, Bethesda, Maryland.,
http://rsb.info.nih.gov/ij/.
Raucher, D., Stauffer, T., Chen, W., Shen, K., Guo, S., York, J., Sheetz, M.,
and Meyer, T. (2000). Phosphatidylinositol 4,5-bisphoshate functions as
a second messenger that regulates cytoskeleton-plasma membrane adhesion.
Cell, 100(2):221–228.
Reigada, R., Mikhailov, A. S., and Sagués, F. (2004). Nonequilibrium orientational
patterns in two-component langmuir monolayers. Physical review
E, Statistical, nonlinear, and soft matter physics, 69(4 Pt 1):041103.
Rentsch, P. and Keller, H. (2000). Suction pressure can induce uncoupling of
the plasma membrane from cortical actin. Eur. J. Cell Biol.
Rosenblatt, C., Pindak, R., Clark, N., and Meyer, R. (1979). Freely suspended
ferroelectric liquid-crystal films: Absolute measurements of polarization,
elastic constants, and viscosities. Phys Rev Lett, 42(18):1220–1223.
Ryskin, G. and Kremenetsky, M. (1991). Drag force on a line defect moving
through an otherwise undisturbed field: Disclination line in a nematic
liquid crystal. Phys Rev Lett, 67(12):1574–1577.
Sancho, J., Miguel, M., Katz, S., and Gunton, J. (1982). Analytical and numerical
studies of multiplicative noise. Phys Rev A, 26(3):1589–1609.
Sekimoto, K., Prost, J., Juelicher, F., Boukellal, H., and Bernheim-
Grosswasser, A. (2004). Role of tensile stress in actin gels and a
symmetry-breaking instability. Eur Phys J E, 13(3):247–259.
Sheetz, M. P. (2001). Cell control by membrane-cytoskeleton adhesion. Nat
Rev Mol Cell Biol, 2(5):392–6.
Sheetz, M. P., Sable, J. E., and Döbereiner, H.-G. (2006). Continuous
membrane-cytoskeleton adhesion requires continuous accommodation
to lipid and cytoskeleton dynamics. Annual review of biophysics and
biomolecular structure, 35:417–34.

REFERENCES 143
Sit, P., Spector, A., Lue, A., Popel, A., and Brownell, W. (1997). Micropipette
aspiration on the outer hair cell lateral wall. Biophys J, 72(6):2812–
2819.
Stone, H. (1995). Fluid motion of monomolecular films in a channel flow
geometry. Physics of Fluids.
Svenˇsek, D. and ˇZumer, S. (2002). Hydrodynamics of pair-annihilating
disclination lines in nematic liquid crystals. Phys Rev E, 66(2):21712.
Tabe, Y., Shen, N., Mazur, E., and Yokoyama, H. (1999). Simultaneous observation
of molecular tilt and azimuthal angle distributions in spontaneously
modulated liquid-crystalline langmuir monolayers. Phys Rev
Lett, 82(4):759–762.
Tabe, Y. and Yokoyama, H. (1994). Novel liquid crystalline structure in langmuir
monolayers of amphiphilic azobenzene derivative. Journal of the
Physical Society of Japan.
Thoumine, O. and Ott, A. (1997). Time scale dependent viscoelastic and
contractile regimes in fibroblasts probed by microplate manipulation. J
Cell Sci, 110 ( Pt 17):2109–16.
Tóth, G., Denniston, C., and Yeomans, J. (2002). Hydrodynamics of topological
defects in nematic liquid crystals. Phys Rev Lett, 88(10):105504.
Trepat, X., Deng, L., An, S. S., Navajas, D., Tschumperlin, D. J., Gerthoffer,
W. T., Butler, J. P., and Fredberg, J. J. (2007). Universal physical
responses to stretch in the living cell. Nature, 447(7144):592–+.
van Oudenaarden, A. and Theriot, J. A. (1999). Cooperative symmetrybreaking
by actin polymerization in a model for cell motility. Nat Cell
Biol, 1(8):493–9.
Vilfan, A., Frey, E., and Schwabl, F. (1998). Elastically coupled molecular
motors. Eur Phys J B, 3(4):535–546.
Wilhelm, J. and Frey, E. (2003). Elasticity of stiff polymer networks. Phys
Rev Lett, 91(10):108103.
Wottawah, F., Schinkinger, S., Lincoln, B., Ananthakrishnan, R., Romeyke,
M., Guck, J., and Käs, J. (2005). Optical rheology of biological cells.
Phys Rev Lett, 94(9):098103.