Modeling cell interaction with the stiffness of the
extracellular matrix
Novikova, E.A.
DOI:
10.6100/IR782821
Published: 01/01/2014
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Novikova, E. A. (2014). Modeling cell interaction with the stiffness of the extracellular matrix Eindhoven:
Technische Universiteit Eindhoven DOI: 10.6100/IR782821
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Modeling Cell Interaction
with the Stiffness of the
Extracellular Matrix
Elizaveta Novikova
A catalogue record is available from the Eindhoven University of Technology Library
ISBN: 978-90-386-3749-5
Cover design: Elena Chochanova www.lateralstripe.com
Printed by Ipskamp Drukkers
c
Copyright 2014
by E.A. Novikova
This work is part of the research programme of the Foundation for Fundamental
Research on Matter (FOM), which is part of the Netherlands Organisation for
Scientific Research (NWO).
Modeling Cell Interaction
with the Stiffness of the
Extracellular Matrix
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de
Technische Universiteit Eindhoven, op gezag van de
rector magnificus, prof.dr.ir. C.J. van Duijn, voor een
commissie aangewezen door het College voor
Promoties in het openbaar te verdedigen
op maandag 15 december 2014 om 14.00 uur
door
Elizaveta Novikova
geboren te Leningrad, USSR
Dit proefschrift is goedgekeurd door de promotoren en de samenstelling
van de promotiecommissie is als volgt:
voorzitter:
prof.dr.ir. G.M.W. Kroesen
promotor:
prof.dr. M.A.J. Michels
copromotor: dr. C. Storm
leden:
prof.dr. C.V.C. Bouten
prof.dr. G.H. Koenderink (VU)
dr.habil. A. Muntean
prof.dr. Th. Schmidt (UL)
prof.dr. U.S. Schwarz (University of Heidelberg)
reserve:
prof.dr.ir. P.P.A.M. van der Schoot (UU)
“One day I will find the right words,
and they will be simple.”
– Jack Kerouac
vi
Contents
1
2
Introduction
1.1 Cells in the ECM . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Mechanical interaction of ECM and the cell: main
players . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Function of mechanical cues in cell life . . . . . . . . . . . .
1.2.1 Adhesion through integrins: why such abundance? .
1.2.2 Cell migration . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Mechanosensing . . . . . . . . . . . . . . . . . . . . .
1.2.4 Goals and organization of this thesis . . . . . . . . .
1
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3
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17
Models and Methods
2.1 Experiments, theory and modeling of mechanics-dependent
processes inside the cell . . . . . . . . . . . . . . . . . . . . .
2.1.1 Applying strain to heterodimers: integrins in action
2.1.2 Catch bonds . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Modeling catch bond . . . . . . . . . . . . . . . . . . .
2.1.4 Modeling cell adhesion . . . . . . . . . . . . . . . . .
2.1.5 Directed migration of cells . . . . . . . . . . . . . . .
2.1.6 Durotaxis . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Models and methods used in this thesis . . . . . . . . . . . .
2.2.1 Bell model . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Stochastic framework . . . . . . . . . . . . . . . . . .
2.2.3 Gillespie algorithm . . . . . . . . . . . . . . . . . . . .
2.2.4 Correlated random walk . . . . . . . . . . . . . . . . .
2.2.5 Measuring directional migration . . . . . . . . . . . .
18
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3 Catch Bonds Under Force
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Single catch bond characteristics . . . . . . . . . . . . . . . .
3.3 Catch bond cluster: fixed force . . . . . . . . . . . . . . . . .
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Integrin Mixtures: Catch and Slip Bonds
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
4.2 Single bond characteristics . . . . . . . . . . . . . . .
4.3 Mixed cluster: fixed force . . . . . . . . . . . . . . .
4.4 Equilibrium solutions: mixed cluster . . . . . . . . .
4.5 Cluster lifetime: catch bonds reinforcement? . . . .
4.6 Integrins in the adhesion: lateral diffusion . . . . .
4.7 Two-spring model, diffusion and substrate stiffness
4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
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3.4
3.5
3.6
3.7
3.8
4
5
Stochastic simulations . . . . . . . . . . . . . . .
Cluster lifetimes: uniform loading . . . . . . . .
Cluster lifetimes: asymmetric loading . . . . . .
Loading by motors pulling on actin stress fibers.
Conclusions . . . . . . . . . . . . . . . . . . . . .
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Spatial Organization of Integrins inside Focal Adhesion
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Integrins in adhesion: basic properties . . . . . . . . .
5.3 Interaction of integrins: simulations . . . . . . . . . .
5.4 Interaction of integrins: equilibrium values . . . . . .
5.5 Interaction of integrins: spatial distribution . . . . . .
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
6 Durotaxis
6.1 Introduction . . . . . . . . . . . . . . .
6.2 Persistent migration: correlated walk .
6.3 Uniform persistence: simulations . . .
6.4 Gradient persistence: simulations . . .
6.5 Biased walk: basic characteristics . . .
6.6 Biased walk: simulations . . . . . . . .
6.6.1 BRW on uniform persistence .
6.6.2 BRW on gradient persistence .
6.7 1D model of durotaxis . . . . . . . . .
6.7.1 Nonuniform persistence . . . .
6.8 Conclusions . . . . . . . . . . . . . . .
viii
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7
Conclusions and Outlook
115
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
List of Abbreviations
123
Bibliography
125
Curriculum Vitæ
137
List of Publications
139
Summary
141
Acknowledgements
145
ix
x
Chapter 1
Introduction
This chapter intends to introduce the reader to the general topic of this thesis: the
interactions that cells have with the mechanical properties of their extracellular
environment. Here we first motivate and outline the overall goals of this work.
We then present an overview of the "main players"; the structures inside and
outside the cell that take part in the interactions with the extracellular matrix.
After reviewing these structures, we describe the biological functions they perform, how they perform these, and present an overview of some evidence - recent
and classic - that shows the crucial role that extracellular mechanical properties
play in determining cellular behavior and, even, fate.
1
2
Introduction
1.1 Cells in the ECM
Very few, if any, living organisms can thrive in the absence of interactions
with their surroundings. Clearly, the environment is essential to survival:
among many, many other functions it is a source of nutrients, a recipient
for waste, provides information in the sense of biochemical signals, may
offer mating opportunities, and so on. Most of these roles hold, virtually
unaltered, at the scale of entire organisms as well as that of individual
cells. For cells inside our bodies, the environment consists, typically, of
extracellular matrix (ECM) material, and many of the clues upon which
cells rely in their decision making processes are imparted to them through
this ECM. So, for each cell, too, processes like proliferation, differentiation
and death depend on how well the cell is able to sense external conditions
of and through the ECM: chemical conditions, temperature but also mechanical properties. The cell must react accordingly and appropriately in
order to properly perform its individual and specifically intended set of
physiological functions inside the organism.
While cells are aware of a multitude of physical characteristics of their surroundings, the sensory apparatus (as well as the downstream processing)
for each differs greatly. For many chemical signals, elaborate biochemical
pathways have been uncovered which are subtly but reliably indicative of
the presence and abundance of extracellular ligands which, accordingly,
may be sensed and processed internally by the cells. Of more direct interest to the physicists, such as us, however is the sensory mechanism that
allows cells to perceive the mechanical characteristics of its surroundings.
Because this is not, in its origin, a chemical property but rather a mechanical one, the process of mechanosensing (a cell’s capability to process
and interpret external mechanical cues) must involve, necessarily, a transduction step where the mechanical signal is somehow converted into a
chemical signal, which may lead to the differential expression of a gene.
This field is not as young as one might think - already in 1926 it was
reported [1] that the migration of fibroblasts is tension-sensitive. More recent examples of the role of mechanics include their ability to guide cell
proliferation [2], migration [3], morphology [3, 4] and even stem cell dif-
1.1 Cells in the ECM
3
ferentiation [5, 6]. The interaction is also very much a two-way process:
the extracellular space is constantly manipulated, and even largely produced, by its inhabitant cells. A wonderful example of cells controlling
their environment mechanically is [7], which shows osteoblasts - the cells
responsible for remodeling of bone - actively and irreversibly compact
collagen gels; a collective process that undoubtedly is used as a group
communication channel of some sort. The responsiveness to mechanical
properties also suggests promising avenues to control the behavior of cells:
By controlling the mechanical component of the ECM, one is for instance
able to direct non-viral gene delivery to the cell [8].
In short, cells are acutely aware of the mechanical properties of their surroundings, and many crucial decision are instructed by such properties.
Those mechanical properties are ultimately a purely physical property
of the environment (for sure, one that may be informed by the chemical
composition and architecture of the environment, but still one that is read
out in a physical manner), and to sense them requires a physical component that is sensitive to stress, strain, stiffness or the direction of stiffness
gradient.
This is the central topic of this thesis: we will attempt to identify and
model molecular mechanisms that allow cells to internally represent, in a
manner amenable to further biochemical processing, the mechanical properties of the extracellular environment. In addition, we will discuss and
model the implications of such processes for one cellular behavior that is
very directly and manifestly sensitive to mechanical properties: cell motility.
1.1.1 Mechanical interaction of ECM and the cell: main players
The first main player is not so much a molecule as it is a concept: stiffness.
Stiffness is a material mechanical property that is determined ultimately
by the constituents and the architecture of a material, but is experienced
purely as a mechanical response coefficient. Throughout this thesis, we
will use the Young’s modulus (Y, dimensions N/m2 = Pa) to quantify the
4
Introduction
L
A
ΔL
F/A
F/A
FIG. 1.1 Cartoon of a piece of material with the length L and cross-section
area A. Tensile force F stretches the material with a stress F/A and the material
elongates by ∆L, experiencing tensile strain of ∆L/L. Young modulus then is
defined as Y = ( F/A) / (∆L/L).
stiffness of a material. For a material that responds linearly, Y is defined
as
Y=
stress
.
strain
(1.1)
See also Fig. 1.1 for illustration. When a material is very stiff it will strain
very little, even for large stresses. Conversely, a soft material will exhibit
relatively large strains even at moderate stress. In practice, the Young’s
modulus (as well as many other and often more fully descriptive ways to
quantify mechanical response) may be measured using micro-indentation
of cellular structures with an AFM-tip, bulk (dynamical) rheology on biological materials, by pulling on the individual polymer fibers, or by analysis of high resolution images of fluctuating filaments. The tissues in our
bodies show a range of stiffnesses, varying from roughly 100’s of Pa for
the softest tissues (found in the brain) to GPa’s for bone. Cells are thus
presented with a broad range of stiffnesses. Also, while the Y we define
1.1 Cells in the ECM
5
above is presented in the context of linear elastic response, many biological materials are very nonlinear. Often, Y (or, more precisely, its nonlinear
equivalent) will increase for higher stresses, a feature called stress stiffening. Finally, the mechanical moduli of soft materials are often strongly
frequency dependent: a collection of very different relaxation modes, each
with its own natural time scale, determine the response over time and
what appears solid at very short timescales may look more viscous over
longer times: generally, biological materials are visco-elastic. While we are
aware of many of the subtleties this introduces in the characterization of
mechanical response, we will focus on the basics in this thesis and use the
simple, linear Young’s modulus to represent external mechanical properties.
In a typical mechanosensing environment, there are three principal distinct components: The cell, the ECM and the structure that connects the
two. For general background, and future reference, we now briefly discuss the composition and mechanical characteristics of each of these three
components.
The mechanical cell
Eukaryotic cells consist of a nucleus that is embedded in cytoplasm, and
contained, together with all the other organelles of the cell, by a compliant
lipid bilayer membrane. This membrane serves not only as a border, that
limits where the cell ends and the outside of the cell begins; it is semipermeable and allows the transport of chemical compounds and other signals
to the inside of the cell. The nucleus contains the DNA and is wrapped
inside a nuclear envelope, that acts as an extra protection for the genetic
information inside it and is the origin of chemical signals to the other organelles guiding cell migration, differentiation and metabolism. The nucleus is an important mechanical entity: Extra reinforcement of the cell
nucleus is achieved by recruiting polymer fibers, mainly lamins, which
make the nucleus significantly stiffer than the rest of the cytoplasmic content: The stiffness of a nucleus is 2-10 times bigger then the organelles of
the cytoplasm and measures in the order of 0.1 to 10kPa [9].
Outside the nucleus is a dynamic multifunctional network of filaments
6
Introduction
called the cytoskeleton. Its three main filamentous constituents are intermediate filaments that provide structure of the cell, microtubules providing signaling and transport, and actin filaments. The latter, together
with myosin motors and crosslinking proteins (e.g., α-actinin) assemble
into various forms in the cytoskeleton, forming disordered networks in
the bulk, quasi-2D laminar structures that form the cortical actin layer
directly below the cell membrane, but actin also aggregates into highly
dynamic actin cables that can actively contract, producing forces in the
range of 1 nN (i.e, on the order of a thousand times higher than what single molecular motors can muster). Mechanically, a single actin polymer
chain possesses a linear stretch stiffness of about of 10’s of pN/nm for 1
µm fibers [10], while their persistence length is about 17 µm. Very coarsely
characterized, the cytoskeleton is a heterogeneous polymer network with
a Young’s modulus in the range of 100-1000 kPa [11]. The stress cable
network (together with the entire acto-myosin aggregate inside the cell)
provides the driving force for cell motility: motors pulling on actin filaments transfer their collectively generated forces through ECM adhesions
to the outside of the cell, and through differential binding and unbinding
at the front and rear ends cells move. Finally, the stress fiber network inside a cell is often strongly polarized, and the direction of this polarization
determines the direction of motion on a substrate [12].
Extracellular matrix
All structures in our body - from bone/connective tissue (Y ∼ GPa [13])
to brain (Y ∼ 0.3 − 1kPa [14]) - owe their mechanical properties mainly to
polymer networks, and the ECM is no different. Extracellular matrix consists chiefly of fibrillar proteins: collagen and elastin, and other fibrillar
structures such as fibronectins and laminins. Filling up the space between
these fibers is the so-called ground substance; a dense gel containing numerous proteoglycans, glycoproteins and glycosaminoglycans. While cells
will generally not develop adhesions on non-biological substrates, substrates may be coated with ECM proteins (often, collagen, vitronectin or
fibronectin) to promote in vitro adhesion. As with the cell’s inside, the
stiffness of the ECM can be measured with rheology experiments, where
a piece of tissue material is fixed between two rotating axially symmetric
1.1 Cells in the ECM
7
plates allowing the dynamic storage and loss modulus to be measured.
The loss modulus corresponds to the (frequency times the) viscosity of
the material, and the storage modulus reports on the elastic properties
of polymer networks. In experiments that probe the cell’s response to
mechanically distinct environments, one may use polyacrylamide (PAA)
or polydimethylsiloxane (PDMS) gels with varying crosslinker density to
sensitively control the modulus of the substrate, while aiming to keep the
adhesive properties constant using uniform (but mechanically, as much
as possible, inert) coatings of collagen or fibronectin. Recent experimental
works were able to match the stiffness of artificially produced tissue to
almost every stiffness that cells may encounter in our body(0.3 − 80kPa
range in [15] and [16]). On an upper limit of stiffness are the coverslips
made of glass (Y ∼ 10sGPa) which are used along with the coverslips
made of plastic for cell culture.
Cell adhesion
Cells can deform the ECM [7], which means that there is a mechanical
connection between the contractile actin stress fibers and the extracellular
matrix. The mechanical connection of the cell and the outside world happens at an intricately structured adhesion locus called the focal adhesion
- the intra-extracellular structure that provides the physical link between
the actin cytoskeleton and the ECM.
In general, a cell’s ability to bind to extracellular entities is vital, which
is probably why cells have such a wide array of abilities to connect to
other things, each with dedicated protein families to ensure optimal binding. Cells adhere to each other using cadherins, to antigens (responsible
for, for instance, the specificity of the immune response) using antibodies,
they adhere to carbohydrates (particularly, sugar containing molecules)
using selectins and finally, they adhere to the fibrous components of the
ECM using integrin proteins. In this work, we will be particularly interested in the latter family, which has a molecular mass of around 250 kDa
[17]. Integrins form the actual connection between inside and outside, but
are part of a much larger adhesive structure called the focal adhesion. As
may be seen in Fig. 1.2, the integrins are in fact dimers. They consist of an
8
Introduction
Cell
Actin
fiber
Cell membrane
ECM
FIG. 1.2 A schematic representation of the organization of cell adhesion, picture inspired by [18]. The integrins (red) directly connect the intracellular focal
adhesion(green) to the ECM outside, and transfer forces generated within the
actin stress bundles (blue), which are bound together by α−actinin(purple), to it.
α and a β subunit, both of which are transmembrane structures that, together, connect to an ECM ligand outside and FA-proteins including talin,
vinculin as well as over a hundred other signaling proteins [19], which
ultimately connect the extracellular matrix, via the integrin, to the contractile machinery of the cell. Focal adhesion provides not only structural
adherence, but is also generally thought to serve as the origin of the signaling pathways that drive the maturation of nascent focal adhesion [20],
actin polymerization [21] and force generation [22]. While the structure
of the adhesion is well studied in experiments like [18], the interaction
between the proteinaceous elements of the adhesion (most importantly,
the integrins) and the contractile intracellular elements are the subject of
much current research [23, 24, 25]. To achieve the goals of this thesis which were first and foremost to identify and model molecular mechanisms that allow cells to internally represent the mechanical properties of
the extracellular environment, we will initially mostly focus on modeling
the behavior of the clusters of integrins under force, but later comment
briefly on their implications and possibilities for signaling.
1.2 Function of mechanical cues in cell life
9
1.2 Function of mechanical cues in cell life
Having introduced to the stage some of the major biochemical components that feature in this work, we will now briefly discuss the mechanical
functions each carries out, and how those are achieved molecularly.
1.2.1 Adhesion through integrins: why such abundance?
Integrins are transmembrane heterodimeric units that provide a direct
connection between the cell and the ECM. There are 24 types of integrins
in mammals, formed from 18 α and 8 β subunit chains [26]. Depending on the type of integrin, they bind to fibronectin, vitronectin or other
extracellular proteins, often possessing the ability to bind not only to a
specific ligand, but to several ligands [27]. Not only mammals, but even
the simplest metazoa - sponges express integrins [26]. Bacteria do not express integrins, and they adhere to integrins through their own adhesion
components, e.g., invasin [28]. Recent experiments, where the types of integrins expressed by cells were controlled [29], have shown that certain
integrin types are responsible for adhesion stability. It was shown that
integrin type may determine the migration of motile cells [30, 31]. Different types of integrins act in concert inside a signaling network to provide
rigidity sensing[21, 32]. Adhesion composition is highly relevant for cell
migration: Inhibiting the action of certain integrin types can impede migration as a whole, or may alter the persistence of cellular migration. In
[33] it is shown that disruption of the αv β 3 integrin function resulted in
short-term (up to 5 hours) loss of persistence, while the inhibition of α5 β 1
integrin binding to Rho kinase (ROCK) was shown, instead, to increase the
persistence of cell migration. Experimental works involving disruption of
integrin function in mice report problems in development or postnatal
functioning for almost every integrin type knockout [26]. Although the
interactions and functioning of integrin types remain the topics of ongoing research, we safely conclude that the abundance of integrin types is
necessary for an effective functioning of a multicellular organism.
10
Introduction
1.2.2 Cell migration
While some cells in our body remain largely or completely immobile during their lives, both the proper functioning (e.g., wound healing, immunity) and malfunctioning (e.g., cancer metastasis) of cells in our bodies is
highly dependent upon the capacity to migrate of certain types of cells.
In the case of wound healing, proper functioning might include faster
motion, while the spreading of cancer cells is a situation in which one
might hope to stop malignant cells from extravasating through the vessel walls into our tissues, to halt or impede metastasis. Precisely what
prompts cell migration varies greatly between different physiological situations, but a common trigger is some gradient of chemical signals, in
which case we call the resultant motion chemotaxis: the ability of cells to
move in response to a chemical (concentration) gradient. Such tactic motion, where cells are prompted by gradients in some external condition
to move in one direction or the other, is much more common: it can also
be triggered by the extracellular flow, density of ligands that cells can adhere to (haptotaxis), by gradients in incident light (phototaxis), or even
by the earth’s gravitational field (gravitotaxis which, surprisingly, also includes upward motion in the gravitational field). As well as moving up the
gravitational field, cells were also found to move up the fluid flow [34].
Although it was shown that, in 3D environments, some cells (leukocytes,
in this case) may migrate without the use of any adhesion just by pushing
outward [35] analogously to microswimmer "pushers", most cell motility
studies place cells on 2D substrates to which the cells adhere using integrins, and transfer forces generated within the cytoskeleton to propel
themselves around. Such integrin-based cell migration involves a cycle
consisting of placing the cytoplasm in the direction of migration, sticking
the forward part (lamellipodium) of the cell to the surface, and contracting the cytoskeleton to release the rear part of the cell. This way cells can
develop migratory speeds of the order of µm/s. Thus, there are three distinct types of fibers that cells employ during the migration: the dendritic
actin in the lamellipodium, the largely uniform cytoskeleton structure in
the bulk of the cell, and finally the polarized contractile actin fibers in the
1.2 Function of mechanical cues in cell life
11
cytoplasm. The latter are the fibers that are responsible for the direction
of cell migration [12].
1.2.3 Mechanosensing
Mechanosensing is the ability of a cell or a tissue to detect the imposition of
a force [11]. Mechanostransduction is the ability of cells to convert such mechanically sensed signals into chemical activity, such as gene expression,
or protein production. The two are intimately linked, as the only measurable changes (i.e., those that produce a signal that may be measured
experimentally) are changes in the behavior of the cell, which presupposes
the presence of a transduced signal. In other words, the sensory mechanism itself need not be directly measurable but the outcome, after transduction, usually is. Indeed, when placed on substrates of varying rigidity
cells may change their phenotype and the morphology of their organelles
[36]: a measurable, but biochemical, response to changes in mechanics
that indicates some sensory functionality is present. Sensing mechanics
can be done on many levels inside the cell, as is elegantly described in
[11] (see also Fig. 1.3), whose reasoning we follow below. Firstly, a part of
the extracellular matrix may be sensitive to tension. An example of this is
fibronectin, a molecule that, when stressed, unfolds to expose previously
hidden binding sites (so called cryptic domains). These sites help it to
form fibrils with its homologous domains, which in turn promote integrin
activation. Secondly, the integrins or adhesion plaque proteins can react to
stress by adopting an extended conformation, providing sensing through
the conformation of the integrins. Finally, stresses may be transferred all
the way to the nucleus, which may be directly mechanically activated to
generate chemical signals, guiding cell behavior. For example, the nucleus
was found to react directly to the external stiffness by scaling the abundance of lamin-A, a nuclear reinforcement filament, in direct accordance
with an increased external stiffness [37].
In general, we may ask what the essential requirements on the design of
mechanical property sensor should be. First, the contractile cell must be
able to strain (deform) its substrate; this is the input. Second, it must be
12
Introduction
FIG. 1.3 A cartoon of a cell(black circle) with a nucleus inside it(green circle), which is connected to the extracellular rigid structure(blue spring) through
the intracellular filaments(red spring). The point of connection is the focal adhesion. Mechanosensory proteins(stars) may occur anywhere along the mechanically connected pathway from outside to inside: extracellular(blue), focal adhesion(red) or intracellular(green) proteins may unfold in response to stresses
or strains and thus report on the mechanical environment (adopted from [11]).
How stiffness sensing is achieved, though, is less clear. In Chapter 3, we propose
a new mechanism where the fraction of bound integrins may reflect the external
stiffness in the vicinity of the focal adhesion, and act as a mechanical property
sensor.
1.2 Function of mechanical cues in cell life
13
able to record the stress; this is the output. In order, however, to be able
to sense gradients it must be able either to compare to some fixed output,
or to operate under conditions of constant input. In the latter case, for the
same input, a stiffer substrate provides an output that will be significantly
different from a soft substrate and thus comparative sensing can be made
possible.
The input must be independent of the stiffness of the substrate, as sensing action must provide this information during the first time that a cell
attaches itself to a substrate of unknown stiffness. In several recent studies, an interesting candidate for such a constant input is suggested: [21]
and more recently [38] measure the traction stress and the mechanical
work exerted on the substrate to be independent of the substrate’s Young’s
modulus - we will make use of this in our modeling to come.
1.2.4 Goals and organization of this thesis
As stated before, our goal is twofold. First; to identify and model molecular mechanisms that allow cells to internally represent, in a manner
amenable to further biochemical processing, the mechanical properties of
the extracellular environment. Second; to assess the implications of such
sensory processes for cell motility.
A large part of this work is focused on simulations of focal adhesions of
cells adhering to the extracellular matrix. In these simulations, we pay
particular attention to the role and force-dependent behavior of the integrins. As we shall see, certain integrin-ligand pairs can form catch-bonds bonds that have a lifetime that increases with increasing force. Opposite to
the conventional slip- bond that tends to dissociate faster under increasing tension, moderate amounts of tension greatly promote the stability
of catch-bonds. We explore the behavior of multiple catch-bonds and the
mixture of catch- and slip bonds under force. This will result in a novel
proposal for a molecular mechanisms to sense and transduce forces and,
importantly, also stiffnesses.
14
Introduction
To address our second objective, the second part of this work is devoted
to the phenomenon of durotaxis - the movement of a cell up a gradient of
substrate stiffness. We analyze and interpret experimental findings, which
show a dependence of the persistence of motion on substrate rigidity. This
may be related to the physical characteristics and force-dependent lifetime
of focal adhesions, and we show how, based on this property of cellular
movement, a novel model for cell durotaxis arises. We discuss the statistical properties of this motion, and how it may be compared to (and
differentiated from) regular directed diffusive cell motion.
This thesis is organized as follows. In Chapter 2 we recall and summarize the main theoretical methods and models that we use in this thesis.
In Chapter 3 we investigate how the focal adhesion, modeled as a cluster
of catch-bonds, reacts to the force, exerted upon it. We incorporate experimental parameters in our model, in a way that allows us to construct
a mechanical property sensor out of a focal adhesion under tension. We
propose how the stiffness of the extracellular matrix can influence the internal properties of the focal adhesion, and show that the resulting system
possesses the requisite properties of a stiffness sensor.
In Chapter 4 we present a more realistic model of a focal adhesion, where
along with the catch bonds slip bonds, too, share the loading force. This results in a force-dependent dynamic behavior. We propose how the macroscopic integrin movement (i.e., diffusion and binding/unbinding kinetics)
inside a focal adhesion depend on the stiffness of the extracellular matrix
and the external force, and suggest ways in which these findings may be
verified experimentally.
In Chapter 5 we describe how the interaction of integrins in mixtures
containing more than one type of integrin affects the spatial distribution
inside a focal adhesion. We explore how lateral coupling between the integrins influences the equilibrium fraction of bound integrins. We show
how two order parameters, quantifying the binding and the distribution
of the integrin mixture depends on the amount of stress it is exposed to,
suggesting physical routes leading to the phenomenon of integrin clustering that is widely reported experimentally.
In Chapter 6 we introduce the phenomenon of durotaxis and show that
1.2 Function of mechanical cues in cell life
15
stiffness-dependent persistence can cause apparently durotactic behavior.
We examine the experimental findings on durotaxis and discuss how those
findings can be interpreted in order to determine whether differential persistence might in part explain the tendency of cells to migrate from softer
to stiffer substrates, perhaps in some combination of gradient - sensing
and stiffness-dependent movement.
We end this work with Chapter 7, which presents our main conclusions
and an outlook.
16
Introduction
Chapter 2
Models and Methods
This chapter gives the overview of main experimental findings that are closely
related to aspects of cell- ECM interactions investigated in this thesis. Here we as
well introduce models and methods that we employ throughout the thesis.
17
18
Models and Methods
2.1 Experiments, theory and modeling of mechanicsdependent processes inside the cell
2.1.1 Applying strain to heterodimers: integrins in action
Integrin binds noncovalently to its ligand. The approach of the ligand influences the integrin in a way that it takes extended conformation [39]
the same activation effect may be achieved by altering the extracellular Ph
[40](see also Fig. 2.1). The characteristic property of integrin-ligand bond
is a force-lifetime curve, that shows how long on average it takes to rupture the bond for each of the values of constant force applied to it. In [41]
such curve was measured experimentally for α5 β 1 − FN pair. In their setup
the atomic force microscope(AFM) tip covered with fibronectin(FN) was
brought in contact with integrin-enriched coverslip. From deflection of
the AFM tip the force-lifetime curves were measured. Other experimental
work quantifying integrin- ligand interactions([40]) uses so-called forceramp experiments. There the mean unbinding rate is measured, depending on the retraction speed of the AFM cantilever tip. Force-ramp experiments may be more coinciding with the events that take place during cell
adhesion than constant force measurements. Knowing enough data, one
can transform the result of one experiment into the other, as it is described
in [42]. Most of the bonds between integrins and ECM components can
support the load of 100 s to 1000 s pN, though the strength αv β 3 -vitronectin
bond can reach slightly above 1000pN [43]. Recent work [44] suggests that
integrin-ligand force (α5 β 1 − FN) may be measured by comparing it to one
of the strongest bonds known in cell adhesion - streptavidin-biotin bond.
2.1.2 Catch bonds
Suppose, one wants to break a bond between the two molecules. We expect that most of the bonds known to us will break faster when there is
a mechanical force, pulling them apart. The higher is the magnitude of
the force, the faster a bond, pulled apart by this force, breaks. This intuitive way of looking at a bond fails when catch bond comes around.
2.1 Experiments, theory and modeling of mechanics-dependent processes
inside the cell
19
Cartoon, illustrating integrin conformational changes in responce to
the extracellular Ph: at low Ph(left) integrin is in a closed conformation and
the ligand biding site is not available, when the acidity rises(right), integrin
takes extended conformation and becomes available to bind the suitable ligand(illustration from [40]).
FIG. 2.1
Catch bond is a bond that gets reinforces by a strain that was aimed to
take it apart. Initially it was a theoretical speculation by [45], who compared it to a children toy nowadays called ’Chinese finger trap’. ’Chinese
finger trap’ captures you when you insert your finger inside it and tightens the grip when you try to pull your finger out. The way to escape
’Chinese finger trap’ is to let go of the tension. The recent experimental
evidence confirms that catch bonds are formed during several processes
in our body. Collective bond-strengthening effects were observed in Lselectin based adhesion of neutrophils [46], then in bacterial FimH-based
adhesion [47]. In search for the explanation for this collective effect, the
discussions in literature proposed two possible reasons for strain-induced
adhesion reinforcement: force-induced rebinding of new bonds, and forceincreased lifetime of individual bonds ([48], [49]). Soon the experimental
results presented in [50] came about with the direct observation of catchbonds formed by P-selectin-covered AFM tip with ligand-coated surface.
Later individual P-selectins ([51]), L-selectins ([52]), FimH ([53]) and finally integrins ([41]) were found do form catch-bonds. The main models
20
Models and Methods
describing catch bonds are treated in section 2.1.3.
2.1.3 Modeling catch bond
Firstly, as it was already mentioned in [54], models can be evaluated by
comparing them to experiments, but every theoretical model represents
an experimentally observed protein-ligand interaction with a certain finite level of detail. As a result of such modeling, experimental data meets
the quantitative description of observed dependencies and the predictions
on the behavior of catch-bonds outside the measured parameter space are
made. These predictions may be in their turn used to model systems containing multiple catch-bonds. Modeling helps us acquire a better understanding of basic principles, underlying catch-bond formation. We enjoy
the beauty of scientific description of this novel phenomenon, and present
a short overview of catch-bond models below.
Let us now describe the approach to modeling catch-bond interaction,
based on Kramers theory [56], that was reviewed in [57]:
Consider the energy landscape U (x), that a bound receptor-ligand conformation explores driven by thermal fluctuations. Let f denote a force,
that pulls a bond apart. Suppose for a moment, that the force is absent.
A stable bound configuration corresponds to a local minimum of U (x) in
a phase space, and the rate of escape from this equilibrium point will be
expressed as [56]:
kU = k0 e
− k∆UT
B
,
(2.1)
where ∆U is the barrier height, which the system has to cross, kB is Boltzmann constant and T is temperature of the system, and k0 is the attempt
rate, that we take to be 1s−1 . From here we are going to consider all of the
rates dimensionless, normalized with respect to k0 .
A finite external force f applied to a bond changes the position and the
height of the energy barrier. The escape rate over this barrier according to
2.1 Experiments, theory and modeling of mechanics-dependent processes
inside the cell
21
a) Schematic picture of energy landscape of a catch bond, without
force(solid lines), and deformed by the force (dashed line). Two planes illustrate
that the energy landsape is shaped around the catch barrier, xc . Dotted arrow
denotes the direction of force f , acting on the bond, which for simplicity forms
the same angle with the two planes. x1 coordinate corresponds to a bound state,
xs denotes a slip barrier. In order to obtain catch bond behavior, the height of the
barrier xc must grow with the applied force, this way causing the reinforcement
of the bond. (graph adapted from [55]). b) Illustration of candidate catch-bond
principles, from top to bottom: ligand must be brought closer to the receptor
in order to unbind; the deformation model; when a ligand gets deformed it fits
better along the receptor; the sliding-rebinding model proposes that force will
change the configuration of the system to provide a better attachment; allosteric
model proposes that pulling on a bond activates a binding domain (depicted
round in active state and rectangular in inactive state) which strengthens the
bond. Illustration from [54].
FIG. 2.2
22
Models and Methods
[58] is:
kU ( f ) = e
−∆U + f ∆x
kB T
.
(2.2)
Here the parameter ∆x shows how strongly the applied force influences
the barrier height. When ∆x > 0 the height of the energy barrier is decreased with the increase of the applied force, and corresponds to what is
called slip-bond behavior. Positive ∆x shows that the force is promoting
the increase of the barrier height and corresponds to catch-bond behavior.
Adjusting the theory above, [57] considers a bond for which slip-and
catch- unbinding pathways coexist forming a multibarrier landscape. In
such a landscape, a bond can dissociate through one of the pathways. Let
a catch-pathway follow the coordinate ∆xc , and a slip pathway follow the
coordinate ∆xs ( see Fig. 2.2a for the illustration). The energy barriers of
catch and slip unbinding without the force are ∆Us and ∆Uc respectively.
According to [57] the escape rate through such multibarrier landscape is:
k cs = e
−∆Uc − f ∆xc
kB T
+e
−∆Us + f ∆xs
kB T
.
(2.3)
The lifetime of this bond is then calculated as the reverse [59] of the unbinding rate:
−∆U − f ∆x
−1
−∆Us + f ∆xs
c
c
τ ( f ) = e kB T
+ e kB T
.
(2.4)
The model described above contains 4 parameters, and in order to achieve
catch-bond behavior it is not necessary to keep them independent, as we
show in Chapter 3.
Another more intricate model is described in [51], this was later called
two-pathway two-state model in [54]. Two-pathway two-state model suggests that a bond can be in two distinct states(catch bond state and slip
bond state), each of which will have their own rupture pathway and thus
lifetime. Finding a bond in a catch bond state becomes more probable
with increasing force acting on the bond. Rupture of a bond from catch
bond state is slower, thus the increased lifetime is achieved with the increase of the force. While the last model is more interesting and might
better describe the actual interactions between the molecules taking part
in bond formation, comparing to two-pathway model it has 2 additional
2.1 Experiments, theory and modeling of mechanics-dependent processes
inside the cell
23
parameters. Two-state two-pathway model can under certain conditions
be matched to two-pathway model, as it is excellently explained in [60]. In
this thesis we use the two-pathway model from [55]. We chose this model
due to its rigorous character and the necessary level of detail that it provides.
A few more structurally bound graphic explanations of catch bond phenomenon are illustrated in Fig. 2.2b. Committing to one or another structurally based explanation may be possible in the future after comparing
the outcome of molecular dynamics simulations of receptor-ligand interactions to the high resolution imaging for single bonds.
2.1.4 Modeling cell adhesion
Cell adhesion is a multiprotein structure that is not only highly dynamic
on the timescale of integrin-ligand bond lifetime, but the dynamic nature
of it is necessary for cells to perform migration and sensing of the extracellular environment. Here, we note, that most of the cells spend their
lives embedded 3D environments, while most of the experiments on cell
culture are performed on a cover slip, and hence in 2D. It is not obvious
that the adhesion structure will not change, when cells are migrating in
2D in comparison to 3D. Some properties of cell adhering in 3D and 2D
are compared in [61] and [62], but it may still be that the structure of adhesion complexes changes from 2D to 3D environments. The community
must await for the results of high resolution imaging in 3D, similar to the
ones presented in[18] for an adhesion in 2D, in order to know the exact
difference between the two cases. Meanwhile we should keep in mind
that adhesions in 2D may be different from adhesions in 3D. The most
adhesion models found in literature match available data and therefore
are focused on treating the adhesion to the flat surface. In this section we
present a brief overview of those models and their classification.
The modeling of adhesion dates back in the 70s started out by Bell [58].
Following the discussion in [63], we would like to mention the challenge
that modeling focal adhesion faces, due to large number of different intracellular components involved in it, and the dynamic nature of it, thus no
matching between physical and biological adhesion must be done in gen-
24
Models and Methods
eral case. We still do so, as for us the quantitative reaction of the adhesion
on force is the main goal. This brings us to the next point that we would
like to make: modeling such multicomponent system should be done in
order to explain certain property, not aiming at capturing the most detailed picture. A known phenomenon is that focal adhesions perform nontrivial movements with respect to the ECM that include stick-slip motion,
growth of focal adhesion and remodeling of adhesion through short and
long cycle endocytosis of integrins. The classification of adhesion models
may be done according to the type of the adhesion dynamics/statics that
the model address. The main processes the adhesion models treat, according to [63] are: stability of stationary adhesion clusters under force, adhesion between moving surfaces, load localization in adhesion and forceinduced growth of adhesion. The model presented in Chapters 3-5 of this
thesis falls in the first group of this classification: we consider an adhesion
cluster of constant size, and investigate, how the force, applied to it will
influence equilibrium adhesion properties and adhesion dynamics.
2.1.5 Directed migration of cells
Cell migration is directed by chemical physical and other cues, for example the gravitational field. The ability to migrate is essential for the
functions of cells in our body: we would like to be able to increase the
migration of fibroblasts and T-cells, which would increase the speed of
wound healing and immunity response respectively. With cancer cells, we
would like to impede their migration, in order to protect our healthy organs from the metastasis of malignant cells. Here we describe the most
studied type of directed migration so far–chemotaxis, and compare it to
the type of migration, guided by the gradient in substrate stiffness.
Chemotaxis
Chemotaxis was discovered half a century ago in E. coli by Adler [64] in
simplest myxamebae. Since then chemotaxis was observed in wide range
of mobile species. Chemotactic behavior is a characteristic of many types
of bacteria and cells and thus there are many reasons of why and how
they perform it. T-cells perform adhesion-based migration in search for
2.1 Experiments, theory and modeling of mechanics-dependent processes
inside the cell
25
pathogens. Bacteria move in search for nutrients, where the type of movement is switched between "run" and "tumble". In shortage of food, myxamoebae spread chemoattractant to attract each other and form a colony
that will then produce spores. The spores will spread around and provide
better possibility for myxamoebae to obtain nutrients. According to [65],
all models of chemotaxis may be divided in macroscopic and microscopic
by the type of systems they treat. Macroscopic model (as an example see
[66]), treats the concentration of motile organisms as a continuous quantity and observes the evolution of concentration depending on the gradient in chemoattractant/chemorepellent. The first microscopic model was
descrived in the 50’s ( [67]). Relatively recent ’infotaxis’ strategy by [68],
treats moving cells as a collection of individually moving but interacting
particles. Microscopic modeling of chemotaxis has been a field for exercising various models of random walks. Recently a universal model for
various types of "taxis" was proposed in [69]. The generalization like this
is an interesting approach, but it is important to not forget that processes
underlying different types of "taxis" may be completely different, as we
also argue in section 2.1.6.
2.1.6 Durotaxis
Cell migration towards stiffer substrate was first mentioned in [70], where
3T3 fibroblasts were placed on a substrate with varied rigidity of 14 −
30kPa over a very short, comparing to the cell size, distance of 50 − 100µm.
When cells were reaching the gradient area from soft substrate, the character of the movement, manifested in the increase of the protrusion towards
the edge and the cell orientation towards the gradient, showed that cells
are much more in favor of going towards stiff substrate, and the opposite
was true: when cells were encountering a softer substrate, their movement
slowed down through the delay in protrusion of lamellipodium. This set
the notion that cells can prefer stiffness of certain substrates, and perform
the movement up the gradient of the preferred stiffness. Durotaxis is the
migration, guided by stiffness of the substrate in absence of gradient of
any other transport factors. The original term "durotaxis" is from latin
26
Models and Methods
which means duro -stiff and taxis- movement-migration, later durotaxis
was confirmed in [16, 71, 72]. It is worth to note that there is a clear difference between durotaxis and chemotaxis: while chemotactic bacteria adapt
to the level of the chemoattractant [73] and renew exactly the same type of
walk after initial migration up the gradient, cells on soft substrates move
distinctively different from cells on hard substrates [3]. This will be the
basis of our modeling in Chapter 6.
Modeling cell durotaxis
As durotaxis was just discovered in 2000’s, there are very few models of
durotaxis found in the literature, and here we give a brief overview of
them.
Mechanics-based modeling of cell migration was performed in [74] where
using the assumptions on the distribution of adhesive receptors or on adhesion receptor trafficking, the biphasic relation between the speed of the
cell and the receptor/ligand bond affinity(adhesiveness) was obtained. In
[74] the force, generated by the cell was considered independent on the
substrate properties.
A model of cell durotaxis in [75] presents a migrating cell as a twodimensional network of visco-elastic fibers. This network has a number
of active nodes, which are situated in the front edge of the cell. Active
forces applied only to these active nodes depend on the substrate rigidity and result in cell motility on substrates with gradient stiffness which
compares well with the experiments in [70]. Other models of cell durotaxis
are based on assumptions on dynamics of stress fiber organization. As an
example is a model described in [76], where cell durotaxis is achieved
through remodeling of stress fibers. The stress fibers attached to a stiffer
area will have a better connection to the substrate, and this way the cells
will advance towards stiffer substrates. More coarsened models start from
the assumptions on the macroscopic parameters of cell movement: speed,
distribution of turning angles, and go into minimal details on stress fiber
organization e.g. in [77]. Another macroscopic model of durotaxis is presented in [78]. We shall mention here that the movement of cell according
to those macroscopic models directly infer the "force" that will guide it
towards a preferred direction. This "force" usually coincides with the di-
2.2 Models and methods used in this thesis
27
rection of stiffness gradient, and represents the ability of a cell to sense
the local changes in stiffness. In Chapter 6 of this thesis, we avoid letting
the cell "know" the direction in which the stiffness change, and the only
information that cells receives in our simulation is the absolute stiffness of
the environment. The persistence of cell migration is stiffness-dependent
in our case and will lead to net cell displacement in the direction of the
increased stiffness.
2.2 Models and methods used in this thesis
2.2.1 Bell model
Model for multiple bonds in adhesion was first proposed by Bell [58] for
adhesion between two cells, which have type A and type B receptors respectively, diffusing inside their membranes. When two receptors that belong to different cells are at a distance smaller than the reaction radius,
they attach to each other with a binding constant r + , and a bond dissociation rate r − . Assuming that the diffusion rate is much faster than binding
constants r + and r − , Bell address the equilibrium number of bound bonds
with the reaction:
k+
−
*
(2.5)
A+B −
)
−
− C.
k−
Where C is the pair of bound receptors, k+ and k− are reverse and forward rates between bound and unbound states. The equilibrium constant
K = k+ /k− does not depend on diffusion constants. When the number
of available bonds on both cells is N1 and N2 respectively, the number of
bound pairs, Nb , evolves as:
dNb
= k + ( N1 − Nb )( N2 − Nb ) − k − Nb .
dt
(2.6)
Bell proposed that when the adhesion is pulled on by a certain force F,
F , which is expressed as:
rate k − becomes a force-dependent rupture rate k −
ΓF
F
k−
= k − e kB TNb ,
(2.7)
28
Models and Methods
where k B T is the thermal energy, Γ is the constant, showing, how much k −
is influenced by the applied force. The shared force F in Eq. 2.7 is divided
by Nb , which means that bound bonds are sharing the load. When the
number of available receptors on one cell is very large, in comparison to
the number of bound bonds, that is N2 >> Nb , we end up with the case
of adhesion of cell to the ECM components. For cell-ECM adhesion, Eq.
2.7 transforms as:
dN
= −k − ( F ) N + k + ( Nt − N ) ,
dt
(2.8)
where N is the number of bound bonds and Nt is the total number bonds
available in a cell, k − ( F ) is a force-dependent rupture rate and k + is the
rebinding rate. Deterministic equation 2.8 solved for N with the condition
dN
dt = 0 provides us with the equilibrium number of bound bonds, and
we use it in Chapter 3 of this thesis.
2.2.2 Stochastic framework
The model described in section 2.2.1 was later elegantly uprooted and
refurbished inside the stochastic framework of [79] by [80]. The modeling
in [80] is based on one-step master equation:
dpi
= r i + 1 p i + 1 + gi − 1 p i − 1 − [ r i + gi ] p i ,
dt
(2.9)
where ri and gi are reverse (unbinding) and forward (rebinding) rates
and pi (t) is the time-dependent probability of having i bound bonds. The
mean value N of the equilibrium number of bound bonds is computed by
solving:
Nt dN
dpi
= ∑i
= −hri i + h gi i ,
(2.10)
dt
dt
i =1
which for ri ( F ) and gi both linear in i averages to the same form as the
deterministic Eq. 2.8. For nonlinear rates the estimate of the solution of Eq.
2.10 can be found by substituting ri and gi with their Taylor expansion in
orders of i around N, as it is demonstrated in [79]. The mean field equation
2.2 Models and methods used in this thesis
29
can still be used when the mean fluctuation size hσ2 i = h(i − hi i)2 is small
compared to the size of the system. Stochastic framework described in
this section allowed [80] to study the timescale, on which the unbinding
of the adhesion cluster happens. Let us denote Ti as a lifetime of a cluster
with i bound bonds, then it will be expressed through the lifetimes of
neighboring states Ti−1 and Ti+1 as:
Ti = Ti+1
gi
ri
1
+ Ti−1
+
gi + r i
gi + r i
r i + gi
(2.11)
This equation expresses the lifetime of state i as the lifetime of state i + 1
multiplied by the probability of reaching state i + 1 from state i, plus the
lifetime of a state i − 1 multiplied by the probability of reaching the state
i − 1, plus the average time it takes to leave state i - the inverse of the sum
of two possible rates. The coupled system of Nt equations on Ti given by
Eq. 2.11 taken the appropriate boundary conditions has a solution which
may be written in a closed form. The lifetime of a cluster of the size Nt ,
starting initially from all of the bonds in the bound state is expressed in
[79] as:
j −1
Nt
g(k)
1 Nt −1 Nt 1
TNt = ∑ + ∑ ∑
.
(2.12)
∏
r
r
(
j
)
r (k)
i =1 i
i =1 j = i +1
k = j −i
First term in Eq. 2.12 corresponds to the cluster unbinding without rebinding, and second term is a sum over all the other ways of the cluster to
unbind, that include rebinding events. In general case, TM - the lifetime of
a cluster unbinding from a state where M bonds are bound - will change
depending on M, such a lifetime is expressed as:
M
TM =
∑
i =1
j −1
Nt
1
g(k)
1
+ ∑
∏
r i j = i +1 r ( j ) k = j − i r ( k )
!
.
(2.13)
Equation 2.13 is used throughout Chapter 3 to determine the lifetime of
a catch bond cluster. We adapt this framework for two types of bonds
sharing the force in Chapter 4 of this thesis.
30
Models and Methods
2.2.3 Gillespie algorithm
As a part of many Monte Carlo methods, Gillespie algorithm [81] allows
us to sample the phase space of the system that we simulate, and find
the points of local equilibrium. The advantage of this kinetic Monte Carlo
method and the main reason why we are using it for our system is its ability to predict not only the static equilibrium points, but also the dynamics
of the system. We set up our simulations as chain of subsequent steps. At
every step, n, we first calculate the total propensity, Pn :
Pn =
∑ ri ,
(2.14)
i
where the summation is over every possible outcome of the next move, i,
and ri is the rate at which this outcome will happen. Having done that,
we find how long it will take till any of these events occur, by calculating
the move time that will have the mean:
hτn i = exp(− Pn ) .
(2.15)
After computing the move time τn , we decide, which event will happen
during this move. In order to do this, we divide all of the possible events
into groups according to their rate and calculate the propensity for each
group. For example, consider a system consisting of a fixed number of
bonds that can bind and unbind, then the two types of moves possible
are: bond rupture and bond rebinding. The rupture of a bond corresponds
to a group a with propensity Pna , bond rebinding will then correspond to
group b, with propensity Pnb . The total propensity of the system will be
Pn = Pna + Pnb . We generate a random number s from interval (0, 1) and
choose the group of events that will act at the next step as:
(
a, s 6
b, s >
Pna
Pn
Pna
Pn
.
(2.16)
Then we simply choose a random event from this group, and make that
move, thus completing one step of our Gillespie algorithm simulation. The
number of groups can vary, as long as every group has at least one event
2.2 Models and methods used in this thesis
31
at some point in the simulation. In this thesis, we use the maximum of
5 groups of events, which are: rupture of catch/slip bond, rebinding of
catch/slip bonds, diffusion of a free bond. The beauty of the method is
that, as it was shown in [81], the time in the simulation will correspond
to the time in the real system, this way we can estimate the diffusion
coefficient of free bonds in a focal adhesion, as it is shown in Chapter 4 of
this thesis.
2.2.4 Correlated random walk
The durotaxis model, that we employ in Chapter 6 is based on the properties of individual walkers. In this section we present the main equations
that govern the behavior of discrete walks used in this thesis.
Let us first describe the behavior of a discrete walk in 1D, for what we
will follow the derivations in [82]. Consider a collection of walkers on a
line making small steps of length δ to the left or to the right every time
interval τ. The population density of the walkers going right at every time
t is: α( x, t), and the population density of walkers going left is β( x, t). The
sum of the two population densities integrated over the spatial coordinate
x must equal 1 to mark the absence of source. The walkers may change the
direction of their movement after time τ, and the probability of turning
will be r1 = λ1 τ for walkers that go to the right, and r2 = λ2 τ for walkers
moving to the left. The movement of such walkers will be governed by:
α( x, t + τ ) = (1 − λ1 τ )α( x − δ, t) + λ2 β( x − δ, t),
β( x, t + τ ) = λ1 τα( x + δ, t) + (1 − λ2 τ ) β( x + δ, t).
(2.17)
Expanding these till the first order in Taylor series around x and t gives:
∂α
∂α
∂α
∂β
= α − δ + λ1 τ α − δ
+ λ2 τ β − δ
,
∂t
∂x
∂x
∂x
∂β
∂α
∂β
∂β
β+τ
= λ1 τ α + δ
+ β + δ − λ2 τ β + δ
,
∂t
∂x
∂x
∂x
α+τ
(2.18)
32
Models and Methods
where α denotes α( x, t) and β denotes β( x, t). Dividing Eq. 2.18 by τ and
regrouping the terms results:
∂α
= λ2 β − λ1 α −
∂t
∂β
= − λ2 β + λ1 α +
∂t
δ ∂α
∂α
∂β
+ λ1 τ − λ2 τ − ,
τ ∂x
∂x
∂x
δ ∂β
∂α
∂β
+ λ1 τ − λ2 τ − .
τ ∂x
∂x
∂x
(2.19)
Assuming τ → 0 and δ/τ = v transform Eqs. 2.19 to the form:
∂α
∂α
= − v + λ2 β − λ1 α ,
∂t
∂x
∂β
∂β
= v − λ2 β + λ1 α .
∂t
∂x
(2.20)
Differentiating these equations with respect to t and to x and subtracting
them one from another given the fact that α + β = p results:
∂2 p
∂p
∂p
∂2 p
+ ( λ1 + λ2 ) + v2 ( λ2 − λ1 )
= v2 2 .
2
∂t
∂t
∂x
∂x
(2.21)
Consider the case, where λ1 = λ2 = λ > 0. Then Eq. 2.21 turns into a
simple Telegraphers equation:
2
d2 p
dp
2d p
+
2λ
.
=
v
dt2
dt
dx2
(2.22)
While the solution for p( x, t) is nontrivial, an easier to approach is the
equation for mean squared displacement, h x2 i, that is obtained from Eq.
2.22 by multiplying it by x2 and integrating over the real axis, with the
additional assumption that the first two derivatives of p vanish as | x | →
∞. Finally, we obtain
d2 h x 2 i
dh x 2 i
+
2λ
= 2v2 .
dt2
dt
(2.23)
2.2 Models and methods used in this thesis
33
The equation 2.23 can be solved for initial conditions p( x, 0) = δ( x ) and
∂p( x, 0)/∂t = 0, coinciding to h x2 (0)i = ∂h x2 (0)i/∂t = 0, resulting in:
v2
1 −2λt
hx i =
t−
.
1−e
λ
2λ
2
(2.24)
For low t we expect a ballistic-like movement, h x2 i ∼ v2 t, while for big
t we expect diffusive motion with h x2 i ∼ vt. For intermediate cases we
would then have a correlated random walk - the direction of movement of
each step will be correlated with the direction of the movement of the previous step. Quantity 1/λ is called persistence, and shows how persistent
the movement is. The resulting Eq. 2.22 includes λ as a constant, independent on x. In Chapter 6 we treat a modified Telegraphers equation where
λ depends of the current position of the walker and even for λ1 = λ2 , the
net displacement of cells towards one side of the substrate arises.
So far we have been considering equations for a 1D system. A model more
suitable for describing the trajectories of cells crawling on a coverslip is a
discrete walk on a plane, and thus in 2D. We consider so-called velocity
jump process([83]), during which a cell moves straight for certain time with
the linear speed S, and then changes the direction of its movement. Let
ϕ(∆t) denote the angle between two parts of cell trajectory, separated in
time by ∆t(see also Fig. 2.3 for illustration). The mean cosine of this angle
will then decay with ∆t as:
hcos ϕ(∆t)iT ∼ e−∆t/τp ,
(2.25)
where τp is the persistence time of a walk, longer values of τp correspond
to a more persistent walk. The mean squared displacement of a persistent
walk in 2D according to [83] is:
2
h x i = 2S
1 Notion
2
h
τp t − τp2
1−e
−t/τp
i
1.
(2.26)
analogous to Eq. 2.26 is also known from polymer physics, [84] has the same
expression for a mean radius of a walk with persistence length l p and contour length L:
h
i
h R2 i = 2l 2p L/l p − 1 − e− L/l p .
34
Models and Methods
Sketch of apparent cell trajectory, where r( T ) denotes the direction of
cell movement at time T and r( T + ∆t) denotes the direction of cell movement at
time T + ∆T. Two adjacent parts of cell trajectory form the angle θ and ϕ denotes
the angle between the two parts of cell trajectory separated in time by a delay ∆t.
FIG. 2.3
The above equation describes persistent random walk in 2D, and we employ it in section 2.2.5 when describing various ways to compute durotaxis
index.
2.2.5 Measuring directional migration
Modern microscopy under controlled environments allows to measure the
trajectories of individual cells. By analyzing cell trajectories it is possible
to conclude whether their movement is directed or random. In this section
we describe the ways in which directional properties of a 2D walk may be
measured.
Consider a trajectory of the cell, consisting of N linear segments. The displacement L of the cell is computed as:
N
L=
∑ xi ,
(2.27)
i =1
where the summation is performed over i - all of the steps in the trajectory,
and xi is the radius-vector of a trajectory segment during a step i. If the
2.2 Models and methods used in this thesis
35
cell moves on a surface with the gradient of stiffness, let the unit vector e g
denote the direction along which the stiffness changes. The projection of
displacement on the direction of the gradient is:
X = e g · L,
(2.28)
and the TI(Tactic Index) can be defined as:
TI =
X
.
|L|
(2.29)
TI shows how far did the cell go in the direction of the gradient. If TI
is positive, then the cell is performing a directional migration along the
direction of the stiffness gradient. On average for a collection of randomly
migrating cells, TI is 0.
Another expression of a tactic index was used in [16] to quantify the durotaxis in movement of vascular smooth muscle cells, we denote TI used
there as TI p to avoid confusion. The formula for TI p proposed in [83] is:
TI p =
−1
X τp t − τp2 1 − e−t/τp
,
|L|
(2.30)
where τp is the persistence time, defined in section 2.2.4. The last factor
in Eq. 2.30 is exactly the same as Eq. 2.26. The details the derivation of
TI p are omitted here and best read in [83], but we want to mention here
that the derivation of Eq. 2.30 assumes that the persistence of the walk
stays constant along the cell trajectory. This assumption on the character
of cell movement does not hold in Chapter 6, where we consider the walk,
persistence of which depends on the coordinate.
Another way of estimating the strength of durotaxis is presented in [15]
and we describe their approach below. Let L denote the number of parts
in the trajectory, during which a cell moves against the direction of the
gradient and R denote the number of trajectory segments where the cell
moves along the direction of the gradient. The durotaxis index will then
be defined as:
R−L
.
(2.31)
DI =
R+L
36
Models and Methods
The above approach is universal for any set of time-dependent coordinates
and does not require any special assumptions on the character of the walk.
For a walk that makes the steps of equal length, definitions 2.29 and 2.31
are equivalent.
Chapter 3
Catch Bonds Under Force
This chapter is about catch bonds - cellular receptor-ligand pairs whose lifetime,
counterintuitively, increases with increasing load. While their existence was initially pure theoretical speculation, recent years have seen several experimental
demonstrations of catch bond behavior in biologically relevant and functional
protein-protein bonds. Particularly notable among these established catch-bond
formers is the integrin α5 β 1 , the primary receptor for fibronectin and, as such,
a crucial determinant for the characteristics of the mechanical coupling between
cell and matrix. In this chapter, we explore the implications of single catch-bond
characteristics for the behavior of a load-sharing cluster of such bonds: these clusters are shown to possess a regime of strengthening with increasing applied force,
similar to the manner in which focal adhesions become selectively reinforced. We
propose a new mechanism, that allows cells to sense and respond to the mechanical
properties of their environment and in particular show how single focal adhesions
may act, autonomously, as local rigidity sensors.
37
38
Catch Bonds Under Force
3.1 Introduction
Most animal cells spend their days embedded in a supporting structure
called the extracellular matrix (ECM). This complex medium is an interconnected, gel-like meshwork of glycosaminoglycans (GAGs) and fibrous proteins (collagen, fibronectin) that provides structural support and anchorage to the cells, as well as mechanical integrity and resilience to the tissue
as a whole. More recently, its additional regulatory function has received
considerable attention in the literature. More than passively anchored to
it, cells actively sense [6, 85] and alter [7] the mechanical properties of their
surroundings which, in turn, may affect the fate of the cells embedded in
it: the mechanical properties of the substrate have been implicated in the
determination of the phenotype of otherwise indistinguishable stem cells
[5, 86]. These findings have spawned considerable and renewed interest
in the physical concepts and foundations underlying cell mechanosensing [87, 88, 89]: How do cells couple to the environment? What information may they glean from it (mechanosensing), how can this information
be internalized (mechanotransmission) and how is it processed (mechanotransduction)?
In this chapter, we consider the force-response of a cluster of catch bonds:
integrin-ligand bonds which display a regime of increasing bond lifetime
with increasing loads [59]. While the molecular mechanisms responsible for this behavior remain debated [90], the behavior itself is by now
firmly established and, in fact, has been demonstrated [41, 50] in individual receptor-ligand pairs. We extract from these experiments operational
parameters, and discuss collective behavior of a macroscopic assembly of
catch bonds. Our results extend previous simulations in [91]: we supply
analytical results, consider the response to loading by molecular motors
and establish a direct link between external stiffness and bound integrin
fraction.
3.2 Single catch bond characteristics
39
3.2 Single catch bond characteristics
The equilibrium binding and unbinding kinetics of unforced noncovalent
molecular bonds is generally summarized in binding and unbinding rates
k0b and k0u . The unbinding rate k0u determines the lifetime of the bond in the
absence of bias. Consider a bond that is closed at time t = 0; the probability that it is still closed after a time t has passed is Ps (t) ∼ exp(−k0u t), and
the expectation value for its lifetime is hti ≡ τ 0 = 1/k0u [59]. If an external force f is applied to the bond, unbinding is enhanced and Kramer’s
rate theory [56] (alternatively known as Bell kinetics [58]) suggests the
unbinding rate be modified as k u ( f ) = k0u exp(+ f ξ/kB T ), with ξ a microscopic length scale characterizing the unbinding transition. Consequently
the bond lifetime is shortened exponentially as τ ( f ) = τ 0 exp(− f ξ/kB T ).
Bonds satisfying this relationship are called slip bonds. Catch bonds, however, behave markedly different when forced: Their bond lifetime initially
increases when a force is applied. The single-bond lifetime of a catch bond
is maximal at some finite force, after which it decays exponentially and
recovers slip-type behavior. While initially a purely theoretical speculation [45], catch bonds are by now a well-established phenomenon. Both
the L- and P-Selectins that feature prominently in neutrophil rolling have
been shown to exhibit catch behavior. The macroscopic phenomenon of
a shear-threshold for the rolling adhesion of neutrophils is generally ascribed to a collective manifestation of individual catch bond connections
[92, 93, 94]. Recent experiments have also demonstrated catch-bond behavior in single biological bonds [41, 50] - in this chapter we present
results obtained by using numerical values for single bonds directly obtained from these experiments. Interestingly, among the receptor-ligand
pairs for which this behavior was observed is the bond between the ECM
component fibronectin FNIII7−10 and the cellular integrin α5 β 1 : the primary receptor for fibronectin [30]. Fig. 3.1 plots the lifetime-force curve
for this particular pair, clearly showing the initial rise in lifetime and the
maximum lifetime at finite force. The solid curve is a fit to the data for
the so-called two-pathway model [55], which considers unbinding of the
receptor-ligand pair via two alternative routes - one ’catch path’ that is
opposed by the applied force, and one ’slip path’ that is promoted by it.
40
Catch Bonds Under Force
ξ c and ξ s are two distinct lengthscales for the two routes, and k c and k s
the associated unforced unbinding rates. The two-pathway model predicts
the lifetime of a catch bond to depend on the pulling as
τ( f ) =
ks e
f ξs
kB T
+ kc e
− kf ξ Tc
−1
B
.
(3.1)
We are interested in the general behavior of coupled clusters of catch
bonds, and choose to work with a simpler model that captures their essential behavior: we set the two unbinding lengths ξ c and ξ s equal, denoting
the single length scale simply ξ. We use this lengthscale to set a force scale
via kB T ≡ f ? ξ. All forces are nondimensionalized using this force scale:
φ ≡ f / f ? . Furthermore, Ref. [55] argues that in order to get proper catch
bond behavior, one needs k c k s : we use this to rewrite k c = k0 exp(φc )
and k s = k0 exp(−φs ) with φs , φc > 0. Note that this is the regime in
which a lifetime peak is encountered, but that positivity of φs and φc is
not a hard constraint: in fact, the slip bond limit may be recovered by
letting φc → −∞. We may set k0 = 1s−1 without loss of generality. With
these reductions and definitions, we shall write the dimensionless force−1
dependent unbinding rate kcb
of the single catch bond
u ( φ ) ≡ ( k 0 τ ( φ ))
as
−(φ−φc )
kcb
+ e(φ−φs ) .
u (φ) = e
(3.2)
We have used this expression to fit the data from Ref. [41] for the FNIII7−10 α5 β 1 bond; the results are plotted in Fig. 3.1. For future reference, we
note that the maximal lifetime for the single bond is attained at φmax =
1
2 ( φs + φc ). We now consider the consequences of this single-bond behavior for a cluster of catch bonds subjected to a fixed external force.
3.3 Catch bond cluster: fixed force
41
tHsL
3.0
2.5
2.0
1.5
1.0
0.5
10
20
30
40
50
60
fHpNL
FIG. 3.1 The catch bond between FNIII7−10 and α5 β 1 : points represent the experimental results from [41], the solid line is a fit to the two-pathway model
from [55], Eq. 4.1, with parameters φc = 4.02, φs = 7.78, f ? = 5.38. In physical
units, this corresponds to a slip-path unbinding rate of k s = 4.2 · 10−4 s−1 , and a
catch-path unbinding rate of k c = 55s−1 .
3.3 Catch bond cluster: fixed force
Following the approach of Schwarz et al. [95] we consider a collection of
Nt bonds, and let i denote the size of the cluster at time t, i.e. the instantaneous number of closed bonds. We may then summarize the evolution of
pi , the probability of having i closed bonds at a given time t, in a one-step
master equation
dpi
= ri+1 ( Ft ) pi+1 + gi−1 pi−1 − [ri ( Ft ) + gi ] pi ,
dt
(3.3)
where ri ( f ) is the (force-dependent) rate at which one bond unbinds from
a cluster of size i, and gi is the rate at which an additional bond is formed
when a cluster of size i is already present. The rate of rebinding may
be assumed to be independent of the applied force, as the bond must
be unstressed in its unbound state. We assume a simple relation of the
form gi = k0 γ( Nt − i ), i.e. rebinding proportional to the number of available, unbound bonds with a uniform dimensionless rebinding rate γ. The
unbinding rate is where the catch bond nature of the individual bonds
¯
is injected into the model. We shall choose ri ( Ft ) = i k0 kcb
u ( f ): propor-
42
Catch Bonds Under Force
tional to the single-bond unbinding rate previously discussed, but evaluated at the force that single bonds actually experience in a cluster. The
cluster is loaded collectively, and depending on the geometry of the cell,
the alignment of the stress fiber attached to the focal adhesion, and the
structure and shape of the substrate individual bonds in the cluster may
experience very different forces. In what follows, we will mostly consider
only the simplest force distribution: a uniform distribution of the load
across all closed bonds. The reasons for this choice is twofold; Firstly,
non-uniform loading requires additional assumptions on the distribution
of forces which in general do not permit analytical treatment, and add
further adjustable parameters to the system - we strive to keep the free
parameters to a minimum. Secondly and more importantly, previous numerical work on non-uniform loading [91] has clearly delineated how
different loading configurations quantitatively modify the binding, but
mostly the unbinding, of integrin clusters. We expect that our uniform
loading configuration will be similarly modified in more complicated settings. In section 3.6 we provide one explicit and analytical confirmation of
this when we present lifetime results for a cluster that exhibits the typical
asymmetric traction force distribution reported in [96].
N
800
Ns
600
Nu
400
200
Ns
Nu
1000
2000
3000
4000
Φ
5000
Equilibrium number of bound bonds in a catch bond cluster as a
function of the applied external force Φ. We graph the two solutions branches
d
of dt
N = 0, and label the stable (solid) and unstable (dashed) branches as Ns
and Nu , respectively (see text). Different curves correspond to different values of
rebinding rate: black curve: γ = 0.1, gray curve: γ = 1. Points on these graphs
correspond to the data obtained from our Gillespie simulations.
FIG. 3.2
3.3 Catch bond cluster: fixed force
43
Assuming a uniformly distributed load, we shall choose f¯ = Ft /i. Under
these assumptions, we may evaluate the temporal evolution of the cluster
size N = hi i:
Nt d
dpi
N = ∑i
= −hri i + h gi i .
(3.4)
dt
dt
i =1
We now pass to a mean-field picture and ignore fluctuation effects, writing
the averages of the i-dependent functions ri and gi , respectively, as their
values at the average of i:
d
N ≈ −rhii + ghii = − N k0 kcb
u
dt
Ft
N
+ k0 γ( Nt − N ) .
(3.5)
In the following, we shall take t to denote the dimensionless time k0 t.
Defining the total scaled force Φ ≡ Ft / f ? , we arrive at the full mean-field
ODE that governs the evolution of a catch-bond cluster sharing a force Ft :
d
N = − N e−(Φ/N −φc ) + e(Φ/N −φs ) + γ ( Nt − N ) .
(3.6)
dt
This equation is to be considered the catch-bond equivalent of the classical
result derived for slip bonds by Bell [58]. Its validity, of course, depends
on the relative magnitude of the contribution of fluctuations. To establish
this, we have checked the ratio between the variance σ2 = h(i − hi i)2 i
and the observed mean N = hi i itself. Interestingly, since we know the
full stochastic structure underlying the mean-field system we can also
compute the variance from an ODE similar to Eq. 3.6 [79, 80]. At very
low forces, and therefore low N (roughly, below N = 10), the mean field
approximation is seen to suffer from discreteness effects. Likewise, for
very high forces the fluctuations become larger and in particular lead to
finite cluster lifetimes as we shall detail in Sec. 5. In the slip bond limit
(φc → −∞), an analytical solution may be found for the saddle node
bifurcation [58] which may be computed to occur at a critical force Φc =
Nt plog(γ/e)2 . In the present case, the implicit equations determining this
d
critical force ( Ṅ = 0, dN
Ṅ = 0) do not permit a closed-form solution. For
some key quantities, however, we do present analytical results also for
2 Function
plog(z) gives the principal solution for w in z = wew .
44
Catch Bonds Under Force
the catch bond system. The equation is a mean-field approximation to the
general, stochastic kinetics encoded by Eq. 3.3, and we will first explore
the resulting behavior at this mean field level before presenting stochastic
simulations of individual trajectories.
Eq. 3.6 is appropriate for large systems, i.e, for adhesion clusters where
many potential bonds are present. It allows us to determine the evolution of the number of closed bonds as a function of time, which of course
asymptotes to the equilibrium number of bound bonds. This equilibrium
d
cluster size is obtained numerically by solving dt
N = 0, and the results are
graphed in Fig. 3.2. This reveals the essential characteristics of the catch
bond cluster: As we increase the constant external force Φ, the number
of closed bonds (normalized in the graph to the total number of bound
bonds to yield the closed fraction) increases as well: the shared load makes
for longer living single bonds. The rebinding rate remaining constant, the
cluster is able to retain its closed bonds better at higher forces - as long
as the average force per closed bond does not exceed φmax . When it does
exceed this maximal value, the equilibrium number of closed bonds drops
rapidly as cluster unbind in avalanche-like fashion; each single bond unbinding resulting in a higher load per bond and hence (as the force per
bond already exceeded φmax ), increased unbinding.
We may rewrite the right hand side of Eq. 3.6 to yield
d
(3.7)
N = −2N cosh (Φ/N − φmax ) α−1 + γ ( Nt − N ) ,
dt
where α = exp 12 (φs − φc ) . This produces the following analytical result
for the curves in Fig. 3.2:
Φ( N ) = Nφmax ± N cosh
−1
αγ( Nt − N )
2N
.
(3.8)
This equation describes both branches in Fig. 3.2 resulting from the saddlenode bifurcation. The upper branch, which we shall call Ns (Φ), is defined
by the minus sign in Eq. 3.8 and is stable: small fluctuations in the number
of bound bonds restore N to its equilibrium value Ns . The lower branch
3.3 Catch bond cluster: fixed force
45
Nu (Φ), corresponding to the plus sign, is unstable: a slightly larger number of bound bonds causes N to shoot up to Ns , while a slightly lower
value causes the cluster to unbind (N → 0). Eq. 3.8 also yields analytical
expressions for the maximal cluster size Nmax and the force at which it is
attained, Φmax :
αγ
αγ
Nmax = Nt
; Φmax = Nt φmax
.
(3.9)
αγ + 2
αγ + 2
Several points are worth noticing here: Firstly, the force at which the maximal cluster size is attained is not simply the product of the total number
of bonds Nt and the single-bond critical force φmax - while this force is
an upper limit there is a prefactor which depends on the binding and rebinding rates, and varies continuously between 0 and 1. This factor is, in
fact, the maximal probability that a single bond is bound. Secondly: while
Φmax is a lower bound on the force Φcrit at which the cluster collectively
unbinds,
we have only found an implicit equation to define Φcrit , given by
d
= 0. Solving it shows that, depending on the value of γ, Φmax
dN Φ Φcrit
is either a good or a poor estimate for Φcrit - in the regime 0 < γ < 2
the error is no larger than 9%. We do note, however, that Φcrit is a biologically significant quantity. For instance, in rolling neutrophils the catch
bonds forged by L-, and to a lesser extent, P-selectins collectively unbind
at a critical shear stress [92, 93, 94] closely related to the critical cluster
unbinding force. A final point of potential biological significance concerns
the extent of the cluster reinforcement. By setting Φ( N ) = 0, we can determine N0 , the equilibrium cluster size at zero force. Simple substitution
yields N0 = (αγNt )(αγ + 2 cosh φmax )−1 . With this unforced occupation,
we compute the ratio between the zero-force and the maximum cluster
size:
1
Nmax
αγ + 2 cosh φmax
1 + γe 2 (φs −φc ) + eφc +φs
=
=
.
(3.10)
1
N0
αγ + 2
γeφs + 2e 2 (φc +φs )
This ratio is independent of the number of available bonds. Its behavior
as a function of φs and φc is telling: while it plateaus for increasing φs , it
may become arbitrarily big as φc is increased. Recalling the definition of
φc , we see that biophysically, this limit corresponds to a high value for
46
Catch Bonds Under Force
k c and in particular, to the limit k c k s : an optimal regime for catch
bond functionality. Note, too, that we have not assumed in our catch bond
model that k c k s ; we simply observe that only in this regime the singlebond lifetime curve displays a maximum at finite force. The present analysis demonstrates that to ensure optimal collective functioning, individual
bonds should show this behavior. And indeed, values reported for k c obtained from fits to single molecule experiments in [55] show that they are
consistently higher than k s , with the ratio k c /k s ranging from 5 to 480.
Interestingly, our fit of the two-state model to the observed lifetime of the
FNIII7−10 -α5 β 1 bond produces an even higher k c /k s ratio of 1.3 × 105 (cf.
Fig. 3.1). We interpret this as evidence that biophysical catch bond links
are designed such that when operating in concert, they provide maximal
collective reinforcement.
So far, we have considered the mean-field, equilibrium properties of a
cluster of catch bonds. We have shown that the equilibrium number of
closed bonds rises with rising force, up to a point, and that aspects of
this process may be understood analytically. In what follows, we analyze
dynamical aspects of such clusters and will, in particular, address the lifetime: through a combination of factors, catch bond clusters may become
extremely long lived.
3.4 Stochastic simulations
We now turn to simulations of the time-dependent behavior of a cluster of
catch bonds under a constant load. We start our runs with a cluster with
some initial number of closed bonds Ni , apply a constant total force to this
cluster, and observe how the fraction of closed bonds evolves in time. The
one-step master equation lends itself well to simulations using the Gillespie algorithm [81], which simulates events in stochastically distributed
time intervals.
The initial value Ni is significant for the typical evolution of the simulation. Consider Fig. 3.2, which shows the stable and unstable branches of
the cluster size. For a fixed force Φ, there are two equilibrium cluster sizes;
3.5 Cluster lifetimes: uniform loading
47
the unstable branch Nu and the stable branch Ns above it. For Ni , there
are consequently three distinct possibilities. First: Ni > Ns . Simulations
starting here are expected to display a decrease in N up until the point
where N attains its stable equilibrium value Ns . Second: Nu < Ni < Ns .
Simulations starting in this regime will tend to show an increase in N until the stable value of Ns is attained. Third: Ni < Nu . Since the flow is away
from the unstable branch, simulations starting in this regime will show
complete detachment of the cluster until N = 0. The first and second
cases are indeed apparent - typical trajectories in this regime are plotted
in Fig. 3.3, where we start simulations from above and below Ns . Fig. 3.3
also shows two typical unbinding events: when the fluctuating number
of bound bonds stochastically drops below Nu , the entire cluster unbinds
in cascade-like fashion. Note, that even in the unbound state there is a finite number of bound bonds due to the random rebinding modeled by γ.
Stochastic simulations, in principle, also offer direct access to the growth
kinetics of a catch bond cluster. This is a topic of great relevance, but in
order to meaningfully address it, it must be considered jointly with the
buildup of the traction force over time and, by extension, the evolution
of the associated stress fiber. We choose here to consider the fixed-force
equilibria and lifetimes and will not address the complex issue of cluster
growth and maturation.
3.5 Cluster lifetimes: uniform loading
How long will a cluster stay bound? We have been calling it stable, but
at best it is metastable as sooner or later, a fluctuation will come along
that is sufficiently large to drive N below Nu and cause the entire cluster
to unbind. There are several ways to compute or determine the lifetime
τ. It is, however, difficult to obtain directly from simulations: as we will
show, the lifetime becomes prohibitively large for realistic parameters and
in simulational practice, the cluster never unbinds. One instructive way
to view the lifetime is to construct the effective potential barrier that the
cluster must cross in order to reach the unstable branch, starting from the
stable region. To this end, we interpret the mean field evolution equation
48
Catch Bonds Under Force
350
i
300
Φ=1450
Ns
250
Nu
200
150
Ns
100
Φ=450
Nu
50
0
0
5
10
15
20
25
t(s)
Stochastic trajectories of i - instantaneous number of bound bonds for
γ = 0.1, Nt = 1024 and two different values of the applied force Φ = 450 and
Φ = 1450 (very close to Φcrit ), with Ni = 60 and 350, respectively. The number of
closed bonds either increases or decreases to reach its stable equilibrium value,
represented by solid lines. The unstable solutions of Eq. 3.6 are represented by
dashed lines. Two out of the three trajectories for the larger value of Φ correspond
to cascade unbinding of a cluster. By compiling statistics on these unbinding
events, we obtain the cluster lifetime from these simulations.
FIG. 3.3
3.5 Cluster lifetimes: uniform loading
49
dU ( N )
(Eq. 3.6) as a gradient flow of the form dN
dt = − dN , which defines the
effective potential U ( N ). Fig. 3.4 graphs the shape of this potential for
various forces, and provides an intuitive interpretation for the unbinding
transition: from the local minimum Umin = U ( Ns ) the cluster passes to the
left over a barrier whose height we shall indicate with U ( Nu ) = Umin +
∆U. Fig. 3.4 b) graphs this potential barrier as a function of the applied
force, demonstrating a generic maximum at finite force.
U
U(Nu)
a)
-200
-400
-600
dN/dt=U'(N)
Umin
10
20
30
40
50
60
N
ΔU
40
b)
30
20
10
-800
50
100
150
Φ
a) The effective potential U ( N ) for Φ = 10, 50, 80, 130, 180, 210, and
γ = 0.1. Arrows indicate U ( Nu ) and Umin for Φ = 50. b) The barrier height
∆U = U ( Nu ) − Umin for γ = 0.1.
FIG. 3.4
The height of the barrier will determine the lifetime of the cluster according to an Arrhenius law [79]:
τ = τ0 e∆U/U ,
?
(3.11)
with τ0 a reverse of the attempt frequency and U ? some reference effective
potential (note that U is dimensionless - the time derivative in the gradient
flow equation is taken with respect to a nondimensionalized time). We
may estimate U ? as the effective potential lost per unbinding bond at zero
force, i.e., U ? ≈ ∆U (Φ = 0)/N0 = 21 γNt . The value τ0 is a timescale on
the order of the single-bond lifetime, i.e., seconds. Combined, this yields
enormous lifetimes and unbinding of the cluster is generally not expected
to occur unless forces very close to Φcrit are applied (cf. Fig. 3.5).
A more direct, but perhaps less intuitive way to obtain the cluster lifetime
is by explicitly constructing the lifetime by summing over all possible dis-
50
Catch Bonds Under Force
sociation paths, weighed with their collective rates [97]:
Ns
τ=
∑
"
i =1
j −1
Nt
1
g(k)
1
+ ∑
r ( i ) j = i +1 r ( j ) ∏
r (k)
k =i
#
.
(3.12)
Fig. 3.5 compares directly the results from a small stochastic simulation
to both the explicit summation method, and a best fit to the potential
method, demonstrating that either works well. As expected, this small
cluster displays a maximum in its lifetime at a finite force. What these
considerations demonstrate is that not only do catch bond clusters grow
with increasing force, they also become longer lived at higher forces. This
enhancement of the lifetime is very pronounced: increasing the number
of available receptors to 1024 (a number we have been using throughout this chapter in simulations) shows why we never see clusters unbind
in previous simulations - the average lifetime at parameter values form
single-bond experiments may become as long as 1020 s(cf. Fig. 3.6).
<τ>(s)
300
250
200
150
100
50
0
0
50
100
150
Φ
Lifetime of a catch bond cluster with Nt = 128, γ = 0.1 (dashed line:
Arrhenius law, solid line: two-pathway model, points: Gillespie simulations) as a
function of force. Parameters in Arrhenius law were τ0 = 3.2 and U ? = 9.2.
FIG. 3.5
3.6 Cluster lifetimes: asymmetric loading
51
<τ>(s)
1020
1016
1012
108
104
200
400
600
800 1000 1200 1400
Φ
FIG. 3.6 Lifetime of a catch bond cluster with Nt = 1024, γ = 0.1 (dashed line:
Arrhenius law, solid line: two-pathway model) as a function of force. Parameters
in Arrhenius law were τ0 = 1.4 and U ? = 59.4.
Slip bond clusters, likewise, may be extremely long lived [91, 95, 98]. The
principal cause for this is that, provided sufficiently many bonds are in
principle available, the rupture of a single closed bond in the slip bond
cluster raises the force-per bond on the remaining closed bonds, and therefore makes them more likely to unbind, but this effect is offset almost
completely for large clusters by the increased rebinding when more unbound bonds are available to do so. The catch bond cluster, however, is
even longer lived as in the stable regime, unbinding of a single bond also
causes a higher force per remaining bond but for catch bonds this may
actually render them even longer lived.
3.6 Cluster lifetimes: asymmetric loading
The loading on a focal adhesion was shown to be nonuniform by recent
findings [96], where the force, acting on a focal adhesion was found to be
concentrated in the center of a focal adhesion with the maximum value
slightly displaced towards the rear edge of it. Focal adhesion anisotropy
was presented in [99]and predicted integrin accumulation at the rear edge
of focal adhesion through treatment of focal adhesion as a system of inter-
52
Catch Bonds Under Force
connected springs. Anisotropy of force distribution in focal adhesion[100]
or a periodic collection of adhesion clusters [101] is characterized through
a dimensionless stress concentration index, which shows, how much the
stress is accumulated at the rim of the focal adhesion. These and other
works applying a coupled stochastic-elastic modeling to cell-matrix adhesion are reviewed in [102].
In this section we consider one specific case of nonuniform loading, corresponding to the concentration of forces along the edges of a focal adhesion as reported in [96]. We divide the FA into distinct zones, and distribute the total force linearly across them(see 3.7 for the illustration of
the nonuniformly loaded cluster). Unpeeling of the FA is implemented by
fully detaching a zone once, for this zone, the average lifetime is exceeded.
The detached zone is then prohibited from rebinding, which accounts for
the spatial separation in the unpeeling region which renders rebinding
highly unlikely. Once a zone is detached, the force that it was supporting
is assigned, again in graded fashion, to the remaining bound zones. As in
[91], these zones are now more likely to also unbind causing a complete
unpeeling of the structure. The total lifetime of the edge-loaded cluster
is now equal to the lifetime of the rear-most zone. We collect our results
in Fig.3.7. The most striking finding is that for the same value of Nt , the
cluster is much shorter lived in edge-loading than in uniform loading.
This is due to our choice of excluding unpeeled bonds from rebinding,
and as we have seen in the preceding section this rebinding is a strongly
stabilizing process in the uniformly loaded cluster. We speculate that cells
may actually use this force-distribution dependence to ensure that for instance filopodia-like processes provide quick and tentative probes of the
external elasticity, but do not remain bound excessively long. The results
we present in Fig. 3.7 are obtained by direct computation of lifetimes according to Eq. 3.12. For more elaborate or dynamic loading conditions
one must turn to numerical simulation. As our findings for non-uniform
loading agree with those reported in [91], we expect qualitatively similar modifications of the adherent behavior under the additional loading
protocols reported there.
3.7 Loading by motors pulling on actin stress fibers.
53
<τt>(s)
▲▲
▲
▲
2.10 5
b)
a)
◆▲
◆
◆
◆
▲
0.38 Φ
0.29 Φ
0.21 Φ
0.12 Φ
Nt/4
Nt/4
Nt/4
▲
◆
◆■
■
●■
●■■
●
10 5
▲
◆
■
●
Unbind
Nt/4
▲
◆
▲
◆
■
●
0
▲
◆
● ■
●
■
▲
◆
● ■
▲
◆▲
◆▲
● ■ ◆▲
▲▲
●●■■ ◆
◆
▲
◆
▲
■■■■
●●●●●●●●●●●●●●●●●●●●●●●●●
▲
◆
▲
◆
▲
◆
▲
◆
▲
◆
▲
■■■■■■■■■■■■■■■■■■■
▲
◆
▲
◆
▲
◆
▲
◆
▲
◆
▲
▲
◆
▲
◆
▲
◆
◆
◆
◆
◆
◆
◆
▲
500
1000
1500
Φ
a) Sketch of a cluster of Nt bonds (drawn in red), divided into four
zones (each of the size Nt /4), sharing the load Φ. Arrows indicate the direction
of the force, the size of blue rectangles, marked with the part of the force that the
zone is bearing, indicates, how the load is initially distributed among the zones.
For each of the zones we compute the lifetime according to Eq. 3.12, and the
zone with the shortest lifetime unbinds. The size of dashed rectangles illustrates
the redistribution of the load after the unbinding of one zone (here it was bearing the load of 0.38Φ): the other bound zones take over the load. We repeat the
procedure until the last zone unbinds, and the cluster lifetime is equal to the lifetime of the last zone bound. b) Lifetime of a catch bond cluster with Nt = 1024,
γ = 0.1 with total force distributed nonuniformly across four zones ( line with
points: 0.05Φ, 0.18Φ, 0.31Φ, 0.45Φ, line with squares: 0.12Φ, 0.21Φ, 0.29Φ, 0.38Φ,
line with diamonds: 0.20Φ, 0.23Φ, 0.27Φ, 0.30Φ, line with triangles: 0.25Φ per
zone) as a function of force. Cluster lifetime shortens considerably with increasing asymmetry, and the unpeeling force diminishes.
FIG. 3.7
3.7 Loading by motors pulling on actin stress fibers.
In this final section, we consider what happens to a catch bond cluster
when the force is not applied externally, but provided by motors actively
pulling from the inside. While motors are not the only source of force in
cellular adhesions - polymerization forces, for instance, may contribute too
- we choose to consider only active, motor-generated forces. For the catch
bonds we consider here, [21] report a strict dependence of FA formation
and stability on myosin II-mediated forces for several catch bonds including α5 β 1 . Thus, we consider focal adhesions connecting the passive ECM
to actively contractile actin stress fibers inside the cell. Inspired by [95,
54
Catch Bonds Under Force
98], we represent the elastic media inside and outside the cell as elements
connected in series(cf. 3.8). In this simplified system, the mechanosensory
question we address is the following: how can the number of closed bonds
in the adhesion cluster - an internal measure report on the extracellular
stiffness, an external property. The effective spring constant of the outsideinside system is given by 1/Keff = 1/KECM + 1/KCSK , and we consider the
force of the myosin motors pulling on the adhesion as given by a simple
force-velocity relation that reflects the basic tendency to slow down upon
increased counterloading. Letting X (t) represent the strain as a function
of time - which equals the displacement against a fixed reference point
of the contractile stress fiber - we express the evolution of the strain with
time as
dX (t)
F (t)
= v0 1 −
.
(3.13)
dt
Fs
v0 is the bare, unloaded velocity of the contractile fiber and Fs is its stall
force: the force at which the fiber can no longer move. The choice for a linear force-velocity relation (identical to the one in [95]) deserves some motivation: While it is well established that collective force-velocity relations
may display complicated, concentration-dependent and nonlinear characteristics we feel that for our purposes and for the sake of transparency
our simplified approach is justified. For instance, in [103] it is shown that
small ensembles of myosin II motors display collective force-velocity relations that start at a finite unloaded velocity and monotonously (though
not exactly linearly) decrease until a collective stall force is reached. In
replacing this with a linearly decreasing force-velocity relation, we have
sought to strike a balance between retaining the essential characteristics
of collective behavior, while introducing a minimum of additional parameters.
In the two-spring system, the force F (t) itself is a simple function of X (t)
through the elastic relation
−1
−1 −1
F (t) = Keff X (t) = (KECM
+ KCSK
) X (t) .
(3.14)
Substituting Eq. 3.14 into Eq. 3.13, we solve for the evolution of the motorsupplied force with time, which expressed in the same dimensionless
3.7 Loading by motors pulling on actin stress fibers.
55
units as before yields a simple exponential approach to a plateau force:
Φ(t) = Φs 1 − e−t/τK ,
(3.15)
where τK = Fs /v0 K is the relaxation time. Note, that the stiffnesses KECM
and KCSK we define here have units of spring constants - force per length
- rather than the force per area (Pascals) appropriate for 3D moduli. In
order to translate our results to experimental stiffnesses, we must factor
in the basic length scale over which these forces are applied. While there
is significant variation in the size and areas of FAs, most studies report
typical dimensions of µm (see, e.g., [104]). We therefore choose to convert
our spring constants to effective moduli with this typical length, which is
how the stiffness axis of Fig. 3.9 was obtained. To mark the distinction,
we label this axis EECM to indicate moduli. This timescale is slow compared to the rapid exchange timescales of the catch-bond cluster, and we
consider the situation where N (t), the instantaneous number of closed
catch bonds, follows Φ(t) adiabatically. At present, regardless of the elasticity of ECM or CSK, Φ(t) will approach the stall force for long times.
As such, this force evolution cannot help discern the extracellular stiffness. The cell as a whole, however, will invest a certain amount of energy
into the contractile fibers. We hypothesize that this is equally distributed
across each of its stress fibers (each receiving the average), in which case
the system is operating under one further constraint: a set total energy
expenditure. Similarly motivated hypotheses of constant work were put
forward in [95] and [89]. The assumption of a fixed contractile energy
investment is further supported by recent experimental determination of
precisely this quantity: [21] measures it to be about 400 fJ for an entire cell
which, assuming between 50 and 100 FAs per cell, comes to a few to tens
of fJ per focal adhesion. We express the elastic work invested in a single
focal adhesion, after a time t, as
W (t) =
Z
F (t) dX (t) =
path
2
1 Fs2 1 − e−t/τK .
2 Keff
(3.16)
Setting a limit on W (t) = Wmax thus effectively terminates this process at
a stop time t? , defined implicitly by W (t? ) = Wmax . The force at this time,
56
Catch Bonds Under Force
Φ(t? ), may be computed to be
Φ(t? ) =
p
2Wmax Keff .
(3.17)
To complete the force sensor, we note that this force, which depends on
the effective and therefore the external stiffness KECM , in turn defines a
fraction of closed bonds N (Φ(KECM )): the asymptotic fraction of bound
bonds - which is an internally observable quantity - directly reports on
the external stiffness. Such mechanosensory relevance of catch bonds, and
specifically α5 β 1 , has been suggested before [90], but our direct correlation
between an intracellular observable and the extracellular stiffness represents a very specific proposal for how this mechanosensory pathway is
organized at the molecular level. As such, the catch bond cluster provides
the functionality of a mechanosensor. Our findings support the conclusions of [96], which argues that each focal adhesion, individually and autonomously, acts as a local rigidity sensor and that the presence of forces
is essential for focal adhesion formation and stabilization - as it is in our
clusters. These findings are summarized in Fig. 3.9, which computes numerically the N (KECM ) characteristics for realistic biophysical parameters,
and different values of the work Wmax . Note that for realistic energies per
FA, the rigidity sensor functions (i.e., does not unbind) within a stiffness
range of roughly 1-50 kPa. Most physiological environments for which
this type of rigidity sensing could be relevant are well within this regime.
3.7 Loading by motors pulling on actin stress fibers.
57
KECM
dX
KCSK
The spring-fiber system where the passive ECM with a spring constant
of KECM is connected by a force bearing FA to the active contractile fiber with a
spring constant KCSK . Rather than have an external force effect a displacement on
one of the springs, this displacement dX is now due to an active contraction of the
stress fiber. We model the evolution of this displacement with the force-velocity
relation Eq. 3.13
FIG. 3.8
Recent experiments have also suggested an important role for fluctuating [96] or cyclically applied [105] forces in dynamical processes such as
persistent motility, durotaxis and cytoskeletal remodeling. One distinct
model of a focal adhesion as a mechanical property sensor was proposed
in [106] and [107], where integrin molecules work as clutches that are
linking F-actin motors bundle to the substrate and at the same time resisting F-actin retrograde flow. This way the two types of adhesions were
achieved depending on the substrate stiffness: a rapidly increasing force
and slipping off adhesion to stiff substrates, and a slowly increasing force
on soft substrates which leads to catastrophic failure of all bonds at longer
timescales. In [107] it was shown that for the intermediate values of substrate stiffness, there is a regime where the motors and the clutches are
used to their fullest and provide the maximum traction force. We do not
consider varying forces in the simulations presented here, but note that
the relaxational timescales in our model (visible, for instance, in Fig. 3.3
as the cluster approaches equilibrium from Ni ) do allow our results to
be applied to relatively slowly varying load scenarios, where the cluster
composition may adiabatically follow the load.
58
Catch Bonds Under Force
N/Nt
0.8 Wmax=13fJ
Wmax=7fJ
Wmax=1fJ
0.4
0.1
0.01
50
100
EECM(kPa)
The fraction of closed bonds as a function of the ECM stiffness EECM .
We consider here 120 parallel filaments, each of which is tensed by 4 · 105 motors.
Each of these motors possesses an effective stall force of 10 pN [108]. The constant
work Wmax varies between the curves(dotted line depicts Wmax = 13 f J, dashed
line: 7fJ, solid line: 1fJ), corresponding to between 0.6 and 8 kB T per motor. If
the external rigidity is too high, a catch bond cluster is unable to hold on and
detaches.
FIG. 3.9
Our model predicts a very specific force-dependence of the diffusion of
catch-bond integrins in focal adhesions. Bound bonds, anchored to the
ECM, will exhibit lower diffusivity at larger applied forces, i.e., for higher
external matrix stiffnesses, and more of them will be bound. This will
result in a lower mobility, both collective and individual, as the force is
raised - until the critical force at which very abruptly the catch bonds collectively detach and become mobile again. These effects should be measurable either by fluorescence recovery after photobleaching, but better
still in experiments based on single molecule tracking or superresolution
imaging such as those reported in [109]. Precise comparison between such
experiments in systems such as [30] requires further modeling of the spatial distribution of catch bonds, particularly in combination with pure slip
3.8 Conclusions
59
bonds. We will treat a system where catch- and slip- bonds are sharing
the load in Chapter 4 and Chapter 5 of this thesis.
3.8 Conclusions
The fraction of bound integrin catch bonds, connecting a focal adhesion
site to the ECM, shows a regime of increase with increasing force. As such,
these clusters provide a stronger adhesive connection to the environment
when the focal adhesion is under tension. In addition to becoming more
tightly connected, the catch bond cluster is also considerably longer lived
at higher forces - absent other cues to cause unbinding it may become,
effectively, indefinitely adherent. In situations where the stress is actively
generated by molecular motors pulling on an actin stress fibre, the fraction
of bound bonds is a one-to-one reporter for the ECM stiffness. This bound
fraction, if properly coupled to further downstream intracellular sensory
processes, may serve as a primary sensory link in a mechanosensing pathway.
60
Catch Bonds Under Force
Chapter 4
Integrin Mixtures: Catch and Slip
Bonds
This chapter is about the behavior of catch and slip bonds in focal adhesion. We
present an extended model for a focal adhesion, where catch- and slip- bonds act
in parallel, thus sharing the load. We explore the implications of single bond
characteristics on the equilibrium parameters of a mixed cluster. We assess if the
mixed cluster may still act as a mechanical property sensor. We show how the
macroscopic parameters of a focal adhesion may change as a result of the force,
exerted on it. Central question of this chapter is: Will the addition of catch bonds
to a slip-bond cluster reinforce its properties?
61
62
Integrin Mixtures: Catch and Slip Bonds
4.1 Introduction
In the previous chapter we have considered how a cluster of catch bond
integrins will react on a force, exerted upon it. While the system treated
there provides us with a design of a mechanical property sensor, we don’t
discuss the composition of adhesion. Cell adhesion is multicomponent
- in focal adhesion various integrin types coexist [110], and the signaling pathways of two most studied integrin types α5 β 1 and αV β 3 interfere
[33], as well as their functions complement each other [29]. Interaction- direct or indirect -between integrins of different types have been implicated
to guide force generation and rigidity sensing [32]. The latter employs a
model where two types of integrins are sharing a load to simulate traction
generation in human breast myoepithelial cells on compliant substrates.
In this chapter we consider a similar system and complete the findings in
[32] with the analysis of mixed cluster stability and a modeling of a integrin mobility inside the focal adhesion. We consider the force-response
of a cluster of catch- and slip- integrin-ligand bonds and investigate the
collective effects induced by the force that they are sharing. We propose
that such a cluster may serve as a mechanical property sensor. Our results
exploit and extend the simulations in the previous chapter: We supply
a more realistic model, we develop a method to determine the stiffnessdependent parameters of the focal adhesion. The central question that we
answer here is: How will the force, exerted on focal adhesion, influence
the macroscopic parameters of focal adhesion?
4.2 Single bond characteristics
Catch-bond behavior in single biological bonds has received considerable
attention in the community, and has been studied in recent experiments
[41, 50, 111] on pulling single integrin-ligand bond with an AFM-tip- in
this chapter we present results obtained by using numerical values for single catch bonds directly obtained from these experiments. As was mentioned in [32], slip-bonds between an integrin and its ligand were not
studied in as much detail. To describe slip-bond behavior, we use an ex-
4.2 Single bond characteristics
63
ponentially decaying curve with varying parameters. Here, as in the previous chapter, we use the two-pathway model from [55], and fit it to the
data from [41], with the reduced number of parameters. Again, we nondimentionalize the forces as in the catch bond rebinding rate, but now we
use a simple exponential for the slip bond unbinding rate. Catch- and slipbond dimensionless rates are thus taken as:
−(φ−φc )
kcb
+ e(φ−φs ) ;
u (φ) = e
(4.1)
(φ/ f ss − f sh )
(4.2)
ksb
u (φ)
=e
,
where f ss determines how fast, compared to the slip-part of our catchbond, the slip- bond lifetime decreases, and f sh is responsible for the lifetime of a single slip-bond in the absence of force. Both catch- and slipbond lifetimes are plotted in Fig. 4.1. In the next Section, we consider the
behavior of a mixed cluster where catch- and slip- bonds act in parallel to
support a load.
Τs ,Τc
3.0
2.5
2.0
1.5
1.0
0.5
2
4
6
8
10
Φ
Average lifetimes of catch- (pink) and slip- bonds as a function of
dimensionless force, φ. Parameter values for catch bonds (described by Eq. 4.1):
φc = 4.02, φs = 7.78; for slip bonds (described by Eq. 4.2): blue - f ss = 1, f sh = 1,
red - f ss = 3.8, f sh = 1, yellow - f ss = 6.6, f sh = 1.
FIG. 4.1
64
Integrin Mixtures: Catch and Slip Bonds
4.3 Mixed cluster: fixed force
Modifying the approach of Schwarz et al. [95], we consider a collection of
total Nt bonds, out of which Nct are catch-bonds, and Nst are slip-bonds.
We let i denote the number of bound catch bonds, and j the number of
bound slip bonds at time t. The evolution of pi,j , the probability of having
i closed catch bonds and j closed slip bonds at a given time t; is a one-step
two-variate master equation:
dpi,j
s
c
= ri,j
+1 ( Ft ) pi,j+1 + ri +1,j ( Ft ) pi +1,j +
dt
h
i
(4.3)
c
s
+ gic−1 pi−1,j + gsj−1 pi,j−1 − (ri,j
+ ri,j
)( Ft ) + gic + gsj pi,j
where r s/c ( f ) is the force-dependent unbinding rate and gs/c is the rate at
which an additional catch/slip bond is attached to an extracellular ligand.
We assume the rate of rebinding to be independent of applied force, and
type of bond to simplify the initial conditions of the system, and assume a
c(s)
relation of the form gi = k0 γ( Nt − i ( j)), i.e., rebinding is proportional to
the number of available, unbound bonds of the same type, with a uniform
rebinding rate γ. Not only is γ independent on the force, but for every
type of bond we assume the same rebinding rate. Of course, it could vary
for the reason that the properties of the integrins approaching a ligand
may differ by type, thus allowing the integrins of one or another type to
come more near to its ligand, resulting into higher rebinding rates. The
force-dependent unbinding rate is where the properties of catch- and slipbond manifest themselves. From now on we consider a total dimensionless
force, Φ = Ft / f ? , and parameter f ? is chosen in such a way that Eq. 4.1
s
sb
holds. We choose ric (Φ) = i k0 kcb
u ( φ̄ ) and r j ( Φ ) = j k 0 k u ( φ̄ ) respectively,
evaluated at the force Φ, that i + j bound bonds in the cluster will share,
resulting in φ̄ = Φ/(i + j). Nonuniformly distributed load may be present
in focal adhesions, but here we start from considering a very simple case
of loading conditions: all of the bound catch- or slip- bonds experience an
equal share of the loading force. The equilibrium number of bound bonds,
4.3 Mixed cluster: fixed force
65
N = hi + ji, will evolve as:
d
N = ∑ (i + j )
dt
{i,j}
dpi,j
dt
c
s
= −hri,j
i + h gi,j i − hri,j
i + h gi,j i ,
(4.4)
where the summation is over all of the possible numbers {i, j} of bound
catch- and slip- bonds in a cluster, and hi denotes the average over all
of the possible configurations of the cluster. Eq. 4.4 can be split into two
separate equations, describing the equilibrium number of catch Nc = hi i
and slip Ns = h ji bonds respectively. We assume that all the rate functions
vary slowly around the equilibrium, and the first terms of Taylor series
c i ≈ rc
s
s
expansion around (hi i,h ji) are hri,j
hi i,h ji and hri,j i ≈ rhi i,h ji . Eq. 4.4
then transforms into a coupled system:

d
Φ
c
cb


 dt Nc ≈rhii,h ji + ghii,h ji = − Nc k u N + N + k0 γ( Nct − Nc )
c
s

d
Φ

 Ns ≈rhsii,h ji + ghii,h ji = − Ns ksb
+ k0 γ( Nst − Ns ) .
u
dt
Nc + Ns
(4.5)
Here the time t is the dimensionless time tk0 . Equilibrium will be achieved
when the right hand sides of both equations in the system above vanish.
Considering Eq. 4.5, and applying the expression for ksb
u , one can compute
an equilibrium number of catch bonds, which depends on the number of
bound slip bonds:
Nc =
Φ
f ss log
γ( Nst − Ns )
Ns
+ f sh f ss
− Ns .
(4.6)
Substituting this expression into equations. 4.5, we compute the numerical
solutions of the system of coupled deterministic equations 4.5, Ns and Nc .
While the analytical solution for a finite force can not be expressed in a
closed form, for Φ = 0 we can compute the mean number of bound catchand slip- bonds:
−1
Nc0 = γNct eφc + e−φs + γ
,
−1
Ns0 = γNst e− f sh + γ
.
(4.7)
(4.8)
66
Integrin Mixtures: Catch and Slip Bonds
One notices that the number of bound slip bonds at vanishing force is
independent of f ss . The expressions for Nc0 and Ns0 have no common parameters and are independent of each other, indicating that the force provides the interaction between two types of bonds, and in absence of force
they are decoupled. In general, the coupled system 4.5 has two solutions
for each value of force. Both of the solutions correspond to zeros of the
derivative of the potential, similar to the ones presented in Chapter 3 (see
Fig. 3.4 for an illustration). One of the solutions is unstable, the other
solution corresponds to the local equilibrium and is stable. In the following section we compare numerical solution of Eq. 4.5 to the results of the
Gillespie algorithm simulation of mixed bonds system.
4.4 Equilibrium solutions: mixed cluster
We use the Gillespie algorithm [81] to simulate the behavior of a mixed
cluster. We start off with a certain total number of bound bonds and adhesion composition (fraction of catch- and slip-bonds in our cluster). As in
[112], the choice of the initial value of bound bonds determines the typical
evolution of the simulation. Starting a cluster with all bonds unbound, as
well as with all bonds in a bound state will lead to cluster rupture before it
is able to achieve the equilibrium number of bound bonds. This situation
does not correspond to a case of our interest, as we are looking for a system that is able to provide a relatively stable connection between cell and
the ECM. When the number of bound slip bonds is very high, they will
make the force, that the bonds are sharing, very low, which promotes unbinding of catch bonds. Our framework allows us to calculate the typical
evolution of the number of bound catch- and slip bonds with time, as Fig.
4.2 demonstrates. We collect the average values over all of the trajectories
in a force-dependent number of bound bonds and examine how they are
connected to each other, via force. As we have mentioned in Chapter 3,
the catch bond cluster started with the number of bound bonds below the
unstable branch of the solution of deterministic equation will not recover
to its equilibrium values; same holds for a mixed cluster. We start our simulation from the maximum number of bound bonds for a zero force. As
4.4 Equilibrium solutions: mixed cluster
67
nc , ns
0.8
0.6
0.4
0.2
0
5
10 15 20 25 30 35
t(s.)
FIG. 4.2 Relative fraction of closed catch and slip bonds as a function of time
for Φ = 1500, f ss = 1 and f sh = 1, 2048 catch bonds and 2048 slip bonds.
Green/pink trajectories correspond to to the result of Gillespie simulations for
instant number of slip-/catch- bonds. Horizontal lines of same color - the solutions of Eq. 4.5.
the simulation progresses, the force increases, and to mimic real adhesion
growth we start every nonzero force value run from the equilibrium number of closed bonds obtained for the previous value of force. As depicted
in Fig. 4.3, catch and slip- bonds keep their tendencies even when coupled
through force in one cluster. Catch bonds will have maximum number of
bound bonds at finite forces, while the equilibrium fraction of bound slip
bonds decays with increasing force.
68
Integrin Mixtures: Catch and Slip Bonds
ns ,nc
0.4
0.3
0.2
0.1
200
400
600
800
1000
1200
1400
Φ
Relative fraction of closed catch and slip bonds as a function of the
force in a cluster of 512 catch and 512 slip bonds. Orange/blue points correspond
to simulation results for catch/slip bonds, starting from all bonds closed for zero
force. Pink/green lines - deterministic solution, obtained by solving equations
4.5, for experimentally derived catch bond force-lifetime curves and slip bonds
with f ss = 1 and f sh = 1. Rebinding rate for both catch and slip bonds is γ = 0.2.
FIG. 4.3
4.5 Cluster lifetime: catch bonds reinforcement?
69
FIG. 4.4 Sketch of a space that cluster with two types of bonds explores.
The parameters of the system are: number of bound catch bonds(x-axis) and
number of bound slip-bonds(y-axis). The example trajectory of unbinding(blue)
starts from (i, j) bound bonds(black point), ends with an absorbing boundary
at (0, 0)(red point) and is subject to reflecting boundaries marked in red along
the red lines. The trajectory is inside the phase space where 0 6 i 6 Nst and
0 6 j 6 Nct .
4.5 Cluster lifetime: catch bonds reinforcement?
The bound bonds vs. force curves for equilibrium values of bound catchand slip- bonds have shown that the mean values of both catch- and slip
bonds follow their own tendencies, but a question still remains: Will the
addition of catch bonds to a slip-bond cluster reinforce its properties? Or, in other
words, will the short lifetime of a catch-bond cluster for small forces (see
Fig. 3.5) increase with the addition of slip-bonds? In this section, we investigate a small cluster of integrins that will experience unbinding as
the time of the simulation goes. It may seem that assuming separation of
timescales, the lifetime of the cluster, Tm , may be calculated from the cluster lifetimes as: Tm = Max [ Tc , Ts ], where Tc/s is the lifetime of catch/slip
cluster, under the same force load. In this section we discuss the validity
of this approach, and present the analytical expression and the simulation
70
Integrin Mixtures: Catch and Slip Bonds
results for the lifetime of a mixed catch/slip cluster. The time, Ti,j , that
it takes a cluster of i bound catch and j bound slip bonds to reach the
point where all catch and slip bonds are unbound, follows the recursive
equation analogous the approach in [79]:
Ti,j = Ti+1,j
+ Ti,j−1
s
ri,j
gi +
c
g j + ri,j
s
+ ri,j
gj
gi
+ Ti,j+1
c
s
c + rs +
gi + g j + ri,j + ri,j
gi + g j + ri,j
i,j
+ Ti−1,j
c
ri,j
gi +
c
g j + ri,j
s
+ ri,j
+
1
c + rs ,
gi + g j + ri,j
i,j
(4.9)
where the last term corresponds to the time that it takes to leave the state
i, j and the first four terms correspond to the product of the lifetimes of
the neighboring states and the respective transition probabilities to those
states. When writing it for all possible combinations of catch- and slipbonds, one will obtain Nc × Ns equations on Ti,j , and one can solve it
as a system of coupled algebraic equations, with the following boundary
conditions:
T0,0 = 0 − absorbing boundary,
(4.10)
T−1,0 = 0 − no negative i,
(4.11)
T0,−1 = 0 − no negative j,
(4.12)
gsNst = 0 − reflecting boundary,
(4.13)
gcNct = 0 − reflecting boundary,
s
ri,0
= 0 − reflecting boundary,
c
r0,j = 0 − reflecting boundary.
(4.14)
(4.15)
(4.16)
Here Eq. 4.10 reflects that the cluster does not rebind after all its bonds
are unbound. Eqns. 4.11 and 4.12 correspond to the condition that the
number of bound bonds is never negative. Eqns. 4.13 and 4.14 take care
that the cluster can not rebind higher than the number of bonds available,
and finally Eqns. 4.15 and 4.16 take care that the rupture rates vanish
4.5 Cluster lifetime: catch bonds reinforcement?
71
when no bonds of that type are bound. See illustration in Fig. 4.4, where
an example unbinding trajectory of a cluster is depicted along with the
boundary conditions that we have described above.
τ(s.)
107
105
1000
10
50
100
150
Φ
200
FIG. 4.5 Lifetime of a cluster consisting of 50 catch and 50 slip bonds depending on force. Each of the points represents the result of 100 simulation trajectories
started out from Nc = 25 and Ns = 25 after 4 · 106 steps each, and f ss = 1 (blue)
f ss = 3.8 (red), f ss = 6.6 (orange), solid line of matching color correspond to
T25,25 - the solution of Eq. 4.9.
The analytical expression for the solution of system 4.9 is quite bulky and
may not be expressed in a compact from for each of the values, Ti,j . Eq. 4.9
is easily solvable using symbolic manipulation software (Mathematica in
our case), for a given total number of catch and slip bonds. The solution
of the system and its comparison to the simulation result is illustrated in
Fig. 4.5, where we calculate the lifetime of a cluster consisting of 50 catch
bonds, and 50 slip bonds with various parameters.
We calculate the lifetime curves for a cluster of 50 catch bonds, and a
cluster of 50 slip bonds separately and plotted them together in order to
72
Integrin Mixtures: Catch and Slip Bonds
1017
τ(s.)
1013
109
105
10
50
100
150
Φ
200
FIG. 4.6 Force-lifetime curve for a cluster, consisting from 50 catch- bonds
(green), 50 slip bonds (pink) and a cluster of 50 catch and 50 slip bonds (blue).
For slip bonds, f ss = 1, f sh = 1. Only for a small range of intermediate forces, a
mixed cluster is longer lived than both pure catch- or slip- bond cluster.
address the question: How does the lifetime of a cluster of mixture of catch- and
slip-bonds benefit from this addition? Do they take turns (catch bond keeps
a cluster of slip bonds stable for higher forces?), is there a match between
the lifetimes at any chosen force?
Fig. 4.6 illustrates the comparison between the lifetimes of pure catch bond
cluste, pure slip bond cluster and a cluster consisting of both catch- and
slip- bonds (mixed cluster). In general, the lifetime of a mixed cluster
is not a summation/maximum value of the lifetimes of pure catch- and
slip-bond clusters. Interestingly, the lifetime of a pure slip bond cluster
decreases with the addition of catch-bonds; the reason for this decrease
may be the change in the shape of the energy landscape, that the addition of catch-bonds to a system leads to. We find that for high values of
force, the lifetime of a catch bond cluster is longer than the lifetime of a
mixed cluster. Addition of slip bonds to catch- bond cluster provides it
with better stability for low force.
4.6 Integrins in the adhesion: lateral diffusion
73
4.6 Integrins in the adhesion: lateral diffusion
In this section the question that we ask is: How will the single bond properties influence the macroscopic parameters of adhesion, e.g. the mean
diffusivity of integrins inside FA, that [109] observed in single-protein
tracking experiments? We model diffusion of unbound bonds on a lattice
and compute the diffusivity, depending on the applied force.
The diffusivity (D, dimensions D = m2 /s) on a squared lattice is computed following [113] as the coefficient of the proportionality between the
mean residence time at a lattice site δt and the size of the lattice, δl 2 :
D = δl 2 /2dδt ,
(4.17)
where d is the dimension of the system: d = 2 in our case. The transition
rate between the nearest neighbor sites, according to [113], is:
r D = D/δl 2 .
(4.18)
We perform a simulation of system of diffusing and binding bonds as
follows: we allow free neighboring bonds to exchange sites with a rate
r D . The size of the lattice, δl, is taken such that the density of integrins
coincides with the results of the experiments in [32], and δl = di−0.5 , where
di is the cumulative density of integrins of both types in the focal adhesion,
yielding a value of 20nm per integrin, and thus the lattice size of δl 2 =
400nm2 . During the simulation, we determine how long each bond spends
in one lattice site before moving to the other site. Averaging over every
bond of one type, we derive the mean time spend in one lattice site, hδtc/s i,
from which according to Eq. 4.17 for a 2D system the diffusion coefficients
may be computed as:
1 δl 2
Dc/s =
.
(4.19)
4 hδtc/s i
The diffusivity in a system with a given number of bound catch- and slip
bonds is determined by two factors: how many bonds are diffusing, and
how many free places are there that those bonds can go to. This way, the
Φ can be estimated from
diffusion of bonds in a cluster under force load Dc/s
74
Integrin Mixtures: Catch and Slip Bonds
free diffusion coefficient, D0 , as:
Φ
Dc/s
Nc (Φ) + Ns (Φ)
= D0 (1 − nc/s (Φ)) 1 −
Nt
,
(4.20)
where nc/s (Φ) is the fraction of bound catch- or slip- bonds, respectively,
Nc/s (Φ) is the equilibrium number of bound catch/slip bonds. Fig. 4.7
demonstrates the agreement between the simulation results and Eq. 4.20
for an example slip-bonds with f ss = 1 and f sh = 3.8.
D(μm2/s)
nc,ns
0.30
0.8
0.6
0.25
0.4
0.2
0.20
5000
10 000
15 000
Φ
0.15
0.10
0.05
5000
10000
15000
Φ
Mean diffusivity, depending on the force, exerted on the adhesion,
f ss = 3.8 and f sh = 1, γ = 1, points: simulation results for pink- catch bonds,
green - slip bonds, solid lines - result of Eq. 4.20. Inset: mean fraction of bound
catch (orange) and slip (blue) bonds, depending on the pulling force, solid
lines(pink- catch, green - slip bonds) corresponds to the stable solution of Eq.
4.5, points(pink - catch, blue - slip bonds) correspond to the Gillespie algorithm
simulation results. Increase in the mean fraction of bound catch bonds with force
corresponds to decrease in their mean diffusivity.
FIG. 4.7
4.7 Two-spring model, diffusion and substrate stiffness
In this section we consider, how the stiffness of the extracellular matrix
affects the lateral diffusion inside the focal adhesion. Following the our
4.7 Two-spring model, diffusion and substrate stiffness
75
approach from [112] (see also Chapter 3), we assume that our cells invest
equal energy into contracting substrates of any stiffness. This assumption is backed up by experimental evidence in [21] and [38], where the
mean energy per cell was measured for cells cultured on substrates of
various stiffnesses. We assume this contraction energy to be distributed
equally among the stress fibers, thus resulting in energy of W s per every FA. We consider the part of the material that the motors feel to be a
cube of dimensions δh = 1µm. We represent this fragment of the ECM
with the Young modulus EECM and the cytoskeleton (spring constant,
KCSK ) as two springs coupled in series [95], then the load from the motors will be invested into contracting one linear spring with the stiffness
of Ke f f = 1/(KSCK + 1/( EECM δh)). This contraction of ECM, in which an
energy of W s is invested, will then comprise a minimal sensing action,
and the force, acting on the adhesion at the end of this contraction will be:
Φ=
q
2W s · Ke f f .
(4.21)
Knowing the force, with the help of Eq. 4.20, we calculate the average diffusivity for the integrins, affected by that force. This way we obtain the
diffusivity of integrins, depending on the stiffness of the extracellular matrix. The resulting curves for various values of W s are depicted in Fig. 4.8.
Our results suggest interesting possibilities for future in vitro experiments: as an example consider experiments in [114], where a cover slip
patterned with adhesion plaque proteins such as talin and vinculin is
brought into contact with contractile acto-myosin network. Supplementing the system developed in [114] by integrins may provide a possibility to
experimentally verify the stiffness- or rather force-diffusivity curves, the
shape of which should be similar to the ones depicted in Fig. 4.8.
Another possible way to test our findings is by using Fluorescence Recovery After Photobleching (FRAP) on a life cell culture expressing fluorescently labeled integrins, placed on varying rigidity substrates. Live FRAP
allows to measure mobility of small particles or proteins. The experimental stiffness-dependent diffusivity curves derived in the setup proposed
above may be compared to the ones depicted in Fig. 4.8.
The shape of the curves depicted in Fig. 4.8 shows that the diffusivity
76
Integrin Mixtures: Catch and Slip Bonds
Ds , Dc
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
20
40
60
80
100
120
140
EECM HkPaL
Mean diffusivity of catch bonds (red) and slip bonds (green), in a
mixed cluster of 1024 bonds in total, quarter of which are catch bonds, diffusivity
at no force applied, D0 = 0.3µm2 /s, W s from left to right: 13fJ, 7fJ, 1fJ. High
diffusivity values for stiff substrates correspond to cluster unbinding and thus
free diffusion of integrins without any interaction with the ECM.
FIG. 4.8
of catch bonds in focal adhesion initially decreases with the increase of
substrate stiffness and then increases because of cluster unbinding. The
diffusivity of slip bonds will increase monotonously with the increase of
the substrate stiffness. This suggests the possibility to potentially use our
findings to deduce whether the bond between certain integrin and its ligand is a catch bond or a slip bond by looking at collective integrin mobility
data.
4.8 Conclusions
The fraction of bound catch bonds in a cluster containing both catch- and
slip- bond forming integrins shows a regime of increase with increasing
force, thus a mixed cluster may still act as a mechanical property sensor. As it was postulated in our previous work [112] (see also Chapter 3),
a very specific force-dependence of the diffusion of catch-bond integrins
in focal adhesions takes place. Catch bonds in a focal adhesion will ex-
4.8 Conclusions
77
hibit lower collective diffusivity at larger applied forces, and for higher
external matrix stiffness, and more of them will be bound. When the total
fraction of bound bonds is high, the movement of all types of bonds will
be restricted, and the mobility will change significantly, depending on the
force. The manner in which it changes can help distinguish diffusing slipbond from diffusing catch-bond. Mixed clusters experience an increase in
its lifetime with respect to pure catch-bond cluster for the same values
of force. For low forces, a cluster consisting of catch- and slip-bonds will
have a shorter lifetime than a cluster composed only of slip-bonds.
78
Integrin Mixtures: Catch and Slip Bonds
Chapter 5
Spatial Organization of Integrins
inside Focal Adhesion
This chapter considers the interactions of integrins inside the focal adhesion.
Through minimal assumptions on the properties of the interactions between integrins, we observe how the strength of integrin interactions affects their spatial
distribution inside a focal adhesion. We conclude that the phase separation of integrin types inside focal adhesion is influenced by the force, pulling on the focal
adhesion. We investigate how the equilibrium fraction of bound bonds of each
type in a mixed cluster in the presence of interactions change as a result of a force
exerted on a cluster. We compare our findings to the results obtained in Chapter
4, and observe that integrin interaction influences the equilibrium parameters of
a focal adhesion.
79
80
Spatial Organization of Integrins inside Focal Adhesion
5.1 Introduction
In Chapter 4 we have described how the properties of a focal adhesion
cluster consisting of catch- and slip-integrins are affected by force. While
our description gives the mean values for the number of bound bonds,
it does not show how the bonds are distributed inside the focal adhesion. It has been shown before that integrins are nonuniformly distributed
throughout the focal adhesion, and that clustering of αv β 3 integrins is impossible without integrin activation [110]. Further by using fibronectin
(FN)-coated beads in [29] and changing the properties of FN polymerization it was determined that decreased attachment of α5 β 1 integrin to the
ECM proteins was responsible for the decrease in the lifetime of the focal
adhesion. The use of a monomeric FN instead of trimers was shown to decrease clustering of integrins and thus the decrease of the force that they
can withstand. The latter suggests that integrin-integrin interactions that
occur in focal adhesion lead to clustering of integrins. In this Chapter we
show, how for simple assumptions on integrin interaction, we can obtain
the change in their spatial distribution, depending on the force.
τ(s.)
3.0
2.5
2.0
1.5
1.0
0.5
φ
2
4
6
8
10
The average lifetime of an individual catch bond (green) and slip
bonds with f sh = 1 and f ss = 3.8 (blue), f ss = 6.6 (red), f ss = 8(yellow), as a
function of dimensionless force, φ.
FIG. 5.1
5.2 Integrins in adhesion: basic properties
In this section we remind the reader of the basic assumptions on integrinligand bonds and the properties of focal adhesion as we modeled it in
5.2 Integrins in adhesion: basic properties
81
Chapter 4. We consider a focal adhesion, consisting of two types of integrins: one of the types forms catch bond with the ligand, the other type
forms slip bonds with the ligand. These bonds will have a force- and typeindependent rebinding rate, γ, and a force- and type-dependent unbinding rates:
−(φ−φc )
kcb
+ e(φ−φs ) ;
u (φ) = e
(5.1)
(φ/ f ss − f sh )
ksb
.
u (φ) = e
(5.2)
Here we use values for φc and φs that were derived from the experimental
data ([41]) and force φ is dimensionless, defined in such a way that Eq. 5.1
holds. Parameter f ss measures how fast compared to the slip-part of our
catch-bond, the slip bond lifetime decreases, and f sh is responsible for the
lifetime of a single slip-bond in the absence of force. With f sh fixed, larger
values of f ss correspond to a cluster that unbinds slower with the increase
of the force. The differences between force-lifetime curves of catch- and
slip- bond are illustrated in Fig. 5.1. Having assumed equal load sharing,
and constant total number of bonds in a cluster, with the approach of
Schwarz et al. [95] we treat a collection of total Nt bonds. Out of these
Nt bonds, Nct are catch bonds, and Nst are slip bonds. The equilibrium
number of bound catch bonds, ( Nc ), and slip bonds, ( Ns ), evolves as:
d
Φ
Nc ≈ − Nc kcb
u
Nc + Ns + k 0 γ ( Nct − Nc )
dt
d
Φ
Ns ≈ − Ns ksb
u
Nc + Ns + k 0 γ ( Nst − Ns ) , .
dt
(5.3)
(5.4)
where t is the dimensionless time, and dimensionless Φ is a total force acting on a cluster. Considering Eq. 5.3 and applying the expression for ksb
u ,
cb
and k u , one can solve this system numerically or through the Gillespie
algorithm simulation, obtaining values of equilibrium number of bound
catch- and slip- bonds in a focal adhesion cluster. The equilibrium number
of bound bonds depends on the force will change, as is illustrated in Fig,
5.2 for various sets of parameters.
As we have seen earlier, the diffusion of integrins will depend on the
force. If coefficient D0 corresponds to the diffusion of a cluster, where all
82
Spatial Organization of Integrins inside Focal Adhesion
of the bonds are free. We set up a simulation where integrins diffuse on
a squared lattice, by means of nearest neighbor exchange. The parameter
that our computation receives is r D , the hopping rate across the neighboring sites of the lattice for a freely diffusing bond:
r D = D0 /δl 2 .
(5.5)
Here D0 is the diffusion of bonds inside the unbound part of the adhesion(we take this experimentally derived value from [32]), and δl is the
linear dimension of the lattice site. We incorporate this value into our
Gillespie algorithm simulation, and use this system as a basis for the interacting adhesion simulation that is studied in the rest of this Chapter.
nc, ns
fss=1
nc, ns
fss=3.8
0.8
0.6
0.6
0.4
0.2
5000
0.4
nc, ns
10 000
15 000
\Φ
fss=6.6
0.8
0.6
0.2
0.4
0.2
5000
5000
10 000
10 000
15 000
15 000
Φ
Φ
The relative fraction of closed catch and slip bonds as a function of
force Φ. Orange/blue points correspond to simulation results for 2048 catch-/
2048 slip- bonds, starting from all bonds closed, green/pink lines - deterministic
solution, obtained by solving equations 5.3 and 5.4 with γ = 1 and f ss = 1. Top
and bottom insets correspond to f ss = 3.8 and f ss = 6.6 respectively.
FIG. 5.2
5.3 Interaction of integrins: simulations
83
5.3 Interaction of integrins: simulations
So far we have investigated only the mean values over the cluster throughout the simulations. Integrins interact and cluster inside focal adhesion
[29]. In the next sections of this Chapter we explore the simplest model of
interactions between integrins, and determine how the force, exerted on
a focal adhesion influences integrin clustering and the unbinding force of
the cluster.
Φ=const
InitiateStheSsystem:S
NtJSNctJSNstJSNcJSNsJSSγ
J>M
ChoseStheSnextSmove:
RuptureSCatch=Slip
RebindingSCatch=Slip
DiffusionSofSfreeSbonds
FindStheSbondJSthatSisSgoingStoSmakeStheSmove
AcceptSwithSBoltzmannSprobability
zinScaseSofSRuptureSandSRebinding(J
AcceptSDiffusion
J=M
AcceptSeverySmove
repeatSnrun)Nt Stimes
CalculateSpropensity:S
u
K sb NsjKucbNcjrS S)NumberSofSDiffusionSmoves
D
C
IfSneededJSadjustStheSNSSandSNS
s
c
CompareSresults
Mathematica
Simulation scheme for a run with and without integrin-integrin interactions. All but one of the steps in our simulation repeat the Gillespie algorithm
steps: the system is initialized, the respective propensities of each type of moves
are calculated, the next move is chosen according to those propensities. The only
difference between our simulation and Gillespie algorithm is that the move is
accepted/rejected with Boltzmann probability depending on the energy of the
proposed configuration.
FIG. 5.3
We attribute a dimensionless coupling constant J to the strength of integrin interaction. The result in Fig. 5.2 will correspond to J = 0. To account
84
Spatial Organization of Integrins inside Focal Adhesion
for nonzero J, we set up a combination of a Gillespie algorithm (described
in previous chapter) with a Monte-Carlo step rejection rule, for simulating
the interactions. When performing a Gillespie algorithm simulation, every
move was accepted. We modify our simulation, by adding a probability of
acceptance for every move, according to the energy change ∆H: the move
will be always accepted if ∆H 6 0, and accepted with a Boltzmann probability exp[−∆H], if ∆H > 0. For the illustration of the scheme used in the
simulation, see Fig. 5.3.
To describe the configuration of the system, we use a vector of length
Nt . Each of its elements characterizes one integrin. Integrin number i has
a type, which we denote by ti : ti = −1 for slip bonds and ti = 1 for
catch bonds. Integrin number i may be in a bound or free state, that
we denote as bi , bi = 0 for a free bond and bi = 1 for a bound bond.
This way we write a configuration σ of one adhesion state in a form
σ = {{t1 , b1 }..{ti , bi }, ..{t Nt , b Nt }}.
We construct the Hamiltonian, H, which assigns an energy to every configuration of the system. When choosing the expression for H, we were
inspired by enriched Ising model system, and chose to use only nearest
neighbor interactions. The conformation of an integrin changes when it
is approached by a ligand, and thus the interactions between two bound
integrins may be different from interactions between two free integrins.
To avoid making more assumptions on the parameters of integrin - integrin interactions, we set up our simulation in such a way that only bound
integrins interact.
The Hamiltonian that we chose to use in our simulations is:
H(σ) = − J
∑ t i t j bi b j
(5.6)
{i,j}
where the sum is over all the pairs of neighbors {i, j}. Introducing this
Hamiltonian makes sure that only bound bonds affect the energy of the
configuration. Dimensionless energy constant, J, compares the energy, corresponding to two bonds of the same type binding next to each other,
with kB T. Value of H will be negative if the bonds of the same type stay
together, and positive if bonds that are bound stay near the different type
5.4 Interaction of integrins: equilibrium values
85
of bonds.
When the move is chosen to unbind a bond i, the move will be accepted
or rejected, depending on the difference between the Hamiltonian of the
proposed configuration and the current configuration:
∆ Hi = − J
∑ t n ( i ) bn ( i ) t i .
(5.7)
n (i )
Here n(i ) are the neighbors of bond i. In a case where bond i has no
bound neighbors, the change in the Hamiltonian will be 0 and the move
is always accepted. If ∆H > 0, the move is accepted with a probability
proportional to exp[−∆H]. In principle, + J for a pair of bonds of the
same type bound next to each other does not have to be the same as − J
- for two bonds of different types, binding next to each other. We do not
make the quantitative difference between those two situations, in order to
avoid adding extra parameters to our system.
5.4 Interaction of integrins: equilibrium values
In this section we observe, how the equilibrium values of bound bonds
are affected by the presence of interactions. Turn to Fig. 5.4 and 5.5 for
an illustration on how the average number of bound bonds changes, depending of the value of J. For small J the change in the number of bound
bonds is insignificant, while as J reaches 0.2 it starts to become noticeable
for small and big forces. We collapse the data depicted in Fig. 5.4 and Fig.
5.4 to the phase diagram shown in Fig. 5.6. In this phase diagram it is
shown, whether both or only one of the types of bonds will get selectively
reinforced, depending on the force and the strength of interactions. For
example, when the force is small, more slip bonds will bind, which will
provide a more beneficial configuration in terms of interactions. For large
values of J both catch- and slip- bonds get reinforced. The increase of J
may correspond to the change in pH or ionic composition of the environment around the focal adhesion, which may result in change of binding of
the integrins, and thus the signaling pathways originating from integrin
86
Spatial Organization of Integrins inside Focal Adhesion
binding will be affected.
a)
b)
J=0.1
ns ,nc
0.4
J=0.2
ns ,nc
0.4
0.3
0.3
0.2
0.2
0.1
0.1
500
1000
1500
Φ
500
1000
1500
Φ
a) Average fraction of bound catch(orange points) and slip(green
points) bonds depending on force, after 800 steps of the simulation for a cluster of 1024 bonds, equal part catch and slip- composition, slip bonds with f sh = 1
and f ss = 3.8, coupling constant J = 0.1, plotted along with the numerical solutions of Eq. 5.3 for equilibrium number of catch(pink solid line) and slip(green
solid line) bonds b) Equilibrium bound bond fraction vs. force curve for the same
cluster parameters and for J = 0.2.
FIG. 5.4
J=1
ns ,nc
J=0.5
ns ,nc
0.7
0.6
0.4
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
a)
500
1000
1500
Φ
b)
500
1000
1500
Φ
a) Average fraction of bound catch(orange) and slip(green) bonds depending on force, after 800 steps of the simulation(points) for a cluster of 1024
bonds, equal part catch and slip- composition, slip bonds with f sh = 1 and
f ss = 3.8, coupling constant J = 0.5, plotted along with the numerical solutions of Eq. 5.3 for equilibrium number of catch(pink solid line) and slip(green
solid line) bonds b) Equilibrium bound bond fraction vs. force curve for the same
cluster parameters and for J = 1.
FIG. 5.5
5.5 Interaction of integrins: spatial distribution
87
Φc
2000
Fall Off
1500
S+C
C
1000
S
500
0
0.0
0.2
0.4
0.6
0.8
1.0
J
FIG. 5.6 a) Sketch of how phase diagram may look like according to our simulations. Letter C marks the region where catch bonds get reinforced(number of
bound bonds is greater than the one with J = 0), S marks the region where slip
bonds get reinforced and S+C denotes where both types of bonds are reinforced.
5.5 Interaction of integrins: spatial distribution
To determine how the integrins are distributed inside the focal adhesion,
we construct parameters that will allow us to quantify it. There are two
ways of characterizing such an enriched Ising system: through the distribution of bound integrins and throught the distribution of integrins of the
same type (see Fig. 5.7a and b, where the spatial distribution of integrins
in the beginning and in the end of our simulation is depicted).
Let us first consider the distribution of integrin types. We define the typeorder parameter ηt to be:
ηt =
1
Nt
∑ ti t j ,
(5.8)
{i,j}
where the summation is over all pairs of neighboring lattice sites {i, j}.
The type-order parameter, ηt is equal to −1 when all bonds are sur-
88
Spatial Organization of Integrins inside Focal Adhesion
rounded by only bonds of a different type. ηt approaches one when integrins phase separate by type; it will never reach it for a cluster with
both types of integrins due to the presence of interfaces between them.
a)
b)
a) The snapshot of the first frame of the simulation run for 512 catchand 512 slip bonds on a squared lattice for Φ = 0 and J = 10. Points denote
bound bonds, lattice site colored in pink corresponds to a catch bond, lattice site
colored in green corresponds to a slip bond. b) Snapshot of the same simulation
run after 1.6 · 107 steps.
FIG. 5.7
The ordering of bound integrins in a focal adhesion is characterized by
the binding order parameter, ηb , defined as:
ηb =
1
Nt
∑ (2bi − 1)(2bj − 1) .
(5.9)
{i,j}
The summation in Eq. 5.9 is over all pairs of neighboring lattice sites {i, j}.
The binding order parameter, ηb depends on the total number of bound
bonds, and ηb = 1 when all of the bonds are bound, ηb = 1 when no
bonds are bound. This order parameter in general case is in the range
(−1; 1). Let ηbu denote the bond parameter for a cluster, where bonds are
uniformly distributed. For nc = Nc /Nt -a fraction of bound catch bonds
and ns = Ns /Nt a fraction of bound slip bonds, ηbu is expressed as:
ηbu = (nc + ns )2 + (1 − (nc + ns ))2 − 2 (nc + ns ) (1 − (nc + ns )) =
= (2 ( n c + n s ) − 1)2 .
(5.10)
5.5 Interaction of integrins: spatial distribution
89
Maximum ηbu = 1 is achieved when nc + ns = 0 or nc + ns = 1, ηb = 0
when ns + nc = 0.5. For a nonuniform distribution of bound bonds, bond
parameter ηb > ηbu when all of the bound bonds are situated close each
other. We denote ηbnu = ηb − ηbu as a nonuniform binding order parameter,
ηbnu > 0 when bound bonds phase separate from free bonds, and ηbnu < 0
when bound bonds have more unbound neighbours than the bonds in a
noninteracting cluster.
ηbη, t
ns ,nc
1.0
0.30
a)
b)
0.25
0.8
0.20
0.6
0.15
0.4
0.10
0.2
0.05
200
400
600
800
1000
t(s.)
1200
t(s.)
200 400 600 800 1000 1200
FIG. 5.8 a) Evolution of order parameters (blue - ηb , purple - ηt ) with time for
of 512 catch- and 512 slip bonds on a squared lattice, Φ = 0, J = 10. b) Evolution
of fraction of bound catch-(red) and slip-(green) bonds with time during the same
simulation run.
To determine, whether the equilibrium spatial distribution is achieved,
one must refer to the order parameters and fraction of bound bonds saturation curves, as shown in Fig. 5.8a and Fig. 5.8b respectively. Judging
from those figures, the ordering mostly takes place when more bonds are
bound of either type, which directly follows from the way that we set up
our simulations: only bound bonds interact and thus only bound bonds
will order.
The equilibrium fraction of bound bonds of either type depends of the
force, acting on the cluster. Thus, the ordering of a mixed cluster must
depend on the load that this cluster experiences. We summarize order parameters dependence on force in Fig. 5.9a and Fig.5.9b. For J > 0.2 the
ordering is significant to show in our ηbnu and ηt measurements. The measurements for J = 10 show that the nonuniform binding order parameter
ηbnu is smaller than the one for J = 1. The reason for this is that ηbu is big,
90
Spatial Organization of Integrins inside Focal Adhesion
as a large fraction of the available bonds is bound, and the interactions do
not organize the bonds to that extent as for J = 1.
a)
ηbnu
b)
0.25
ηt
0.4
0.20
0.3
0.15
0.10
0.2
0.05
0.1
500
1000
1500
Φ
500
1000
1500
Φ
a) Order parameter ηbnu for a cluster of 512 catch- and 512 slip bonds
at the end of the simulation of 1.6 · 107 steps, f ss = 3.8, f sh = 1, and J = 0.2 (red),
J = 0.5 (orange), J = 1 (green) and J = 10 (blue). One can see that the maximally
ordered state according to our measurements is reached around J = 1 b) Order
parameter ηt for J = 0.2(red), J = 0.5 (orange), J = 1 (green) and J = 10 (blue) for
a cluster of 512 catch- and 512 slip bonds at the end of the simulation of 1.6 · 107
steps, ( f ss = 3.8). The order parameter increases with the increase of the force,
and the shapes of the curves of ηb − ηbu and ηt for the same J are similar, which
suggests that bond ordering and type ordering may be correlated.
FIG. 5.9
5.6 Conclusions
Integrin-integrin interaction may play role in the clustering of catch- and
slip-bonds inside focal adhesion. Their coupling constant determines to
which extent the order parameter changes compared to a uniform bond
distribution. With the presence of interactions, the fractions of bound
catch-/slip- bonds saturate at significantly different levels than for noninteracting system. For moderately high coupling constant, catch-/slipbonds will get selectively reinforced: the number of bound catch bonds
will increase for higher forces, the number of bound slip bonds will increase for lower values of force. At high coupling constant both cacthand slip- bond fraction show the increase compared to the non interacting
cluster.
Chapter 6
Durotaxis
This chapter is about cell durotaxis - the ability of cells to migrate in stiffness
landscapes. We incorporate recent experimental findings, that show that the persistent characteristics of cell movement changes depending of the stiffness of the
underlying substrate, into a novel mechanism that may explain one way in which
cells migrate in the direction of increasing stiffness. We compare our findings to
previous models of cell durotaxis, and point out how one may distinguish our
model from biased migration. We use a simple 1D model simulation to illustrate
how the mechanism that we propose may be qualitatively analyzed and understood.
91
92
Durotaxis
6.1 Introduction
Durotaxis is the tendency of an organism to migrate when confronted with
a gradient in the mechanical properties of its environment. In the context
of cells, it is generally reserved for a motility directed towards the stiffer
end of a substrate (to which the cell is adhering) that possesses a gradient
in its elastic modulus - an early example of this was demonstrated in
[15], which reported that fibroblasts placed in the vicinity of the interface
between a soft and a stiff substrate tend to form a longer trailing edge
when crossing from a more compliant towards a stiffer substrate, and that
the speed of migration would decrease when they approached the more
compliant substrate from the side of the stiffer one. Subsequent studies
[16, 72] have furthermore suggested that the steeper the stiffness gradient
is, the faster the cells will migrate towards the stiffer substrate.
The name durotaxis is introduced in analogy to the well-known and seemingly similar ability of cells to move in response to chemical gradients
known as chemotaxis. Chemotaxis equips cells with the exceedingly useful ability to seek out environments that are optimal - either in terms of
the availability of nutrients, or the absence of toxins [64].
Using modern imaging techniques it is possible to record the position
of the cell at fixed intervals of time, thus resolving its full 2D trajectory.
Recent analysis of mesenchymal stem cells on gradient stiffness polyacrylamide (PAA) gels demonstrates that these trajectories are not pure random walks: depending on the substrate, the path of the cell may be more
or less persistent, while the speed of the cell (the magnitude of its velocity
along the path) appears to remain roughly constant.
Persistence measures the tendency of cells to continue moving in the same
direction. It is not, itself, a directional effect but, as we will see, under
certain conditions may result in net directed motion. Particularly relevant
to this is the experimental observation [3, 15] that the persistence of a
cell is stiffness dependent, and thus that a gradient in stiffness effects
nonuniform persistence.
6.1 Introduction
93
Φ
b
xb
Schematic representation of a cell trajectory. Assuming the velocity
along the path to be constant, the position of the cell at equally spaced temporal
intervals is recorded, and these positions are connected by straight segments. ϕ is
the angle between two segments of the trajectory, connecting translation vectors
r (t) and r (t + ∆t) recorded ∆t apart in time, θ is the angle between two adjacent
segments of cell trajectory. Note that both θ and ϕ are relative angles - for future
use we will also consider Φb , the instantaneous orientation of the motion with
respect to some fixed spatial direction (for instance, the direction of a gradient in
stiffness xb .)
FIG. 6.1
In this Chapter, we address the general statistical properties of durotactic
motion in a stiffness gradient, and explore its potential biological significance to cells. We begin by defining the persistence of a trajectory, how it
can be measured in cell trajectories, and summarize the main conclusions
we may draw from experiments in literature. Then, we present simulations
that emulate cell migration in a gradient stiffness substrate, and investigate which factors determine the migration velocity. It the last section, we
consider a simplified 1D model for persistent motion, which gives us a
glance at the origin of persistence-based drift.
94
Durotaxis
6.2 Persistent migration: correlated walk
In this section we introduce the main quantitative measures that parametrize
a cell trajectory that we are going to use throughout this Chapter.
Consider a cell that is executing a walk (random or nonrandom) on a surface. The coordinates of the cell’s position are measured at fixed intervals
in time, where the intervals are chosen sufficiently short to justify the assumption that between two measurements, the cell moves in a straight
line, with a constant velocity Vc . Fig. 6.1 depicts a typical cell trajectory,
and defines the angles that we are going to employ in the rest of this
Chapter. Let x(t) denote the position of the cell at time T, and x(t + ∆t)
its position at time (t + ∆t). We will call the difference vector between
these two subsequent positions r(t + ∆t) = x(t + ∆t) − x(t). If we let ϕ
denote the angle between those two segments of the trajectory, then cos ϕ
is computed as:
r (t) · r (t + ∆t)
cos ϕ(∆t) =
.
(6.1)
|r (t)||r (t + ∆t)|
An increased time ∆t between two segments that are compared will decrease the correlation of their directions, and hcos ϕ(∆t)i −→ 0. Gener∆t→∞
ally, the correlations between the directions of cell movements will decay
exponentially with time;
hcos ϕ(∆t)it ∼ e−∆t/τp ,
(6.2)
where τp is called the persistence time of the walk - it measures the
amount of time over which the cell ’remembers’ its orientation (and h..i
denotes the average over the entire trajectory). The mean squared displacement (MSD), similarly ensemble averaged, for such persistent random walks is then readily computed to be [84]:
2
h R i(t) =
2Vc2 τp2
t
+ e−t/τp − 1
τp
.
(6.3)
We will drop the angular brackets for the MSD, and simply call it R2 (t). At
long timescales, every walk with finite persistence tends to a true random
6.3 Uniform persistence: simulations
95
√
√
walk, √
with R2 ∼ Vc t. For small t (small compared to the persistence
time) R√2 ∼ Vc t: persistence dominates and the motion is ballistic. In
general, R2 evolves as:
√
R2 ∼ Atα(t) ,
(6.4)
where A is a constant that depends on the parameters of the walk (velocity,
etc.) and α(t) is called the migration exponent (introduced in [15]).
The migration exponent shows how fast a trajectory turns from diffusive √to ballistic, and may be computed directly form the MSD as α(t) =
d log R2 (t)
.
d log t
For a persistent walk, as described by Eq. 6.3, the migration
exponent is
t
−t/τp
1
−
e
τp
.
(6.5)
α(t) = 2 e−t/τp − 1 + τtp
Following the approach of Raab et. al. in [15], we expand Eq. 6.5 in terms
of (t/τ ) as:
1 1
α(t) ≈ + e−t/(3τp ) + O((t/τp ))3 .
(6.6)
2 2
For long times α(t) is equal 1/2, and the walk is diffusive. We use Eq. 6.6
in the rest of this Chapter to determine the persistence of the walk from
the collective migration data obtained in our simulations.
6.3 Uniform persistence: simulations
We now set up the simulation of a cell on a plane with constant persistence. At every simulation step, cells go straight for a time ∆t, with linear
velocity Vc , covering equal lengths of Vc ∆t at every time step. We simulate
cell migration as a one-step memory process, which means that the direction
of the current step only depends on the direction of the previous step.
At every subsequent step the cell will alter the direction of its movement
with respect to the previous direction, at an angle θ (see Fig 6.1 for the
illustration). The magnitude of this turning angle defines how much the
direction of the cell movement changes, and is determined by the variance
96
Durotaxis
σp of what we chose to be a Gaussian distribution of turning angles:
P (θ ) =
1
√
σp 2π
e
.
−θ 2 2σp2
.
(6.7)
The persistence time t p will then have a one-to-one relation with the variance of this distribution:
∆t
σp2 =
.
(6.8)
tp
Walks with large values of σp are less persistent, walks with small values
of σp are more persistent. We verify that our simulations produce walks
with the correct persistence time, τp by fitting Eq. 6.6 to the mean squared
displacement (MSD) data obtained, as it is illustrated in Fig. 6.2a and 6.2b.
In the next section we set up a simulation where the persistence depends
on the position of the cell, and assess the directional properties of the
walk.
α(t)
t
1.1
b) τ
a)
30
25
20
15
10
5
1.0
0.9
0.8
0.7
0.6
0.5
0
20
40
60
80
t
0.21+3.01tp
2
4
6
8
√
10
tp
FIG. 6.2 a) Evolution of the migration exponent α(t) = d log R2 (t)/d log t
with time for 103 cells, with linear speed Vc = 0.2, time step ∆t = 0.1. Color of
the solid curves correspond to the simulation data for substrate with uniform
persistence from left to right: t p = 0.2, 8.2, 16.1, 24.1, 32, 40; dashed lines correspond to the fits of the shape α(t) = 0.5 + 0.5e−t/τ to the simulation data; b)
Solid line: transition time τ resulting from the fit of the shape α = 0.5 + 0.5e−t/τ
for the migration exponent simulation data on the substrates with uniform persistence t p ; dashed line: the fit of the shape τ = a + bt p shows that τ = 3t p and
confirms that the walk we simulated is correctly described by Eq. 6.3.
6.4 Gradient persistence: simulations
97
6.4 Gradient persistence: simulations
tp
max
tp
tpmin
x
Gw
Gw
FIG. 6.3 Persistence landscape cross section in X direction. All cell trajectories
start at X = 0, in the middle of the linear persistence gradient area that has a
width of 2Gw . Outside the gradient area, the maximum persistence time is tmax
p
at X > Gw , and the minimum persistence is tmin
at
X
6
G
.
w
p
So far we have investigated migration (or lack thereof) on substrates with
constant persistence. Let us consider here the case where the persistence
depends on one of the cell coordinates, X as: t p = t p ( X ), and is linear on
an area of the width 2Gw . The persistence at the left border of this gradient
area is t p (− Gw ) = tmin
and maximum persistence is achieved at the right
p
border of the area t p ( Gw ) = tmax
(see Fig. 6.3 for illustration). Cells start at
p
X = 0 and make Nstep consecutive steps with linear velocity Vc . Each step
takes time ∆t. The length of the cell path, L, is then equal to Vc Nstep ∆t. To
measure the strength of a directed migration, we compute the durotaxis
index, DIX (t = Nstep ∆t ), which is expressed as:
DIX (t) =
h X (t)i
h X (t)i
=
,
L(t)
Vc t
(6.9)
where X (t) is the mean displacement along X direction at time t. We set
up a simulation with Gw = 2, Vc = 2, tmin
= 0.2 and varying tmax
p
p . Examples of cell trajectories resulting from these simulations are depicted in
Fig. 6.4a. We record the average displacement across X and Y, depending
on time, example curves of which are shown in Fig. 6.4b.
98
Durotaxis
a) Y
b) X
12
20
10
10
8
10
10
20
20
30
X 6
4
2
10
20
30
40
t
a) Trajectories of 20 cells on a uniform gradient of width Gw = 2, linear
= 2. Cells start
= 0.2 and tmax
speed Vc = 2, length of timestep ∆t = 0.1, tmin
p
p
from the origin (red dot at { X, Y } = {0, 0}), and take Nstep = 200 steps following
the persistence of their current position. Black points mark the positions of the
cells at the end of the simulation. Cells that ended up to the right of the origin
move farther from their initial position, which corresponds to higher persistence.
Cells that finish to the left of the origin have made the same number of steps,
but remain, on average, nearer to the origin, and thus have been walking less
persistently. b) Mean values of X (red) and Y (blue) coordinates averaged over
103 cells, with linear speed Vc = 2, timestep ∆t = 0.1, started at a substrate with
Gw = 2 (dashed line), tmin
= 0.2, tmax
= 10.2 after Nstep = 400 steps. The mean
p
p
coordinate Y averages to zero, which means that there is not drift in Y-direction.
The mean displacement along the X-direction keeps growing, even after it has
passed the border of the gradient area (X = 2).
FIG. 6.4
6.4 Gradient persistence: simulations
99
In Fig. 6.4 we show that cells drift towards the area with the high persistence time. One may suggest that the drift would stop after all of the
cells cross the gradient area, and thus that the durotaxis index reaches a
maximum value at a finite time. We vary tmax
p , and record the evolution
of durotaxis index with time; a collection of time-dependent durotaxis index curves is depicted in Fig. 6.5a. Indeed, the durotaxis index reaches a
maximum, after a time that depends on the parameters that we chose. We
measure the maximum value of durotaxis index, DIXmax , achieved during
max in Fig. 6.5b.
our simulations with varying tmax
p , and plot it against t p
DIXmax increases with the increase of the maximum persistence tmax
p .
For the walks that we simulate here, changing Gw is effectively the same as
changing Vc , and thus we choose to keep Vc constant. We simulate walks
on a gradient of a varying width, Gw , and quantify the gradient strength
as:
!
min
tmax
−
t
p
p
Sg = log
.
(6.10)
Gw
We then assess how Sg affect the maximum durotaxis index, and plot
DIXmax values in Fig. 6.6. As it is seen in Fig. 6.6, DIXmax increases with the
increase of Sg , we fit a line of a shape DIXmax = aSg + b to the DIXmax vs.
Sg curves and show how it is affected by Sg . We see that an increase of
tmax
leads to an increase in the durotaxis index, and that the maximum
p
durotaxis index value for fixed tmax
and tmin
is larger for smaller values of
p
p
the gradient width.
We study the evolution of the MSD of walks on a gradient persistence
substrate using the migration exponent, α(t) as in Eq. 6.5. We calculate
the migration exponents for several simulation trajectories, as it is depicted in Fig. 6.7a. The walkers start at X = 0, from the area with the
persistence takes on its mean value, and thus - while the walker is still
in the gradient area - α(t) slowly changes during the walk. As the walk
progresses, the walkers will stumble upon the substrate of increased persistence and, likely, will continue walking the same direction, this way
α(t) decays slower for higher t corresponding to the migration on a substrate with persistence tmax
p .
Fig. 6.7b taken from [15] depicts the migration exponent for experimen-
100
Durotaxis
DIX
0.4
0.5
0.3
0.4
DImax
X
b)
a)
0.3
0.2
0.2
0.1
0.1
5
10
max
t 0.0
20
0
15
5
10
15
20
tp
FIG. 6.5 a) Mean durotaxis index of 104 cells during 200 steps of ∆t = 0.1, with
linear velocity Vc = 2, Gw = 3.7, tmin
= 0.2 and tmax
= 0.2 . . . 20.2. Solid lines
p
p
correspond to the substrates with the same tmax
,
points
on each curve mark the
p
maximum DIX , DIXmax ; b) DIXmax for the simulated walks, as a function of tmax
p .
The maximal durotaxis index increases with increasing tmax
.
p
DImax
X
a
0.7
0.110
0.105
0.6
0.100
0.095
0.090
0.5
0.5 1.0 1.5 2.0 2.5 3.0 3.5
Gw
0.4
b
0.22
0.20
0.18
0.16
0.14
0.12
0.3
0.2
0.5 1.0 1.5 2.0 2.5 3.0 3.5
0
1
2
3
4
Gw
5
Sg
Maximum durotaxis index DIXmax vs. gradient strength Sg , obtained
from simulations (solid lines) with tmin
= 0.2, tmax
= 0.2..20.2, and Nstep =
p
p
300. Each color corresponds to one value of Gw , from orange to blue Gw =
0.2, 0.7, 1.2, 1.7, 2.2, 2.7, 3.2, 3.7. Dashed lines: linear fits (DIXmax = aSg + b) to the
simulation results; Inset a): Fitted values of a vs. Gw , DIXmax increase slightly
faster with the increase of Gw , Inset b): Fitted values of b vs. Gw , DIXmax start from
a slightly higher point for higher Gw .
FIG. 6.6
6.5 Biased walk: basic characteristics
101
α(t)
a)
1.0
0.9
0.8
0.7
0.6
0.5
0
5
10
15
20
t
a) Migration exponent α vs. t averaged for 105 simulated trajectories,
= 0.2, tmax
= 4.2, Vc = 1.6, dt = 0.1 and Gw = 0.2 (green), 2.2 (blue), 4.2
p
(purple); fit of Eq. 6.6 for the last 50 points (dashed curves) of the trajectories,
and for the first 12 points (dotted lines) of cell trajectories; b) Experimentally observed migration exponent α vs. t averages for MSCs on a gradient substrate with
rigidity changing from 1kPa(soft substrate) to 34kPa (rigid substrate), the position
was recorded every 15 minutes during 10 hours (figure taken from supplementary material in [15]): for soft (blue), stiff (black) and gradient (red) substrate: the
curves to the left have the same shape as α(t) on the gradient persistence substrate: initially cells start on the substrate with low persistence, then they progress
to the substrate with high persistence and keep moving in it for longer times.
FIG. 6.7
tmin
p
tally derived cell migration data. There the plot for a gradient persistence
is supplemented with the migration data on soft and stiff substrates. While
the shape of the curves is similar, an attentive reader would notice that our
simulated curve, started from low persistence, never cross the line, corresponding to high persistence, while the experimentally derived curve
approaches high persistence curve from above. This observation brings us
to the question, that we are going to address in the next sections: Can we
tell a biased random walk from a gradient-persistent walk? Will the characteristics of the migration change when we enable the cells to sense the
local (at the scale of the cell) change in the substrate stiffness?
6.5 Biased walk: basic characteristics
We have considered persistent walks, and now we would like to know,
how persistence may combine with local stiffness sensing. In this section
we model cell migration based on sensing of the local (on the scale of the
102
Durotaxis
cell) substrate stiffness gradient.
Theoretical description in [115] shows how the change in substrate stiffness across the cell area provides for stronger adhesion in the part of the
cell situated on the stiffer part of the substrate. Thus cells may be able to
recognize the change in substrate stiffness on the scale of cell size, and
perform a motion driven directly by this stiffness gradient. This is fundamentally different form the scenario we considered previously: a change
in persistence is a non-directional change in behavior, while a local sensing of the direction of the gradient is directional.
A few gradient sensing-based models of durotaxis may be found in literature: for example, a recent [77] work applies the approach of [116] and
models the invasion of fibroblasts inside an artificially created wound as
a biased random walk (BRW). The key property of a BRW is the presence
of a chosen direction - the direction of bias, which we denote as xb , and
the direction of every step of the walk is chosen in a way that it prefers
to move parallel to xb . The angle Φb (t) between xb and the direction of
the walk, r(t), is nonuniformly distributed with the mean value at 0. The
shape of the distribution can vary as described in [116]. We set up a substrate, where the stiffness changes along the X-axis and thus the direction
of bias coincides with the X-direction. As in the section 6.4, we chose a
Gaussian distribution of angles Φb . We express the variance of this distribution, σb , as:
−1
Emax − Emin
σb = B Vc
,
(6.11)
2Gw
where B is the bias coefficient, Emax and Emin correspond to maximal and
minimal dimensionless ECM stiffnesses respectively, and the transition
between them occurs at a distance 2Gw . The bias coefficient measures
how strong the effect of a stiffness change on the change in bias is, and
( Emax − Emin )−1 indicates that the variance increases with a decrease of
the gradient strength. The typical simulated trajectories of a collection of
biased walkers are depicted in Fig. 6.8.
6.6 Biased walk: simulations
103
6.6 Biased walk: simulations
Firstly, we want to observe how the durotaxis index of a walk with a
constant bias depends on the value of σb that we supply to our simulation.
One may expect that the smaller σb is, the narrower is the distribution of
the steps around xb and thus larger values of DIX would occur. For a
perfectly biased walk, DIX = 1, and the length of the walk is equal to
the displacement in X-direction. In the limit of very large values of σb , the
walk has to become purely diffusive again.
We set up a wide gradient area, and make sure that the length of every
Y
1.5
1.0
0.5
0.5
1.0
1.5
2.0
2.5
3.0
X
-0.5
Trajectories of 20 biased walkers on a plane with Emin = 0.1, Emax =
100.2, Gw = 4, Vc = 2, Nstep = 20, and B = 1.
FIG. 6.8
cell path never exceeds the width of the gradient area, this way the bias
stays constant. The constant bias causes the durotaxis index to saturate
at a certain value, DIS , after several steps. The mean displacement in the
direction of the bias normalized to the length of one step is equal to the
mean cosine of the angle Φb :
hcos Φb i =
1
√
σb 2π
Z ∞
−∞
cos(Φb ) e
.
−Φ2b 2σ2
dΦb = e−
σ2
b
2
.
(6.12)
We simulate walks with constant bias, varying Emax , and collect the values of DIX in Fig. 6.9a. We plot DIS against σb , as illustrated in Fig. 6.9b,
where it is shown that durotaxis index saturates exactly as exp(−σb2 /2) as
predicted by Eq. 6.12.
To investigate the effect of the boundaries of the bias area on the migra-
104
Durotaxis
DIX
1.0
0.8
DIS
1.0
b)
0.8
0.6
0.6
0.4
0.4
0.2
0.2
a)
0.5
1.0
1.5
2.0
t
0
1
2
3
4
σb
a) The durotaxis index of 104 walkers on a substrate with, Emin = 0.2,
Gw = 200, Nstep = 400, Vc = 2. Each color corresponds to an average value
over the ensemble of walkers with one value of Emax , that we take in a range
0.21..1000.2, and hence a value of σb ranging from 0.2 (cyan) till 20000 (orange).
Small values of σb correspond to highly biased walks. b) Saturated value of durotaxis index, DIS vs. σb for these simulations (points), plotted along with the shape
DIS = exp(−σb2 /2) (solid line).
FIG. 6.9
tion, we set up a simulation where the width of the gradient, Gw = 10
allows the walkers to escape the region where the bias applies on the
timescales of the simulation. As it is illustrated in Fig 6.10a, for biased
walkers DIX plateaus quickly and then decreases, which shows that when
exposed to purely biased cues, cells will quickly turn parallel to the direction of the bias. DIX for walkers with persistence, which we also plotted
in Fig. 6.10a, increases slowly and then decreases, not showing a plateau
area. Fig. 6.10b compares the migration exponents, α(t) for biased walk
and the walk on the gradient persistence substrate. One can notice that
for a walk in a persistence gradient α(t) initially decays with time, reaching a local minimum, then increases, and after that saturates to 0.5. For
a BRW, α(t) starts out increasing, reaches a maximum and saturates to
α < 0.5, which corresponds to crowding at the border, where cells lose the
bias cue.
The case of a purely biased walk corresponds to cells that exhibit no persistence in their walk, and perform the sensing of the stiffness gradient
purely locally, i.e. between their front to rear edge. Cell migration is persistent, and thus differential persistence walk and biased walk must combine in cell migration trajectories. The question that we ask for the next
section is: How do the persistent walk and the biased walk influence each
6.6 Biased walk: simulations
105
α(t)
a) DIX
1.0
b)
0.30
X
0.25
0.8
40
30
20
10
0.20
0.15
0.6
20 40 60 80
0.10
t
0.4
0.2
0.05
20
40
60
80
t
0
20
40
60
80
t
a) Evolution of the durotaxis index DIX for walkers with Vc = 2,
dt = 0.1 on a gradient of width Gw = 10 during Nstep = 800 steps. For biased
walk (blue): Emin = 0.2, Emax = 20, σb = 1.5, B = 3; persistent walkers(red):
tmin
= 0.2 and tmax
= 20. Inset: Displacement along X-axis for biased walkers
p
p
(blue) and persistent walkers (red). b) Time evolution of the migration exponent
averaged over 104 biased(blue) walkers on a plane without persistence and persistent walkers(red), tpmin = Emin = 0.2, tpmax = Emax = 20, Gw = 10, Vc = 2,B = 3.
For persistent walkers, the migration exponent saturated at 0.5, which shows that
at longer timescales their migration turns diffusive. Biased non-persistent walkers acquire subdiffusive characteristics with time(α < 0.5), which corresponds to
the crowding at the right border of the area with the biased cue. This subdiffusive behavior is likely to saturate to α = 0.5 at t → ∞, the verification of which is
impeded by the growth of the fluctuations.
FIG. 6.10
other when the effects are combined? Does adding persistence to a walk
influence the process of durotaxis?
6.6.1 BRW on uniform persistence
When modeling the one-step memory process with bias, one must take
into account two factors influencing the direction of the next step. Let
Φi denote the angle that the walker forms with the x-axis at step i. The
direction of next step, Φi+1 , is then expressed through Φb -the angle that
corresponds to biased random walk with the corresponding variance computed in Eq. 6.11, and the angle related to persistence, Φ p which depends
on Φi for persistent walk. We compute Φi+1 as:
Φi+1 = κΦ p + (1 − κ )Φb ,
(6.13)
106
Durotaxis
σp
σb
θ
κ·
1
2
3
Φb
-1
-3
-2
-1
-2
-3
Φi
Cartoon of a combination of two cues in a persistent biased trajectory.
Here the direction of bias coincides with the direction of X- axis(black arrow). The
angle that the cell trajectory forms after step i + 1 with the X-axis is calculated
as a sum of two components, taken with respective weights κ and 1 − κ: The first
component is the sum of the previous angle Φi and the turning angle θ, which is
distributed according to the persistence of the walk with the variance in Eq. 6.8;
the second component is the bias angle Φb with the variance σb from Eq. 6.11.
The coefficient κ measures the relative contribution of persistence to the cell’s
movement.
FIG. 6.11
where Φ p = Φi + θ is the angle that the walk would form with x-axis if the
walk was not biased; κ shows how much the bias influences the walk: for
κ = 0 the walk is only biased, for κ = 1 the cell trajectory corresponds to
a non-biased walk with persistence. For intermediate values, this model
interpolates and combines the two effects. The illustration of Eq. 6.13 is
given in Fig. 6.11.
In this section we set up a simulation of a walk with constant persistence
that has a bias in a x-direction proportional to the steepness of the gradient. Starting out at κ = 0, we increase κ, gradually bringing more persistence in. We measure how DIX and DIY change with time. The result of
our findings can be summarized as follows: for κ < 0.55 an increase of the
presence of persistence increases the speed of durotaxis and the value of
DImax , as persistent walk once directed parallel to xb will continue going
that direction for longer times than an uncorrelated walk. For bigger κ on
the other hand, DIX increases more slowly, hence the persistence of the
walk becomes a stronger effect than the bias in the movement. We ask
the reader to turn to Fig. 6.12 for the illustration of the simulation results.
6.6 Biased walk: simulations
107
Next, we set up a simulation where a persistently moving cell is walking
DIX
0.35
DImax
0.35
0.30
0.25
0.20
0.15
0.10
0.30
0.25
0.20
Κ
0.20.40.60.81.0
0.15
0.10
0.05
10
20
30
40
t
Directional durotaxis index, DIX vs. time, averaged over 104 walkers
on a plane with constant persistence(t p = 0.2), Emin = 0.2, Emax = 2, Gw = 4,
Vc = 2; green: from low to high κ = 0..0.44, blue from hight to low κ = 0.56..1
Inset: maximum durotaxis index DImax vs κ: for intermediate values of κ, DIX
reaches higher values, which means that persistence and bias cooperate.
FIG. 6.12
in the area with the bias.
6.6.2 BRW on gradient persistence
A BRW on a substrate with differential persistence is based on two cues:
the bias cue and persistence cue. When splitting them into two components, one must take care that they may be distributed in space noncoincidentally: the area with gradient persistence will not necessarily coincide with the area on which the biased cue is present. Here we investigate
the case where those two areas are overlapping, and focus on that part of
the substrate where the bias is constant and the persistence time t p varies
linearly in space. Without loss of universality, we choose Emin = tmin
and
p
max
max
E
= t p . Then, the parameter B in Eq. 6.11 measures how a change
in persistence translates into the strength of the bias. The bigger B is, the
more the change in t p narrows the variance of the distribution of Φb as in
Eq. 6.11. We set up a simulation on a substrate where persistence/stiffness
108
Durotaxis
DImax
DIX
a)
0.6
0.5
0.4
0.3
0.2
0.1
b)
0.6
0.5
0.4
0.3
0.2
0.1
10
20
30
40
t
κ
0.2
0.4
0.6
0.8
1.0
FIG. 6.13 a) Durotaxis index along x-axis depending on t for the ensemble of
104 walkers, Emin = tmin
= 0.1 till Emax = tmax
= 2 along the gradient width
p
p
Gw = 4 with Vc = 2, B = 1 and dt = 0.1, and κ in a range from 0 till 0.6(green)
and from 0.7 till 1(blue). b) Durotaxis index along x-axis depending on κ for the
same ensemble of walkers(solid line) is comparison for DImax from Fig. 6.12.
starts from 0.1 and ends at 2.1 with B = 1. Our system has four more pamax
rameters: tmin
p , t p , Gw and κ-the proportion of bias in cell movement. We
present a simulation for several values of κ in Fig. 6.13. The durotaxis index reaches higher maximum values than in Fig. 6.12, which shows that
having additional differential persistence helps cells durotax, and that the
combination of differential persistence and biased migration may lead to
a more efficient motility mechanism.
We have assessed the migration of cells guided with biased cues and differential persistence on 2D substrates. We now turn to a more illustrative
case of migration in 1D, to investigate the spatial distribution of the walkers on the gradient persistence substrate.
6.7 1D model of durotaxis
109
6.7 1D model of durotaxis
We consider now an ensemble of discrete walkers on a line, with a fixed
step length, s, and a position xi corresponding to the i-th step of the walk.
For a walk with a certain rate of turning λ, the probability of a walker to
turn, pturn , during one step of duration τ is:
pturn = λτ.
(6.14)
From now on we assume unit time step, and unit velocity: τ = 1 and s = 1,
then pturn = λ. For λ = 0.5 Eq. 6.14 corresponds to a general random walk.
For λ = 0 the walkers move in a perfectly persistent manner: Starting from
x = 0 they reach the coordinate x = +/ − N. Depending on the direction
of the first step, walkers that started off going right cover a distance of
+ N, walkers that started off to the left all reach the point − N. While the
mean displacement for the walk for λ = 0 will be 0, the mean squared
displacement, h X 2 (i )i = i2 corresponds to a ballistic movement in 1D. For
the case of λ = 0.5 the walkers perform a diffusion and MSD is h X 2 (i )i =
2Di and D = 1/2λ. In the intermediate case where 0 < λ < 0.5, the MSD
after the i-th step of the walk is computed as:
h Xi2 i =
o
1
1 {i −
1 − e−2λi ) ,
λ
2λ
(6.15)
which corresponds to the MSD computed from the solution of the onedimensional Telegrapher’s equation for velocity jump processes:
d2 p
d2 p
dp
+
2λ
=
.
dt2
dt
dx2
(6.16)
Here p( x, t) is the probability distribution for the walkers in space and
time. For specific boundary conditions: p(0, 0) = δ( x ) and p( x, 0) x = 0,
Eq. 6.16 has a closed form (though quite unwieldy) solution, which is
derived in [83].
Notice, that Eq. 6.15 is equivalent to Eq. 6.3, when τp = 1/2λ and Vc = 1.
We first set up a simulation to verify that the migration exponent α(t)
from Eq. 6.5 matches the migration exponent α(i ) in this 1D case. The
110
Durotaxis
walkers will have a turning probability per unit time equal to the value
of λ, and the probability of not turning, consequently, will be 1 − λ. We
compute the migration exponents after a certain number of steps in Figs.
6.14a and 6.14bm which illustrate the analysis as in section 6.4.
The persistence of a walk in 1D is thus quantified by λ, and in the next
section we mimick the gradient in persistence by treating the case where
λ depends on the current position of the walker.
α(i)
τp
1.1
12
1.0
10
0.9
8
0.8
0.7
6
0.6
4
2
0.5
a)
0
50
100
150
b)
i
200
5
10
15
20
1/λ
√
d log h Xi2 i
a) Migration exponent α(i ) =
vs. t for 104 walkers on
d log i
a line with λ from 0.1 (orange solid line) to 0.5 (red solid line). The dashed
lines of corresponding color illustrate the fit of the full derivative in Eq. 6.5; b)
Persistence time τp from these simulation curves vs. 1/λ (solid line), fitted to the
shape τp = a/λ(dashed line) and the fit is τp = 0.47/λ.
FIG. 6.14
6.7.1 Nonuniform persistence
When the walkers described above move on the substrate with varying
persistence, their probability distribution function will be guided by the
following inhomogeneous Telegrapher’s equation:
d2 p
d2 p
dp
+
2λ
(
x
)
=
.
dt2
dt
dx2
(6.17)
The questions that we ask are: Will there be a drift, when the ensemble of
walkers, guided by Eq. 6.17 is exposed to a landscape with nonuniform
persistence? How does the steepness of the persistence gradient influence
the mean displacement?
6.7 1D model of durotaxis
a)
111
X
b) DIX
200
0.30
150
0.25
0.20
100
0.15
0.10
50
0
0.05
50
100
150
200
i
20
40
60
80
i
100
a) Evolution, over simulation step i, of the positions of 102 cells peforming a walk on a line with turning probability scaling from λmin = 0.02 to
λmax = 0.5, length of the gradient interval Gw = 10 b) Durotaxis index DIX , for
gradient of width Gw = 10, λmax = 0.5 and three values of minimal turning probability: λmin = 0.02(green), λmin = 0.25(blue) and flat λmin = λmax = 0.5(red).
For the same set of other parameters, a steeper gradient corresponds to a higher
durotaxis index.
FIG. 6.15
We set up the landscape with evolving turning probability, λ that starts
from λmin on a substrate with high persistence(stiff) and goes till λmax on
a more compliant substrate, over the distance Gw - the gradient width.
We record the displacement for the walkers with time, as illustrated in
Fig. 6.15a. Fig. 6.15 shows that the walkers displace further into positive
values of X, which corresponds to the section of the line that that has a
lower local turning probability. After i steps, we measure the durotaxis
index through the mean displacement h Xi i as:
DIX =
h Xi i
.
i
(6.18)
The time evolution of DIX for various turning probabilities is shown in
6.15b.
We now want to determine how the value of λmin influences the maximum
durotaxis index. Fig. 6.16a shows the evolution of the mean durotaxis index for trajectories of varying λmin . We observe the highest value of durotaxis reached, DImax , and how it will depend of the extreme values of λ in
Fig. 6.16b. As expected, the larger the difference between the persistence
on one side of the gradient the other, the greater DImax .
112
a) DI
Durotaxis
b) 0.4
X
0.30
0.3
0.25
DImax
0.20
0.15
0.2
0.1
0.10
0.05
20
40
60
80
i
100
0
0.5
1.0
1.5
2.0
2.5
log(-1/λmin+1/λ max)
3.0
3.5
a) Average values of DIX (t) during 100 steps of the walkers on a
gradient substrate with λmax = 0.5. Each curve corresponds to the averaged values on a substrate with a value of λmin ranging from 0.02(orange) to 0.5(red). b)
DImax as a function of the difference between the inverse of turning probabilities
on the either side of the gradient area log (−1/λmin + 1/λmax ). The durotaxis
index increases with the increase in gradient steepness.
FIG. 6.16
b)
a)
b
2.5
0.5
DImax
0.4
0.3
2.0
0.2
1.5
0.1
0.0
2
1
0
1
2
log(-1/λmin+1/λmax)
3
4
4
6
8
G
10 w
FIG. 6.17 a) Maximum durotaxis index, DImax vs. log (−1/λmin + 1/λmax )
for 104 walkers with λmax = 0.5 and λmin ranging from 0.02 till 0.5. One
curve corresponds to one value of gradient width, from orange to red: Gw =
2, 3, 4, 5, 6, 7, 8, 9, 10. DImax decreases with increasing gradient width. Dashed
lines correspond to linear fits (y = a( x + b)2 ) to the simulation curves. b) The
parameter b from the fitted curves collected in one plot (black), plotted against
the length of the gradient interval Gw . The solid blue line, which corresponds to
y = 2.8 − 0.2x, shows that the wider the gradient area is, the lower is the value
of the maximum durotaxis index.
Then, for the same sets of values of λmin and λmax , we vary the width of
the gradient interval, and show that the steeper the transition from λmin
to λmax , the higher are the DImax values achieved. Our final findings are
6.7 1D model of durotaxis
113
illustrated in Fig. 6.17a and Fig. 6.17b.
To assess the drift towards the area with higher persistence, we now turn
to the analysis of the spatial distribution of the walkers. We collect the
positions of the walkers at every time step, and bin them into spatiallyresolved intervals, thus computing a probability distribution p( x, t) of
walkers on the line. Fig. 6.18 depicts snapshots of p( x, t) at different times
of the simulation. The shape of p( x, t) with two peaks shows that the
walkers propagate in two wave fronts; one going right, and the other one
going left. The walkers situated to the right of the origin propagate much
faster than the ones that are situated to the left. We track the positions of
the maximums of p( x, t) and compare their evolution with time in Fig.
6.19. The walkers move farther during the same number of steps when
they are situated in the area with small λ than the walkers which are in
the area with high λ. This way h X i tends to increase and the durotaxis is
achieved.
a) 1.0
0.8
0.6
0.4
0.2
0
b) 0.10
p(x)
100
50
0
50
x
100
c) 0.07
0.06
p(x)
100
50
0
50
p(x)
x
0
50
x
100
0
100
50
0.03
p(x)
0.03
0
50
x
100
p(x)
0.02
0.01
0.010
50
p(x)
f)
0.02
100
0.05
0.04
0.03
0.02
0.01
0
100
e)
d)
0.04
0.03
0.02
0.01
0
0.08
0.06
0.04
0.02
0
100
50
0
50
x
100
0
100
50
0
50
x
100
FIG. 6.18 a)-f) Snap shots of simulated averaged probability distribution of
the positions of 105 cells on a line with λ scaling from λmin = 0.02 to the right
of the origin till λmax = 0.5 to the left of the origin, Gw = 10. a) We start the
walkers at x = 0 with equal probability of making the first step up or down the
gradient of persistence, and see at step 21, 41, 61, 81 (b)-e) respectively two wave
fronts propagating towards the right and towards the left of the origin. f)The
distribution of the walkers after step 101: a fraction of the walkers that was only
making steps up the X −axis, reaches X = 100, which corresponds to the peak in
p( x ) to the right of the origin. The wave front going to the right is moving faster
than the wave front going to the left.
114
Durotaxis
a) x
b)
max(R)
xmax(L)
100
20
80
40
60
80
i
100
2
60
40
4
20
6
20
40
60
80
i
100
8
FIG. 6.19 a) The evolution of the value xmax ( R) at which the maximum of p( x )
is achieved to the right of the origin, in a simulation where 105 cells walk on
a line with λ varying from λmin = 0.02 to λmax = 0.5 over a gradient interval
of length Gw = 10(points). The solid line is a fit of the simulation results to a
shape xmax = ai, with a = 1.001 which corresponds to a ballistic motion; b) The
evolution of the position of the wave front xmax ( L) for walkers situated to the left
of the origin during the same sumulation(points). Dashed line fits the simulation
results to xmax = ai with parameter a = −0.08, which shows that the distance
between the origin and this wave front is very slowly growing, and that the net
motion is to the right.
6.8 Conclusions
Durotaxis can be achieved through differential persistence, while biased
migration mechanism combined with persistent migration results in higher
maximum values of durotaxis index. The nature of directed migration
may be concluded from the shape of the migration exponent: for purely biased migration, the migration exponent initially increases with time, then
goes down to a value below 0.5. For a purely persistence-guided migration, the migration exponent initially decreases with time, corresponding
to the walk on a substrate with low persistence, and then the migration
exponent saturates at the level corresponding to the highest persistence
value. When migrating on substrates with a gradient in persistence, a net
drift towards stiffer part of the substrate is achieved as a result of the
otherwise undirected walkers moving much faster in the direction of the
gradient on the high persistence zone of the substrate, compared to the
low-persistence zone.
Chapter 7
Conclusions and Outlook
In this chapter we remind the reader of the main questions that were addressed in
this work and summarize the main results achieved in this thesis. We conclude
with an overview of possible directions for further research.
115
116
Conclusions and Outlook
7.1 Conclusions
The main questions around which we frame the work, presented in this
thesis are:
How do the mechanical properties of the environment affect the processes inside
the cell and cells migration? What are the molecular mechanisms that allow cells
to sense the extracellular stiffness?
In this chapter we summarize the work that we did while trying to answer the questions stated above, and present the main conclusions of this
thesis. We end this chapter with suggesting directions for further research.
In Chapter 3 we introduced the model of cell-ECM adhesion we were
going to work with, we stated in detail how we incorporated the experimental findings in our model. We assessed how the cluster of catch-bond
forming integrins reacts to the load exerted upon it. The main conclusions
of Chapter 3 are:
• The steady-state number of bound catch bonds increases with increasing
force. At sufficiently high forces the catch-bond cluster is unstable and
unbinds.
• Initially, for small loads, the lifetime will increase with increasing force,
and reach its maximum at a finite force. This feature of a collective forcelifetime curve leads us to a conclusion that catch bond clusters are more
stable with a force than without a force.
• Assuming that independent of the ECM stiffness, cell invests a fixed amount
of work in contracting each of the individual stress fibers that transfers
the load to a focal adhesion, we constructed a mechanical property sensor
from a catch-bond enriched focal adhesion. We have shown that the number of bound catch bonds increases with increasing extracellular stiffness.
The work in Chapter 3 is based on the approach in [95], where a collection
of slip bonds sharing the load was treated. We incorporate the experimental results found in [41] into this framework. The analytical findings in
7.1 Conclusions
117
Chapter 3 complement and extend the results in [91], we provide a hypothesis for a mechanical property sensor constructed out of catch bonds
sharing the load. The central result of Chapter 3 is: Catch-bond cluster
with a constant work invested in the contraction of a single stress fiber
acts as a mechanical property sensor.
A logical extension of our work on a pure catch bond cluster is a model
of focal adhesion where catch- and slip bonds are present in the same
cluster and act in parallel, which was first assessed in Chapter 4. The
system of catch and slip bonds supporting the load together inside one
focal adhesion, was designed in order to represent a more realistic setup
similar to the structure of a functioning focal adhesion. We also investigate
if our hypothesis of minimal sensing action would hold in this extended
setup. The main results of Chapter 4 are:
• The number of bound catch bonds in a mixed cluster will increase with
increasing force. Catch- and slip bonds will react according to their individual force-lifetime curves when acting in parallel: the number of bound
catch bonds will increase with increasing force, and the number of bound
slip bonds will decrease with the increasing force.
• The number of bound catch bonds in a mixed cluster, similarly to purely
catch bond cluster, will, under constant work assumption, reflect the extracellular stiffness. This way a cluster of catch- and slip bonds acting in
parallel can still act as a mechanical property sensor.
• We presented the first survey of the spatial organization of integrins inside the focal adhesion: we investigate how the mobility of different types
of integrins is affected by the presence of force acting on the adhesion.
The type of the bond that an individual integrin forms with its ligand
manifests itself in a macroscopic property of focal adhesion - integrin mobility. Integrin mobility can be recorded during FRAP experiments and
we proposed that it may serve as an indirect way to measure individual
integrin-ligand bond properties.
118
Conclusions and Outlook
The system treated in Chapter 4 was previously simulated in [32], where
the model of two types of integrins in a focal adhesion was treated. There
the assumption on the cluster response to tension was that the density of
one type of integrins increases after a certain force threshold generated by
actin retrograde flow was achieved. Our findings complement the modeling in [32] with the analysis of a mixed cluster stability, we compute how
the diffusion of integrins inside the focal adhesion is influenced by the
force, exerted upon it. The main result of Chapter 4 is: Catch- and slip
bond forming integrins in a mixed cluster keep their individual properties, and a mixed cluster may still act as a mechanical property sensor.
In Chapter 5 we look deeper into the spatial organization of integrins
inside focal adhesion by adding interaction to our system. Based on the
notion that only bound integrins interact, we concluded that:
• A mixed cluster of catch bonds and slip bonds acts a non-interacting cluster for low coupling constant. As we increase the coupling constant, the
number of bound bonds in an interacting cluster saturates at higher values
for higher values of force, compared to the cluster without interactions.
Slip bonds will get similarly reinforced at low values of force experienced
by a cluster. Further increase of the coupling constant leads to both catchand slip bond reinforcement.
• The spatial organization of integrin types in an interacting cluster will
be guided by the force, acting on that cluster. Bonds of the same type
form aggregations, depending on the level of force. For low forces, high
level or aggregation is achieved due to slip bond clustering. For high load,
the bound catch bonds will cluster, for intermediate values of force, the
ordering will take place to a lesser extent.
• The force at which the cluster looses stability increases with the increase
of the coupling constant, and we conclude that integrin interaction may
provide for a more resilient cell-ECM adhesion.
In Chapter 5 we probed the topic of force-induced phase separation between integrins inside focal adhesion. The systems of interacting stickers
7.1 Conclusions
119
were treated before in [117] when modeling T-cell adhesion. Two types
of sticker of unequal length were placed on a rigid membrane, the elastic energy of which would prevent the two stickers of different types to
be placed near each other. There as well, a tangential force, bringing the
stickers towards the center of a focal adhesion was introduced, and this
way the pattern formation was achieved. The model we treat in Chapter
5 do not imply any tangential force and the phase separation of integrins
in guided by the force acting perpendicular to the plane of the adhesion.
The central conclusion of Chapter 5 is: Ordering of different types of interacting bonds in a mixed adhesion cluster is selectively promoted by
a tensile force acting on this cluster.
In Chapter 6 we investigate another stiffness-dependent property of the
cell behavior - durotaxis. The questions that we asked are: is the ability
of an individual cell to sense the gradient in stiffness of the substrate
necessary for cells ability to perform durotaxis? In other words, is there
a mechanism that will not involve local gradient sensing (on the scale of
a cell) but will guide cell migration in a way that will result in a net cell
displacement towards a stiffer part of the substrate? The main results of
Chapter 6 are:
• When cells migrate more persistently on a stiff substrate in comparison
to a more compliant substrate, placing them on a substrate with gradient
persistence will cause durotaxis to occur. This novel durotaxis mechanism
does not involve sensing the local change in substrate stiffness.
• The durotaxis index that we compute in our simulations will increase with
the increase of the steepness of the gradient.
• In order to compare persistence-based simulation trajectories with the trajectories of cells performing biased walk, we compute how the average
migration exponent evolves with time for both biased and persistent walk.
The differences in the shapes of migration exponent curves, obtained for
the two types of migration, are crucial to infer the mechanisms underlying
cell durotaxis from experimentally derived data. The migration exponent
120
Conclusions and Outlook
for a biased walk will initially grow with time and then saturate at the
level that corresponds to subdiffusive walk. On the other hand, the collection of walkers on a gradient persistence will have on average a migration
exponent starting out at low levels, initially decreasing for a short time
and for longer simulation time increasing to reach the level that matches
persistent walk on stiff substrate.
• Combining persistence and bias in the same walk leads to the overall
increase in the average speed at which cells migrate towards a stiffer substrate, and in particular results in a higher values of maximum durotaxis
index.
Chapter 6 exploits the experimental findings in [15] and [3], and models
cell migration as a velocity jump process, which was also used by [78]
to model cell durotaxis. We propose a novel mechanism of cell durotaxis,
which is distinctly different from the biased migration, treated in [78] and
provide the way to experimentally distinguish cell migration data between
biased and persistent cell. The main conclusion of Chapter 6 is: Stiffnessdependent persistence of cell migration may alone result in cell durotaxis.
7.2 Outlook
The findings presented in this thesis suggest novel methods according to
which cell-ECM interactions may be treated. As in every model of a biological system, some of the parameters of the cell crucial for its functioning
were omitted when developing the minimal system which suits the goals
of this work. In this section we outline the main directions that the future
research may take.
The properties of catch bond cluster under nonuniform load were briefly
assessed in Chapter 3. Further simulation of a mixed cluster of integrins,
experiencing nonuniform loading may provide interesting results concerning the spacial organization of integrins inside the focal adhesion.
7.2 Outlook
121
The nonuniform loading of a focal adhesion may cause slip integrins to
aggregate in the area where small force is applied. Catch-bond integrins
will aggregate in the adhesion area experiencing larger force. This may
be achieved by applying the approach in [100] where the loading on the
focal adhesion is characterized by the stress distribution parameter and
the continuous equations are applied. This approach applied to a mixture
of catch and slip bonds and the assumptions that free integrins would get
internalized, may result in the conclusion that the load influence the distribution of catch and slip bonds inside the focal adhesion.
Other interesting issue concerns the stability of an interacting cluster: our
simulations in Chapter 5 are designed in a way that the dynamic properties of a focal adhesion with interacting integrins are not possible to
assess. The simulations using kinetic Monte Carlo method can provide
the information on the lifetime of a cluster, where catch- and slip bonds
interact.
Concerning Chapter 6, a logical continuation of this work is examining
the experimentally derived trajectories in order to compute the realistic
simulation parameters of our walks. In particular it would be interesting
to know to which extent the migration is guided by persistence. If biased
and persistent walks in cell durotaxis are combined, how large is the contribution of the bias in guiding cell migration? This lies outside the goal
of this work but may be done using large sets of trajectories which may
be obtained experimentally in the future.
Another more broad subject, that this thesis does not cover, is bridging the
length scales from individual adhesion to a full cell simulation. Integrins
are known to influence the persistence of cell migration [118]. How do the
individual properties of integrin-ligand bonds affect the persistence of cell
walk? It would be instructive to investigate the physical mechanisms underlying cell persistence. The final goal of this direction to future research
would be incorporating the ECM-stiffness in the model of cell motility.
122
Conclusions and Outlook
Why is it so that cells are more persistent on stiff substrates and not the
other way around?
In this thesis, the individual binding properties of catch - and slip- bonds
were proven to affect the macroscopic parameter of focal adhesion - integrin diffusivity. Experimentally measuring integrin diffusivity and the
force exerted on a focal adhesion with the help of our modeling, may
enable us to indirectly measure the individual binding properties of integrins. This may present us with a way of measuring the properties of
integrin-ligand interaction, and on the basis of this information to benefit
the design of biomaterials.
The idea that durotaxis is persistence-based, gives us potentially a new
way of directing cells towards certain areas of the substrate: which can
be done by supplementing the parts of substrates with the coating that
will increase cell persistence. These new findings, applied to, for example,
increase the migration of fibroblasts towards the wounded area, may be
relevant for the future of medicine.
List of Abbreviations
AFM - atomic force microscope
BRW - biased random walk
CRW - correlated random walk
CSK - cytoskeleton
ECM - extracellular matrix
FA - focal adhesion
FN - fibronectin
FRAP - fluorescence recovery after photobleaching
MSC - mesenchymal stem cell
MSD - mean squared displacement
123
124
List of Abbreviations
Bibliography
[1]
C. Bunting and C. C. Eades. The effect of mechanical tension upon the
polarity of growing fibroblasts. The Journal of Experimental Medicine
44, 147 (1926). (page 2)
[2]
H.-B. Wang, M. Dembo, and Y.-L. Wang. Substrate flexibility regulates growth and apoptosis of normal but not transformed cells. American
Journal of Physiology-Cell Physiology 279, C1345 (2000). (page 2)
[3]
P. W. Oakes, D. C. Patel, N. A. Morin, D. P. Zitterbart, B. Fabry,
J. S. Reichner, and J. X. Tang. Neutrophil morphology and migration
are affected by substrate elasticity. Blood 114, 1387 (2009). (pages 2, 26,
92, 120)
[4]
T. Yeung, P. C. Georges, L. A. Flanagan, B. Marg, M. Ortiz, M. Funaki, N. Zahir, W. Ming, V. Weaver, and P. A. Janmey. Effects of substrate stiffness on cell morphology, cytoskeletal structure, and adhesion.
Cell Motility and the Cytoskeleton 60, 24 (2005). (page 2)
[5]
A. J. Engler, S. Sen, H. L. Sweeney, and D. E. Discher. Matrix elasticity directs stem cell lineage specification. Cell 126, 677 (2006). (pages 3,
38)
[6]
B. Trappmann, J. Gautrot, J. Connelly, D. Strange, Y Li, M. Oyen,
M. Cohen Stuart, H. Boehm, B Li, V Vogel, J. Spatz, F. Watt, and
W. Huck. Extracellular-matrix tethering regulates stem-cell fate. Nature
Materials 11, 642 (2012). (pages 3, 38)
[7]
P. Fernandez and A. R. Bausch. The compaction of gels by cells: a
case of collective mechanical activity. Integrative Biology 1, 252 (2009).
(pages 3, 7, 38)
125
126
Bibliography
[8]
H. J. Kong, J. Liu, K. Riddle, T. Matsumoto, K. Leach, and D. J.
Mooney. Non-viral gene delivery regulated by stiffness of cell adhesion
substrates. Nature Materials 4, 460 (2005). (page 3)
[9]
K. N. Dahl, A. J. Ribeiro, and J. Lammerding. Nuclear shape, mechanics, and mechanotransduction. Circulation Research 102, 1307 (2008).
(page 5)
[10]
H. Kojima, A. Ishijima, and T. Yanagida. Direct measurement of stiffness of single actin filaments with and without tropomyosin by in vitro
nanomanipulation. Proceedings of the National Academy of Sciences
USA 91, 12962 (1994). (page 6)
[11]
P. A. Janmey and D. A. Weitz. Dealing with mechanics: mechanisms
of force transduction in cells. Trends in Biochemical Sciences 29, 364
(2004). (pages 6, 11, 12)
[12]
P. Hotulainen and P. Lappalainen. Stress fibers are generated by two
distinct actin assembly mechanisms in motile cells. The Journal of Cell
Biology 173, 383 (2006). (pages 6, 11)
[13]
J. Y. Rho, R. B. Ashman, and C. H. Turner. Young’s modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements. Journal of Biomechanics 26, 111 (1993). (page 6)
[14]
Y. Feng, R. J. Okamoto, R. Namani, G. M. Genin, and P. V. Bayly.
Measurements of mechanical anisotropy in brain tissue and implications
for transversely isotropic material models of white matter. Journal of
the Mechanical Behavior of Biomedical Materials 23, 117 (2013).
(page 6)
[15]
R. Matthew, J. Swift, D. P. C. Dingal, P. Shah, J.-W. Shin, and D. E.
Discher. Crawling from soft to stiff matrix polarizes the cytoskeleton and
phosphoregulates myosin-II heavy chain. The Journal of Cell Biology
199, 669 (2012). (pages 7, 35, 92, 95, 99, 101, 120)
[16]
B. C. Isenberg, P. A. DiMilla, M. Walker, S. Kim, and J. Y. Wong.
Vascular smooth muscle cell durotaxis depends on substrate stiffness gradient strength. Biophysical Journal 97, 1313 (2009). (pages 7, 26, 35,
92)
Bibliography
127
[17]
D. Boal. Mechanics of the Cell (Cambridge University Press, 2002).
(page 7)
[18]
P. Kanchanawong, G. Shtengel, A. M. Pasapera, E. B. Ramko, M. W.
Davidson, H. F. Hess, and C. M. Waterman. Nanoscale architecture of
integrin-based cell adhesions. Nature 468, 580 (2010). (pages 8, 23)
[19]
A. Bershadsky, M. Kozlov, and B. Geiger. Adhesion-mediated mechanosensitivity: a time to experiment, and a time to theorize. Current Opinion
in Cell Biology 18, 472 (2006). (page 8)
[20]
C. K. Choi, M. Vicente-Manzanares, J. Zareno, L. A. Whitmore,
A. Mogilner, and A. R. Horwitz. Actin and α-actinin orchestrate the
assembly and maturation of nascent adhesions in a myosin II motorindependent manner. Nature Cell Biology 10, (2008). (page 8)
[21]
H. B. Schiller, M. R. Hermann, J. Polleux, T. Vignaud, S. Zanivan,
C. C. Friedel, Z. Sun, A. Raducanu, K. E. Gottschalk, M. Thery, M.
Mann, and R. Fassler. β 1 - and αν -class integrins cooperate to regulate
myosin II during rigidity sensing of fibronectin-based microenvironments.
Nature Cell Biology 15, 625 (2013). (pages 8, 9, 13, 53, 55, 75)
[22]
K. A. Beningo, M. Dembo, I. Kaverina, J. V. Small, and Y.-l. Wang.
Nascent focal adhesions are responsible for the generation of strong propulsive forces in migrating fibroblasts. The Journal of Cell Biology 153,
881 (2001). (page 8)
[23]
K. A. DeMali, K. Wennerberg, and K. Burridge. Integrin signaling to
the actin cytoskeleton. Current Opinion in Cell Biology 15, 572 (2003).
(page 8)
[24]
H. Wolfenson, A. Bershadsky, Y. I. Henis, and B. Geiger. Actomyosingenerated tension controls the molecular kinetics of focal adhesions. Journal of Cell Science 124, 1425 (2011). (page 8)
[25]
U. Theisen, E. Straube, and A. Straube. Directional persistence of migrating cells requires Kif1C-mediated stabilization of trailing adhesions.
Developmental Cell 23, 1153 (2012). (page 8)
[26]
R. O. Hynes. Integrins: bidirectional, allosteric signaling machines. Cell
110, 673 (2002). (page 9)
128
Bibliography
[27]
R. O. Hynes. Integrins: Versatility, modulation, and signaling in cell
adhesion. Cell 69, 11 (1992). (page 9)
[28]
J. Pizarro-Cerda and P. Cossart. Bacterial adhesion and entry into host
cells. Cell 124, 715 (2006). (page 9)
[29]
P. Roca-Cusachs, N. C. Gauthier, A. del Rio, and M. P. Sheetz.
Clustering of α5 β 1 integrins determines adhesion strength whereas αv β 3
and talin enable mechanotransduction. Proceedings of the National
Academy of Sciences 106, 16245 (2009). (pages 9, 62, 80, 83)
[30]
H. Truong and E. H. Danen. Integrin switching modulates adhesion
dynamics and cell migration. Cell Adhesion & Migration 3, 179 (2009).
(pages 9, 39, 58)
[31]
M. R. Morgan, A. Byron, M. J. Humphries, and M. D. Bass. Giving off mixed signals–distinct functions of α5 β 1 and αv β 3 integrins in
regulating cell behaviour. IUBMB Life 61, 731 (2009). (page 9)
[32]
A. Elosegui-Artola, E. Bazelliares, M. D. Allen, I. Andreu, R. Oria,
R. Sunyer, J. J. Gomm, J. F. Marshall, J. L. Jones, X. Trepat, and
P. Roca-Cusachs. Rigidity sensing and adaptation through regulation of
integrin types. Nature Materials 14, 6 (2014). (pages 9, 62, 73, 82, 118)
[33]
D. P. White, P. T. Caswell, and J. C. Norman. αv β 3 and α5 β 1 integrin
recycling pathways dictate downstream Rho kinase signaling to regulate
persistent cell migration. The Journal of Cell Biology 177, 515 (2007).
(pages 9, 62)
[34]
M.-P. Valignat, O. Theodoly, A. Gucciardi, N. Hogg, and A. C.
Lellouch. T lymphocytes orient against the direction of fluid flow during LFA-1-mediated migration. Biophysical Journal 104, 322 (2013).
(page 10)
[35]
T. Lämmermann, B. L. Bader, S. J. Monkley, T. Worbs, R. WedlichSöldner, K. Hirsch, M. Keller, R. Förster, D. R. Critchley, R. Fässler,
et al. Rapid leukocyte migration by integrin-independent flowing and
squeezing. Nature 453, 51 (2008). (page 10)
[36]
D. B. Lovett, N. Shekhar, J. A. Nickerson, K. J. Roux, and T. P. Lele.
Modulation of nuclear shape by substrate rigidity. Cellular and Molecular Bioengineering 6, 230 (2013). (page 11)
Bibliography
129
[37]
J. Swift, I. L. Ivanovska, A. Buxboim, T. Harada, P. D. P. Dingal, J.
Pinter, J. D. Pajerowski, K. R. Spinler, J.-W. Shin, M. Tewari, et al.
Nuclear lamin-A scales with tissue stiffness and enhances matrix-directed
differentiation. Science 341, 1240104 (2013). (page 11)
[38]
P. W. Oakes, S. Banerjee, M. C. Marchetti, and M. L. Gardel. Geometry regulates traction stresses in adherent cells. Biophysical Journal
107, 825 (2014). (pages 13, 75)
[39]
J. A. Askari, C. J. Tynan, S. E. Webb, M. L. Martin-Fernandez, C.
Ballestrem, and M. J. Humphries. Focal adhesions are sites of integrin
extension. The Journal of Cell Biology 188, 891 (2010). (page 18)
[40]
R. K. Paradise, D. A. Lauffenburger, and K. J. Van Vliet. Acidic extracellular pH promotes activation of integrin αv β 3 . PloS One 6, e15746
(2011). (pages 18, 19)
[41]
F. Kong, A. J. Garcia, A. P. Mould, M. J. Humphries, and C. Zhu.
Demonstration of catch bonds between an integrin and its ligand. J. Cell.
Biol. 185, 1275 (2009). (pages 18, 19, 38–41, 62, 63, 81, 116)
[42]
O. K. Dudko, G. Hummer, and A. Szabo. Intrinsic rates and activation
free energies from single-molecule pulling experiments. Physical Review
Letters 96, 108101 (2006). (page 18)
[43]
M. Lee, H.-K. Yang, K.-H. Park, D.-K. Kang, S.-I. Chang, and I.-C.
Kang. Measurement of interaction force between nanoarrayed integrin alphavbeta3 and immobilized vitronectin on the cantilever tip. Biochemical
and Biophysical Research Communications 362, 935 (2007). (page 18)
[44]
C. Jurchenko, Y. Chang, Y. Narui, Y. Zhang, and K. S. Salaita. Integringenerated forces lead to streptavidin-biotin unbinding in cellular adhesions. Biophysical Journal 106, 1436 (2014). (page 18)
[45]
M. Dembo, D. Torney, K. Saxman, and D. Hammer. The reactionlimited kinetics of membrane-to-surface adhesion and detachment. Proceedings of the Royal Society B: Biological Sciences 234, 55 (1988).
(pages 19, 39)
[46]
E. B. Finger, K. D. Puri, R. Alón, M. . Lawrence, U. H. von Andrian,
and T. A. Springer. Adhesion through L-selectin requires a threshold
hydrodynamic shear. Nature 379, 18 (1996). (page 19)
130
Bibliography
[47]
W. E. Thomas, E. Trintchina, M. Forero, V. Vogel, and E. V. Sokurenko.
Bacterial adhesion to target cells enhanced by shear force. Cell 109, 913
(2002). (page 19)
[48]
K.-C. Chang and D. A. Hammer. The forward rate of binding of surfacetethered reactants: effect of relative motion between two surfaces. Biophysical Journal 76, 1280 (1999). (page 19)
[49]
R. R. Isberg and P. Barnes. Dancing with the host: flow-dependent bacterial adhesion. Cell 110, 1 (2002). (page 19)
[50]
B. T. Marshall, M. Long, J. W. Piper, T. Yago, R. P. McEver, and C.
Zhu. Direct observation of catch bonds involving cell-adhesion molecules.
Nature 423, 190 (2003). (pages 19, 38, 39, 62)
[51]
E. Evans, A. Leung, V. Heinrich, and C. Zhu. Mechanical switching
and coupling between two dissociation pathways in a P-selectin adhesion bond. Proceedings of the National Academy of Sciences of the
United States of America 101, 11281 (2004). (pages 19, 22)
[52]
K. K. Sarangapani, T. Yago, A. G. Klopocki, M. B. Lawrence, C. B.
Fieger, S. D. Rosen, R. P. McEver, and C. Zhu. Low force decelerates
L-selectin dissociation from P-selectin glycoprotein ligand-1 and endoglycan. Journal of Biological Chemistry 279, 2291 (2004). (page 19)
[53]
O. Yakovenko, S. Sharma, M. Forero, V. Tchesnokova, P. Aprikian,
B. Kidd, A. Mach, V. Vogel, E. Sokurenko, and W. E. Thomas.
FimH forms catch bonds that are enhanced by mechanical force due to allosteric regulation. Journal of Biological Chemistry 283, 11596 (2008).
(page 19)
[54]
W. E. Thomas, V. Vogel, and E. Sokurenko. Biophysics of catch bonds.
Biophysics 37, (2008). (pages 20–22)
[55]
Y. V. Pereverzev, O. V. Prezhdo, M. Forero, E. V. Sokurenko, and
W. E. Thomas. The two-pathway model for the catch-slip transition in
biological adhesion. Biophysical Journal 89, 1446 (2005). (pages 21,
23, 39–41, 46, 63)
[56]
H. A. Kramers. Brownian motion in a field of force and the diffusion
model of chemical reactions. Physica 7, 284 (1940). (pages 20, 39)
Bibliography
131
[57]
E. Evans. Probing the relation between force-lifetime-and chemistry in
single molecular bonds. Annual Review of Biophysics and Biomolecular Structure 30, 105 (2001). (pages 20, 22)
[58]
G. I. Bell. Models for the specific adhesions of cells to cells. Science 200,
618 (1978). (pages 22, 23, 27, 39, 43)
[59]
W. Thomas. Catch bonds in adhesion. Annual Review of Biomedical
Engineering 10, 39 (2008). (pages 22, 38, 39)
[60]
Y. V. Pereverzev, E. Prezhdo, and E. V. Sokurenko. The two-pathway
model of the biological catch-bond as a limit of the allosteric model. Biophysical Journal 101, 2026 (2011). (page 23)
[61]
E. Cukierman, R. Pankov, D. R. Stevens, and K. M. Yamada. Taking
cell-matrix adhesions to the third dimension. Science 294, 1708 (2001).
(page 23)
[62]
A. Jayo and M. Parsons. Imaging of cell adhesion events in 3D matrix environments. European Journal of Cell Biology 91, 824 (2012).
(page 23)
[63]
U. S. Schwarz and S. A. Safran. Physics of adherent cells. Reviews of
Modern Physics 85, 1327 (2013). (pages 23, 24)
[64]
J. Adler. Chemotaxis in bacteria. Science 153, 708 (1966). (pages 24,
92)
[65]
D. Horstmann et al. From 1970 until present: the Keller-Segel model in
chemotaxis and its consequences. Jahresbericht der Deutschen MathematikerVereinigung 105, (2003). (page 25)
[66]
E. F. Keller and L. A. Segel. Model for chemotaxis. Journal of Theoretical Biology 30, 225 (1971). (page 25)
[67]
C. S. Patlak. Random walk with persistence and external bias. The Bulletin of Mathematical Biophysics 15, 311 (1953). (page 25)
[68]
M. Vergassola, E. Villermaux, and B. I. Shraiman. ’Infotaxis’ as a
strategy for searching without gradients. Nature 445, 406 (2007). (page 25)
[69]
S. J. Mousavi, M. H. Doweidar, and M. Doblaré. 3D computational
modelling of cell migration: A mechano-chemo-thermo-electrotaxis approach.
Journal of Theoretical Biology 329, 64 (2013). (page 25)
132
Bibliography
[70]
C.-M. Lo, H.-B. Wang, M. Dembo, and Y.-l. Wang. Cell movement
is guided by the rigidity of the substrate. Biophysical Journal 79, 144
(2000). (pages 25, 26)
[71]
Y. K. Cheung, E. U. Azeloglu, D. A. Shiovitz, K. D. Costa, D. Seliktar, and S. K. Sia. Microscale control of stiffness in a cell-adhesive
substrate using microfluidics-based lithography. Angewandte Chemie
International Edition 48, 7188 (2009). (page 26)
[72]
L. G. Vincent, Y. S. Choi, B. Alonso-Latorre, J. C. del Álamo, and A.
J. Engler. Mesenchymal stem cell durotaxis depends on substrate stiffness
gradient strength. Biotechnology Journal 8, 472 (2013). (pages 26, 92)
[73]
U. Alon, M. G. Surette, N. Barkai, and S. Leibler. Robustness in bacterial chemotaxis. Nature 397, 168 (1999). (page 26)
[74]
P. DiMilla, K. Barbee, and D. Lauffenburger. Mathematical model for
the effects of adhesion and mechanics on cell migration speed. Biophysical
Journal 60, 15 (1991). (page 26)
[75]
I. V. Dokukina and M. E. Gracheva. A model of fibroblast motility on
substrates with different rigidities. Biophysical Journal 98, 2794 (2010).
(page 26)
[76]
B. Harland, S. Walcott, and S. X. Sun. Adhesion dynamics and durotaxis in migrating cells. Physical Biology 8, 1 (2011). (page 26)
[77]
A. D. Rouillard and J. W. Holmes. Mechanical boundary conditions
bias fibroblast invasion in a collagen-fibrin wound model. Biophysical
Journal 106, 932 (2014). (pages 26, 102)
[78]
F. Stefanoni, M. Ventre, F. Mollica, and P. A. Netti. A numerical
model for durotaxis. Journal of Theoretical Biology 280, 150 (2011).
(pages 26, 120)
[79]
N. V. Kampen. Stochastic Processes in Physics and Chemistry (NorthHolland Physics Publishing, 1987). (pages 28, 29, 43, 49, 70)
[80]
T. Erdmann and U. S. Schwarz. Stochastic dynamics of adhesion clusters under shared constant force and with rebinding. Journal of Chemical Physics 121, 8997 (2004). (pages 28, 29, 43)
Bibliography
133
[81]
D. T. Gillespie. Exact stochastic simulation of coupled chemical reactions.
Journal of Physical Chemistry 81 (25), 2340 (1977). (pages 30, 31, 46,
66)
[82]
E. A. Codling, M. J. Plank, and S. Benhamou. Random walk models in
biology. Journal of the Royal Society Interface 5, 813 (2008). (page 31)
[83]
H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. Journal of Mathematical Biology 26, 263 (1988).
(pages 33, 35, 109)
[84]
M. Rubinstein and R. Colby. Polymers Physics (Oxford, 2003). (pages 33,
94)
[85]
D. Discher, P. Janmey, and Y.-L. Wang. Tissue cells feel and respond to
the stiffness of their substrate. Science 310, 1139 (2005). (page 38)
[86]
N. D. Evans, C. Minelli, E. Gentleman, V. LaPointe, S. N. Patankar,
M. Kallivretaki, X. Chen, C. J. Roberts, and M. M. Stevens. Substrate stiffness affects early differentiation events in embryonic stem cells.
European Cells & Materials 18, 1 (2009). (page 38)
[87]
S. Walcott and S. X. Sun. A mechanical model of actin stress fiber formation and substrate elasticity sensing in adherent cells. Proceedings of
the National Academy of Sciences USA 107, 7757 (2010). (page 38)
[88]
Ph. Marcq, N. Yoshinaga, and J. Prost. Rigidity sensing explained by
active matter theory. Biophysical Journal 101, L33 (2011). (page 38)
[89]
I. B. Bischofs and U. S. Schwarz. Cell organization in soft media due
to active mechanosensing. Proceedings of the National Academy of
Sciences USA 100, 9274 (2003). (pages 38, 55)
[90]
J. C. Friedland, M. H. Lee, and D. Boettiger. Mechanically activated
integrin switch controls α5 β 1 function. Science 323, 642 (2009). (pages 38,
56)
[91]
L. Sun, Q. H. Cheng, H. J. Gao, and Y. W. Zhang. Effect of loading
conditions on the dissociation behaviour of catch bond clusters. J. Roy.
Soc. Interface 9, 928 (2012). (pages 38, 42, 51, 52, 117)
134
Bibliography
[92]
K. K. Sarangapani, T. Yago, A. G. Klopocki, M. B. Lawrence, C. B.
Fieger, S. D. Rosen, R. P. McEvers, and C. Zhu. Low force decelerates
L-Selectin dissociation from P-Selectin Glycoprotein Ligand-1 and endoglycan. Journal of Biological Chemistry 279, 2291 (2004). (pages 39,
45)
[93]
K. E. Caputo, D. Lee, M. R. King, and D. A. Hammer. Adhesive dynamics simulations of the shear threshold effect for leukocytes. Biophysical Journal 92, 787 (2007). (pages 39, 45)
[94]
C. D. Paschall, W. H. Guilford, and M. B. Lawrence. Enhancement
of L-Selectin, but not P-Selectin, bond formation frequency by convective
flow. Biophysical Journal 94, 1034 (2008). (pages 39, 45)
[95]
U. S. Schwarz, T. Erdmann, and I. Bischofs. Focal adhesions as mechanosensors: The two-spring model. Biosystems 83, 225 (2006). (pages 41, 51,
53–55, 64, 75, 81, 116)
[96]
S. V. Plotnikov, A. M. Pasapera, B. Sabass, and C. M. Waterman.
Force fluctuations within focal adhesions mediate ECM-rigidity sensing
to guide directed cell migration. Cell 151, 1513 (2012). (pages 42, 51,
52, 56, 57)
[97]
U. S. Schwarz and T. Erdmann. Impact of receptor-ligand distance on
adhesion cluster stability. European Physical Journal E 22, 123 (2007).
(page 50)
[98]
T. Erdmann and U. S. Schwarz. Stability of adhesion clusters under
constant force. Physical Review Letters 92, 108102 (2004). (pages 51,
54)
[99]
A. Nicolas, B. Geiger, and S. A. Safran. Cell mechanosensitivity controls the anisotropy of focal adhesions. Proceedings of the National
Academy of Sciences of the United States of America 101, 12520
(2004). (page 51)
[100]
J. Qian, J. Wang, and H. Gao. Lifetime and strength of adhesive molecular bond clusters between elastic media. Langmuir 24, 1262 (2008).
(pages 52, 121)
Bibliography
135
[101]
J. Qian, J. Wang, Y. Lin, and H. Gao. Lifetime and strength of periodic
bond clusters between elastic media under inclined loading. Biophysical
Journal 97, 2438 (2009). (page 52)
[102]
H. Gao, J. Qian, and B. Chen. Probing mechanical principles of focal
contacts in cell–matrix adhesion with a coupled stochastic–elastic modelling framework. Journal of The Royal Society Interface , rsif20110157
(2011). (page 52)
[103]
T. Erdmann and U. S. Schwarz. Stochastic force generation by small
ensembles of myosin II motors. Physical Review Letters 108, 188101
(2012). (page 54)
[104]
S. J. Han, K. S. Bielawski, L. H. Ting, M. L. Rodriguez, and N.
J. Sniadecki. Decoupling substrate stiffness, spread area, and micropost
density: a close spatial relationship between traction forces and focal adhesions. Biophysical Journal 103, 640 (2012). (page 55)
[105]
B. Chen, R. Kemkemer, M. Deibler, J. Spatz, and H. Gao. Cyclic
stretch induces cell reorientation on substrates by destabilizing catch bonds
in focal adhesions. PLOS One 7, e48346 (2012). (page 57)
[106]
C. E. Chan and D. J. Odde. Traction dynamics of filopodia on compliant
substrates. Science 322, 1687 (2008). (page 57)
[107]
B. L. Bangasser, S. S. Rosenfeld, and D. J. Odde. Determinants of
maximal force transmission in a motor-clutch model of cell traction in
a compliant microenvironment. Biophysical Journal 105, 581 (2013).
(page 57)
[108]
J. Howard. Mechanics of Motor Proteins and the Cytoskeleton (Sinauer
Associates, Sutherland, Massachusets, 2001). (page 58)
[109]
O. Rossier, V. Octeau, J.-B. Sibarita, C. Leduc, B. Tessier, D. Nair,
V. Gatterdam, O. Destaing, C. A. s Rizo, R. Tampé, L. Cognet, D.
Choquet, B. Lounis, and G. Giannone. Integrins β 1 and β 3 exhibit
distinct dynamic nanoscale organizations inside focal adhesions. Nature
Cell Biology 14, 1057 (2012). (pages 58, 73)
136
Bibliography
[110]
C. Cluzel, F. Saltel, J. Lussi, F. Paulhe, B. A. Imhof, and B. WehrleHaller. The mechanisms and dynamics of αv β 3 integrin clustering in
living cells. The Journal of Cell Biology 171, 383 (2005). (pages 62,
80)
[111]
D. Nordin, L. Donlon, and D. Frankel. Characterising single fibronectinintegrin complexes. Soft Matter 8, 6151 (2012). (page 62)
[112]
E. A. Novikova and C. Storm. Contractile fibers and catch-bond clusters: a biological force sensor? Biophysical Journal 105, 1336 (2013).
(pages 66, 75, 76, 139)
[113]
J. W. Haus and K. W. Kehr. Diffusion in regular and disordered lattices.
Physics Reports 150, 263 (1987). (page 73)
[114]
C. Ciobanasu, B. Faivre, and C. Le Clainche. Actomyosin-dependent
formation of the mechanosensitive talin-vinculin complex reinforces actin
anchoring. Nat. Commun. 5, (2014). (page 75)
[115]
R. Bruinsma. Theory of force regulation by nascent adhesion sites. Biophysical Journal 89, 87 (2005). (page 102)
[116]
E. A. Codling, R. N. Bearon, and G. J. Thorn. Diffusion about the
mean drift location in a biased random walk. Ecology 91, 3106 (2010).
(page 102)
[117]
T. R. Weikl and R. Lipowsky. Pattern formation during T-cell adhesion.
Biophysical Journal 87, 3665 (2004). (page 119)
[118]
E. H. Danen, J. van Rheenen, W. Franken, S. Huveneers, P. Sonneveld, K. Jalink, and A. Sonnenberg. Integrins control motile strategy
through a Rho–cofilin pathway. The Journal of Cell Biology 169, 515
(2005). (page 121)
Curriculum Vitæ
Elizaveta Novikova was born on 16 May 1986 in Leningrad, USSR. In
the same city, she went to Physics and Mathematics Lyceum number 30.
She graduated in 2003 and started her studies in Physics at the SaintPetersburg State University. She got her Bachelors Degree in 2007, after
completing a thesis project on x-ray scattering on smectic-A films with
V.P. Romanov in the group of Statistical Physics. After that, Elizaveta pursued a Masters Degree in the same group, which she was awarded in
2010 after successfully finishing a graduation projected under supervision
of S.V. Ulyanov, on dynamics and fluctuations in free-standing smectic-A
films. Later in 2010, Elizaveta moved to Eindhoven as a PhD candidate
in the group Theory of Polymers and Soft Matter, with prof. dr. M.A.J.
Michels as promotor and dr. C. Storm as copromotor. The main results of
her PhD research are presented in this thesis.
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Curriculum Vitæ
List of Publications
S. Akerboom, A. Darhuber, S. van Driested, S. Elmendorp, M. Kamperman, T. van der Loop, Ch. Luo, F.V. Mackenzie, M.-J. van der Meulen, E.
A. Novikova, Y. Yanson. Develop the self adhesive to stick on moist and icy
substrates. Proceedings Physics with Industry, 3, 2011.
E. A. Novikova and C. Storm. Contractile fibers and catch-bond clusters: a biological force sensor? Biophysical Journal 105, 1336 (2013).
E.A. Novikova and C. Storm Catching up on slip: focal adhesion composition
and mechanosensing. Biophysical Journal 106, 173a (2014).
E.A. Novikova and C. Storm Catch and slip: adhesion composition and mechanosensing soon to be submitted.
E.A. Novikova and C. Storm Catch- and slip- bonds inside focal adhesion: interactions and spatial organization, in manuscript.
E. A. Novikova, D. E. Discher, C. Storm. Durotaxis through persistence-based
migration: theory and simulations compared with experiments, in manuscript.
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140
List of Publications
Summary
Modeling Cell Interaction with the Stiffness of the
Extracellular Matrix
This work focuses on modeling the interaction between the cell and the
mechanical properties of the extracellular environment. Cells in our body
spend most of their life embedded in an external polymer network called
the extracellular matrix (ECM). Cells actively interact with the ECM, and
sense its properties. The ability of the cells to react to changes in ECM
chemical composition, temperature and stiffness is essential for their proliferation, migration and death. While the biochemical pathways through
which the cell reacts to chemical signals from the ECM have been extensively studied, and are - to a large extent - understood, the topic of cell
interaction with the stiffness of the ECM is still presents promising and
largely unexplored opportunities for further investigation. The huge range
of stiffnesses that cells encounter, adapt and react to in our bodies alone
suggests that there must be a very precise and fine-tuned mechanism that
allows cells to interpret the mechanical properties of the ECM. We build
our work around the notion that the cell interacts with the stiffness of the
ECM. We devote a part of this thesis to modeling the molecular mechanism that allows cells to quantitatively interpret the mechanical stiffness
of the ECM internally. We also model the macroscopic parameters inside
the cell affected by the stiffness of the ECM, and finally present a coarsegrained model of cell migration on substrates with gradients in their stiffness.
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142
Summary
The first point of contact for external mechanical properties is the focal
adhesion - a multiprotein structure that connects the ECM with the intracellular network of actin filaments: Through the focal adhesion, the force
generated by the cell is transferred to the ECM. The transmembrane proteins that directly connect the inside to the outside of the cell are called integrins. In Chapter 3 we set up a simple model of a focal adhesion, which
consists of a collection of multiple integrins that are sharing the force
exerted upon them. We incorporate into our model recent experimental
findings on the single-bond mechanical properties between integrins and
their ECM ligands. We propose a novel mechanism for stiffness sensing,
by assuming fixed the amount of work that the cell invests into contraction
of its actin stress fibers. This model presents a first step towards modeling
the mechanosensitivity of a multicomponent focal adhesion, which follows in Chapter 4. We assess the force-dependent macroscopic properties
of a two-component focal adhesion, and show that two-component focal
adhesions can act as a mechanical property sensor.
Our model of a focal adhesion is, by then, more accurate but still incomplete. Integrins inside the focal adhesions are known to exhibit nonuniform spatial organization. In Chapter 5, we further develop the model
system by the inclusion of integrin-integrin interactions. We report that
integrins of different types get selectively reinforced by a force acting on
the focal adhesion. Chapters 3 to 5 are built around the properties of individual adhesions, and mostly focus on sensing the mechanical properties
of the ECM.
Sensing the ECM stiffness, and reacting to it, are separated by several
multidisciplinary multiscale modeling steps, which we skip and go to a
macroscopic model of cell migration in Chapter 6 where we investigate
another action that is guided by stiffness: Durotaxis. This recently discovered phenomenon has already received a decent amount of attention
from the modeling community, but existing models are all based on the
assumption that cells are directly sensing the gradient in the stiffness and
reacting on it in an directional manner. We ask: What if cells are able
143
to sense and incorporate into their signaling networks only one value of
ECM stiffness averaged over the cell area? Is there a mechanism that may
lead to durotaxis then? We disentangle the role of gradient-sensing and
stiffness-sensing in the mechanism of durotaxis, and propose a manner in
which the spatial trajectory of a stiffness-sensing cell may be distinguished
from that of a gradient-sensing cell.
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Summary
Acknowledgements
Writing this thesis would have been less pleasurable or not even possible
without the help and support of several people.
First of all I would like to thank my promotor prof.dr. Thijs Michels for
his comments on my manuscript and reading of my stellingen.
I would like to thank the other members of my defense committee: prof.
dr. Carlijn Bouten, prof. dr. Gijsje Koenderink, prof.dr. Thomas Schmidt,
prof.dr. Ulrich Schwarz, dr. habil. Adrian Muntean for their willingness
to accept the task, for the reading of my thesis and for their comments
that made it so much better work. I thank my supervisor dr. Kees Storm
for the support and encouragement that he provided me with during the
whole 4 years, it was a great pleasure working with you. I would like to
thank you for going out of your way to help me reach my not always well
placed deadlines.
I thank prof. dr. Dennis Discher for welcoming me in his lab at UPenn
during 3 months in 2013, for giving me the opportunity to explore there
and for providing me with the interesting questions that I answer in Chapter 6 of this thesis. I thank the colleagues at UPenn for their patience and
care with showing me the things that they do at the lab and letting me
try things there. Special thanks goes to Jerome and Dave - thank you for
taking me to your climbing practices, because of you I felt at home in
Philly. I thank my housemate in IHouse Aigul for keeping me company
and taking me out for trips to Washington, New York and Atlantic City!
Back to Eindhoven: I would like to thank Helmi van Lieshout - our TPS
group secretary: whatever I ask her she would help me with it and at the
same time have her cheerful attitude.
I would like to thank dr. Henk Huinink and dr. Alexey Lyulin whom I
assisted for Thermische Fysika II for their tips in my first teaching experience.
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146
Acknowledgements
I thank all of my current and former colleagues at TPS group that made
my days with their fun discussions during lunches and coffee breaks.
Without calling each of your names, many of you have contributed to
making this happen. You helped me with my code, listened to my practice presentations, expressed sincere interest in my research work, took
care of things that had to be done when I was out of the office and gave
me a hand with many other things. You did this all with the willingness
that I always admire, you are a great group!
I am grateful to my fellow students during this last years of my PhD, for
taking me outside to our non-smoking breaks where I could literally and
figuratively see the light.
I would like to thank my former colleagues from Saint-Petersburg State
University who were or still are pursuing PhD in Russia or abroad: Tanya,
Vasya, Pasha, Anya - talking about our experiences helped me understand
that the issues that I encountered were somehow universal.
I thank my friends in Saint-Petersburg - Sasha, Sasha and Renata for giving me a warm welcome every time that I was there, which helped me feel
like home after a long time away. Aleksandra, you are my great friend of
15 years, and I always enjoy your energy and determination: it is contagious! I thank you for always being there for me!
I thank my housemates Divi and Alex and my new houseys Danna and
Anna for the great atmosphere that they have provided to the house,
and many evenings that we spent together with relaxing discussions: you
made me remember that there is a world outside science and PhD.
Rotem, my housemate and my yoga friend, I thank you for being there for
me when I most needed it, I enjoyed our interactions, thank you for your
willingness to listen to me and accepting me for who I am!
Katya Makhu, I thank you for always being so cheerful and for taking me
out of the house, thank you for welcoming me in your house and cooking
for me in mine!
Katya, now pursuing her PhD in Groningen, we’ve known each other for
some ten years now, I cherish our friendship. Experiencing your honest
and fresh mind benefited so greatly to the quality of my being, you always challenge me to become better.
No words can explain how I’m grateful to my family for providing me
147
with the support and freedom, never asking for anything, but being there
for me anytime I needed it.
ß õî ÷ ó ï î á ë à ã îä à ð è ò ü ì î þ ñ å ì ü þ : ì à ì à , ï à ï à , á à á ó ø êà ,
ñ ï à ñ è á î â à ì ç à ï î ä ä å ð æ ê ó, ê î ò î ð ó þ â û ì í å î ê à ç û â à ë è í à
ï ð î òÿ æå í è è ý ò è õ ÷ å ò û ð ¼ õ ë å ò ! Á å ç â à ø å ã î ó ÷ à ñ ò è ÿ è
ì óä ð î ñ ò è ÿ í à â å ð í ÿ ê à á û í å ä î á ð à ë à ñ ü ä î ý ò î ã î ì î ì å í ò à â
ìîåé áèîãðàôèè!
I would like to thank family Scolari for welcoming me in their house and
taking care of me even when I was not the most conscious guest.
Dear Vittore! Last but not least, I thank you for contributing in so many
ways to the writing and editing of this thesis and for discussing with me
the research presented here! I always felt your support, and having you
in my life made all this so much more of a dynamic and joyful experience!
Elizaveta,
Eindhoven, November 2014