On the mechanics of actin and intermediate filament networks and

On the mechanics of actin and intermediate
filament networks and their contribution to
cellular mechanics
Björn Fallqvist
Doctoral thesis no. 92, 2015
KTH School of Engineering Sciences
Department of Solid Mechanics
Royal Institute of Technology
SE-100 44 Stockholm Sweden
TRITA HFL-0583
ISSN 1104-6813
ISRN KTH/HFL/R-15/19-SE
ISBN 978-91-7595-752-4
Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm
framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredagen
den 29 januari 2016, Kungliga Tekniska Högskolan, Kollegiesalen, Stockholm.
Abstract
The mechanical behaviour of cells is essential in ensuring continued physiological function,
and deficiencies therein can result in a variety of diseases. Also, altered mechanical response
of cells can in certain cases be an indicator of a diseased state, and even actively promoting
progression of pathology. In this thesis, methods to model cell and cytoskeletal mechanics
are developed and analysed.
In Paper A, a constitutive model for the response of transiently cross-linked actin networks
is developed using a continuum framework. A strain energy function is proposed, modified
in terms of chemically activated cross-links. The constitutive model was compared with experimental relaxation tests and it was found that the initial region of fast stress relaxation
can be attributed to breaking of bonds, and the subsequent slow relaxation to viscous effects
such as sliding of filaments.
In Paper B, a finite element framework was used to assess the influence of numerous geometrical and material parameters on the response of cross-linked actin networks, quantifying the
influence of microstructural properties and cross-link compliance. Also, a micromechanically
motivated constitutive model for cross-linked networks in a continuum framework was proposed.
In Paper C, the discrete model is extended to include the stochastic nature of cross-links. The
strain rate dependence observed in experiments is suggested to depend partly on this, but
a more complex mechanism than mere cross-link debonding is suggested, as there are some
discrepancies in computed and experimental results.
In Paper D, the continuum model for cross-linked networks is extended to encompass more
general mechanisms, and a strain energy function for a composite biopolymer network is proposed. Favourable comparisons to experiments indicate the interplay between phenomenological evolution laws to predict effects in biopolymer networks. Importantly, by fitting the
model to experimental results for singular network types, those parameters can be used in
the composite formulation, with the exception of two parameters.
In Paper E, experimental techniques are used to assess influence of the actin cytoskeleton on
the mechanical response of fibroblast cells. The constitutive model developed for composite
biopolymer networks can be used to predict the indentation and relaxation behaviour. The
influence of cell shape is assessed, and experimental and computational aspects of cell mechanics are discussed.
In Paper F, the filament-based cytoskeletal model is extended with an active response to
predict the force generation observed at longer time-scales in relaxation experiments. An additional strain energy function is implemented, and the influence of contractile and chemical
parameters are assessed. The method is general, and can be used to extend other existing
filament-based constitutive models to incorporate contractile behaviour. Importantly, experimentally observed stiffening of cells with applied stress is predicted.
Sammanfattning
Cellers mekaniska beteende är av största vikt för deras fortsatta fysiologiska funktion, och
ett defekt sådant beteende kan resultera i flertalet sjukdomar. Dessutom kan den mekaniska
responsen hos celler vara en indikator på ett sjukdomstillstånd, och även underlätta spridning av sjukdomen. I denna avhandling utvecklas och utvärderas metoder för att modellera
cellmekanik.
I Paper A utvecklas en konstitutiv kontinuummekanisk modell för responsen hos aktinnätverk
med transienta bindningar. En töjningsenergifunktion definieras, modifierad för att ta hänsyn
till aktiverade bindningar. Jämförelser med relaxationsexperiment visade att den initiella
snabba relaxationen kan tillskrivas bindningar som släpper, och den efterföljande långsammare
relaxationen beror på viskös deformation.
I Paper B används finita element-metoden för att bestämma inverkan hos flertalet parametrar
relaterade till mikrostrukturens geometri, material och bindningsegenskaper. En kontinuummekanisk modell för nätverk med bindningar presenteras också.
I Paper C utökas den diskreta modellen till att innefatta stokastiska bindningar. Det experimentellt uppvisade beroendet på töjningshastighet föreslås delvis bero på detta, men en
mer komplex mekanisism existerar antagligen, då skillnader uppvisas mellan beräknade och
experimentella resultat.
I Paper D utvecklas den kontinuummekaniska modellen för nätverk vidare för att inkludera mer allmänna mekanismer, och en töjningsenergifunktion för kompositnätverk föreslås.
Jämförelser med experiment visar att beroendet mellan evolutionslagarna för interna variabler
kan användas för att beräkna experimentellt uppvisade effekter i nätverk av biopolymerer.
Genom att anpassa modellen till experimentella data för enstaka nätverkstyper kan dessa
även användas för kompositnätverk, med undantag för två parametrar.
I Paper E används experimentella mättekniker för att undersöka inverkan av aktinets cytoskelett på den mekaniska responsen hos fibroblaster. Den konstitutitiva modellen utvecklad för kompositnätverk av biopolymerer kan användas för att beräkna indentations- och
relaxationsbeteendet med samma materialparametrar som för aktin och intermediära filamentnätverk. Inverkan hos cellgeometri bestäms, och experimentella samt beräkningsmässiga
aspekter på cellmekanik diskuteras.
I Paper F utökas den tidigare konstitutiva modellen till att även innefatta kontraktilt beteende. Detta åstadkomms genom att inkludera en töjningsenergifunktion för den kontraktila responsen. Inverkan av kontraktila och kemiska parametrar utvärderas. Metoden kan
även användas för andra filament-baserade konstitutiva modeller. Experimentellt observerat
cellhårdnande för celler predikteras.
Preface
The work presented in this thesis has been performed between April 2011 and November 2015
at the department of Solid Mechanics at KTH (Royal Institute of Technology). The research
was made possible with funding by project grant No. A0437201 of the Swedish Research
Council, which is gratefully acknowledged.
Thank you - To my friends and family, for your support and love.
Jacopo, Prashanth and Belinda, for being stalwart friends and helping me become the person
I am today.
Stockholm, November 2015
List of appended papers
Paper A: Mechanical behaviour of transiently cross-linked actin networks and a theoretical
assessment of their viscoelastic behaviour
Björn Fallqvist and Martin Kroon
Biomechanics and Modeling in Mechanobiology, vol. 12, 2013, pp. 373-382
Paper B: Network modelling of cross-linked actin networks - Influence of network parameters
and cross-link compliance
Björn Fallqvist, Artem Kulachenko and Martin Kroon
Journal of Theoretical Biology, vol. 350, 2014, pp. 57-69
Paper C: Cross-link debonding in actin networks - influence on mechanical properties
Björn Fallqvist, Artem Kulachenko and Martin Kroon
International Journal of Experimental and Computational Biomechanics, vol. 3, 2015, pp.
16-26
Paper D: Constitutive modelling of composite biopolymer networks
Björn Fallqvist and Martin Kroon
Report 577, Department of Solid Mechanics, KTH Engineering Sciences, Royal Institute of
Technology, Stockholm, Sweden
Submitted for publication in Journal of Theoretical Biology
Paper E: Experimental and computational assessment of F-actin influence in regulating
cellular stiffness and relaxation behaviour of fibroblasts
Björn Fallqvist, Matthew Fielden, Torbjörn Pettersson, Niklas Nordgren, Martin Kroon and
Annica Gad
Report 578, Department of Solid Mechanics, KTH Engineering Sciences, Royal Institute of
Technology, Stockholm, Sweden
Submitted for publication in Journal of the Mechanical Behavior of Biomedical Materials
Paper F: Implementing cell contractility in filament-based cytoskeletal models
Björn Fallqvist
Report 582, Department of Solid Mechanics, KTH Engineering Sciences, Royal Institute of
Technology, Stockholm, Sweden
Submitted for publication in Cytoskeleton
In addition to the appended papers, the work has resulted in the following publications and
presentations1 :
Transiently cross-linked actin networks
Björn Fallqvist and Martin Kroon
Presented at World Congress on Computational Mechanics, Sao Paolo 2012 (P)
Transiently cross-linked actin networks
Björn Fallqvist and Martin Kroon
Presented at Svenska Mekanikdagar, Lund 2013 (P)
Mechanical response of cross-linked actin networks - Influence of geometry and
cross-link compliance
Björn Fallqvist, Artem Kulachenko and Martin Kroon
Presented at World Congress on Computational Mechanics, Barcelona 2014 (P)
1
Ea = Extended abstract, P = Presentation, Pp = Proceeding paper
List of individual contributions to appended papers
Paper A: Mechanical behaviour of transiently cross-linked actin networks and a theoretical
assessment of their viscoelastic behaviour
Performed computational analyses, evaluated model results. Wrote the manuscript.
Paper B: Network modelling of cross-linked actin networks - Influence of network parameters
and cross-link compliance
Set up numerical model and study, performed computations of discrete network models and
evaluated results. Also modified the (previously disregarded by us) continuum-model to
investigate its predictive behaviour and evaluated the results. Wrote the manuscript.
Paper C: Cross-link debonding in actin networks - influence on mechanical properties
Set up the study with idea from first paper, and performed numerical computations. Wrote
the manuscript.
Paper D: Constitutive modelling of composite biopolymer networks
Developed and extended network continuum model into general composite form with internal
variables and evolution laws. Set up computational study, performed analyses and evaluation
of results. Wrote the manuscript.
Paper E: Experimental and computational assessment of F-actin influence in regulating
cellular stiffness and relaxation behaviour of fibroblasts
Initiated collaboration with Karolinska Institutet and other KTH researchers. Decided upon
type of experimental procedure (relaxation and influence of F-actin) for verification of material
model. Wrote material routine for numerical computations in Abaqus. Set up and performed
numerical analyses of cell indentation and study. Wrote majority of the manuscript.
Paper F: Implementing cell contractility in filament-based cytoskeletal models
All work performed by the author.
Contents
Introduction
11
The importance of cell mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
The cell is a complex heterogenous structure . . . . . . . . . . . . . . . . . . . . .
12
Experimental methods in cell mechanics
17
Passive microrheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Cell poker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Micropipette Aspiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Microplates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Magnetic Twisting Cytometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Atomic Force Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
Mechanical models of biological cells
21
Cell models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Liquid drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Theory of SGR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Continuum models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Tensegrity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Cytoskeletal network characteristics . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Actin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Intermediate filaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
Microtubules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Cytoskeletal models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Elasticity of semiflexible polymer networks . . . . . . . . . . . . . . . . . . . .
27
Discrete network models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
Conclusions and contribution to the field
9
29
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
Modelling mechanical cytoskeletal and cell behaviour by use of continuum and
discrete models
31
A continuum model for transiently cross-linked actin networks
. . . . . . . . . . .
31
A discrete network model to assess the influence of geometrical parameters and
cross-link compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
A discrete network model to assess the influence of cross-link debonding on mechanical properties of cross-linked actin networks . . . . . . . . . . . . . . . . . . .
40
A constitutive model for composite biopolymer networks - Irreversible effects in the
large strain regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Experimental and computational assessment of F-actin influence in regulating cellular stiffness and relaxation behaviour of fibroblasts . . . . . . . . . . . . . .
44
Implementing cell contractility in filament-based cytoskeletal models . . . . . . . .
50
Conclusions
55
Summary of appended papers
57
Bibliography
60
Errata
67
10
Introduction
The importance of cell mechanics
Cells are fundamental building blocks of life, not only containing and reproducing information necessary for the continued existence of our species, but with a multitude of tasks to
be fulfilled to ensure physiological function of our bodies. Hereditary information is stored
in the form of chromosomal DNA in the cell nucleus, passed on to the cell’s progeny during
reproduction in a continuous cycle, ensuring the survival of our genes. During this reproductive cycle, evolution of species is also made possible due to errors - mutations - of the DNA
sequence, in which favourable properties increase the likelihood of survival, and vice versa.
Apart from this responsibility of maintaining and propagating the hereditary information of
a species, more immediate tasks for ensured continued physiological function of organisms
also lie with the responsibility of many various types of cells. To name a few, circulation of
red blood cells (erythrocytes) is the principal means by which vertebrates deliver oxygen to
body tissues, endothelial cells line the interior surface of blood vessels as a filtration barrier
and fibroblasts synthesise collagen and the extracellular matrix, useful in wound healing.
For continued physiological function, the mechanical response of the cell must be tuned such
that integrity is preserved to avoid damage or cell death, while maintaining capability to fulfill
required tasks. In the examples mentioned above, the red blood cell must be able to accomodate large deformations to squeeze through narrow capillarie. In the case of endothelial cells,
they must adapt to being continuously subjected to blood pressure, however arterial stiffening
affects endothelial function and may lead to heart failure. In the case of fibroblasts, their role
in wound healing involves migrating to the injured area and using intracellular contractile
machinery to close it. Thus, their mechanical behaviour must be adapted to not only migrate
efficiently, but also adapt cellular shape to such an extent as to make this possible.
Apart from diseases directly related to an inability to perform their tasks necessary for continued proper physiological function, improper mechanical behaviour of cells also contribute to
a major medical area of research, cancer. Many types of cancers result in altered mechanical
properties of the affected cells, although the nature of alteration is not unambiguous for all cell
types, see (Suresh, 2007). When investigated, intracellular components are often expressed
by a phenotype differing from that of healthy cells. For example, the vimentin network of
11
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
intermediate filaments has been observed to collapse around the nucleus (Rathje et al, 2014;
Rönnlund et al, 2013; Suresh et al, 2004) in transfeced cells, and altered expression of nuclear lamins is believed to be instrumental in facilitating movement through narrow passages
of metastasising cells (Denais and Lammerding, 2014). It has further been shown that cell
spreading, an important factor for metastasising cells, is instrumental in determining cellular
stiffness (Coughlin et al, 2013).
Taken together, the importance of cellular ability to properly adapt its mechanical behaviour
in response to its environment or specific tasks, can not be understated. Evident from the
numerous diseases resulting from lack of this capability, it is important to understand not only
the macroscopic, but also the mechanical response from structural elements at a molecular
level. Thus, characterisation of intracellular components to properly reflect their behaviour
is essential, before characterising the cell as a whole. In this manner of multi-scale modelling, it is possible to develop a microstructural model as the basis of cytoplasmic response
in a numerical cell model, characterising cellular response in terms of parameters related to
intracellular components (here the cytoskeleton).
The cell - a complex heterogenous structure
The cell is not only a passive composite structure of various components, but a living entity
actively responding to biochemical and mechanical signals, sometimes transforming the latter
into the former or vice versa. One illustrative example of this active characteristic is the
contractile machinery of the cell, which is instrumental in fundamental cellular processes such
as cytokinesis and cell migration (Pardee, 2010). During cytokinesis, the protein myosin II
forms a contractile ring, cleaving the cell at the furrow. The same protein type is fundamental
in cell migration, in which myosin I slides polymerised filaments along the cell membrane at
the leading edge of the cell. At the rear of the cell, myosin II slides filaments relative to each
other, creating a contractile motion that pushes the cell forward. During this cycle, the cell
membrane must be able to accomodate deformations due to extension of filaments, filopodia,
in the migration direction while maintaining structural integrity. Obviously, the cell must
consist of a number of interconnected structural components to accomplish this. A schematic
of a typical cell with prominent constituents are shown in Fig. 1, together with a photo of
a fibroblast cell with cytoskeletal components visualised. In the figure above, the various
components all have their own areas of responsibilities. For example in ribosomes, biological
proten synthesis takes place, mitochondria generate Adenosine TriPhosphate (ATP), used as
a source of chemical energy in the cell, and the nucleus is where hereditary information is
stored as tightly packed chromatin in chromosomes. The enclosing cell membrane is a lipid
bilayer with channels to regulate transport in and out of the cell. The main intracellular entity
cell governing not only the mechanical behaviour but also cell shape, in an active manner,
12
Cytosol
Endoplasmic reticulum
Cell membrane
Nucleus
Golgi apparatus
Actin filaments
Microtubule
Focal adhesion complex
Mitochondrion
Figure 1: a) Schematic of cell with prominent constituents, reprinted from (Bao and Suresh, 2003) with
permission from Macmillan Publishers Ltd. b) Fibroblast cell visualised by immunofluorescence
staining under Zeiss AxioVert 40 CFL microscope, using Zeiss AxioCAM MRm digital camera and
AxioVision software (Carl Zeiss AG) (Rathje et al, 2014). Cytoskeletal networks are coloured as
blue - actin, green - intermediate filaments (vimentin) and red - microtubules.
is the interconnected network of polymerised filaments called the cytoskeleton. Three main
types of filaments can be identified, all with distinctly dissimilar structure and function - actin
filaments (also called microfilaments), intermediate filaments and microtubules. A schematic
picture of these is shown in Fig. 2. Actin is well-studied, abundant and found in every
type of cell, responsible for a multitude of tasks. Actin filaments are formed by globular
actin (G-actin) polymerising into filaments (F-actin), a helical structure with a diameter of
approximately 6nm. Lengths in the cell is typically on the order of a micron, although their
contour length in vitro is typically on the order of their persistence length 10-17μm (Gittes
et al, 1993; Kasza et al, 2010), i.e. they behave as semiflexible polymers (Kamm and Mofrad,
2006; Xu et al, 2000, 1998; Kasza et al, 2010). Actin is one of the main providers of stiffness for
the cell, evident in many experimental investigations where disrupting the actin cytoskeleton
with agents such as cytochalasin D significantly increased cell compliance (Maniotis et al,
1997; Wu et al, 1998; Wakatsuki et al, 2000; Thoumine and Ott, 1997; Kamm and Mofrad,
2006). Actin is found in numerous forms in the cell, for example the densely cross-linked Factin network below the plasma membrane (cortical actin), the distributed lattice throughout
the cytoplasm, the dense meshwork in the leading edge of the cell (lamellipodium) and bundles
of filaments termed stress fibres. Stress fibres are formed from focal adhesions and provide the
actin cytoskeleton through integrins with a connection to the ExtraCellular Matrix (ECM).
An important aspect in the formation of F-actin networks is their propensity to assemble
into various types of cross-linked networks (Lieleg et al, 2010) by cross-linking proteins, or
Actin Binding Proteins (ABPs). Examples of such proteins are the rod-like α-actinin, found
at focal adhesions, in stress fibres and cortical actin, and filamin, a hinge-like protein that
binds filaments at orthogonal angles into a gel (Kamm and Mofrad, 2006; Alberts et al, 2004).
Actin filaments are dynamic, continuously polymerising and depolymerising, resulting in a
13
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
a
b
F-actin
Negative end (–)
Dep
oly
riz
me
atio
n
Intermediate filament
N
C
Polypeptide
N
C
Dimer
N
Tetramer
C
C
Pol
y
me
riz
N
Protofilament
atio
n
Positive end (+)
Filament (D = 10 nm)
7.9 nm
c
Microtubule
α-tubulin
β-tubulin
25 nm
14 nm
a
Figure 2: Cytoskeletal filaments. Image reproduced from (Mofrad, 2009) with permission from Annual Reviews for use in academic titles.
machinery able to adapt to biochemical and mechanical signals.
Intermediate filaments are a family of various proteins of which the expression depends on
the type of cell and function. For example, endothelial cells express different kinds of keratin,
and vimentin is prominent in mesenchymal cells such as fibroblasts (Herrmann et al, 2007).
These filaments are more flexible than actin, and often have a highly curved appearance when
observed in the cell (Ingber et al, 2014), probably due to their shorter persistence length ∼1μm
(Kamm and Mofrad, 2006; Schopferer et al, 2009; Rammensee et al, 2007; Mücke et al, 2004;
Herrmann et al, 2007). They are further more resistant to high salt concentrations, and
much more stable, than actin filaments or microtubules (Kamm and Mofrad, 2006; Herrmann
et al, 2007). They have been proposed to be responsible for enabling the cell to withstand
larger forces, as entangled networks typically exhibit strong strain hardening, and single
filaments can be stretched far beyond their original length, unlike actin filaments which break
at smaller strains (Herrmann et al, 2007; Janmey et al, 1991). Intermediate filaments connect
to the nuclear envelope through lamins, and radiate towards the surface of the cell where
they through interaction with plectins might provide a connection to the ECM and other
cells (Goldman et al, 1986; Herrmann et al, 2007; Na et al, 2008; Wiche, 1989; Svitkina et al,
1996). Improper expression of intermediate filaments are known to greatly affect physiological
function. For example, in mice, knocking out the vimentin gene in mice demonstrated that
vimentin regulates the mechanical response to blood flow and pressure in arteries (Herrmann
et al, 2007). Further, cancer cells have been shown to exhibit a disrupted intermediate filament
network, collapsed around the nucleus, along with distinctly different mechanical properties
14
(Rathje et al, 2014; Suresh et al, 2004). It can also be mentioned that mutations in the
intermediate filament type desmin can cause muscular dystrophies (Herrmann et al, 2007),
and in keratin severe blistering diseases (Lane and McLean, 2004).
Microtubules, the third type of cytoskeletal filaments is formed from tubulin into a hollow
cylinder with an outer diameter of roughly 25nm and consequently a far higher bending
stiffness than that of either actin or intermediate filaments (Kamm and Mofrad, 2006). Their
persistence length is thus far greater, about 6mm (Kamm and Mofrad, 2006; Gittes et al,
1993). With a high bending stiffness and rod-like appearance, they are useful in aiding the
formation of long slender structures. Microtubules are often found in an arrangement in
which they radiate from the center of the cell (Alberts et al, 2004), and perturbation of the
microtubule network has been observed to collapse the intermediate filament network in turn,
suggesting a role in keeping this open and stabilising it against compression (Rathje et al,
2014; Maniotis et al, 1997; Herrmann et al, 2007). Microtubules are highly dynamic, allowing
for remodelling and adaptation of cell structure and response in a matter of minutes (Kamm
and Mofrad, 2006).
Apart from the cytoskeleton, the cell nucleus is a prominent and stiff organelle which can be
expected to greatly influence the mechanical response of the cell. The nucleus is anchored in
place by the cytoskeletal filaments, which connect to the nuclear envelope through nesprin
proteins, providing a link to not only the cytoplasm but also the ECM and other cells through
focal adhesions, hemidesmosones and adherens junctions.
An image thus emerges of the cell as a complicated structure in which numerous building
blocks assemble into structural elements that together enable it to fulfill physiological function
and respond properly when subjected to mechanical stimuli. One example of this cooperative
effect is the aforementioned interdependence of micrutubules and intermediate filaments, a
requirement for a well-distributed cytoskeleton in the cell. Further, it has been noted how a
composite network of actin and intermediate filaments such as neurofilaments (Wagner et al,
2009) or vimentin (Esue et al, 2006) together form a stiffer structure better able to cope
with mechanical deformations over a larger order of strains than individually, supported by
experimental observations (Maniotis et al, 1997).
Properly characterising such a complex structure is a challenge, and several approaches have
led to various cellular models being proposed over the years. For a review, see (Lim et al,
2006).
In this thesis, avenues of characterising the mechanics of cells are explored by continuum
modelling and discrete network models. First, a few widely used experimental methods
to characterise cell behaviour are presented, followed by selected established cell models.
Governing the macroscopic cell response, cytoskeletal network characteristics are subsequently
described, along with chosen modelling techniques. Finally, the contribution of this thesis to
the field of cellular mechanics is described in terms of the appended papers.
15
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
16
Experimental methods in cell
mechanics
Probing the mechanical behaviour of cells can be done using a wide variety of methods, each
with the aim of measuring a particular response. Three general classifications can be made;
local cell deformation, global cell deformation or mechanical loading of a group of cells (Bao
and Suresh, 2003). A few important techniques used to probe single cells are shortly summarised here; for details see (in Cell Biology, 2007; Kamm and Mofrad, 2006). Both passive
and active measurement methods exist, in which results obtained by the former are obtained
by examining thermally induced motion of intracellular componente or introduced particles
(Mofrad, 2009). The latter types make use of actively deforming the cell and measuring the
response.
Passive microrheology
In passive microrheology, the displacement of an introduced probe due to thermal fluctuations
is measured. Beads on a micron scale are introduced into the cytoplasm and monitored by
video or particle tracking methods (Mofrad, 2009). Circumventing the need for an applied
force or torque ensures minimal active response due to perturbation of intracellular structures
such as the cytoskeleton.
Cell poker
The cell poker is an early effort to apply local forces to the cell surface. A cell is suspended in
fluid above a poker tip of a glass needle, which is attached by a wire to an actuator generating
vertical displacement. The difference in displacement between ends of the wire is evaluated,
and by knowing the stiffness of the wire (Kamm and Mofrad, 2006), the applied force can be
computed. This technique can reveal local deformation specific to one area in the cell, and
has been instrumental in evaluating the viscoelastic behaviour of cells (Kamm and Mofrad,
2006). A schematic is shown in Fig. 3.
17
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
Figure 3: Schematic of cell poker apparatus. Reprinted from (Goldmann et al, 1998) with permission from
Elsevier.
Micropipette Aspiration
In a micropipette aspiration experiment, a cell is sucked into a micropipette by applying a
suction pressure, see Fig. 4.
Figure 4: Schematic of micropipette aspiration experiment. A suction pressure P is applied in the pipiette.
Cells can be a) suctioned freely, b) adherent to a substrate or c) closely fitting inside the pipette.
Reprinted from (Hochmuth, 2000) with permission from Elsevier.
This is a popular technique to deform the membrane of many cells, such as the red blood
cell with its membrane-related viscoelastic protein network and lack of ordered cytoskeleton
(Kamm and Mofrad, 2006; Hochmuth and Evans, 1976). The liquid drop model was developed
with this type of cells and experiment in mind (Yeung and Evans, 1989). A disadvantage is
that the influence of cell adhesion is not accounted for, limiting the number of appropriate
cells.
18
Microplates
Adherent cells can be sheared or compressed by letting them attach at top and bottom top
microplates, and displacing one of these, see Fig. 5.
flexible microplate
cell
fixed
support
nucleus
rigid microplate
piezo-driven
translation
Figure 5: Schematic of parallell microplates. The microplates may be used to impose shear, tensile or compressive deformation on the cell. Reprinted from (Caille et al, 2002) with permission from Elsevier.
This method has been used to, for example, assess the contribution of the nucleus to
adherent cells during compression (Caille et al, 2002). Plates have also been used to measure
the stiffness of cancer cells subjected to tensile loading (Suresh et al, 2004), and characterise
cell properties by computational modelling (McGarry, 2009).
Magnetic Twisting Cytometry
Magnetic beads can be used to deform either the exterior or interior of the cell by subjecting
it to a magnetic field. The beads can either be attached to the surface, injected or engulfed
by phagocytosis (Mofrad, 2009). For example, Wang et al. bound ferromagnetic beads to
endothelial cells and showed that cytoskeletal stiffness increases in proportion to the applied
stress (Wang et al, 1993). An illustration is shown in Fig. 6.
Figure 6: Schematic of magnetic twisting cytometry, in which a magnetic field is applied and the bead is
subjected to a torque as a result. Reprinted figure from (Fabry et al, 2001) with permission from
the American Physical Society.
19
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
Atomic Force Microscopy
Similar to the cell poker, AFM induces a local deformation of the cell by pressing a tip,
attached to a beam, into the surface. Key components are a flexible cantilever beam and a
tip fastened to one end of it, often in the shape of a pyramid or sphere (in Cell Biology, 2007).
The cantilever is displaced by a piezoeletric device and during tip indentation, the deflection
of a laser off the upper side of the cantilever is measured by a photodetector. The indentation
of the sample is determined by subtracting the cantilever deflection due to bending from the
total displaced distance of the motor. The choice of tip and cantilever is important when
performing AFM, as induced sample strain fields are dependent on the geometry, and higher
cantilever stiffness is advantageous for soft samples (in Cell Biology, 2007). This equipment
is schematically shown in Fig. 7.
Figure 7: Schematic of AFM apparatus. The laser beam reflected off the upside of the cantilever beam due to
bending is recorded during indentation. Reprinted from (Meyer and Aber, 1988) with permission
from AIP Publishing LLC.
It is also possible to use AFM in imaging mode, in which the tip over a sample is scanned
and simultaneously recording the deflection to produce an image of the surface profile (Kamm
and Mofrad, 2006).
20
Mechanical models of biological
cells
Proper models of cellular mechanics require a viable approach to characterise not only the
cell as a whole, but the cytoskeletal networks governing it. Important models for both are
presented in this chapter.
Cell models
Numerous approaches to modelling the mechanical behaviour of cells have been developed,
for example phenomenological solid or visco-elastic continuum models or those taking into
account cell heterogenity. All with their respective advantages and drawbacks. This is by
no means an attempt to list all of them, but merely highlight a few that has received much
attention over the years. We focus here on the models that aim to capture the passive cellular
response, i.e. not cell contractility or migration.
Liquid drop
One of the simplest cell models, originally developed to account for rheology of neutrophils
(Lim et al, 2006), is the liquid drop model, schematically shown in Fig. 8. In this manner,
the cell is considered to behave as a liquid surrounded by an elastic membrane subjected to
tension, generated by the contractile machinery of the cell. Assuming the fluid to behave like
Membrane is under sustained tension.
Interior is modelled as fluid.
Figure 8: Schematic image of the liquid drop model. The elastic membrane is under sustained tension,
surrounding a liquid interior.
21
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
f
f0
E
Figure 9: Energy landscape of different states. Image reproduced from (Mofrad, 2009) with permission from
Annual Reviews for use in academic titles.
a Newtonian liquid, and the membrane to carry tension but without bending resistance, the
model was developed to simulate the observed experimental flow of cells into micropipettes
(Yeung and Evans, 1989). This basic idea can be extended to consider the liquid cell interior
as non-Newtonian, or compartmentalise it to mimic the nucleus and other rheological models
(Lim et al, 2006).
Theory of SGR
An important characteristic of living cells is their frequency-dependent stiffness and dissipation, observed during mechanical testing by Magnetic Twisting Cytometry (MTC) (Kamm
and Mofrad, 2006; Fabry et al, 2001, 2003). Typically, one can observe a power-law dependence of the storage and loss modulii for cells, with a fixed exponent for the former over all
frequencies. This can be motivated by thermal exchange between cytoskeletal filaments and
the surrounding cytoplasm; for high frequencies there simply is not sufficient time for them to
disentangle and orient themselves according to the deformation field. For lower frequencies,
the filaments will have more time to change to alternative configurations as they are essentially polymers with a number of possible conformations.
Soft Glassy Materials (SGMs) is a class of materials that share some characteristics, for example being very soft and obeying a weak power-law of frequency. Bouchaud proposed that
these materials can be envisioned as a rough energy landscape with many local minima corresponding to metastable states (Mofrad, 2009; Kamm and Mofrad, 2006; Bouchaud, 1992).
The elements of the material can then escape from its state to another well, provided with
enough energy f to cross the barrier E, visualised in Fig. 9. This energy is provided by
random thermal fluctuations. However, it was argued that thermal activation compared to
actual depth of the well in which elements are trapped, is small (Sollich et al, 1997). In
an extension to the theory, an effective temperature or noise level, representing interaction
22
between elements was introduced along with strain degrees of freedom to describe material
behaviour. This Soft Glassy Rheology (SGR) model replicates power law behaviour for the
storage and loss modulii.
Continuum models
Continuum models are appropriate if the length scale of interest is significantly greater than
the dimensions of the intracellular of components. In their simplest form, they are useful in
the sense that they can often be implemented with ease to phenomenologically characterise
cell components when studying the macroscopic properties in a numerical framework such as
the Finite Element Method (FEM). Obviously, this lack of easily interpretable physical parameters is a drawback, as relating microstructural arrangements to the mechanical behaviour
then proves a challenge. Nevertheless, despite a possible lack of transcluency in governing
material parameters, it is a popular approach as the influence of other factors (e.g. viscous
deformation or cell morphology) can be assessed.
Studies performed in this manner include investigating the effect of viscoelasticity and cell
compressibility (McGarry, 2009), verification and parametric study of red blood cell subjected
to deformation by optical tweezers (Dao et al, 2003) and contribution of nucleus (Caille et al,
2002) to cell mechanical properties. While many continuum models aim to capture primarily
the passive cellular response, models also exist that predict the influence of active cellular
response, such as the model proposed by Ronan et al., a constitutive formulation to capture
the remodelling and contractile behaviour of the actin cytoskeleton (Ronan et al, 2012).
Mechanical cell models developed in a continuum framework span from very simple (such as
the hyperelastic neo-Hooke formulation) to rather complex, for example the one mentioned
above. Despite the drawbacks, they offer an indispensable tool, in that they are very versatile. For example, they can be formulated as anisotropic, and in terms of microstructural
variables, then homogenised by transition to a macroscopic scale. This multi-scale modelling
approach allows for characterisation of the cell as a whole, with material parameters related
to microstructural properties.
Tensegrity
The close association between various cytoskeletal components led to the proposal of the
cellular tensegrity model, in which the contracile machinery of the cell generate a prestress
carried by the actin and intermediate filament networks. This prestress is then balanced
by compression of microtubules and adhesion traction (Ingber and Jamieson, 1985; Ingber
et al, 2014). Evidence supporting this theory exist, such as proportionality between prestress
and cellular stiffness, typical of tensegrity structures (Stamenovic and Ingber, 2002; Wang
et al, 1993, 2001, 2002; Volokh and Vilnay, 1997), and a significant increase of traction forces
following disruption of the microtubule network (Stamenovic et al, 2002). A consequence
23
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
of this model is a global response to perturbations of cytoskeletal components, observed
experimentally by investigators (Maniotis et al, 1997) in endothelial cells. Contrary to this,
results wholly inconsistent with the tensegrity model by way of exclusively local response was
observed when glass needles were used to poke fibroblasts, both uncoated and lamin-treated
to ensure connection to the underlying cytoskeleton (Heidemann et al, 1999).
Cytoskeletal network characteristics
In a multi-scale approach to cell modelling, proper characterisation of underlying structural
and mechanical properties is essential. Assuming the response to be governed by intracellular
entities (e.g. cell cortex, nucleus and cytoskeleton), a first step in modelling the macroscopic
response is finding ways to describe that of microstructural components. While each of
these is a complex system on its own, the cytoskeleton is the focus of this section. First,
characteristics of each cytoskeletal network type is presented, before presenting two oftenused modelling methods.
Actin
Actin, as mentioned previously, is the most well-studied of the three main types of cytoskeletal proteins. In cells, it is instrumental in providing the cell with a contractile machinery
(Pardee, 2010), forming the branched network known as the lamellipodium and probing the
ECM during cell migration (Alberts et al, 2004; Small et al, 2002; Insall and Machesky, 2009),
assembling into stress fibres (Pellegrin and Mellor, 2007), providing the cell cortex with stiffness (Kamm and Mofrad, 2006; Dao et al, 2003; Evans, 1998) and in general contributing to
cellular stiffness (Wu et al, 1998; Maniotis et al, 1997; Kamm and Mofrad, 2006; Wang et al,
1993). It has even been observed that F-actin networks cross-linked with filamin qualitatively
replicate the response of cells (Gardel and Nakamura, 2006). With such a diverse repertoire
of known responsibilities, it is no surprise that the dynamics of actin ensures its capability to
assemble into structures with vastly dissimilar mechanical behaviour.
Actin is a semiflexible polymer, i.e. the contour length of a filament observed in vitro ∼15μm
is typically on the order of the persistence length ∼17μm (Kasza et al, 2010; Xu et al, 2000,
1998), although in physiological conditions the contour length is more typically on the order
of a micron (Kamm and Mofrad, 2006). However, for complete characterisation, semiflexible
polymer theory is then needed.
Actin in the abscence of other chemical constituents form quite soft entangled networks that
break at moderate strains, exhibit strong viscoelastic behaviour and with a stiffness strongly
dependent on filament length (Janmey et al, 1994). Network rupture is preceded by either a
stress-stiffening (Semmrich et al, 2008; Janmey et al, 1994; Xu et al, 2000) or stress-softening
(Xu et al, 1998, 2000) behaviour. This depends on whether the network is weaky connected,
24
Figure 10: Different types of cross-linker proteins allow structural variety of F-actin networks when crosslinked. Reproduced from (Lieleg et al, 2010) with permission of The Royal Society of Chemistry.
a concentration-dependent factor (Gardel et al, 2008) which also affects the network stiffness
and frequency-dependent behaviour (Boal, 2002).
Actin can assemble into bundles such as stress fibres and filopodia (Pellegrin and Mellor, 2007;
Small et al, 2002; Insall and Machesky, 2009), or various types of network structures. These
can consist either of purely filamentous networks, bundled networks or a composite structure
of both (Lieleg et al, 2010; Gardel et al, 2004, 2008). To achieve this structural diversity,
cells make use of a number of different cross-linker proteins, which define the network characteristic as a function of cross-linker concentration, see Fig. 10 (Lieleg et al, 2010). These
cross-linker proteins have further been observed to determine the structure and mechanical
behaviour of actin, for example by regulating filament length (Biron and Moses, 2004; Biron
et al, 2006). It has been proposed that the concentration-dependent average cross-linker
distance is an intrinsic parameter that governs the microstructure of cross-linked networks
(Luan et al, 2008). A relevant concern is how the behaviour of networks in the presence of
various types of cross-linker is determined. It has been observed for the cross-linkers fascin
and filamin, the behaviour is determined by the highest relative concentration, although this
might not be independent of cross-linker type (Schmoller et al, 2008).
Importantly, it has been found that cross-linker binding affinity affects the network rheological
behaviour (Xu et al, 1998; Wachsstock et al, 1993, 1994). For an extremely strong cross-linker
with low dissociation rate constant, no frequency dependence is observed, and the network
behaves like a solid gel, while a cross-linker with high dissociation rate constant yields a more
viscous behaviour. This is further complicated by the microstructure, as certain cross-linkers
25
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
Nonaffine
Affine
entropic
Log(L )
Affine
Affine
mechanical
No
naf
fine
Solution
Log(c)
Figure 11: Illustration of affine and non-affine deformations. Filament deformations of non-affine shear are
not similar to the macroscopic direction, typically occuring in sparse networks. The schematic
on the right indicates dependence on polymer contration and molecular weight. Reprinted from
(Gardel et al, 2008) with permission from Elsevier.
have a tendency to form bundles that can more easily slip by eachother as opposed to an
isotropically cross-linked network (Wachsstock et al, 1993, 1994). Thermal unbinding has in
this context been shown to, at least partly, be responsible for stress relaxation and energy
dissipation (Lieleg et al, 2008), and strong rate-dependence of mechanical behaviour has been
attributed to forced cross-linker unbinding (Lieleg and Bausch, 2007). A comprehensive study
of thermal influence on entangled and cross-linked F-actin solutions and were conducted by
Xu et al. (Xu et al, 2000), in which the enhanced stiffness at lower temperatures was attributed to the longer life-time of of the bonds.
The unbinding characteristics of cross-link proteins are not unique in governing network response, however. In a study, it was shown that filament length as well as cross-linker stiffness
tunes the mechanical behaviour of cross-linked F-actin networks (Kasza et al, 2010). Rigid
cross-links exhibited a length-dependent stiffness qualitatively similar to that of entangled solutions, in which a plateu is approached as filament length increases and network connectivity
is fulfilled (Janmey et al, 1994; Kasza et al, 2010). Contrary to this, compliant cross-linkers
such as filamin yielded an increasingly stiffer response as the length increased, evidently also
concentration-dependent (Kasza et al, 2010).
The origin of elasticity in F-actin networks depends on all these factors. It is important to
bear in mind the semiflexible nature of actin filaments, as their stiffness can be of either
entropic nature due to reduced number of configurations, or enthalpic. In general, a sparse
network facilitates non-affine bending deformation, and a more dense network promotes affine
filament stretching. The latter may be either entropic or mechanical in nature, see Fig. 11.
Intermediate filaments
Intermediate filaments, also of semiflexible nature (Wagner et al, 2009; Schopferer et al, 2009;
Rammensee et al, 2007), have a role to play in both cytoskeletal and nuclear mechanics.
26
They radiate from their lamin-facilitated connections at the nuclear envelope and provide
connections to the cytoplasm and other cells (Denais and Lammerding, 2014; Herrmann et al,
2007; Goldman et al, 1986). Vimentin intermediate filaments have been observed to exhibit
significant strain hardening and rupture at far higher strains than actin or microtubules
(Janmey et al, 1991; Herrmann et al, 2007). The concentration dependence of intermediate
filament networks appears less pronounced than F-actin networks (Janmey et al, 1991; Lin
et al, 2010). Together with F-actin, they have been observed to assemble into composite
networks with greater stiffness and load-bearing capability than either network is individually
capable of (Wagner et al, 2009; Esue et al, 2006). In endothelial cells, this functional synergy
was observed to be especially pronounced as the actin cytoskeleton effectively mediates force
transfer to the nucleus, but ruptures at small strains, while the intermediate filament network
does so at increasing deformation magnitude (Maniotis et al, 1997). Experimental evidence
of intermediate filament contribution to cellular stiffness is, for example, a more compliant
behaviour and lower proliferation rate shown by mechanical testing (Wang and Stamenovic,
2000) of vimentin-deficient fibroblasts.
Microtubules
Microtubules, the shortest and stiffest of the cytoskeletal filaments, exhibit rod-like features
as their persistence length is many times greater than the actual cell dimensions (Kamm and
Mofrad, 2006). Being very stiff allows them to act against compression, and their important
role in keeping the intermediate filament network from collapsing has been observed experimentally (Rathje et al, 2014; Maniotis et al, 1997), and these networks are also believed to
mutually stabilise each other (Herrmann et al, 2007). The mechanical behaviour of microtubules in vivo has been observed to be similar to that of a viscoelastic solid (Heidemann
et al, 1999).
Cytoskeletal models
Elasticity of semiflexible polymer networks
As mentioned, actin and intermediate filaments are semiflexible polymers and therefore, their
stiffness is both entropic and mechanical in origin. For a network of inextensible filaments, a
model was proposed by defining an energy functional consisting of thermal bending undulations and applied tension of the polymer chain (Mackintosh et al, 1995). Letting the chain
conformation be represented by a Fourier series and utilising the equipartition theorem to
compute the corresponding amplitudes, the chain end-to-end displacement can be computed
as function of the applied tension. The assumption of inextensibility is, however, rather strict
and not necessarily reflective of semiflexible polymers in general (Storm et al, 2005; van Dillen
et al, 2008). To address these shortcomings, the theory was extended to account for finite
27
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
extensibility of the polymer chain (Storm et al, 2005).
By this theory, the shear modulus of weakly cross-linked and entangled F-actin network solutions is expected and verified to obey an exponential scaling law of monomer concentration
(Mackintosh et al, 1995). This exponent is however not general, and depends on the microstructure which regulates whether affine or non-affine, and entropic or enthalpic origins of
stiffness dominates (Gardel et al, 2004; Kamm and Mofrad, 2006).
Interestingly, the shear modulus of intermediate filaments exhibit less dependence on concentration than actin, especially evident in vimentin and desmin (Schopferer et al, 2009).
It has also been observed for neurofilaments, although the difference is not as significant
(Rammensee et al, 2007).
Discrete network models
In a discrete model, the effect on the macroscopic network response of individual filament
properties such as stiffness, length, shape and orientation, are taken into account. Most
often, this is done by discretising the network filaments by a numerical method such as FEM,
subjecting the whole model domain to mechanical forces and computing the response.
By doing this, it has been shown that stiffening of actin networks with both undulated and
straight filaments is a result primarily of a rearrangement of filaments. At low strains, nonaffine bending of filaments dominate, but for increasing strain the filaments rearrange into
a deformation mode governed by affine stretching, a filament-density dependent transition
(Onck et al, 2005; Huisman et al, 2007; Head et al, 2003).
The role of cross-linker dynamics has also been elucidated on. For example, it was shown
that when comparing unfolding and unbinding of ABPs, stress relaxation was accomplished
by unbinding only (Kim et al, 2011). Investigating the influence of stochastic unbinding on
stress-strain curve, stiffness and strain rate behaviour emphasised the importance of cross-link
characteristics (Abhilash et al, 2012, 2014).
These computational methods facilitate the investigation of microstructural elements more
directly, but are in general greatly limited by considerations of computational efficiency. Large
deformation formulations, complicated material models and interactions such as contact all
make simulation of larger samples or macroscopic structures unviable. Primarily, equation
system size restricts the applicability to model entities such as the entire cell.
28
Conclusions and contribution to the
field
A substantial body of litterature is available on experimental and modelling approaches to
cellular mechanics, of which the main aspects have been summarised in the preceding chapters. It is clear that creating a unifying model to characterise mechanical behaviour, even only
in terms of a passive response, for various cell phenotypes and physiological conditions, is an
immense task. Developing new material models, or merely taking a new approach to existing
ones, can however provide methods to characterise behaviours specific either to the cell as a
whole, or intracellular component such as the cytoskeleton. This thesis, and the appended
papers contained herein, do not claim to consitute an all-compassing model, but rather provide new insights of intracellular components such as the cytoskeleton. For example, most
mechanical models are primarily used for small deformations, quantifying the storage or loss
modulus. Few material models in cell mechanics are formulated in a large-strain continuum
framework, which is introduced in this thesis for the underlying cytoskeletal networks. Further, the viscoelastic behaviour is often modelled for the cell as a whole (Lim et al, 2006;
Kamm and Mofrad, 2006), whereas in this thesis, it is introduced as viscous evoluion of microstructural variables for proper representation in a finite element framework. While focus
lies in modelling passive cytoskeletal response, in Paper F, cell contractility is introduced as
part filament-based models.
In conclusion, in this thesis the fields of solid mechanics, computational modelling and biology
come together to elucidate the role of microstructural cytoplasmic elements in cell mechanics.
The work performed herein aims to provide versatile tools for computational modelling of
cells, utilising principles of continuum mechanics. Importantly, dissipative effects such as viscoelasticity are introduced in the constitutive formulations for easy implementation in finite
element frameworks.
29
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
30
Modelling mechanical cytoskeletal
and cell behaviour by use of
continuum and discrete models
A continuum model for transiently cross-linked actin networks
Given the importance of the actin cytoskeleton in providing stiffness to the cell (Maniotis
et al, 1997; Wu et al, 1998; Wakatsuki et al, 2000; Thoumine and Ott, 1997; Kamm and
Mofrad, 2006) and the distinct relaxation behaviour of it (Xu et al, 2000, 1998; Janmey et al,
1991), proper characterisation of cell mechanics over a range of time scales dictates the need
for modelling actin dynamics.
It has been shown that cross-link dynamics are instrumental in governing the viscoelastic
response of actin networks (Wachsstock et al, 1993, 1994; Xu et al, 1998, 2000). Inspired by
this, and the concept of SGR (Sollich et al, 1997), we let a bound cross-link be represented as
trapped in energy well according to Fig. 12. Here, ψ is the strain energy (per mole α-actinin)
stored in the actin network, β ∈ [0,1] is the fraction of energy available to break the bond
and Eb is the bond strength. Then, we know from Arrhenius law that
Eb
kd = A · exp −
RT
(1)
where kd is the dissociation rate constant, A is a factor, R is the universal gas constant and T
is absolute temperature. Thus, for a network experiencing mechanical stress, the term (Eb )
is replaced by (Eb − βψ), and the dissociation rate constant is scaled by a factor φ, defined
as
φ = exp
βψ
RT
.
(2)
We use a chemical model developed by Spiros to model α-actinin dynamics (Spiros, 1998).
The cross-links and actin filaments can be thought of as in different states, Fig. 13. State d is
considered the load-bearing state of actin filaments. Evolution equations for concentrations
31
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
Eb -βψ
Eb
Loaded
bound state
βψ
Unloaded
bound state
Figure 12: Bound state for cross-linked molecules. To escape the potential well in the loaded state, the
energy required to break the bond is Eb -βψ. Image reproduced from (Fallqvist and Kroon, 2012),
with kind permission from Springer Science.
a
c
b
d
Figure 13: Schematic view of the four different possible structures in the network. State d is considered the
load-bearing state of filaments. Image reproduced from (Fallqvist and Kroon, 2012), with kind
permission from Springer Science.
32
na - nd of corresponding states can be derived as
ṅa = kd nc − ka na nb
(3)
ṅb = kd nc − ka na nb + 2kd nd
(4)
ṅc = −kd nc + ka na nb + 2kd nd − νka nb nc
(5)
ṅd = −2kd nd + νka nb nc ,
(6)
where ka is the association rate constant and ν is a factor to account for slower reaction from
state c to d. All instances of kd are scaled with the exponential factor φ defined in Eqn. 2.
Developing the constitutive model, we introduce a multiplicative split of the deformation
gradient F into an elastic and viscous part Fe and Fv , respectively, according to
F = Fe Fv .
(7)
We now define a modified neo-Hookean strain energy function as linearly dependent on the
concentration of cross-links as
Ψ = Nd ψ = Nd
μ
2
(Ie1 − 3) − p(Je − 1)
(8)
where Nd is the concentration of cross-links in state d in the reference configuration, μ is the
shear modulus of the network and the Lagrange multiplier p is included to enforce elastic
incompressibility, i.e. Je ≡ 1.
The strain energy function in Eqn. 8 contains information about the elastic and viscous
deformation in the network in terms of the elastic invariant. Thermodynamically consistent
evolution laws are introduced for viscous deformation in terms of the inverse of the right
Cauchy-Green tensor of viscous deformation A−1 . Second Piola-Kirchhoff stresses are computed as
S = Nd μ(A − A33 C33 C−1 ).
(9)
The model was assessed by computing its response to shear deformation with step strain and
subsequent relaxation. Using a set of material parameters fit to predict experimental results,
the cross-link dynamics in response to various magnitudes of applied step shear strain γ was
computed as in Fig. 14. Predicted stress relaxation curves compared to experimental results,
obtained from the litterature for F-actin networks isotropically cross-linked by α-actinin (Xu
et al, 2000) are shown in Fig. 15. The model could be successfully used to predict the
mechanical response, although the parameter β must be reduced for higher strain values.
We attribute this to an overestimate of strain energy available to break cross-link bonds,
computed from the strain energy. Indeed, scaling the computed energy with fraction β yields
33
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
24
γ18
γ30
γ40
γ50
23
nb [ mmol
]
m3
na [ mmol
]
m3
0.15
0.10
22
0.05
21
0.0
19.5
0.2
0.4 0.6
Time [s]
0.8
4.0
0.100
3.0
0.075
0.0
0.2
0.4 0.6
Time [s]
0.8
0.0
0.2
0.4 0.6
Time [s]
0.8
nd [ mmol
]
m3
nc [ mmol
]
m3
0.0
2.0
0.050
1.0
0.025
0.0
0
0.0
0.2
0.4 0.6
Time [s]
0.8
Figure 14: Concentration of chemical states as functions of time after applied shear deformation. Image
reproduced from (Fallqvist and Kroon, 2012), with kind permission from Springer Science.
γ18
γ40
γ30
101
101
γ50
100
10−1
10−2
P12 [Pa]
P12 [Pa]
10−1
Time [s]
100
100
10−1
10−2
10−1
Time [s]
100
Figure 15: Stress relaxation of isotropically cross-linked F-actin networks. Image reproduced from (Fallqvist
and Kroon, 2012), with kind permission from Springer Science.
34
a)
b)
Figure 16: Boundary conditions for network model. a) VC and b) NVC. Reprinted from (Fallqvist et al,
2014) with permission from Elsevier.
values (≈2 RT , R is the universal gas constant) in good agreement with estimates of bond
strength between F-actin and α-actinin (Miyata et al, 1996; Biron and Moses, 2004).
An important observation is the presence of an initial rapid stress relaxation which we propose
is dominated by cross-link debonding, facilitating filament movement that governs subsequent
slower decay of stress at longer time scales, when the cross-links have bound to filaments once
again. The constitutive model developed herein is able to predict this behaviour properly
and furthers our understanding of isotropically cross-linked actin networks and the role of
cross-link dynamics in governing their mechanical behaviour.
A discrete network model to assess the influence of geometrical
parameters and cross-link compliance
The length distribution of actin filaments are known to we roughly of exponential type (Nunnally et al, 1980; Janmey et al, 1994; Xu et al, 1998; Yin et al, 1980), and affected by factors
such as the presence of cross-linkers (Biron and Moses, 2004; Kasza et al, 2010) and interfilament attractions (Biron et al, 2006). Cross-linkers are themselves proteins with distint
mechanical behaviours, and in order to elucidate the effect of F-actin network architecture
and interfilament protein response, we perform numerical computations by use of a discrete
network model.
The finite element software ANSYS is utilised to generate random network structures, given
certain characteristic parameters such as filament length, distribution parameters and crosslink compliance. This computational network is then analysed by use of a custom code with
user-programmable features due to the degree of customisation required with regards to, for
example, adaptive stabilisation, improved time step control and implementation of zero-length
unidirectional springs to represent cross-links.
An obvious choice boundary conditions for a representative network structure is not obvious.
We define two types of boundary conditions, termed VC (Vertically Constrained) and NVC
(Non-Vertically Constrained), see Fig. 16. At small deformations, the response is identical,
35
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
b)
a)
Figure 17: a) Initial shear modulus as computed for variations in length and length distribution. The parameter β can be interpreted as the mean value of filament length. Circles are an assumption
of equal filament length, and triangles are for exponential distribution. b) Influence of cross-link
stiffness and filament length on the initial shear modulus. Reprinted from (Fallqvist et al, 2014)
with permission from Elsevier.
and therefore we perform analyses with both types of boundary conditions only when investigating large deformations. For small deformations involving computation of merely the shear
modulus G0 , we use NVC boundary conditions. Mesh sensitivity analyses are performed for
random seeding of various filament lengths to determine the appropriate size of the representative network to minimise variation of results.
To represent the cross-links, a hyperbolic function is introduced, with which the parameters
are defined to represent three stiffnesses; Type A (representative of filamin A, a compliant
cross-linker), Type B (intermediate stiffness) and Type C (representative of very stiff or rigid
interfilament connections). Actin filament characteristics are defined from litterature.
We assume the qualitative behaviour between computations of stiff and compliant cross-links
to be independent for variations in filament length, and perform computations for stiff crosslinks. In Fig. 17 a) a comparison between an exponential distribution and assumption of no
distribution is shown. Apparently, neglecting the actual distribution of filament length has
a significant effect on the dependence of the shear modulus. Not only the magnitude, but
the increase with respect to increasing filament length seems less pronounced than for the
equal assumption. For simplicity, we perform also other computations in this study with the
equal length assumption. The influence of cross-link stiffness is determined by normalising
the shear modulus for stiff cross-links G0,s with that obtained for compliant ones G0,c , Fig.
17 b). For an increasing length, the influence of stiff cross-links is greater, possibly due to
more numerous filament intersections.
For large-strain computations, we visualise the influence of cross-link compliance with a stressstrain (τ12 -
12 ) curve, in which the stress is computed as the sum of nodal forces at the top
36
Figure 18: Stress-strain curves for different cross-link types. Solid lines and dashed lines represent VC and
NVC boundary conditions, respectively. Reprinted from (Fallqvist et al, 2014) with permission
from Elsevier.
a)
b)
Figure 19: Filament bending, stretching and cross-link energies in the network. a) VC network and b) NVC
network. Reprinted from (Fallqvist et al, 2014) with permission from Elsevier.
of the model divided by the area, Fig. 18. As expected, the cross-link compliance is thus
instrumental in tuning the network mechanical response by shifting the onset of hardening
to larger strains. The origin of this hardening is interesting to investigate further with our
cross-link dominated behaviour in mind.
We expect that the previously determined transition from non-affine bending to affine stretching (Onck et al, 2005; Huisman et al, 2007; Head et al, 2003) is also affected by cross-link
characteristics. In Fig. 19 this is shown in terms of total network filament bending, stretching and cross-link energies Eb , Es and Ec , respectively. It can be seen that the transition is
shifted to larger strains for more compliant cross-links, and also that the influence of boundary
conditions is a main determinant in governing this behaviour. To further hightlight the importance of boundary conditions in discrete network models, we found that when comparing
37
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
Figure 20: One-dimensional model of cross-linked filament networks. Reprinted from (Fallqvist et al, 2014)
with permission from Elsevier.
the differential modulus at large strains, a computational model that is vertically constrained
will yield an overly stiff response as compared to experimental observations.
With the importance of filament length established, we formulate a continuum model for use
in future studies. We start by defining a one-dimensional model for cross-linked networks with
filament length La and cross-link length Lc subjected to force f resulting in a displacement
ua due to filament bending and stretched cross-link length lc , see Fig. 20. The strain energy
function Ψ for the representative unit can be split in two parts:
Ψ = Ψc + Ψa − p(J − 1),
λc
μc
y(λc )dλc ,
Ψc =
2
(10)
(11)
1
μa
(λa − 1)2 ,
Ψa =
4
(12)
where Ψc is the contribution from cross-links, Ψa from the actin filaments and λa and λc is
filament and cross-link stretch, respectively. The terms μc and μa are stiffness parameters, y
is the function we choose to approximate the force-displacement relationship of the cross-link
with, p is a Lagrange multiplier to enforce incompressibility and J is the Jacobian of the
deformation gradient. The second Piola-Kirchhoff stresses S can be computed as
dλ
Lc −1
1 a
μc · y + μa (λa − 1)
+2
S=
6λ
dλc
La
· I − C33 C−1 ,
38
(13)
Figure 21: Filament length dependence and cross-link stiffness influence on initial shear modulus for continuum model. Triangles and circles are experimental results corresponding to rigid and compliant
cross-links, respectively (Kasza et al, 2010). Reprinted from (Fallqvist et al, 2014) with permission
from Elsevier.
where
μ c La
dλa
=
·
dλc
μa 2Lc
dy
dλc
(14)
and dy/dλc is found from the assumed force-displacement function for the cross-link. The
model is generalised to three-dimensional macroscopic behaviour by defining an eight-chain
model similar to the well-known Arruda-Boyce rubber model (Arruda and Boyce, 1993), and
relating the total stretch of the representative unit λ to the first invariant of the deformation
tensor.
The influence of cross-link stiffness and filament length is qualitatively in line with that found
experimentally, see Fig. 21, in which an increasing C1 defines increasing cross-link stiffness.
While a single set of model parameters could not predict experimental results, the qualitative
predictive capability of one of the main determinants of network architecture, the filament
length, is a promising indicator that the one-dimensional model is a good starting point for
more refined models.
To conclude, a discrete network model was used to assess the influence of cross-link compliance
on network response. We found that the cross-links govern the behaviour by shifting the
transition between non-affine bending and affine stretching regimes, and thereby also the
onset of strain hardening. Further, we developed a micromechanically motivated continuum
model that predicts the length-dependence in a qualitative manner.
39
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
10
× ts
o ts
× ts
o ts
8
τ12 6
[Pa]
4
×
×
×
×
×
= 0.01s, model
= 0.01s, Xu et al.
= 1.00s, model
= 1.00s, Xu et al.
×
×
2
0
×
× ×
×
×
× × ×
0.0
× ×
× ×
× 0.1
0.2
0.3
0.4
12
Figure 22: Shear stress τ12 at two time scales ts for various shear strain magnitudes 12 . Model predictions
(crosses) compared to experimental results (circles) (Xu et al, 2000).
A discrete network model to assess the influence of cross-link
debonding on mechanical properties of cross-linked actin networks
The influence of possible cross-link debonding was not taken into account in the previous
study, as we assumed all bonds to be permanent. We further investigate the mechanical
response of cross-linked actin networks by taking into account the stochastic nature of the
bonds. Similar work has been undertaken before (Kim et al, 2011; Abhilash et al, 2012, 2014)
to quantify this effect, but we introduce a slightly more general approach than the classical
Bell model (Bell, 1978), in which the exponential is modified by the applied tensile force of
the bond.
According to Arrhenius’ law, Eqn. 1, the exponential function defines the probability that a
debonding event will occur. We revisit Eqn. 1, and modify the probability of debonding Δp
during a time interval Δt by including the fraction β of elastic strain energy ψ stored in the
cross-link as
Δp = Δt·exp(− (EbkB−βψ)
T )
(15)
with kb being the Boltzmann constant. Utilising the numerical method for a discrete network
model as described in the previous paper, we implement this possible debonding event in
the formulation for the cross-link elements. This allows us to study the influence of bond
strength, cross-link stiffness and fraction of strain energy, which in our case is mechanical.
Finding parameters for the cross-link reproducing force-extension behaviour of α-actinin
(M.Rief et al, 1999), model predictions and experimental results for α-actinin cross-linked
actin networks (Xu et al, 2000) are according to Fig. 22. We may consider β a free parameter
40
30
50
a)
40
— Red. Eb
— Red. C1
20
τ12
[Pa]
— ts = 0.01s, β = 1.0
- - ts = 0.01s, β = 0.1
— ts = 1.00s, β = 1.0
- - ts = 1.00s, β = 0.1
b)
— Baseline
τ12 30
[Pa]
20
10
10
0
0
0.0
0.1
0.2
0.3
0.0
0.4
0.1
0.2
0.3
0.4
12
12
Figure 23: Influence of parametric variation with respect to cross-link properties. In a), Solid lines are for a
time scale of 0.01s and dashed lines for 1s.
for which a clear interpretation is elusive, but dependent on the magnitude of deformation.
Values of 1.0 and 0.1 are used for the short and long time scales, respectively, in the experimental comparison. The results suggest that the relaxation mechanism is more complex than
merely stochastic debonding of cross-links.
We studied the interplay between mechanical properties and stochastic debonding by variation of cross-link properties and time scale from a reference, baseline, model with nominal
values. This is shown in Fig. 23. As expected, the response with respect to bond strength
is either independent of, or characteristic of a softening behaviour, for shorter and longer
time scales, respectively. Such a softening behaviour has also been observed experimentally
(Xu et al, 2000). In accordance with this, it can also be observed that the parameter β is
only governing the behaviour at large strains and short time scales, i.e. only when subjected
to such a magnitude of strain is there a noticeable effect on the stress from the number of
debonded cross-links.
We thus conclude that the mechanical response of actin networks as characterised by their
cross-links originates either from their stiffness or stochastic nature. At short time scales,
only at large strains do the dynamical properties of the bond play a role, and the cross-link
stiffness governs the mechanical behaviour. Conversely, at longer time scales, the bond properties are quite influential and could be responsible for observed phenomena such as strain
softening, although the model by no means provide the whole picture.
41
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
A constitutive model for composite biopolymer networks - Irreversible effects in the large strain regime
Proper modelling of the intracellular cytoplasm must account for the cytoskeleton, composed
of the actin, intermediate and microtubule filament networks. Actin and intermediate filaments have been indicated to provide stiffness to the cell, and we therefore focus on developing
a continuum model for use with composite biopolymer networks.
As a starting point we use the one-dimensional filament model developed in Paper B, although
we now let the filament length be determined by the end-to-end distance of a polymer, dependent on temperature, contour length and persistence length.
We split the deformation into isochoric and volumetric parts, and perform a multiplicative
split of the one-dimensional unit stretch λ̂ into an elastic (λe ) and viscous (λv ) part on the
form
λ̂ = λe λv .
(16)
To derive a composite model, we postulate the existence of a strain-energy function Ψ which
can be assumed to be composed of an isochoric (Ψich ) and volumetric (Ψvol ) part. This can
be further decomposed into the contribution from various constituents as
Ψ = Ψich + Ψvol ,
Ψich = Ψa ωa + Ψb ωb
(17)
where Ψa ωa and Ψb ωb are the contributions from two networks a and b, respectively. The
introduced variables ωa and ωb are damage variables that take into account effects such as
debonding and filament rupture in the networks. Adopting a strain energy function with a
high enough bulk modulus K should enforce incompressibility efficiently in a future numerical
implementation, by defining it on the form
Ψvol =
K
(J − 1)2 ,
2
(18)
where J is the jacobian of the deformation gradient.
We now replace the eight-chain model by integration of filaments over the unit sphere, following Miehe et al. (Miehe et al, 2004), enabling us to define the second Piola-Kirchhoff stress
S for a (this form can also be made anisotropic, see Paper 4) constituent network as
S = 2ω
m j=1
wj μf (λf,j − 1)
dλs,j ∂λe,j
,
+ μs fs (λs,j )
dλe,j ∂C
dλf,j
+
dλs,j
(19)
42
8
28
— P 12
a)
6
b)
23
o Xu (2000)
— P 12
o Tseng (2009)
18
4
P12 [Pa]
P12 [Pa]
2
0
13
−2
0.1
γ12
0.0
0.2
3
0.0
8
−2
0.1
0.2
γ12
0.3
Figure 24: First Piola-Kirchhoff stress P12 for cyclic shear of actin networks, model predictions and experiments. Experimental observation from a) (Xu et al, 2000) and b) (Tseng et al, 2004).
and the total stress for the composite network is computed as the sum of constituent network
stresses. In the above equation, wj is the weight factor corresponding to integration direction
j on the unit sphere used in the transition from micro-to macro response when integrating
over the unit sphere (Miehe et al, 2004). The parameters μf and μs are stiffness parameters
and fs an introduced effective interaction function to describe filament interaction (e.g. crosslink behaviour). These stiffness parameters are scaled phenomenologically in the case of a
composite network. Filament and interaction stretch λf and λs , respectively, are defined with
respect to a particular integration direction j. For details on computation of the differentials,
see Paper 4. The volumetric stress Svol is computed from the strain energy function as
Svol = KJ(J − 1)C−1 ,
(20)
where C is the right Cauchy-Green deformation tensor. This is added to the composite network stresses.
By defining thermodynamically consistent evolution laws for damage variables and evolution
of viscous stretches, the model can be verified with respect to experimental results. As a first
step, excellent model predictions for actin and intermediate filament networks are obtained
by comparisons to litterature. Of much importance is the definition of introduced evolution
laws, and we fit the model to cyclic experiments of actin networks with dissimilar cross-links,
Fig. 24. The model reproduces the hysteresis behaviour fairly well, although not perfectly.
Parametric studies show an interdependence between the damage and viscous variables, by
which the viscous parameter has a regulating effect on the damage due to the choice of evolution laws. This turns out to be favourable, as rupture of biopolymer networks need not
be immediate, but preceded by slow strain softening. Computations by fitting the model
43
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
b) 60
a) 5
— P 12
4
— P 12
50
o Wagner (2009)
o Wagner (2009)
40
3
2
30
P12 [Pa]
P12 [Pa]
20
1
10
0
0.0
0
0.2
0.4
12
0.6
0.8
1.0
0.0
0.3
0.6
12
0.9
1.2
Figure 25: First Piola-Kirchhoff stress P12 for large deformation in shear, model predictions and experiments
(Wagner et al, 2009). Results are for a) actin, and b) intermediate filament networks.
to experiments for actin and intermediate filament networks are shown in Fig. 25. With
these parameters for each network defined, we can now perform computations for a composite network, and find out that merely by changing the effective interaction parameter and
evolution parameter β (here also taken as energy available to facilitate chemical reactions) of
each network, the strain softening behaviour could be fairly well reproduced qualitatively, see
Fig. 26. While the predictions are far from perfect, we find it interesting that the choice of
evolution laws allows the network to strain soften in a manner similar to in the experiments,
as opposed to a critical failure of the load-bearing capacity. Indeed, this type of behaviour
has also been observed (Gardel et al, 2004; Janmey et al, 1994), and can be mimicked by
appropriate choice of parameters.
By extending the model in Paper B to a more general framework and introducing thermodynamically consistent evolution laws for viscous and damage variables, we have developed a
phenomenological model that can predict dissipative processes such as relaxation and damage
evolution of composite biopolymer networks.
Experimental and computational assessment of F-actin influence in regulating cellular stiffness and relaxation behaviour of
fibroblasts
As vimentin and actin are two of the main components which provide stiffness to the cell, we
performed colloidal-probe AFM to assess the predictive capability of the constitutive model
and simultaneously investigate the influence of F-actin on regulating the stiffness and relaxation behaviour of fibroblast cells. To do this, experiments were performed on BjhTERT
fibroblast cells incubated with Latrunculin B (LatB), which results in depolymerisation of
44
15
— P 12
o Wagner (2009)
P12 [Pa]
10
5
0
0.0
0.2
0.4 0.6
12
0.8
1.0
Figure 26: First Piola-Kirchhoff stress P12 for shear of composite actin and intermediate filament network,
model predictions and experiments (Wagner et al, 2009).
1
2
Figure 27: Illustration of the analysis of cell stiffness using colloidal probe force-mode AFM, indicating the
position of analysis over the cell nucleus or periphery, 1 and 2, respectively.
F-actin, or DMSO control.
The AFM cantilever was pressed into the cell surface with an indentation speed of 1μm/s
until the measured force reached a pre-determined level of approximately 5nN, and the height
was then kept constant for 30s. In this regime, no active response of fibroblast cells is to be
expected, and we can thus expect to quantify only the passive cellular response (Thoumine
and Ott, 1997). This indentation δ was performed over the nucleus, and in the peripheral
region of the cell, though far enough from the edge to expect influence from the substrate,
shown schematically in Fig. 27. For three control cells, the measured indentation (until the
pre-determined magnitude was reached) force finit and subsequent relaxation force fr is shown
in Fig. 28 and 29, respectively.
Apparently, measuring the response to AFM indentation
over the peripheral region yields a stiffer response than the nuclear region, in line with previous investigations (Sato et al, 2000).
After incubation with LatB, ∼40% of the actin cytoskeleton can be expected to depolymerise
(Gronewold et al, 1999; Gibbon et al, 1999), and we thus expect a significant effect on the
measured mechanical response. The measured force curves analogous to the above are shown
45
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
6
6
6
b)
a)
c)
4
4
4
fl
[nN]
fl
[nN]
fl
[nN]
2
2
2
— Nucleus
— Cytoplasm
— Nucleus
— Cytoplasm
0
— Nucleus
— Cytoplasm
0
0.0
0.25 0.50 0.75 1.00 1.25
0
0.0
0.25 0.50 0.75 1.00 1.25
Time [s], δ [μm]
0.0
0.25 0.50 0.75 1.00 1.25
Time [s], δ [μm]
Time [s], δ [μm]
Figure 28: Measured force indentation curves for colloidal-probe AFM of three DMSO control cells a)-c).
6
6
a)
6
b)
— Nucleus
— Cytoplasm
c)
— Nucleus
— Cytoplasm
4
4
4
fr
[nN]
fr
[nN]
fr
[nN]
2
2
2
0
0
0.0
5
10
15
Time [s]
20
25
30
— Nucleus
— Cytoplasm
0
0.0
5
10
15
Time [s]
20
25
30
0.0
5
10
15
20
25
30
Time [s]
Figure 29: Measured force relaxation curves for colloidal-probe AFM of three DMSO control cells a)-c.
46
6
6
6
b)
a)
c)
4
4
4
fl
[nN]
fl
[nN]
fl
[nN]
2
2
2
— Nucleus
— Cytoplasm
— Nucleus
— Cytoplasm
0
— Nucleus
— Cytoplasm
0
0.0
5
10
0
0.0
5
Time [s], δ [μm]
10
0.0
5
Time [s], δ [μm]
10
Time [s], δ [μm]
Figure 30: Measured force indentation curves for colloidal-probe AFM of three LatB-incubated control cells
a)-c).
6
6
a)
6
b)
— Nucleus
— Cytoplasm
c)
— Nucleus
— Cytoplasm
4
4
4
fr
[nN]
fr
[nN]
fr
[nN]
2
2
2
0
0
0.0
5
10
15
20
25
30
— Nucleus
— Cytoplasm
0
0.0
5
Time [s]
10
15
Time [s]
20
25
30
0.0
5
10
15
20
25
30
Time [s]
Figure 31: Measured force relaxation curves for colloidal-probe AFM of three LatB-incubated control cells
a)-c).
in Fig. 30 and 31.
When incubated with LatB, we observed that some cells appeared to
lose their peripheral adhesions and round up, and is the reason for the lack of a peripheral
measurement of the second cell. We performed additional measurements and found similar
behaviour to that exhibited by the first cell, and we therefore choose that as representative
for our future computational predictions.
A numerical model in the finite element software ABAQUS was created, and the material
model developed in the previous paper was implemented in the subroutine VUMAT for use
in ABAQUS/Explicit solver. The geometry is defined as characteristic from previous observations of these cells (Rönnlund et al, 2013). An axisymmetric model is used for indentation
over the nucleus, and a doubly symmetric solid model is used for indentation over the peripheral region, see Fig. 32. The developed material model is used for the cytoplasm with the
47
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
a)
b)
Figure 32: Meshed model assemblies with rigid indenter for indentation over a) nucleus, and b) cytoplasm.
6
6
a)
— Computation
— Experiment
b)
4
4
f
[nN]
f
[nN]
2
2
0
0
0.0
5
10
15
20
25
30
0.0
Time [s]
— Computation
— Experiment
5
10
15
20
25
30
Time [s]
Figure 33: Predicted force curves for indentation and relaxation as compared to experimental result for
DMSO control cell over a) nucleus, and b) cytoplasm.
simplification of neglecting the damage parameter, and shear modulus found in the litterature
(Caille et al, 2002) is used for the nucleus, considered as a neo-Hookean hyperelastic material.
Interestingly, computations with the material model and identical parameter values as in
Paper D, with the exception of scaling the stiffness parameters to a higher magnitude, (presumably to account for the cell cortex and cell adhesion) fit the experimental results for DMSO
control cells well, see Fig. 33. By merely scaling the value of the stiffness parameters (to
account for depolymerisation of the actin cytoskeleton), predictions in line with experimental
results for LatB-incubated cells are also promising, Fig. 34. That is, the two distinct regimes
of relaxation can be predicted by utilising the developed material model with time constants
representative of actin and intermediate filament networks, respectively.
We also show that the stiffer peripheral region need not be a result of a small thickness, as
proposed previously (Caille et al, 2002). Our type of cells tend to be 3-4μm thick in this
region, and any influence from the substrate would probably be more evident in the force
48
6
6
— Computation
— Experiment
a)
b)
4
4
f
[nN]
f
[nN]
2
2
0
— Computation
— Experiment
0
0.0
5
10
15
20
25
30
0.0
Time [s]
5
10
15
20
25
30
Time [s]
Figure 34: Predicted force curves for indentation and relaxation as compared to experimental result for
LatB-incubated cell over a) nucleus, and b) cytoplasm..
a)
b)
RP
RP
Figure 35: Meshed elliptical model assemblies for a) axisymmetric and b) symmetrical quarter model.
indentation curve. Instead, we find from defining a more elliptical cell model, Fig. 35, that
the measured indentation force in the peripheral region is smaller, and conversely larger over
the nucleus, Fig. 36. We thus attribute a stiffer peripheral region mainly to geometrical
effects. We also find that the rather strict assumption of incompressibility, often used in
computational analyses, provides a significant contribution, the magnitude of which depends
on cell shape, to the computed stress (and force).
Further, for our model cell we also find the influence of the cell nucleus to be minor. However,
its size or proximity to the cell surface is an important factor in general, and we deduce that
to ensure that nuclear properties are correctly measured by use of AFM, combined visualisation and measurement methods must be used to ensure this. An example of this is combined
AFM-confocal microscopy (Krause et al, 2013).
In this paper, we were able to predict the mechanical response and relaxation behaviour by
using the developed constitutive model for biopolymer networks, with merely scaling the stiff49
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
250
250
a)
b)
— Nucleus
— Cytoplasm
200
— Nucleus
— Cytoplasm
200
150
150
f
[nN]
f
[nN]
100
100
50
50
0
0
0.0
5
10
15
20
25
30
0.0
Time [s]
5
10
15
20
25
30
Time [s]
Figure 36: Predicted force curves for indentation and relaxation for a) elliptic and b) original model geometries.
ness coefficients to account for other cellular constituents such as the cell cortex. Not only is
this promising for the formulation for the underlying mechanical model, but also an indication
of well-defined evolution laws for the viscous response. There are a number of ways by which
the model can be improved in terms of pure network modelling, e.g. reducing the number
of phenomenological parameters, implementing active filament dynamics, taking into account
the prestress of the network and formulating a damage evolution law in terms of chemical
parameters. However, given the complexity of a cellular context as opposed to mere network
modelling, the effective approach used to take into account the multitude of responses that
may occur provides a conventient tool with which to study cell mechanics.
Implementing cell contractility in filament-based cytoskeletal
models
All the models presented so far focus on capturing the passive response of cytoskeletal networks and relating it to cellular mechanics. However, it is well known that cells also actively
generate force by using their contractile machinery (Pardee, 2010). Depending on the cell of
interest, this contractile behaviour may manifest itself over different time-scales. For example,
while fibroblasts generate force only after several minutes (Thoumine and Ott, 1997), muscle
cells that require an immediate response exhibit contractile behaviour much faster (Gunst
and Fredberg, 2003). In general, activation of cell contraction may result from mechanical or
biochemical signals, triggering a series of reactions within the cell. Typically, stretch-sensitive
integrins trigger mechanosensitive ion-channels in the membrane, increasing the intracellular
concentration of Ca2+ , leading to activation of contractile actin filaments (Gunst and Fred50
berg, 2003; Murtada et al, 2010).
The cytoskeletal filament network model developed in Paper D is merely an attempt to capture
the passive response of cytoskeletal networks. It was extended to encompass the contractile
response of the actin cytoskeleton by making use of prior work on smooth muscle cell activation (Murtada et al, 2010).
By making use of seven rate constants k1 to k1 , a four-state model as taken by Hai and
Murphy (Hai and Murphy, 1988) enabled the computation of inactive and active contractile
units, with nc and nd being the latter.
The strain energy function for one contractile unit was defined as
Ψc =
μc
(nc + nd )(λ̂ + ūrs − 1)2 ,
2
(21)
where μc is a stiffness parameter and ūrs is relative sliding of cross-linked actin filaments, i.e.
contraction, governed by a thermodynamically consistent evolution law. The stretch λ̂ is the
isochoric stretch, as defined previously.
The macroscopic contractile stresses Sc were computed in the same manner as the filament
model, i.e. by integration over the unit sphere as
m ∂Ψc
=2
wj 2μc (nc + nd (λ̂+
Sc = 2
∂C
j=1
∂ λ̂
,
+ ūrs − 1)
∂C
(22)
with notation as before.
The modified strain energy function was defined as
Ψ = Ψp + Ψc ,
(23)
where Ψp and Ψc denote the passive and active response, respectively. Thus, the total stress
is a sum of the passive and active parts. Stress relaxation computations were performed in
MATLAB with the composite network parameters defined in Paper D. A shear strain of 50%
was applied and then held constant, computing the stress in terms of first Piola-Kirchhoff
stress P 12 . Stress relaxation results with varying contractile parameters, and different values
of k1 are shown in Fig. 37. The model can clearly exhibit active force generation of different
magnitudes, over different time-scales. This is desirable, as the contractile behaviour can
be expected to vary between cell types. The model was further tested in a finite element
framework by extending the VUMAT in Paper E, for use in ABAQUS/Explicit. Predicted
force f from indentation and relaxation of the cell is shown in Fig. 38. Experimental evidence
indicates a mechanism that provides cells with an ability to enhance their stiffness when in a
stressed state (Wang and Ingber, 1994; Wang et al, 2002; Pourati et al, 1998). To assess the
51
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
a)
b)
102
103
—
—
—
—
—
101
10−2
P12 [Pa]
P12 [Pa]
103
Set 1
Set 2
Set 3
Set 4
Passive
10−1
100
101
102
102
—
—
—
—
101
10−2
k1
k1
k1
k1
=
=
=
=
0.001964
0.019640
0.196400
1.964000
10−1
100
Time [s]
101
102
Time [s]
Figure 37: Predicted stress relaxation for a) various contractile parameters, b) different chemical rate constant k1 .
200
— Passive
— μc = 0.0375 MPa
— μc = 0.1000 MPa
— μc = 0.3000 MPa
150
f [nN]
100
50
0
0.0
10
t [s]
20
30
Figure 38: Predicted force relaxation for indentation of the cell.
52
20
o - Passive
× - μc = 0.0375 MPa
- μc = 0.1 MPa
Seff [mN/m]
15
10
×
×
×
5
10
5
0
0.0
[%]
Figure 39: Effective cell stiffness for increasing levels of strain and contractile parameter.
model predictions in this aspect, the finite element model was complemented with an elastic
substrate which was stretched before the cell was subjected to indentation. An effective stiffness Seff , defined as the ratio of maximum force and indentation depth for increasing levels of
substrate strains and contractile paramater μc are shown in Fig. 39. The model is thus able
to predict this important feature of strain-dependent cell stiffening by incorporating active
force generation.
It has been experimentally observed that the stiffness in a stretched state is also timedependent (Mizutani et al, 2004). Qualitatively, the model predicts this, where the observed
plateau arises from a balance of evolution of internal viscous variables and active force generation, see Fig. 40.
A final model assessment showed that the cell stiffening due to contractile machinery alone,
according to a prestressed actin cytoskeleton envisioned in the tensegrity model, can be reproduced. In a previous investigation, adherent cells were treated with contractile (histamine)
and relaxing (isopropenol) agents, and a proportional relation between prestress (due to
cytoskeletal contraction) and stiffness (Wang et al, 2002). By linearly increasing μc , a proportional increase in stiffness was evident in the finite element model as well, see Fig. 41.
In conclusion, the model shows promise not only in its ability to actively generate force over
different time-scales, but in its simplicity. Any constitutive model based on one-dimensional
filament units may be extended to encompass also contractile behaviour with this approach.
However, it should be mentioned that the current lack of data, primarily in terms of chemical
parameters for different cells, renders a robust assessment of the predictive model capability
a task for the future.
53
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
×
×
×
×
×
×
×
Seff [mN/m]
×
11
7
0
25
50
75
t [s]
Figure 40: Computation of effective stiffness Seff over time. The active part of the model takes precedence
for increasing μc , and the force-generating effect thereby supersedes that of relaxation. Model
representations as follows: ◦ - passive, - μc = 0.01 MPa, × - μc = 0.0375 MPa.
60
Seff [mN/m]
45
30
15
0
0.0
0.5
1.0
Prestress [Pa]
Figure 41: Proportional cell stiffening due to increasing μc , mimicking stimulation of contractile activity in
the cell.
54
Conclusions
In conclusion, a variety of methods have been utilised to elucidate the role of the cytoskeleton
in cell mechanics. By developing a chemo-mechanical constitutive model for actin networks,
a framework is provided in which the initial rapid relaxation experimentally observed is taken
to be merely due to cross-link debonding. The subsequent slower relaxation is viscous deformation due to mechanisms such as filament sliding. This model proved reliable in predicting
the initial response of cross-linked actin networks, and the reasoning provided the basis of extending the idea to discrete network models. By using the finite element model, the influence
of cytoskeleton architecture and cross-link compliance was extensively studied. The reduced
stiffness and shift of strain hardening regime is due to cross-link mediated reorientation of
filaments, a previously determined effect which is now shown to depend on the characteristics
of the binding protein. Further, by revisiting the chemo-mechanical model, it was hypothesised that the strain rate dependence of actin networks is much due to stochastic debonding
of cross-links. Verifying this was partly successful, by comparing computational results with
experiments, although some discrepancies between our formulation of mechanically induced
cross-link debonding and observations indicate a more complex mechanism at longer time
scales. Developing a microstructural model of cross-linked filament networks proved successful in qualitatively reproducing the interplay between cross-link characteristics and filament
length. This encouraged us to further generalise the model and extend it to a composite formulation taking into account dissipative mechanisms which we regard as damage and viscous
effects. Although an unambiguous physical meaning of these remain elusive, they are quite
successful in predicting the mechanical response of both singular and composite networks,
in an effective sense. We are able to replicate the mechanical response of fibroblast cells in
AFM experiments by a finite element numerical model with this composite formulation. The
results indicate a possibility that the relaxation behaviour for fibroblasts over two time scales
are governed by the mechanical response of actin and intermediate filament networks. We
further quantify the influence of cell shape on the measured force by AFM. Finally, the passive response of the filament-based cytoskeletal model is extended to encompass contractile
behaviour over different time-scales, by successfully adopting a previously suggested strain
energy function for contractile actin units.
55
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
56
Summary of appended papers
Paper A: Mechanical behaviour of transiently cross-linked actin networks and a theoretical
assessment of their viscoelastic behaviour
Biological materials can undergo large deformations and also show viscoelastic behaviour.
One such material is the network of actin filaments found in biological cells, giving the cell
much of its mechanical stiffness. A theory for predicting the relaxation behaviour of actin
networks cross-linked with the cross-linker α-actinin is proposed. The constitutive model is
based on a continuum approach involving a neo-Hookean material model, modified in terms
of concentration of chemically activated cross-links. The chemical model builds on work done
by Spiros (Doctoral thesis, University of British Columbia, Vancouver, Canada, 1998) and has
been modified to respond to mechanical stress experienced by the network. The deformation
is split into a viscous and elastic part, and a thermodynamically motivated rate equation is
assigned for the evolution of viscous deformation. The model predictions were evaluated for
stress relaxation tests at different levels of strain and found to be in good agreement with
experimental results for actin networks cross-linked with α-actinin.
Paper B: Network modelling of cross-linked actin networks - Influence of network parameters
and cross-link compliance
A major structural component of the cell is the actin cytoskeleton, in which actin subunits
are polymerised into actin filaments. These networks can be cross-linked by various types
of ABPs (Actin Binding Proteins), such as Filamin A. In this paper, the passive response
of cross-linked actin filament networks is evaluated, by use of a numerical and continuum
network model. For the numerical model, the influence of filament length, statistical dispersion, cross-link compliance (including that representative of Filamin A) and boundary
conditions on the mechanical response is evaluated and compared to experimental results. It
is found that the introduction of statistical dispersion of filament lengths has a significant
influence on the computed results, reducing the network stiffness by several orders of magnitude. Actin networks have previously been shown to have a characteristic transition from an
initial bending-dominated to a stretching-dominated regime at larger strains, and the crosslink compliance is shown to shift this transition. The continuum network model, a modified
57
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
eight-chain polymer model, is evaluated and shown to predict experimental results reasonably well, although a single set of parameters can not be found to predict the characteristic
dependence of filament length for different types of cross-links. Given the vast diversity of
cross-linking proteins, the dependence of mechanical response on cross-link compliance signifies the importance of incorporating it properly in models to understand the roles of different
types of actin networks and their respective tasks in the cell.
Paper C: Cross-link debonding in actin networks - influence on mechanical properties
The actin cytoskeleton is essential for the continued function and survival of the cell. A
peculiar mechanical characteristic of actin networks is their remodelling ability, providing
them with a time-dependent response to mechanical forces. In cross-linked actin networks,
this behaviour is typically tuned by the binding affinity of the cross-link. We propose that
the debonding of a cross-link between filaments can be modelled using a stochastic approach,
in which the activation energy for a bond is modified by a term to account for mechanical
strain energy. By use of a finite element model, we perform numerical analyses in which we
first compare the model behaviour to experimental results. The computed and experimental
results are in good agreement for short time scales, but over longer time scales the stress is
overestimated. However, it does provide a possible explanation for experimentally observed
strain-rate dependence as well as strain-softening at longer time scales.
Paper D: Constitutive modelling of composite biopolymer networks
The mechanical behaviour of biopolymer networks is to a large extent determined at a microstructural level where the characteristics of individual filaments and the interactions between them determine the response at a macroscopic level. Phenomena such as viscoelasticity
and strain-hardening followed by strain-softening are observed experimentally in these networks, often due to microstructural changes (such as filament sliding, rupture and cross-link
debonding). Further, composite structures can also be formed with vastly different mechanical properties as compared to the indivudal networks. In this present paper, we present
a constitutive model presented in a continuum framework aimed at capturing these effects.
Special care is taken to formulate thermodynamically consistent evolution laws for dissipative effects. This model, incorporating possible anisotropic network properties, is based on a
strain energy function, split into an isochoric and a volumetric part. Generalisation to three
dimensions is performed by numerical integration over the unit sphere. Model predictions indicate that the constitutive model is well able to predict the elastic and viscoelastic response
of biological networks, and to an extent also composite structures.
Paper E: Experimental and computational assessment of F-actin influence in regulating cellular stiffness and relaxation behaviour of fibroblasts
In biomechanics, a complete understanding of the structures and mechanisms that regulate
cellular stiffness at a molecular level remain elusive. In this paper, we have elucidated the role
58
of filamentous actin (F-actin) in regulating elastic and viscous properties of the cytoplasm
and the nucleus. Specifically, we performed colloidal-probe atomic force microscopy (AFM)
on BjhTERT fibroblast cells incubated with Latrunculin B (LatB), which results in depolymerisation of F-actin, or DMSO control. We found that the treatment with LatB not only
reduced cellular stiffness, but also greatly increased the relaxation rate for the cytoplasm in
the peripheral region and in the vicinity of the nucleus. We thus conclude that F-actin is a
major determinant in not only providing elastic stiffness to the cell, but also in regulating
its viscous behaviour. To further investigate the interdependence of different cytoskeletal
networks and cell shape, we provided a computational model in a finite element framework.
The computational model is based on a split strain energy function of separate cellular constituents, here assumed to be cytoskeletal components, for which a composite strain energy
function was defined. We found a significant influence of cell geometry on the predicted mechanical response. Importantly, the relaxation behaviour of the cell can be characterised by a
material model with two time constants that have previously been found to predict mechanical
behaviour of actin and intermediate filament networks. By merely tuning two effective stiffness parameters, the model predicts experimental results in cells with a partly depolymerised
actin cytoskeleton as well as in untreated control. This indicates that actin and intermediate
filament networks are instrumental in providing elastic stiffness in response to applied forces,
as well as governing the relaxation behaviour over shorter and longer time-scales, respectively.
Paper F: Implementing cell contractility in filament-based cytoskeletal models
Cells are known to respond over time to mechanical stimuli, even actively generating force
at longer times. In this paper, a microstructural filament-based cytoskeletal network model
was extended to incorporate this active response, and a computational study to assess the
influence on relaxation behaviour was performed. The incorporation of an active response
was achieved by including a strain energy function of contractile activity from the cross-linked
actin filaments. A four-state chemical model and strain energy function was adopted, and
generalisation to three dimensions and the macroscopic deformation field was performed by
integration over the unit sphere. Computational results in MATLAB and ABAQUS/Explicit
indicated an active cellular response over various time-scales, dependent not only on contractile parameters, but also chemical constants. Important features such as force generation and
increasing cell stiffness due to prestress are qualitatively predicted. The work in this paper
can easily be extended to encompass other filament-based cytoskeletal models as well.
59
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
60
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On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
66
Errata
Paper A: Mechanical behaviour of transiently cross-linked actin networks and a theoretical
assessment of their viscoelastic behaviour
Eqn. 13, pp. 376, should be
ṅa = kd nc − ka na nb .
67
On the mechanics of actin and intermediate filament networks and their contribution to
cellular mechanics
68

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