Mechanical response of cross-linked actin networks
Bj¨orn Fallqvist
Licentiate thesis no. 120, 2013 KTH School of Engineering Sciences
Department of Solid Mechanics Royal Institute of Technology SE-100 44 Stockholm Sweden
TRITA HFL-0549 ISSN 1104-6813
ISRN KTH/HFL/R-13/19-SE ISBN 978-91-7501-875-1
Akademisk avhandling som med tillst˚and av Kungliga Tekniska H¨ogskolan i Stockholm framl¨agges till offentlig granskning f¨or avl¨aggande av teknisk licentiatexamen fredagen den 25 oktober, Kungliga Tekniska H¨ogskolan, Teknikringen 8, Stockholm.
Abstract
The ability to predict the mechanical properties of cells should be seen in the light of the close connection between abnormal cell states and a change in the cell response to stimuli. For example, it has been found that the stiffness of cancer cells is much lower than their healthy counterparts, influencing metastasis and cell migration. On the contrary, malaria cells have been found to exhibit a significant increase in stiffness.
The major structural entity of the cell is called the cytoskeleton, an interior network consisting of three types of protein filaments - actin filaments, intermediate filaments and microtubules.
The remodelling ability of the cytoskeleton through polymerisation provides the cell with the ability to adapt its response to external forces accordingly. The properties of interfilament cross-links in terms of stiffness and ability to detach can be expected to influence the mechan- ical response. The work presented herein focuses on the mechanical response of cross-linked actin networks. The results indicate a strong dependence of the mechanical properties on cross-link dynamics and characteristics.
In Paper A, a constitutive model for the response of transiently cross-linked networks is devel- oped using a continuum framework. The deformation is split into viscous (representing sliding of filaments) and elastic deformation. A strain energy function is proposed in the form of a neo-Hookean model, modified in terms of chemically activated cross-links. The disassociation rate constant is modified in terms of an exponential function taking into account the amount of strain energy available to break bonds. The constitutive model was compared with exper- imental 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 sliding of filaments.
In Paper B, a finite element framework was used to assess the influence of numerous geomet- rical and material parameters on the response of cross-linked actin networks. It was shown that considering the presence of a statistical dispersion in filament lengths has a significant effect on the mechanical properties of the network. Further, the compliance of the cross- links was shown to influence the stress-strain curve and shift the region of strain hardening.
The influence of boundary conditions and the effect of network parameters on experiments in terms of local and global effects were also addressed. Finally, a micromechanically moti- vated constitutive model in a continuum framework was presented, capturing some essential characteristic features of cross-linked actin networks.
Sammanfattning
Den n¨ara kopplingen mellan abnorma tillst˚and hos celler och dess respons p˚a extern stimuli medf¨or att m¨ojligheten f¨or prediktion av cellens mekaniska egenskaper ¨ar av stort intresse.
Som exempel kan n¨amnas att cancerceller ¨ar mer komplianta, medan malariaceller tv¨artom uppvisar en st¨orre styvhet j¨amf¨ort med sina respektive friska celler.
Den huvudsakliga intracellul¨ara strukturen som ger cellen dess mekaniska egenskaper ¨ar cy- toskelettet, ett n¨atverk best˚aende av tre typer av proteinfilament - aktinfilament, intermedi¨ara filament samt mikrotubuli. Den kontinuerliga polymeriseationen av filamenten inneb¨ar att cytoskelettet omstruktureras och cellens respons p˚a yttre krafter blir tidsberoende. Styvheten och tendens att sl¨appa hos bindningar mellan filament kan f¨orv¨antas p˚averka responsen hos det makroskopiska n¨atverket. Arbetet som presenteras i denna avhandling fokuserar p˚a aktinn¨atverks mekaniska egenskaper. Resultaten indikerar ett starkt beroende p˚a bind- ningsegenskaperna.
I Paper A presenteras en konstitutiv kontinuum-mekanisk modell f¨or responsen hos ak- tinn¨atverk med transienta bindningar. Deformationen delas upp i en visk¨os (representa- tiv f¨or glidande filament) och elastisk del. En t¨ojningsenergifunktion definieras formulerad som en neo-Hookean modell, men modifierad f¨or att ta h¨ansyn till antal aktiva bindningar.
Hastigheten med vilken bindningarna sl¨apper modifieras med en exponentialfunktion beroende p˚a m¨angden t¨ojningsenergi. J¨amf¨orelse med relaxationsexperiment visade att den initiella snabba relaxationen kan tillskrivas bindningar som sl¨apper, och den efterf¨oljande l˚angsammare relaxationen beror p˚a visk¨os deformation.
I Paper B anv¨ands finita element-metoden f¨or att best¨amma inverkan hos flertalet parame- trar relaterade till s˚av¨al geometri som material. Resultaten visade att en statistisk sprid- ning i filamentl¨angd drastiskt kan p˚averka de mekaniska egenskaperna hos n¨atverket. Vidare visades ¨aven att styvheten hos bindningarna hade en stark p˚averkan p˚a n¨atverkets sp¨annings- t¨ojningskurva och att regionen i vilken h˚ardnande intr¨affar kan f¨orflyttas genom att modi- fiera bindningsstyvheten. ¨aven inverkan av randvillkor och effekten av n¨atverksparametrar diskuterades i samband med lokala och globala n¨atverkseffekter. Slutligen presenterades en kontinuum-mekanisk modell som f˚angar vissa karakt¨aristiska drag hos aktinn¨atverk.
Preface
The work presented in this thesis has been performed between April 2011 and August 2013 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.
I would like to express my sincere gratitude to my supervisor, Associate Professor Martin Kroon, for his encouragement and help during the project during which I gained many valuable insights.
I am also immensely thankful to my family and friends who have always been a source of inspiration and encouragement. Especially to my father, who led me down this road in the first place, and my mother and sisters who have been encouraging me from the start and being ever-present.
Stockholm, September 2013
List of appended papers
Paper A: Mechanical behaviour of transiently cross-linked actin networks and a theoretical assessment of their viscoelastic behaviour
Bj¨orn Fallqvist and Martin Kroon
Biomechanics and Modeling in Mechanobiology, 2013, 373–382
Paper B: Network modelling of cross-linked actin networks - Influence of network parameters and cross-link compliance
Bj¨orn Fallqvist and Artem Kulachenko and Martin Kroon
Report 548, Department of Solid Mechanics, KTH Engineering Sciences, Royal Institute of Technology, Stockholm, Sweden
Submitted for publication
In addition to the appended papers, the work has resulted in the following publications and presentations1:
Transiently cross-linked actin networks Bj¨orn Fallqvist and Martin Kroon
Presented at World Congress on Computational Mechanics, Sao Paolo 2012 (P) Transiently cross-linked actin networks
Bj¨orn Fallqvist and Martin Kroon
Presented at Svenska Mekanikdagar, Lund 2013 (P)
1Ea = Extended abstract, P = Presentation, Pp = Proceeding paper
Contents
Introduction 11
The actin cytoskeleton . . . 11
Network characteristics . . . 12
Continuum models for cross-linked actin networks . . . 13
A constitutive model for transient networks . . . 13
A micromechanically motivated continuum model . . . 15
Numerical model - Finite element analysis of a two-dimensional cross-linked actin network . . . 15
Conclusions 19
Bibliography 19
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Mechanical response of cross-linked actin networks
10
Introduction
Passive biological tissues are very different from classical engineering materials - often anisotropic and able to withstand large deformations before rupture, they prove themselves a challenge to model. In the context of cell mechanics, the situation is further complicated by the ability of the cell to respond to mechanical and biochemical signals - the cell is a dynamic structure with chemical and mechanical properties that cannot be characterised by a fixed quantity. The tightly integrated system between biological functions and mechanical properties means that structural changes in the cell might result from foreign organisms or biochemical factors, and these structural changes in turn alters the ability of the cell to function normally. A striking example can be seen in epithelial cells treated with spingosylphosphorylcholine (SPC), which influences cancer metastasis. These cancer cells exhibit a significant reduction in stiffness and an increase in dissipated energy when subjected to stretching (Suresh et al, 2004). This increased deformability implies an increased risk of metastasis, as the cell is more probable to migrate. Due to the inadequacies of cell extraction methods in terms of patient safety and quantifying cell invasiveness (R¨onnlund et al, 2013), an attractive alternative to identify cancer cells is characterising the mechanical behaviour theoretically in a way that allows for comparisons with small cell samples. A good starting point is to look at the load-bearing microstructures inside the cell.
The actin cytoskeleton
A major intracellular structure is the cytoskeleton, a network of polymerised protein fila- ments. This filamentous network functions as both a passive mechanical structure and a way of providing an active response to external stimuli, as well as the basis of normal biological functions such as cell migration.
There are three major types of protein filaments in the cytoskeleton - actin filaments, inter- mediate filaments and microtubules. Each of these filaments is associated with a specific type of protein that may be added or removed from a filament. This continuous remodelling is the determining factor in the ability to respond to signals and partake in cell functions on different time scales.
Actin in particular has numerous functions within the context of cell mechanics and the cy- 11
Mechanical response of cross-linked actin networks
toskeleton. During cell migration, actin in the shape of spike-like structures (filopodia) are formed in the leading edge, pushing the lamellipodum forward. Meanwhile, tight bundles of actin filaments - stress fibres - are formed as anchoring points between the ExtraCellular Matrix (ECM) and the cell membrane, pulling the cell forward. Actin can also be found as a thin mesh layer - the actin cortex - underneath the plasma membrane, maintaining cell shape.
There is also a distributed actin lattice throughout the cell, presumably providing some of the cell stiffness (Kamm and Mofrad, 2006). It is clear that in understanding cell mechanical properties, it is essential to have a good representation of the subcellular structures such as the actin cytoskeleton, which is the focus of the work presented herein.
Network characteristics
There are many factors governing the microstructural shape of the actin network. In some instances, a network property is related to a certain protein, such as the reduction of statistical dispersion by the presence ofα-actinin (Biron and Moses, 2004) or the reduction of filament length by capping proteins such as gelsolin and CapZ (Xu et al, 1999; Yin et al, 1980). The network is not necessarily a solution of entangled filaments, but cross-links (ABP - Actin Binding Proteins) can create interfilament bonds. The characterisation of the macroscopic network response then needs to include contribution from these bonds.
There are many types of ABPs, differing in mechanical properties, structure and function within the cytoskeleton. From observations, it is clear that the architecture of cross-linked actin network is dependent on ABP type and cross-link concentration (Lieleg et al, 2009). For increasing concentrations of cross-links, the reconstituted network can appear as an entangled solution, an istropically cross-linked network, a composite network with bundles or a network of bundle clusters, see Fig. 1.
Entangled Cross-linked Composite Bundled
Increasing relative cross-linker concentration
Figure 1: Different types of actin network structures. For an increasing relative concentration of cross-linker (red dots), the structure goes from an entangled or isotropically cross-linked network to a composite network, and finally a network of bundles.
12
In general, an isotropically cross-linked network behaves much like a solid where the shear modulus is determined by bending and sliding of the filaments. Cross-linked bundles are stiffer than the filaments themselves, but are also able to slide between each other, causing a more fluid-like behaviour (Wachsstock et al, 1993, 1994). For the work presented herein, the network was assumed to be one of cross-linked filaments, which is in accordance with experimental results for relative ABP concentration of referred litterature (Wachsstock et al, 1993, 1994; Xu et al, 2000).
The ABP α-actinin is a rod-like protein which tightly bundles the actin filaments of stress fibres and is also found at adhesion sites where transmembrane proteins link to the ECM (Stphanou, 2006; Kamm and Mofrad, 2006). It also functions as a cross-linker for cortical actin, just beneath the cell membrane. Filamin A is yet another ABP, which is prominent in the distributed actin lattice throughout the cell cytoplasm (Kamm and Mofrad, 2006).
Its hinge-like structure promotes the formation of three-dimensional gel-like actin networks (Kamm and Mofrad, 2006). The hinge-like shape gives Filamin A a characteristically com- pliant mechanical response due to unfolding of the protein (Furuike et al, 2001).
It is the aim of this thesis to present two distinct approaches to modelling cross-linked actin networks. Experimental results of actin networks cross-linked withα-actinin and Filamin A (Xu et al, 1998, 2000; Kasza et al, 2010; Wachsstock et al, 1993, 1994) were used to validate model predictions and constituted the basis of discussion.
Continuum models for cross-linked actin networks
An attractive approach thanks to its straight-forwardness is to view the biological tissue, or in this case network, as a continuum. With basic continuum mechanical principles, the pre- dicted stresses are easily computed, although the model may vary in complexity. However, these models require finding appropriate parameters and subsequent validation against ex- periments. Two possible constitutive models in the context of cytoskeletal mechanics are here presented. The first is a phenomenological model developed to consider the transient nature of interfilament cross-link and filament sliding in stress relaxation behaviour. The second is a microstructural model with the aim to assess the influence of cross-link characteristics and geometrical parameters.
A constitutive model for transient networks
For the development of an appropriate continuum model for transiently cross-linked actin networks, two key considerations must be taken into account. First, a computational approach considering the cross-link dynamics of the network is needed. Secondly, the sliding of filaments in terms of viscous deformation must be considered.
In this specific case ofα-actinin, the binding kinetics are taken into account by utilising the 13
Mechanical response of cross-linked actin networks
four-structure model presented by Spiros (Spiros, 1998) and assuming that only a doubly bound filament contributes to the load-bearing part of the network. The rate at which α- actinin molecules detach cannot be considered to be independent from the forces applied however, and the disassociation rate constant for the chemical rate equations is modified by an exponential factor. The physical reasoning for this exponential factor is explained by considering a bond trapped in an energy well, see Fig. 2.
Eb Eb-βψ
βψ Unloaded
bound state Loaded bound state
Figure 2: Bound state for cross-linked molecules. To escape the potential well in the loaded state, the energy required to break the bond isEb-βψ.
In accordance with Arrhenius’ law, the disassociation rate constant is basically the proba- bility of a bond escaping from an energy well of depth Eb modified by a prefactor. Assuming that the fraction β of strain energy ψ due to mechanical loading is also available to break bonds, the depth is reduced to Eb - βψ, equivalent to modifying the disassociation rate con- stant with an exponential factor.
To account for viscous sliding of filaments, the deformation gradient is split into a viscous and elastic part. A strain energy function is then proposed which depends only on the elastic deformation, in terms of the first invariant of the elastic right Cauchy-Green tensor. A ther- modynamically consistent evolution law is proposed for the evolution of viscous deformation.
Model predictions for the initial shear relaxation of a two-dimensional actin network com- pared to experiments (Xu et al, 2000) prove a fairly good fit for a single set of parameters, see Fig. 3, for stress relaxation curves of different applied shear γ. The model suggests a mech- anism by which the network experiences initial relaxation due to debonding events, followed by relaxation at longer time scales due to viscous deformation, i.e. sliding of filaments owing to thermal motion and applied deformation.
14
10−1 100 101
10−2 10−1 100
P12[Pa]
Time [s]
γ18
γ30
10−2 10−1 100
Time [s]
γ40 γ50
Figure 3: Comparison of model predictions and test results.
A micromechanically motivated continuum model
While the constitutive model presented in the previous section is theoretically sound, it is a phenomenological model lacking the sophistication required to accurately model the cytoskele- ton in terms of geometrical and material parameters. In order to assess this inadequacy, a simple micromechanically motivated continuum model was developed, following the method presented by Arruda & Boyce (Arruda and Boyce, 1993), but modified to account for bending of actin filaments and stretching of the cross-link. The initial shear modulus predicted by this continuum model for different filament lengths is presented in Fig. 4. Unfortunately, the model is too simple to accurately predict all sets of experiments for a single set of parameters.
However, the tendency of the cross-linked network to approach a plateau for increasing fila- ment lengths and stiffer cross-links agrees well with experimental observations (Kasza et al, 2010), as well as the more exponential behaviour in the case of compliant cross-links.
Numerical model
A discrete network model has advantages over a continuum model thanks to greater degree of network customisation and the possibility to directly assess the influence of geometrical and material parameters on the network response. A two-dimensional numerical model in a finite element framework was used to assess the influence of geometrical and material properties on the macroscopic network response. A two-dimensional network consisting of beam elements
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Mechanical response of cross-linked actin networks
0 0.5 1.0 1.5 2.0
1.0 3 5 7 9 11 13 15
G0 [Pa]
L [μm]
Figure 4: Shear modulus for different filament lengths and cross-link stiffness. Triangles (rigid) and circles (filamin A) are experimental results from Kasza et al (Kasza et al, 2010), shown together with predicted model results for one set of parameters (solid lines). Red, blue and green are weak, intermediate and large cross-link stiffness,respectively.
were generated, with unidirectional springs created at filament intersections. With previous investigations assuming rigid cross-links, (Huisman et al, 2007; van Dillen et al, 2008) these springs were used to reproduce cross-links with varying degrees of stiffness. Numerous com- putations for a sheared network were performed with the aim of clarifying the influence of factors such as statistical dispersion of filament lengths, boundary conditions and cross-link compliance.
Periodic boundary conditions were used, but one issue to consider is whether constraining the top boundary is accurately reproducing physiological and/or experimental conditions.
The deformed configuration of the two cases is shown in Fig. 5. The influence of boundary conditions must be expected to have an impact on the mechanical response on the network,
Figure 5: Network in deformed shape at 12= 0.25. In Fig. a) the top boundary is constrained, whereas in Fig. b) it is not.
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0×103 25×103 50×103
0.0 0.1 0.2 0.3 0.4
τ [Pa]
12
0×102 25×102 50×102
0.0 0.1 0.2 0.3 0.4
τ [Pa]
12
Figure 6: Stress τ vs. shear strain 12 for networks with increasingly compliant cross-links. In Fig. a) the top boundary is constrained vertically, in Fig. b) the corresponding boundary nodes are coupled but able to move vertically.
especially when subjected to large deformations. The computational results indicate that while neither case may be a perfect representation of reality, the vertically unconstrained network is more prone to replicate certain network tendencies experienced during mechanical testing.
As shown in Fig. 6 for two sets of boundary conditions, the cross-link compliance has a sig- nificant effect on the stress-strain curve. An interesting find is the effect of network activation on predicted results. The initial stress response from computations does not reproduce exper- imental results, which results from the transition between local and global effects, depending on network geometry.
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Mechanical response of cross-linked actin networks
18
Conclusions
The importance of cross-link dynamics and their characteristics have been shown to influence the mechanical properties of cross-linked actin networks in a multitude of ways. It appears that the initial stress relaxation observed in experiments is accurately predicted by includ- ing cross-link debonding, and the subsequent viscous deformation is governed by sliding of filaments. However, at times longer than 1s (presented here) the stress predicted decreases too rapidly, so clearly further modification is needed for the model to be valid over a larger span of time scales. One may conjecture that an improved evolution law with an explicit dependence on the concentration of load-bearing filaments might improve the viability. It is reasonable to expect that as filaments form cross-links after the initial perturbation and associated debonding, filament sliding will be more difficult. The model predictions were performed specifically with the binding kinetics ofα-actinin in mind, but the concept should be valid also for other types of cross-links.
The micromechanically motivated continuum model is an attractive alternative to the phe- nomenological model first proposed. It is more microstructurally sound with relevant filament and cross-link properties defined accordingly, but is ultimately too simple. The inadequacy of the model lies in predicting several sets of experimental results for a single set of parame- ters. One possibility is that the length dependence introduced makes no consideration to the increase in network connectivity that will occur as the filament length increases.
The mechanical characteristics of the cross-links appear to have a major role in regulating the macroscopic network stiffness, as an increasingly compliant cross-link will shift the region of strain hardening to larger strains. This implicates a role of cross-links in regulating the cytoskeletal stiffness in different parts of the cell. The strong influence of statistical dispersion in filament lengths indicates that proteins regulating this dispersion, such asα-actinin (Biron and Moses, 2004), have dual functions as they cross-link the network on one hand, but also govern the network stiffness.
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