On the mechanics of actin and intermediate filament networks and their contribution to cellular mechanics

Bj¨orn 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 tillst˚and av Kungliga Tekniska H¨ogskolan i Stockholm framl¨agges till offentlig granskning f¨or avl¨aggande av teknologie doktorsexamen fredagen den 29 januari 2016, Kungliga Tekniska H¨ogskolan, 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 ex- perimental 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 geomet- rical 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 pro- posed. 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 pro- posed. Favourable comparisons to experiments indicate the interplay between phenomeno- logical 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 me- chanics 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 ad- ditional 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, exper- imentally observed stiffening of cells with applied stress is predicted. Sammanfattning

Cellers mekaniska beteende ¨ar av st¨orsta vikt f¨or deras fortsatta fysiologiska funktion, och ett defekt s˚adant beteende kan resultera i flertalet sjukdomar. Dessutom kan den mekaniska responsen hos celler vara en indikator p˚a ett sjukdomstillst˚and, och ¨aven underl¨atta sprid- ning av sjukdomen. I denna avhandling utvecklas och utv¨arderas metoder f¨or att modellera cellmekanik. I Paper A utvecklas en konstitutiv kontinuummekanisk modell f¨or responsen hos aktinn¨atverk med transienta bindningar. En t¨ojningsenergifunktion definieras, modifierad f¨or att ta h¨ansyn till aktiverade bindningar. J¨amf¨orelser med relaxationsexperiment visade att den initiella snabba relaxationen kan tillskrivas bindningar som sl¨apper, och den efterf¨oljande l˚angsammare relaxationen beror p˚avisk¨os deformation. IPaperBanv¨ands finita element-metoden f¨or att best¨amma inverkan hos flertalet parametrar relaterade till mikrostrukturens geometri, material och bindningsegenskaper. En kontinuum- mekanisk modell f¨or n¨atverk med bindningar presenteras ocks˚a. IPaperCut¨okas den diskreta modellen till att innefatta stokastiska bindningar. Det ex- perimentellt uppvisade beroendet p˚at¨ojningshastighet f¨oresl˚as delvis bero p˚a detta, men en mer komplex mekanisism existerar antagligen, d˚a skillnader uppvisas mellan ber¨aknade och experimentella resultat. I Paper D utvecklas den kontinuummekaniska modellen f¨or n¨atverk vidare f¨or att inklud- era mer allm¨anna mekanismer, och en t¨ojningsenergifunktion f¨or kompositn¨atverk f¨oresl˚as. J¨amf¨orelser med experiment visar att beroendet mellan evolutionslagarna f¨or interna variabler kan anv¨andas f¨or att ber¨akna experimentellt uppvisade effekter i n¨atverk av biopolymerer. Genom att anpassa modellen till experimentella data f¨or enstaka n¨atverkstyper kan dessa ¨aven anv¨andas f¨or kompositn¨atverk, med undantag f¨or tv˚a parametrar. IPaperEanv¨ands experimentella m¨attekniker f¨or att unders¨oka inverkan av aktinets cy- toskelett p˚a den mekaniska responsen hos fibroblaster. Den konstitutitiva modellen utveck- lad f¨or kompositn¨atverk av biopolymerer kan anv¨andas f¨or att ber¨akna indentations- och relaxationsbeteendet med samma materialparametrar som f¨or aktin och intermedi¨ara fila- mentn¨atverk. Inverkan hos cellgeometri best¨ams, och experimentella samt ber¨akningsm¨assiga aspekter p˚a cellmekanik diskuteras. IPaperFut¨okas den tidigare konstitutiva modellen till att ¨aven innefatta kontraktilt be- teende. Detta ˚astadkomms genom att inkludera en t¨ojningsenergifunktion f¨or den kontrak- tila responsen. Inverkan av kontraktila och kemiska parametrar utv¨arderas. Metoden kan ¨aven anv¨andas f¨or andra filament-baserade konstitutiva modeller. Experimentellt observerat cellh˚ardnande f¨or 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 Iamtoday.

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 Bj¨orn 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 Bj¨orn 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 Bj¨orn 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 Bj¨orn 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 Bj¨orn Fallqvist, Matthew Fielden, Torbj¨orn 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 Bj¨orn 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 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)

Mechanical response of cross-linked actin networks - Influence of geometry and cross-link compliance Bj¨orn Fallqvist, Artem Kulachenko and Martin Kroon Presented at World Congress on Computational Mechanics, Barcelona 2014 (P)

1Ea = 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 Theimportanceofcellmechanics...... 11 Thecellisacomplexheterogenousstructure...... 12

Experimental methods in cell mechanics 17 Passivemicrorheology...... 17 Cellpoker...... 17 MicropipetteAspiration...... 18 Microplates...... 19 MagneticTwistingCytometry...... 19 AtomicForceMicroscopy...... 20

Mechanical models of biological cells 21 Cellmodels...... 21 Liquiddrop...... 21 TheoryofSGR...... 22 Continuummodels...... 23 Tensegrity...... 23 Cytoskeletalnetworkcharacteristics...... 24 Actin...... 24 Intermediatefilaments...... 26 Microtubules ...... 27 Cytoskeletalmodels...... 27 Elasticityofsemiflexiblepolymernetworks...... 27 Discretenetworkmodels...... 28

Conclusions and contribution to the field 29

9 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 Acontinuummodelfortransientlycross-linkedactinnetworks...... 31 A discrete network model to assess the influence of geometrical parameters and cross-linkcompliance...... 35 A discrete network model to assess the influence of cross-link debonding on mechan- icalpropertiesofcross-linkedactinnetworks...... 40 A constitutive model for composite biopolymer networks - Irreversible effects in the largestrainregime...... 42 Experimental and computational assessment of F-actin influence in regulating cel- lularstiffnessandrelaxationbehaviouroffibroblasts...... 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 informa- tion 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 reproduc- tive 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 accomo- date 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 contin- ued 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; R¨onnlund et al, 2013; Suresh et al, 2004) in transfeced cells, and altered expression of nu- clear 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 mod- elling, 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 F- actin 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 F-actin b Intermediate filament Negative end (–) NCPolypeptide

Dimer Depolymerization NC

NCTetramer

C N

Polym erization Protofilament

Positive end (+)

7.9 nm Filament (D = 10 nm)

c Microtubule α-tubulin β-tubulin 14 nm 25 nm

a

Figure 2: Cytoskeletal filaments. Image reproduced from (Mofrad, 2009) with permission from Annual Re- views 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; M¨ucke 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 sum- marised 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 nucleus fixed support rigid microplate piezo-driven translation

Figure 5: Schematic of parallell microplates. The microplates may be used to impose shear, tensile or com- pressive 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

Interior is modelled as fluid. Membrane is under sustained tension.

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 dissipa- tion, 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 depen- dence 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 essen- tially polymers with a number of possible conformations. Soft Glassy Materials (SGMs) is a class of materials that share some characteristics, for ex- ample 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 cor- responding 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 pa- rameters 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 ver- satile. 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 often- used modelling methods.

Actin

Actin, as mentioned previously, is the most well-studied of the three main types of cytoskele- tal 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 stiff- ness (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 cross- linked. 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 char- acteristic 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 Affine )

L mechanical Nonaffine Affine Log(

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 at- tributed to the longer life-time of of the bonds. The unbinding characteristics of cross-link proteins are not unique in governing network re- sponse, 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 so- lutions, 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 experi- mentally (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 micro- tubules 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 undula- tions 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 so- lutions 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 mi- crostructure 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 con- centration 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, non- affine 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 chap- ters. 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 pro- vide 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. Fur- ther, 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 mi- crostructural 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 vis- coelasticity 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 trappedinenergywellaccordingtoFig.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   E k A · − b d = 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   βψ φ . =exp 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 b c 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

n˙ a = kdnc − kananb (3)

n˙ b = kdnc − kananb +2kdnd (4)

n˙ c = −kdnc + kananb +2kdnd − νkanbnc (5)

n˙ d = −2kdnd + νkanbnc, (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 = FeFv. (7)

We now define a modified neo-Hookean strain energy function as linearly dependent on the concentration of cross-links as μ  Ψ=N ψ = N (I − 3) − p(J − 1) (8) d d 2 e1 e 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 com- puted as

−1 S = Ndμ(A − A33C33C ). (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 . 0 15 γ40 23 ] ] γ50 3 3 m m . mmol mmol 0 10 22 [ [ a b n n 0.05 21

0.0 19.5 0.00.20.40.60.8 0.00.20.40.60.8 Time [s] Time [s] 4.0 0.100

3.0 0.075 ] ] 3 3 m m . . mmol mmol 2 0 0 050 [ [ c d n n 1.0 0.025

0.0 0 0.00.20.40.60.8 0.00.20.40.60.8 Time [s] Time [s]

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  1  γ 1  γ 10 30 10 50

        

[Pa]  [Pa]        12   12    P P 0 0      10 10                                     

10−1 10−1 10−2 10−1 100 10−2 10−1 100 Time [s] Time [s]

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 (≈2RT , 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 (Nun- nally 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 inter- filament 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 cross- link 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

a) b)

Figure 17: a) Initial shear modulus as computed for variations in length and length distribution. The pa- rameter 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 investi- gating 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 represen- tative 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 cross- links. 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 stress- strain (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 stretch- ing (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, stretch- ing 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 im- portance 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), (10) λc μ Ψ = c y(λ )dλ , (11) c 2 c c 1 μ Ψ = a (λ − 1)2, (12) a 4 a 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 L −1 1 μ · y μ λ − a c S = λ c + a( a 1) dλ +2L  6 c a −1 · I − C33C , (13)

38 Figure 21: Filament length dependence and cross-link stiffness influence on initial shear modulus for contin- uum 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   dλ μ L dy a = c · a (14) dλc μa 2Lc dλc 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 = 0.01s, model  8 o ts = 0.01s, Xu et al. × ts = 1.00s, model o ts = 1.00s, Xu et al. τ12 6 [Pa] 4 ×

× 2 × × × × × × ×× × ×   ××    × ×     0 × ×  0.00.10.20.30.4 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 net- works

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

· − (Eb−βψ) Δp =Δt exp( kBT ) (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 t β a)— Baseline b) — s = 0.01s, = 1.0 --ts = 0.01s, β = 0.1 —Red.Eb 40 — ts = 1.00s, β = 1.0 --ts = 1.00s, β = 0.1 20 —Red.C1 τ12 τ12 30 [Pa] [Pa] 20 10 10

0 0 0.00.10.20.30.4 0.00.10.20.30.4

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 experi- mental 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 varia- tion 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 prop- erties 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 - Ir- reversible 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 fila- ments 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, de- pendent on temperature, contour length and persistence length. We split the deformation into isochoric and volumetric parts, and perform a multiplicative ˆ split of the one-dimensional unit stretch λ into an elastic (λe)andviscous(λv)partonthe form

ˆ λ = λeλv. (16)

To derive a composite model, we postulate the existence of a strain-energy function Ψ which canbeassumedtobecomposedofanisochoric(Ψich) and volumetric (Ψvol)part.Thiscan be further decomposed into the contribution from various constituents as

Ψ=Ψich +Ψvol, Ψich =Ψaωa +Ψbωb (17) where Ψaωa and Ψbωb 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 K Ψ = (J − 1)2, (18) vol 2 where J is the jacobian of the deformation gradient. We now replace the eight-chain model by integration of filaments over the unit sphere, follow- ing 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  m λ ω μ λ − d f,j S =2 wj f( f,j 1) λ + j d s,j =1  dλs,j ∂λe,j + μsfs(λs,j) , (19) dλe,j ∂C

42 8  28    P  a)— 12 b) P   — 12   23  6 o Xu (2000) o Tseng (2009)     18  4     [Pa]  [Pa] 13   12 12 P 2 P    8     

  0    3           −2 −2 0.00.10.2 0.00.10.20.3 γ12 γ12

Figure 24: First Piola-Kirchhoff stress P12 for cyclic shear of actin networks, model predictions and experi- ments. 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. cross- link 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

−1 Svol = KJ(J − 1)C , (20) where C is the right Cauchy-Green deformation tensor. This is added to the composite net- work 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 evo- lution 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

5 60 a) b)  — P 12 — P 12 50   4 o Wagner (2009) o Wagner (2009)  

 40 3    [Pa] [Pa]    30 12  12 P 2 P   20  

 1    10         0 0        0.00.20.40.60.81.0 0.00.30.60.91.2 12 12

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 compos- ite 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 thermo- dynamically 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 influ- ence 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 relax- ation 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) 10  

  [Pa]

 12 P 5 

        0   0.00.20.40.60.81.0 12

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 pre- vious 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 a) b) c)

4 4 4

fl fl fl [nN] [nN] [nN] 2 2 2 — Nucleus — Nucleus — Nucleus —Cytoplasm —Cytoplasm —Cytoplasm

0 0 0 0.00.25 0.50 0.75 1.00 1.25 0.00.25 0.50 0.75 1.00 1.25 0.00.25 0.50 0.75 1.00 1.25 Time [s], δ [μm] 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 6 a) — Nucleus b) — Nucleus c) — Nucleus —Cytoplasm —Cytoplasm —Cytoplasm

4 4 4

fr fr fr [nN] [nN] [nN] 2 2 2

0 0 0 0.0 5 10 15 20 25 30 0.0 5 10 15 20 25 30 0.0 5 10 15 20 25 30 Time [s] Time [s] Time [s]

Figure 29: Measured force relaxation curves for colloidal-probe AFM of three DMSO control cells a)-c.

46 6 6 6 a) b) c)

4 4 4

fl fl fl [nN] [nN] [nN] 2 2 2 — Nucleus — Nucleus — Nucleus —Cytoplasm —Cytoplasm —Cytoplasm

0 0 0 0.05 100.05 100.05 10 Time [s], δ [μm] Time [s], δ [μm] Time [s], δ [μm]

Figure 30: Measured force indentation curves for colloidal-probe AFM of three LatB-incubated control cells a)-c).

6 6 6 a) — Nucleus b) — Nucleus c) — Nucleus —Cytoplasm —Cytoplasm —Cytoplasm

4 4 4

fr fr fr [nN] [nN] [nN] 2 2 2

0 0 0 0.0 5 10 15 20 25 30 0.0 5 10 15 20 25 30 0.0 5 10 15 20 25 30 Time [s] Time [s] 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 obser- vations of these cells (R¨onnlund 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 periph- eral 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 b) — Computation — Experiment — Experiment

4 4 f f [nN] [nN] 2 2

0 0 0.0 5 10 15 20 25 30 0.0 5 10 15 20 25 30 Time [s] 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, (pre- sumably 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 a) — Computation b) — Computation — Experiment — Experiment

4 4 f f [nN] [nN] 2 2

0 0 0.0 5 10 15 20 25 30 0.0 5 10 15 20 25 30 Time [s] 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 visualisa- tion 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 stiff-

49 On the mechanics of actin and intermediate filament networks and their contribution to cellular mechanics

250 250 a) — Nucleus b) — Nucleus —Cytoplasm —Cytoplasm 200 200

150 150 f f [nN] [nN] 100 100

50 50

0 0 0.0 5 10 15 20 25 30 0.0 5 10 15 20 25 30 Time [s] Time [s]

Figure 36: Predicted force curves for indentation and relaxation for a) elliptic and b) original model geome- tries. 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 net- works 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 Fred-

50 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 activa- tion (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 μ Ψ = c (n + n )(λˆ +¯u − 1)2, (21) c 2 c d rs where μc is a stiffness parameter andu ¯rs is relative sliding of cross-linked actin filaments, i.e. contraction, governed by a thermodynamically consistent evolution law. The stretch λˆ 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 Sc =2 =2 wj 2μ (n + n (λˆ+ ∂C c c d j=1  ∂λˆ +¯u − 1) , (22) rs ∂C 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)

103 103 [Pa] —Set1 [Pa] 12 2 —Set2 12 2 P 10 P 10 —Set3 — k1 = 0.001964 —Set4 — k1 = 0.019640 — Passive — k1 = 0.196400 — k1 = 1.964000

101 101 10−2 10−1 100 101 102 10−2 10−1 100 101 102

Time [s] Time [s]

Figure 37: Predicted stress relaxation for a) various contractile parameters, b) different chemical rate con- stant k1.

200

— Passive μ 150 — c = 0.0375 MPa — μc = 0.1000 MPa — μc = 0.3000 MPa f [nN] 100

50

0 0.0102030 t [s]

Figure 38: Predicted force relaxation for indentation of the cell.

52 20 o - Passive × - μc = 0.0375 MPa μ  15 - c =0.1MPa

 × 10  [mN/m] × × eff   S 5

0 0.05 10 [%]

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 stiff- ness 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 time- dependent (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 gen- eration, 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 repro- duced. 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,apro- portional 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

× × × × × × ×

11 × [mN/m]



eff      S   

       7 0 255075 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 =0.01MPa,× - μc = 0.0375 MPa.

60

 45  

 30 [mN/m] eff S 15

0 0.00.51.0 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 defor- mation 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 ex- tending 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 hypothe- sised 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 success- ful 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 for- mulation 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 pas- sive 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 dis- persion, 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 magni- tude. 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 cross- link 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 reason- ably 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 signi- fies 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 mi- crostructural level where the characteristics of individual filaments and the interactions be- tween them determine the response at a macroscopic level. Phenomena such as viscoelasticity and strain-hardening followed by strain-softening are observed experimentally in these net- works, often due to microstructural changes (such as filament sliding, rupture and cross-link debonding). Further, composite structures can also be formed with vastly different mechan- ical 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 dissipa- tive 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 in- dicate 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 cel- lular 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 depoly- merisation 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 con- stituents, 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 me- chanical 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 stiff- ness 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 contrac- tile 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

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65 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

n˙ a = kdnc − kananb.

67 On the mechanics of actin and intermediate filament networks and their contribution to cellular mechanics

68