Cell Shape and Durotaxis Follow from Mechanical Cell-Substrate Reciprocity and Focal Adhesion Dynamics: A Unifying Mathematical Model
CCell Shape and Durotaxis Follow from MechanicalCell-Substrate Reciprocity and Focal AdhesionDynamics: A Unifying Mathematical Model
Elisabeth G. Rens (1,2,3) and Roeland M.H. Merks (1,2,4) (1) Scientific Computing, CWI, Science Park 123, 1098 XG Amsterdam, TheNetherlands(2) Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden,The Netherlands(3) Present address: Mathematics Department, University of British Columbia,Mathematics Road 1984, V6T 1Z2, Vancouver, BC, Canada, [email protected] (4) Present address: Mathematical Institute and Institute for Biology, LeidenUniversity, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands, [email protected]
Abstract
Many animal cells change their shape depending on the stiffness ofthe substrate on which they are cultured: they assume small, roundedshapes in soft ECMs, they elongate within stiffer ECMs, and flatten outon hard substrates. Cells tend to prefer stiffer parts of the substrate,a phenomenon known as durotaxis. Such mechanosensitive responses toECM mechanics are key to understanding the regulation of biological tis-sues by mechanical cues, as it occurs, e.g., during angiogenesis and thealignment of cells in muscles and tendons. Although it is well establishedthat the mechanical cell-ECM interactions are mediated by focal adhe-sions, the mechanosensitive molecular complexes linking the cytoskeletonto the substrate, it is poorly understood how the stiffness-dependent ki-netics of the focal adhesions eventually produce the observed interdepen-dence of substrate stiffness and cell shape and cell behavior. Here we showthat the mechanosensitive behavior of single-focal adhesions, cell contrac-tility and substrate adhesivity together suffice to explain the observedstiffness-dependent behavior of cells. We introduce a multiscale computa-tional model that is based upon the following assumptions: (1) cells applyforces onto the substrate through FAs; (2) the FAs grow and stabilize dueto these forces; (3) within a given time-interval, the force that the FAsexperience is lower on soft substrates than on stiffer substrates due to thetime it takes to reach mechanical equilibrium; and (4) smaller FAs arepulled from the substrate more easily than larger FAs. Our model com-bines the cellular Potts model for the cells with a finite-element modelfor the substrate, and describes each FA using differential equations. To- a r X i v : . [ q - b i o . CB ] J un ether these assumptions provide a unifying model for cell spreading, cellelongation and durotaxis in response to substrate mechanics. Classification: Physical Sciences, Applied MathematicsKeywords: mechanobiology, cell biophysics, multiscale mathematical biology,cell-based modeling 2 ignificance statement
Besides molecular signaling, mechanical cues coordinate cell behavior duringembryonic development and wound healing; for example, in tendons cells andcollagen fibers align to optimally support the forces that the tendon experiences.To this end, cells actively probe their environment and respond to its mechanics.Key building blocks for such active mechanosensing are the contractile actincytoskeleton and the extracellular matrix (ECM), the matrix of fibrous proteinsthat glues cells together into tissues (e.g., collagen). Actin is linked to theECM by structures called focal adhesions, which stabilize under force. Herewe present a novel mathematical model that shows that these building blockssuffice to explain the cellular responses to ECM stiffness. The insights advanceour understanding of cellular mechanobiology.3
Introduction
Embryonic development, structural homeostasis and developmental diseases aredriven by biochemical signals and biomechanical forces. By interacting with theextracellular matrix (ECM), a network of fibers and proteins that surrounds cellsin tissues, cells can migrate and communicate with other cells, which contributesto tissue development. Mechanical interactions between cells and the ECM arecrucial for the formation and function of tissues. By sensing and respondingto physical forces in the ECM, cells change their shape and migrate to otherlocations. Here, we show a single, unifying mechanism that suffices to explainsuch ECM-mechanics induced cell shape changes and cell migration.The shape of a wide range of mammalian cell types depends on the stiffness ofthe ECM. In vitro, cells cultured on top of soft, two-dimensional ECM substratesbecome relatively small and rounded, whereas on top of a stiffer ECM the cellsassume elongated shapes. On ECM of high rigidity, like glass, cells spread outand flatten. This behavior has been observed for a wide range of cell types,including endothelial cells [1], fibroblasts [2, 3], smooth muscle cells [4], andosteogenic cells [5]). Secondly, cells tends to migrate towards stiffer parts ofthe ECM, a phenomenon known as durotaxis. Such behavior also occurs for awide range of mammalian cell types, including fibroblasts [6], vascular smoothmuscle cells [7] and mesenchymal stem cells [8].It is still poorly understood what molecular mechanisms regulate such cel-lular response to ECM stiffness [9]. Cells are able to sense matrix stiffnessthrough focal adhesions (FAs), multi-molecular complexes consisting of integrinmolecules that mediate cell-ECM binding and force transmission, and an esti-mated further 100 to 200 protein species that strongly or more loosely associatewith focal adhesions [10, 11, 12]. Among these are vinculin and talin, whichbind integrin to actin stress fibers. Manipulations of FA assembly affects cellspreading and motility. For instance, a lack of vinculin directly decreases cellspreading [13] through FA stabilization, even when actin stress fibers are notaffected. Generally, changes in cell polarization were associated with alteredFA response to substrate rigidity (through gene knock-downs) [14]. FA assem-bly and disassembly has also been associated to cell migration and orientation[15, 16, 17, 18] in response to the ECM. So, the mechanosensitive growth of FAsis key to our understanding of how cells respond to ECM stiffness.Focal adhesions dynamically assemble and disassemble, where the disassem-bly rate is highest on soft ECMs and is lower on stiffer ECMs [2] leading toFA-stabilization. This mechanosensitivity of FA dynamics is regulated by FA-proteins, such as talin and p130Cas. These proteins change conformation inresponse to mechanical force [10, 9]. For instance, stretching the structural pro-tein talin reveales vinculin binding sites, allowing additional vinculin to bindto focal adhesions [19] and stabilize the FA [20]. Furthermore, integrins suchas α β , behave as so-called “catch-bonds” [21]: bonds whose lifetime increaseunder force [22]. However, how different mechanosensitive proteins in FAs andcytoskeletal forces work together to regulate cell spreading, cell shape, and duro-taxis is still to be elucidated. 4revious mathematical models have proposed various explanations for themechanosensitive behavior of cells. A central idea in these explanations is dy-namic reciprocity [23]: the cell pulls on the ECM to probe its mechanical re-sponse. The mechanical resistance of the matrix can either lead the cell to pullmore strongly, or it can change the stability of the focal adhesions. Ni et al.[24] assume that cells exert stronger traction forces on stiffer ECMs than onmore soft ECMs. The resulting balance between intracellular and extracellularstresses leads the cell to deform; the deformation is attenuated by an interfacialenergy due to cell-ECM adhesion. This model predicts that cells deform only ifthe cell and substrate stiffness are roughly equally stiff, whereas cell spreadingforces are not included in the model. To show how stresses in the ECM canaffect cell polarization, Shenoy et al. [25] argue that deformation of cells canlead to stress-induced cell polarization of contractility. They show that stress-induced recruitment of myosin to the actin cytoskeleton can locally increasescontractility, leading to further intracellular stress. The resulting positive feed-back loop can polarize cells, contributing to cell elongation and durotaxis. Inour own work, we have proposed that cells spread and elongate because of a pos-itive feedback loop between protrusion forces and strain stiffening of the matrix[26], a mechanism motivated by the stabilization of focal adhesions on stiff ma-trices. Similarly, previous models have explained durotaxis. The prevalent ideain these mathematical models is increased stabilization of focal adhesions atthe stiff side of the cell, which in turn drives intracellular dynamics, leading toincreased traction force [27], modified stress fiber dynamics [28, 29], a bias invelocity [30], viscous forces and cell stiffening [31], motor protein recruitment[25], membrane tension [32], enhanced persistence time in stiffer matrices [33],or cell polarization [34, 35].Some models integrated explicit kinetics of focal adhesions [36] in cell-basedmodels to study how force dependent FA assembly regulates cell spreading [37,38]. In these models the size of the focal adhesion (i.e., the number of cell-substrate adhesions within it) determines how much force cells apply on thematrix. These models assumed that on stiff substrates, the experienced stresspromotes stress fiber formation in the cell, allowing it to apply more force onthe focal adhesions. As a result, the focal adhesions grow and the cell canapply even more force, stabilizing the stress fibers even more, etc. In both thesemodels, such a feedback between force and focal adhesions makes the cell spreadout. Another hybrid cell and focal adhesion model (that integrated the focaladhesion growth model in ref. [39]) where the size of the focal adhesions did notaffect the magnitude of cell forces, could not explain increased cell spreadingon stiff matrices [40]. These results suggest that cell spreading is regulatedby a feedback between focal adhesions and traction force. These models couldhowever not predict how cells elongate as a function of substrate stiffness.With a hybrid cell and focal adhesion model we propose a focal adhesionmechanism that unifyingly explains three ECM stiffness-dependent cell behav-iors: cell spreading, cell elongation, and durotaxis. The model is based on thefollowing assumptions: (1) focal adhesions are discrete clusters of integrin-ECMbonds; (2) new bonds are added to the FAs at a constant rate; (3) the unbinding5ate is suppressed by the tension in the FA, which is due to pulling of stressfibers [41]; (4) on soft ECMs it takes more time for the tension in the FA tobuild up to its maximum value, than on stiff ECMs [42]; therefore, on average,the unbinding rate in FAs is higher on soft ECMs than on stiff ECMs. (5) Asa result, FAs grow larger on stiff ECMs than on softer ECMs; (6) thus the FAsdetach less easily from the ECM on stiff matrices than on softer ones. (7) Planarstress reinforces FAs due to recruitment of stabilizing proteins such as vinculin.We show that, apart from the substrate-stiffness-dependent cell tractionforces proposed in the previous works, an alternative explanation for cell spread-ing on stiff substrates is that cells build up their forces with a faster rate on stiffsubstrates. Interestingly, in our model, a feedback between focal adhesions andforce magnitude, regulates cell elongation (and not cell spreading itself). Themodel shows that the range of stiffness on which cells elongate, depends on thevelocity of myosin motor proteins. Finally, simulated cells exhibit durotaxis andconsistent with experimental observations, the durotaxis speed increases withthe slope of the stiffness gradient [7, 8] . Results
Model development : We based our model on a recently developed hybridCellular Potts - Finite Element framework that has been tested on collectivecell behavior driven by ECM forces [26, 43]. We extended and adapted thismodel to include explicit descriptions of focal adhesion dynamics. With thismodel, we propose that focal adhesion dynamics can explain cell spreading, cellelongation and durotaxis in response to substrate stiffness. Figure 1 gives anoverview of the model, showing the flow and feedback between the cell, its focaladhesions and the elastic substrate it adheres to. Here, we will give a briefexplanation of the model. We refer to the methods section for more details.
Cells : A cell is described as a collection of discrete lattice sites in a cellular Pottsmodel (CPM), see Figure 1A. Cells in the CPM change shape by iterativelymaking extensions and retractions, modeling the formation and break down ofadhesions with the substrate during so-called Monte Carlo time steps (MCS).We assume that retractions from the substrate are less likely at sites with largefocal adhesions.
Cell traction forces : The cell applies a contractile force upon focal adhesionsthat adhere to the matrix. We use the shape of the cell to calculate the contrac-tile force based on a First Moment of Area (FMA) model [44], see Figure 1B.This model assumes that the cell acts as a single contractile unit, so that theresulting forces are pointed towards the center of mass and forces are propor-tional to the distance to the center of mass. To describe the force dynamics intime, we adopt a model of Schwarz et al. [42]: F ( t ) = F s (1 − exp( − t · v KF s )) , where v is the free velocity of the motor proteins and K the substrate stiffness6igure 1: Flowchart of the multiscale CPM. (A) CPM calculates cell shapes inresponse to focal adhesions and substrate stresses; (B) calculation of cellulartraction forces based on cell shape and force build-up dynamics; (C) focal adhe-sion grow according to dynamics of catch-slip bond clusters; and (D) calculationof substrate stresses due to cellular traction forces.7nd F s is the stall force. This model assumes that forces build-up towards thestall force (here, given by the FMA model) with a rate proportional to thevelocity of the myosin motors (that are responsible for force generation) andthe matrix stiffness. So, the more compliant the substrate is, the longer it takesfor a cell to build up this force. Focal adhesions : We model focal adhesions as clusters of catch-slip bonds,as proposed by Novikova and Storm [41]. This model is based on experimentaldata of one single α − β Simulation : A simulation proceeds as follows. We initiate cells in the CPM(part A of our model). Then, in part B of the model, we let those forces buildup in time for t FA seconds. At the same time, part C of the model is executed.So as the forces build up, we let the integrin clusters grow simultaneously.After these t FA seconds, we let the cells move, i.e. perform one timestep in theCPM. After one timestep of the CPM, we again let the forces build up and theintegrin clusters grow for t FA , and so forth. Parameter values were, if possible,chosen based on literature and if not, chosen arbitrarily. For the arbitrarilychosen parameters, we performed parameter variations to study their effects(see Supplementary Material), which showed that deviations from the defaultvalues do not qualitatively change our results. The default parameter values aregiven in Supplementary Table S1. ‘Minimal’ model M1 : In a minimal version of our model (called model M1),we only follow the loop with arrows 1 and 2 in the flowchart, thus excluding thefeedback with matrix stress as depicted in Figure 1D (which is not describedhere, but will be describe in a later section where it will be used for the firsttime). We start out with model M1, to study how this minimal model translatesto cell spreading. Catch-bond cluster dynamics suffices to predict cell area asa function of substrate stiffness
In this section, we show with model M1, that catch-bond dynamics of integrinclusters suffices to explain cell spreading on elastic substrates. We set up thesimulations as described above and varied the substrate stiffness. Figure 2Ashows the response of cells on substrate of size 500 µ m by 500 µ m with astiffness of 1 kPa, 5 kPa, and 50 kPa after 2000 MCS ( ≈ µ m on the softest substrate and plateaus at around6500 µ m at a stiffness of 50 kPa. Thus, the cell area has increased more than2.5 fold on the stiffest substrate compared to the softest substrates. This factoris consistent with experimental observations [1, 45, 46]. We also investigated ifthe model could qualitatively predict spreading dynamics. Figure 2C plots thecell area as a function of time. The cells quickly increase in size and and reachtheir final size after 30 to 60 minutes. Experimental curves of cell area versustime follow a similar trend [47, 48].We also investigated the distribution of the integrin cluster sizes. Figure 2Dplots the distribution of the cluster sizes and the median cluster size (averagecluster size is roughly the same). The median cluster size and variance areunaffected by substrate stiffness, in contrast with experimental observations[49]. We performed a more detailed analysis of the distribution of integrinclusters and describe two observations. 1) On stiffer substrates, there are morelarger clusters. For instance, the percentage of focal adhesions with N >
Focal Adhesion strengthening due to matrix stress inducescell elongation
After having captured, at a qualitative level, the rate of spreading as a functionof substrate stiffness, we now set out to explain the ability of mammalian cellsto elongate on stiff enough substrates. Because the first version of the model(M1) could not yet explain cell elongation, we aimed to find an additional focaladhesion mechanism that can explain cell elongation.Since cell traction forces are transferred to the matrix through the integrins,stresses develop in the matrix. Such stresses have been observed to affect focaladhesion assembly [17]. We therefore hypothesized that such a feedback mayexplain cell shape changes. ‘Extended’ model M2 : We extended our model to Model M2, that includes afinite element model (FEM) to calculate the matrix stress (Figure 1D) as a result9igure 2: Model M1 predictions: Cell area increases with increasing substratestiffness. (A) Example configurations of cells at 2000 MCS on substrates of 1,50and 50 kPa; (B) Cell area as a function of substrate stiffness, shaded regions:standard deviations over 25 simulations; (C) Timeseries of cell area, shadedregions: standard deviations over 25 simulations; and (D) distribution of N,the number of integrin bonds per cluster, all clusters at 2000 MCS from 25simulations were pooled. We indicate the median. Color coding (C and D): Seelegend next to (D). 10f the cell traction forces [26, 43]. So, model M2 follows arrows 1, 3 and 4 inFigure 1D. We assume that matrix stresses reinforces cell-matrix adhesions. Wemodel focal adhesion strengthening by reducing the probability of retractionsfrom the matrix due to matrix stress, i.e. we multiply the energy it takes for acell to make a retraction with1 + p g ( σ ( (cid:126)x (cid:48) )) σ h + g ( σ ( (cid:126)x (cid:48) )) . Here, parameter p regulates the strengthening and σ h its saturation and g ( σ ( (cid:126)x (cid:48) ))denotes the hydrostatic stress on the lattice site of retraction. Such a strength-ening due to matrix stress can have various molecular origins. We hypothesizethat this strengthening is due to stretching of the structural protein talin exposesbinding sites for vinculin, which binds to the cytoskeleton and thus strengthensthe actin-integrin linkage [19, 20].Figure 3A shows representative configurations of cells after running modelM2 (see also Video S2). Similar to model M1, on the most soft substrate (1 kPa),the cell stays small and round. However, for stiffer substrates, such as 50kPa,the cell elongates. To quantify cell elongation, we measured the eccentricityof cells as (cid:112) (1 − b a ) with a and b the lengths of the cell’s major and minorsemi-axes, calculated as the eigenvalues of the inertia tensor. Figure 3B showsthat cells start to slightly elongate on substrates with a stiffness of 10 kPa/20kPa. On stiffer matrices, 50 kPa and 100 kPa, cells are very much polarizedin shape and large focal adhesions have grown at the tips of the cell. On thevery rigid substrate, the cell is more circular again. So, the eccentricity of cellshas a biphasic dependence on substrate stiffness. We also again quantified thedistribution of the integrin cluster sizes. Figure 3C shows the distribution ofthe cluster sizes for the different elastic substrates. The median (which is againroughly the same as the mean) cluster size does not vary much between substratestiffness. The shape of the distributions, however, are more flat and with higherstandard deviation on the substrates where cells have elongated compared toround cells. The kurtosis is around 2.0 for elongated cells, compared to a kurtosisaround 2.3 for round cells. The standard deviation is around 2700 for elongatedcells, compared to around 1500 for round cells. This more flat distribution offocal adhesion sizes can be explained as follows. An elongated shape results inlarge traction force at the tip of the cells, such that the focal adhesions growlarger in size at these tips, while at the sides of the cell, the forces are muchsmaller and focal adhesion stay small there.The model explains the process of cell elongation as follows. On sufficientlystiff matrices, the cell initially starts to spread. The cell continuously makes ran-dom protrusions, allowing the cell shape to become slightly anisotropic. Aroundthese cell protrusions, matrix stresses develop, which strengthens cell-matrix ad-hesion in this region. So, the cell can continue to build up forces, allowing thefocal adhesion to grow larger. In contrast, at site of lower matrix stress, focaladhesions are more likely to disassemble. At protruding sites, cell traction forcesincrease due to an increased distance from the cell centroid. This results in abreaking of symmetry and the cell starts to elongate due to a positive feedback11igure 3: Model M2 predictions: Cells elongate on substrates of intermediatestiffness. (A) Example configurations of cells at 2000 MCS on substrates of 1,50and 50 kPa. Colors: hydrostatic stress; (B) Cell eccentricity as a function ofsubstrate stiffness, shaded regions: standard deviations over 25 simulations; (C)distribution of N, the number of integrin bonds per cluster, all focal adhesionat 2000 MCS from 25 simulations were pooled. We indicate the median. Colorcoding (C): See legend next to (C). 12oop of force build-up, focal adhesion growth and matrix stress induced adhe-sion strengthening. On soft matrices, matrix stresses are not high enough toinitiate a symmetry breaking. On the most rigid surface, matrix stresses are toohigh, allowing focal adhesions to strengthen equally well such that no symmetrybreaking can occur. So, cell elongation is only possible on substrates with anoptimal rigidity. Similar dynamics of cell spreading followed by a symmetrybreaking has also been observed experimentally [14]. Note that the spindle-likeshapes that cells obtain in our model resembles those observed in vitro [50].Some parameters, such as p and σ h were chosen arbitrarily. So, we tested thesensitivity of our model M2 to these parameters. Increasing p which regulatesthe extent of focal adhesion strengthening by matrix stress, enables cells tostart elongating on softer matrices and also induces cell elongation on the mostrigid surface (Supplementary Figure 2). Variations in σ h , which regulates thesaturation of stretch exposed binding sites for vinculin does not greatly affectmodel behavior (Supplementary Figure 3). Other parameters might be cell typespecific, such as the lifetime of protrusions t FA (Supplementary Figure 4), extentof random motility T (Supplementary Figure 5) and the magnitude of tractionforces µ (Supplementary Figure 6). The qualitative behavior is conserved forvariations of these parameters, but all parameters affect the range of substratestiffness on which the cell can elongate. Second version of ‘extended’ model M2 : Since previous models proposedthat cell spreading is regulated by a feedback between focal adhesions andforces, we also investigate what happens if focal adhesions start to apply moreforce when subject to matrix stresses. So, instead of letting matrix stressesstrenghten focal adhesions by increasing the detachment energy, we let matrixstress strengthen the focal adhesions by locally increasing cell traction forces.To model this, we assume that the stall force increases as a function of matrixstress, i.e. (cid:126)F s = (cid:126)F s · (cid:16) p g ( σ ( (cid:126)x (cid:48) )) σ h + g ( σ ( (cid:126)x (cid:48) )) (cid:17) . This other feedback mechanism givessimilar results to Figure 3 (see Supplementary Figure 7 and Video S3 for theresults of Model M2-version 2). Such a mechanism can have various molecularorigins. For instance, addition of vinculin through talin stretching can induceincreased traction forces [51]. Stretching forces also induces α -smooth muscleactin recruitment to stress fibers [52], and myosin motor binding [53].In conclusion, our model suggests that by applying a force on the matrix, cellsdevelop an anistropic matrix stress field that can induce a symmetry breakingof the cell by reinforcing focal adhesion sites. This allows a cell to elongate onsubstrates of intermediate stiffness. Such a matrix stress reinforcement can befrom various molecular origins, such as a matrix stress induced focal adhesionstrengthening or increased traction forces. Motor protein velocity changes stiffness regime on whichcells elongate
The regime of substrate stiffness on which cells spread and elongate varies percell type. For instance, neutrophils do not respond to changes of substrate13tiffness in the range of substrate stiffness where both fibroblasts and endothelialchange in area and shape [48]. To try and understand why this is the case, wecan vary cell related parameters in our model. One cell specific parameter isthe velocity of the myosin motors, which regulates the speed of force build-up.Many cells express non-muscle myosin II, which exists in isoforms A,B and C[54]. Other cell types also expresses myosin isoforms such as skeletal, cardiac andsmooth muscle myosin [54]. Different cell types may have different expressionprofiles of myosin isoforms [55] and since the velocity of myosin motors variesamong isoforms [56, 57], this may impact the response of cells to matrix stiffness.Using our model, we study how myosin motor velocity, v , can impact cell shape.We study a range from 10 nm/s, corresponding to non-muscle myosin II B [56])to 1000 nm/s, corresponding to muscle myosin [58].Figure 4A and B shows the cell configurations for a slow (10 nm/s) andfast motor velocity (1000 nm/s), compared to the default value of 100 nm/s asshown in Figure 3A, respectively. This shows that cells with slow motors do notspread significantly and do not elongate, even on stiffer substrates. In contrast,cells with fast motors already spread more and elongate on softer matrices.We quantified this further by running 25 simulations for each combination ofsubstrate stiffness and motor velocity. Figure 4B and Figure 4C plot the cellarea and eccentricity, respectively, as a function of motor protein velocity. Withthe fastest velocity tested here (1000 nm/s), cell area saturates already at 5kPa and cells elongate on a larger stiffness regime (5 kPa - 100 kPa). With theslowest motor velocity (10 nm/s), cells do not elongate at all, while they stillspread well on stiff matrices. This is explained as follows. Decreasing v is verysimilar to decreasing the stiffness of the substrate, because they both contributeto the rate of force build-up in the same way, given by | (cid:126)F s | v K . So, in terms ofcell area, cells with slower motor proteins would obtain a larger spreading areaat stiffer matrices. However, they are not able to elongate because forces arenot built up fast enough to generate high enough matrix stress that induces thefocal adhesion strengthening.So, in summary, we predict that cells with faster motor proteins start spread-ing/elongating at softer substrates, while cells with slower motor proteins needa stiffer substrate to instigate a response. Durotaxis explained by a bias in integrin clustering
On substrates with a stiffness gradient, cells move up the stiffness gradient, aphenomena called durotaxis. Cells may durotact by sending out protrusionswhich better stick to stiff substrates because focal adhesions grow on stiff sub-strates [6, 59]. Here, we investigate if force induced focal adhesion growth issufficient to reproduce durotaxis. We simulated durotaxis by placing an initialcircular cell with its center at x = y =250 µ m on a lattice of 1250 µ m by 500 µ mfor 10000 MCS ( ≈ h ). In the x -direction, we let the stiffness increase from1 kPa to 26 kPa, so with a slope of 20 Pa/ µ m. Figure 5A plots ten differenttrajectories of the cell, showing that most cells have moved significantly in the14igure 4: Range of stiffness on which cells elongate depends on myosin motorvelocity. Model M2 was used. (A) Example configurations of cells at 2000 MCSon substrates of 1,50 and 50 kPa with motor velocity 10 nm/s; (B) Exampleconfigurations of cells at 2000 MCS on substrates of 1,50 and 50 kPa with motorvelocity 1000 nm/s. Colors (A-B): hydrostatic stress; (C) Mean cell area as afunction of motor velocity, error bars: standard deviations over 25 simulations;(D) Mean cell eccentricity as a function of motor velocity, error bars: standarddeviations over 25 simulations. 15igure 5: Durotaxis as a result of integrin catch-bond dynamics. (A) Tentrajectories of durotacting cells on a matrix with slope 20 kPa/ µ m; (B) Cellspeed as a function of the slope of the stiffness gradient. x -direction, up the stiffness gradient. One realization is shown in Video S4.Cells, on average, move in the x -direction with a constant speed of around 4.3 µ m/h, measured as the slope of the x -coordinate of the cell from 25 simulations.Vincent et al. [8] found speeds of 6.2 µ m/h with gradient slope 10 Pa/ µ m invitro for mesenchymal stem cells. In our model, how far cells can move up thegradient, depends on the flexibility and motility of the cell. We varied λ , theLagrangian multiplier of the area constraint, controlling cell flexibility, and thecellular temperature T , and found that both affect cell speed (SupplementaryFigure 8).In the CPM, cell movement is a result of subsequent protrusions and re-tractions. In stiffer areas the focal adhesions grow larger, so that retraction aremore likely to be made at more flexible parts of the matrix. As a result, the cellmoves up the stiffness gradient. So, naturally, one would expect that durotaxisdepends on the slope of the stiffness gradient. Figure 5B shows the speed of thecell as a function of the slope of the stiffness gradient. Indeed, simulated cellsmove faster up the gradient if the slope is steeper, as observed in experimentalconditions [8, 7]. This is because the difference in focal adhesion growth betweenthe front and the back of the cell is larger with a higher slope, causing a largerbias. We suspect that the durotaxis speed saturates at steep slopes, becausethe growth rate of focal adhesions is limited.In conclusion, durotaxis is an emergent behavior in our model, cells exhibitdurotaxis as a result of a biased growth of focal adhesions. A cell can build upforces faster on stiffer matrices, allowing focal adhesions to grow larger here.So, the cell better attaches at the stiffer part and will retract at the softer side.As a result, the cell moves up the stiffness gradient.16 iscussion We have presented a multiscale computational model to show that force inducedfocal adhesion dynamics can explain 1) cell area increasing with substrate stiff-ness (Figure 2A-B), 2) cell elongation on substrates of intermediate stiffness(Figure 3A-B) and 3) durotaxis (Figure 5A). The model described cells spread-ing on an elastic substrate via focal adhesions, which are modelled as integrinclusters. Cells applied traction forces on integrin clusters, which grow accord-ing to catch-slip bond dynamics [41]. In this model, the disassembly of focaladhesions decreases with force. How fast a cell in our model can build up thisforce was assumed to depend on the stiffness of the matrix, based on a model bySchwarz et al. [42]. On soft matrices, forces build up slowly such that integrinclusters do not have enough time to grow, while on stiff matrices forces buildup fast such that integrin clusters can grow in size. Because we assumed thatlarger focal adhesions detach less likely from the substrate than smaller ones,cell spreading area increased on stiffer substrates (Figure 2B). If we includeda feedback between matrix stresses and cell-matrix adhesive forces, simulatedcells were able to elongate (Figure 3A-B). The model suggests that the range ofsubstrate stiffness on which cells elongate depends on the velocity of the myosinmolecular motors, which determine the rate of force build-up. Cells with highermotor protein velocity started to elongate on softer matrices (Figure 4). Finally,our model explains durotaxis as a bias in focal adhesion growth on stiffer matri-ces. Because extensions are more likely to stick at these regions and retractionsare more likely to be made on the softer side, cells obtain a bias in cell motilityup the stiffness gradient (Figure 5A). Our model predicted that cell velocityincreases with the slope of the stiffness gradient (Figure 5B), which compareswell with experimental data [7, 8]. The spreading dynamics in our model alsoqualitatively compare well with in vitro dynamics: the spreading dynamics inFigure 2C are similar to spreading area curves found in vitro [47, 48] and thedynamics of cell elongation (Movie S1) resemble in vitro observations [14].
Mechanisms driving cell elongation
We hypothesized that the stabilization of focal adhesions by matrix stress isdue to stretching of talin. Stretching of talin exposes vinculin binding sites[19] and vinculin in turn binds the focal adhesion to the cytoskeleton, whichstrengthens cell-matrix adhesion [20]. Our model suggests that this might reg-ulate cell elongation. In agreement with this observation, vinculin regulates cellelongation on glass substrates [13]. We could attempt to further unravel howvinculin drives cell elongation by studying vinculin depleted cells on substratesof different stiffness, or by adapting talin in such a way that vinculin cannotbind as a result of talin stretching.Interestingly, our model suggests that cells can also elongate if matrix stressinduces an increase in cell traction forces (Supplementary Figure 7). This mech-anism can be justified by two experimental observations; 1) vinculin increasescell traction forces [51] and 2) stressing focal adhesions induces α -smooth mus-17le actin recruitment to stress fibers that in turn increases traction forces [52].Experimental testing can be done to elucidate which mechanism might be re-quired for cell elongation, since our model does not differentiate between thesetwo and vinculin adhesion strengthening.Our model also predicts that cells elongate on different ranges of substratestiffness, due to different velocities of their myosin motors (Figure 4). This couldexplain why different cell types elongate on different stiffness regimes [48, 60], asthey might express different isoforms of myosin motors. Many studies of differenttypes of cells on compliant substrates have been performed, but often either therange of substrate stiffness tested differs or the type of matrix ( i.e. type ofligand, ligand density, or gel type) is different. Therefore, spreading of differentcell types cannot be compared one to one. Model validation would benefitfrom more systematic in vitro experiments of different cell types on compliantmatrices. To then confirm this model prediction, it could be measured whichisoform of myosin the cells express. There are some experiments that seem tosupport our model prediction. For instance, cell elongation is promoted in Dlc1deficient ovarian tumour [61]. Dlc1 leads to increases of phosphorylation levelof nonmuscle IIA mysosin [61], which suggests that an increase in motor proteinvelocity indeed enables cells to elongate more. Furthermore, cells treated withblebbistatin on stiff matrices obtain phenotype as if they are on a soft matrix[62], while upregulating myosin gives opposite results. In this paper by Jiang etal. [62] it was suggested that the actomyosin pulling speed produce has a similareffect on integrin stem cell lineage specification (which is highly associated withcell shape [63]) as the effective spring constant of the substrate. Focal adhesion regulation of cell spreading
In our model, we have differentiated between integrin size dependent focal adhe-sion strength and focal adhesion strength reinforcement by structural proteins,which have been observed experimentally to be different mechanisms [64]. Thestrength of our model is that we can directly associate cell response to matrixstiffnes with mechanisms at the level of focal adhesions. Previous mathematicalmodels were often based on how matrix stiffness influences mechanisms at thecellular level. The assumptions on how matrix stiffness affect cellular dynamicswere however often motivated by a change in focal adhesion dynamics (stiffnesssensing: [65, 24, 66, 67], durotaxis: [27, 30, 32, 34].) Similarly, the mechanismfor cell spreading proposed in our previous work [26], was based on focal ad-hesion dynamics and depends on cell traction and cell adhesion [68]. In thisprevious model, we suggested that protrusions are more likely to stick and pro-trude to highly strained matrices that have strain-stiffened. This was motivatedby the experimental observation that cells more efficiently build up forces onstiff matrices, which enables stabilization of focal adhesions [2].Cell based models including mechanosensitive focal adhesions have been usedto study cell behavior (for instance, stepping locomotion [69] and cell migrationunder the influence of external cues [70]). In terms of cell spreading, hybridcell-focal adhesion models suggested that cell spreading requires focal adhesions18o upregulate cell forces [37, 38]. In our model, such a feedback is only requiredfor cells to elongate but not for an increase in cell spreading area. In line withthis view, the model by Stolarska et al. [40] suggested that the mechanosensitivegrowth of focal adhesions alone could not explain increased cell spreading onstiff matrices [40] because increased cell contraction on stiff matrices resists cellspreading. In contrast to these previous models, our model suggests that a morerapid build up of forces on stiff matrices, allows focal adhesions to stabilizeenough to enable cell spreading and no increased traction force is required.Cells in our model are able to spread on rigid matrices without focal adhesionreinforcement of cell traction forces, because the adhesion strength of large focaladhesions resist cell retractions on stiff matrices.
Focal adhesion regulation of durotaxis
We also compare our results to recent hybrid cell-focal adhesion models thatwere used to study durotaxis. In the model by Yu et al. [34], the number offocal adhesions was assumed to be higher on stiff substrates and the distributionof focal adhesions was assumed to be more narrow on stiff substrates. Both thenumber and distribution of focal adhesion then controlled the deviation fromthe direction of motion: on stiff matrices cells move more persistent, causing itto durotact. Recently, it was also proposed that cells durotact by tugging onthe matrix and then polarizing towards areas that the cell perceives as stiff [35].So, both these models suggest that the mechanosensitivity of focal adhesionsdrive durotaxis by polarizing the cell. Feng et al. [71] showed that if focal ad-hesion degradation is higher in the back than in the front and focal adhesionsmature under applied force, then a cell can durotact. Based on experimentalobservations, Novikova and et al., presumed that cells move more persistentlyon stiffer substrates and showed that a persistent random walk can reproducedurotaxis [33]. In contrast to previous models, our model suggests that duro-taxis emerges from the mechanosensitive growth of focal adhesions and that noinherent polarized or persistent cell migration is required.
Model limitations
A limitation of our model is that it cannot accurately predict increasing focaladhesion sizes as a function of substrate stiffness (see Figure 2D), while this hasbeen observed experimentally [14]. This may be explained by modeling choices.In the CPM, cells only make retractions at the boundary of the cell, so in themiddle of the cell, integrin clusters continue to grow even on soft matrices.Also, there is a fixed pool of free integrin bonds, making the growth rate ofnew focal adhesions to go down due to existing focal adhesions. Furthermore,our lattice based model does not define spatial effects in integrin clustering.In reality, small clusters may merge into larger adhesions and the availabilityof integrins that can bind to ECM, active integrin, is spatially and temporallyregulated. Cells produce integrins, that diffuse and are activated within thecell. This activation of integrin depends on interaction with other proteins,19uch as talin [72] and vinculin [73]. Furthermore, Stretching of p130cas inducesits phosphorylation, which in turn activates the small GTPase Rap1 [74] whichactivates integrins [75]. So, to better reproduce focal adhesion growth in futuremodels, we can include other relevant mechanisms such as diffusion and theactivation of integrins [36, 76, 72, 38]. However, because we were interested incell shape in this work, which can be predicted with our model, we find the levelof detail of focal adhesion dynamics sufficient at the moment.
Conclusion
In summary, we propose that the mechanosensitive response of molecules infocal adhesions suffice to explain the response of cells to matrix stiffness. Inagreement with experimental observations, cells spread more on stiff matricesand obtain an elongated shape if the matrix is stiff enough. Furthermore, cellsdurotact and move faster with steeper stiffness gradients. This model paves theway to study how specific molecular mechanisms within focal adhesions impactcell and tissue level responses to matrix mechanics. This can give rise to newtargets of treatment and the design of tissue engineering experiments.
Methods
We developed a multiscale model where cell movement depends on force inducedfocal adhesion dynamics. The model couples a cell-based model, substrate modeland focal adhesion model in the following way. The Cellular Potts Model (CPM)describes cell movement. The shape of the cell is used to describe the stallforces that the cell exerts on the focal adhesions attached to a flexible substrate.These forces affect the growth of the focal adhesions. We assume that focaladhesions are clusters of integrins that behave as catch-slip bonds. Its dynamicsare described using ordinary differential equations (ODEs). Finally, we assumethat the cell-matrix link is strengthened by matrix stresses, which we calculateusing a finite element model (FEM). The default parameter values are describedin Supplementary Table S1.
Cellular Potts Model
To simulate cell movement, we used the Cellular Potts Model (CPM) [77]. TheCPM describes cells on a lattice Λ ⊂ Z as a set of connected lattice sites. Sincethe simulations in this article are limited to one cell, we describe the CPM herefor a single cell. To each lattice site (cid:126)x ∈ Λ a spin s ( (cid:126)x ) ∈ { , } is assigned. Thisspin value indicates if (cid:126)x is covered by the cell, s ( (cid:126)x ) = 1, or not, s ( (cid:126)x ) = 0. Thusthe cell is given by the set, C = { (cid:126)x ∈ Λ | s ( (cid:126)x ) = 1 } . (1)The cell set C evolves by dynamic Monte Carlo simulation. During one MonteCarlo Step (MCS), the algorithm attempts copy a spin value s ( (cid:126)x ) from a source20ite (cid:126)x into a neighboring target site (cid:126)x (cid:48) from the usual Moore neighbourhood.Such copies mimic cellular protrusions and retractions. During an MCS, N copyattempts are made, with N the number of lattice sites in the lattice. Whethera copy is accepted or not depends on a balance of forces, which are representedin a Hamiltonian H .A copy is accepted if H decreases, or with a Boltzmann probability otherwise,to allow for stochasticity of cell movements: P (∆ H ) = (cid:40) H + Y < e − (∆ H + Y ) /T if ∆ H + Y ≥ . (2)Here ∆ H = H after − H before is the change in H due to copying, and the cellulartemperature T ≥ Y denotes a yield energy, an energy a cell needs to overcome to make a move-ment. Finally, to prevent cells from splitting up into disconnected patches, weuse a connectivity constraint that always rejects a copy if it would break aparta cell in two or more pieces.Following Ref. [78], we use the following Hamiltonian: H = λA (cid:124)(cid:123)(cid:122)(cid:125) contraction + (cid:88) neighbours( (cid:126)x,(cid:126)x (cid:48) ) J ( s ( (cid:126)x ) , s ( (cid:126)x (cid:48) )) (cid:124) (cid:123)(cid:122) (cid:125) line tension − λ C AA h + A (cid:124) (cid:123)(cid:122) (cid:125) cell-matrix adhesion . (3)The first term of H denotes cell contraction, where A is the area of the cell and λ is the corresponding Lagrange multiplier. In the second term, J ( s ( (cid:126)x ) , s ( (cid:126)x (cid:48) ) arethe adhesive energy between two sites (cid:126)x and (cid:126)x (cid:48) with spins s ( (cid:126)x ) and s ( (cid:126)x (cid:48) ). Whentaking a sufficient large neighborhood, the second term describes a line tension,as it approximates the perimeter of a cell [79]. We take a neighborhood radiusof 10 for this calculation. The third term describes the formation of adhesivecontacts of cells with the substrate, where the bond energies lower the totalenergy [78], causing the cells to spread. The parameter λ C is the correspondingLagrange multiplier. The energy gain of occupying more lattice sites saturateswith the cell area, as the total number of binding sites is limited. The parameter A h regulates this saturation.To describe cell-matrix binding via focal adhesions, we implement the fol-lowing yield energy in the CPM Y = λ N N ( (cid:126)x (cid:48) ) − N N h + N ( (cid:126)x (cid:48) ) · s ( (cid:126)x (cid:48) )=1 · s ( (cid:126)x )=0 , (4)where N ( (cid:126)x (cid:48) ) is the size of the focal adhesion at the target site. This modelsthat a retraction is energetically costly for a cell to make, because it needsto break the actin-integrin connection. We assume that the size of the actin-integrin link is proportional to the size of the focal adhesion, i.e. the numberof integrin bonds [80], and that the strength of a focal adhesion saturates [20]with a parameter N h . The substraction of N represents that a focal adhesion21nly creates extra linkage if it is greater than a nascent adhesion. Note thatthe Y can not become negative, because we assume that focal adhesions smallerthan N , a nascent adhesion, breaks down due to its short lifetime, see section‘Focal Adhesions’. So, only focal adhesions larger than N create a yield energy.In section ‘Substrate stresses’, we further adapt this yield energy to describe amatrix stress induced focal adhesion reinforcement. Cell traction forces
Following Schwarz et al. [42], we assume that traction forces are generated bymyosin molecular motors on the actin fibers, of which the velocity is given by v ( (cid:126)F ) = v (cid:16) − (cid:126)F / (cid:126)F s (cid:17) , (5)where v is a free velocity. The traction forces are applied to the ECM, whichwe assume is in plane stress. The constitutive equation is given by h(cid:126) ∇ σ = (cid:126)F where σ is the ECM stress tensor and h is the thickness of the ECM. We assumethat the ECM is isotropic, uniform, linearly elastic and we assume infinitesimalstrain theory. We solve this equation using a Finite Element Model (FEM) (seesection ‘Substrate stresses’). In the FEM, traction field (cid:126)f and ECM deformation (cid:126)u are related by: K (cid:126)u = (cid:126)f , (6)where K is the global stiffness matrix given by assemblying the local stiffnessmatrices K e for each lattice site e K e = h (cid:90) B T E − ν ν ν − ν B , (7)where B is the conventional strain-displacement matrix for a four-noded quadri-lateral element [81] and E is the Young’s modulus and ν is the Poisson’s ratioof the ECM. For more details on this part of the model, we refer to our previouswork [26, 43].Following Schwarz et al. [42], the force build-up is given by the ODE: K (cid:126)v ( (cid:126)f ) = d (cid:126)fdt , (8)The matrix K describes force interactions between neighbouring nodes in theECM. However, since solving this equation is expensive, we ignore the interac-tions between neighbouring sites, i.e. , we reduce K to a scalar for each site (cid:126)x .This gives us, (cid:126)F ( (cid:126)x, t ) = (cid:126)F s ( (cid:126)x ) + ( (cid:126)F ( (cid:126)x ) − (cid:126)F s ( (cid:126)x )) exp( − t/t k ) , (9)where (cid:126)F is the force already exerted by the actin and t k = | (cid:126)F s | v K . Here, K is givenby the diagonal entry of K at site (cid:126)x , i.e. the stiffness of this node, neglecting22hanges in local stiffness due to connections to neighbouring nodes describedin the off-diagonal entries of K . Since the cell configuration and therefore thetraction forces change each MCS, the tension on the focal adhesions does notbuild up from zero, but from the tension that was built up during the previousMCS: (cid:126)F at the current MCS is given by (cid:126)F ( t FA ) of the previous MCS.To calculate the stall force of the actin fibers, (cid:126)F s , we employ the empiricalfirst-moment-of-area (FMA) model [44]. This model infers the stall forces fromthe shape of the cell of the CPM, based on the assumption that a network ofactin fibers in the cell acts as a single, cohesive unit, (cid:126)F s ( (cid:126)x ) = µA (cid:88) { (cid:126)y ∈ C | [ (cid:126)y(cid:126)x ] ⊂ C } (cid:126)x − (cid:126)y. (10)So, the force at site (cid:126)x is calculated as the sum of forces between (cid:126)x and all othersites (cid:126)y within cell C that are connected to (cid:126)x (this sum excludes line segments[ (cid:126)y(cid:126)x ] running outside the cell that occur if the shape of cell C is non-convex).The force is assumed to be proportional to the distance between the sites. Wedivide over the cell area A such that force increases roughly linear with cell area,as experimentally observed [50]. Focal adhesions
At each lattice site occupied by the cell, (cid:126)x ∈ C , a focal adhesion is modeledas a cluster of bound integrin bonds N . Each individual integrin bond behavesas a catch-slip bond, whose lifetime is maximal under a positive force [41].Accordingly, the growth of a cluster of such bonds is described by the ODE-model derived by Novikova and Storm [41], dN ( (cid:126)x, t ) dt = γN a ( t ) (cid:18) − N ( (cid:126)x, t ) N b (cid:19) − d ( φ ( (cid:126)x, t )) N ( (cid:126)x, t ) (11)with γ is the binding rate of integrins to the ECM, N a the number of free bonds,and N b the maximum number of bound bonds a lattice site can contain. Thislogistic growth term is a slight adaptation compared to Novikova and Storm [41].This additional term was added to avoid packing more integrins in a lattice site,than the size of a lattice site (∆ x ) can accomodate for. The degradation of thefocal adhesions d ( φ ) depends on the tension φ on the focal adhesion N . Thisdegradation rate is given by d ( φ ( (cid:126)x, t )) = exp (cid:18) φ ( (cid:126)x, t ) N ( (cid:126)x, t ) − φ s (cid:19) + exp (cid:18) − φ ( (cid:126)x, t ) N ( (cid:126)x, t ) + φ c (cid:19) (12)where φ s and φ c describe the slip and catch bond regime in N/m , respectively.Here, φ ( (cid:126)x, t ) = | (cid:126)F ( (cid:126)x,t ) | ∆ x is the stress applied to the lattice site of the focal adhe-sion. We assume that the number of free bonds N a is limited by the numberof available integrin receptors in the entire cell, N m . These N m receptors canbe recruited to each focal adhesion site and enable binding of a bond. Thus,23 a ( t ) = N m − (cid:80) (cid:126)x ∈ C N ( (cid:126)x, t ). We let the focal adhesions grow after each MCSfor t FA seconds with time increments of ∆ t FA . If there is no pre-existing focaladhesion at site (cid:126)x ∈ C , we set N ( (cid:126)x ) = N , so that at this site, a new initial ad-hesion is formed. This assumption represents the generation of focal complexesor nascent adhesions, precursors of focal adhesions that contain a small amountof integrins and have a very short lifetime[82]. Also, after the focal adhesionswere allowed to grow, i.e. after t = t FA seconds, we set all N ( (cid:126)x ) < N back to N ( (cid:126)x ) = N , again modeling the quick (re)generation of nascent adhesions.When a site (cid:126)x is removed from the cell C after a retraction, such that s ( (cid:126)x ) = 0, we set N ( (cid:126)x ) = 0 reflecting the destruction of the focal adhesion. Weassume that if a cell extends, i.e. a site (cid:126)x is added to the cell C , a nascentadhesion is formed: we set N ( (cid:126)x ) = N . Substrate stresses
The forces that were build up during a MCS, (cid:126)F ( t FA ) are applied as planar forcesto a finite element model (FEM). The FEM calculates the stress tensor σ ( (cid:126)x ) oneach lattice site. We assume that the integrin-cytoskeletal adhesion strengthensas a result of stress. We define g ( σ ) = (cid:40) ( σ xx + σ yy ) if ( σ xx + σ yy ) ≥
00 if ( σ xx + σ yy ) < Y = λ N N ( (cid:126)x (cid:48) ) − N N h + N ( (cid:126)x (cid:48) ) · (cid:18) p g ( σ ( (cid:126)x (cid:48) )) σ h + g ( σ ( (cid:126)x (cid:48) )) (cid:19) · (cid:126)x (cid:48)∈ C · (cid:126)x/ ∈ C (14)We thus assume that stress strengthens the focal adhesions, with parameter p and that this strengthening saturates with parameter σ h . Stiffness gradient
To study durotaxis, we model a stiffness gradient in the x -direction on a latticeof 1250 µ m by 500 µ m. The Young’s modulus of the substrate E ( P a ) is givenby E ( x ) = max { , x − · slope } , with x in µ m , such that the Young’smodulus at the center of the cell at time t = 0 is 6000 Pa and is nonzero. Thedefault value for the slope is 20 Pa/ µ m. This work was part of the research program Innovational Research IncentivesScheme Vidi Cross-divisional 2010 ALW with project number 864.10.009 toRMHM, which is (partly) financed by the Netherlands Organization for Scien-tific Research (NWO). 24 eferences [1] Califano JP, Reinhart-King CA (2010) Substrate stiffness and cell areapredict cellular traction stresses in single cells and cells in contact.
Cellularand Molecular Bioengineering
Proceedings of the National academy ofSciences of the United States of America
Soft Matter
Biophysical Journal
Biophysical Journal
Biophysical Journal
Biophysical Journal
Biotechnology Journal
Biochimica et Biophysica Acta
Annual Review Cell Developmental Biology
Nature Cell Biology
Nat Rev Mol CellBiol
Experimental Cell Research
Nature Cell Biology
Current Opinion in Cell Biology
Cell
Proceedings of the National academy of Sciences of the UnitedStates of America
Development
Science
Molecular Biology of the Cell
The Journal of Cell Biology
Proceedings ofthe Royal Society of London. Series B, Biological sciences
Journal of Theoretical Biology
Soft Matter
Interface Focus
PLoS Computational Biology
Biophysical Journal
Journal of Biomechanics
Physical Biology
Journal of Theoretical Biology
Physical Bi-ology
MathematicalBiosciences
Physical ReviewLetters
Physical Review E
Proceedings of the National academy of Sciences ofthe United States of America
Journal of the Mechanics and Physics of Solids
Biomechanics and Modeling in Mechanobi-ology
Journal of MathematicalBiology
Biophysical Journal
PLOS ONE
Biophysical Journal
Biosystems
Biophysical Journal
Biophysical Journal
Journal of Cell Science
Physiological Reports
Biophysical Journal
Cell Motility and the Cytoskeleton
Cancer Cell
Cellularand Molecular Bioengineering
Proceedings of the National academy of Sciences of the United States ofAmerica α -smooth muscle actin to stress fibers. The Journal of CellBiology
PLOS ONE
Frontiers in Bioscience
The International Journal of Biochemistry
Journal of Biological Chemistry
Journal ofCell Biology eLife
Proceedings of the Nationalacademy of Sciences of the United States of America
Journal of Applied Physiology: Respiratory, Environmentaland Exercise Physiology
Biology Open
ACS Nano
Developmental Cell α β α v β Proceedings of the National academy of Sciences of theUnited States of America
Proceedings of the National academy of Sciences ofthe United States of America
Nature Physics
PLoSComputational Biology
Physical Review E
Biophysical Journal arXiv:1810.11435. Preprint,posted Oct 26, 2018. [71] Feng J, Levine H, Mao X, Sander LM (2018) Stiffness sensing and cellmotility: Durotaxis and contact guidance. bioRxiv Preprint posted May11, 2018 .[72] Welf ES, Naik UP, Ogunnaike BA (2012) A spatial model for integrin clus-tering as a result of feedback between integrin activation and integrin bind-ing.
Biophysical Journal
Journal of Cell Biology
Cell
Biochemical Society Transactions
Biophysical Journal
Physical Review Letters
BiophysicalJournal
BMC Biophysics
Methods in Enzymology
The Finite Element Method: An Introduction With Par-tial Differential Equations . (Oxford University Press, Oxford).3082] Changede R, Sheetz M (2016) Prospects & overviews integrin and cadherinclusters: A robust way to organize adhesions for cell mechanics.
Bioessays
SIAM Journalon Applied Mathematics
Soft Matter upporting Material
Supplementary methods
In the main text, we proposed that matrix stress induces focal adhesion strength-ening but noted that matrix stress might also reinforce cell contractility. Sup-plementary Figure7 shows the results of having (cid:126)F s = (cid:126)F s · (cid:16) p g ( σ ( (cid:126)x (cid:48) )) σ h + g ( σ ( (cid:126)x (cid:48) )) (cid:17) instead of focal adhesion strengthening as described in the main text. Sincematrix stresses are defined on the lattice sites while forces are defined on thenodes of the lattice, we needed to assume some interpolation. We choose to take (cid:126)F s = (cid:126)F s · (cid:88) surrounding4nodes (cid:18) p g ( σ ( (cid:126)x (cid:48) )) σ h + g ( σ ( (cid:126)x (cid:48) )) (cid:19) . (15) Supplementary videos
Video S1
Cell spreading on substrates of 1,5 and 50kPa (Model 1).These are time series of Figure 2A of 500 MCS,.
Video S2
Cell spreading on substrates of 1,5 and 50kPa (Model 2-version1).These are time series of Figure 3A of 500 MCS,.
Video S3
Cell spreading on substrates of 1,5 and 50kPa (Model 2-version2).These are time series of Figure S7A of 500 MCS.
Video S4
Cell durotacting on substrate with rigidity gradient 20 Pa/ µ m .32arameter description value unit value was CPM ∆ x lattice site width 2.5 µ m chosen λ area constraint/cellstiffness 0.0002 N/m per latticesite chosen J (0 , cell) adhesive energy 3000 Nm per latticesite chosen nbo neighbourhood ra-dius for adhesiveenergy 10 - estimated basedon accuracy of linetension [79] λ C adhesion strength 600 Nm per latticesite chosen A h area saturation 1000 lattice sites chosen λ N focal adhesionstrength 4 Nm chosen p actin-integrinstrength 1 - chosen σ h saturation actin-integrin binding 5000 N/ m chosen T cellular tempera-ture 2 Nm chosen Forces µ traction magnitude 0.001 Nm per latticesite estimated based onendothelial tractionstresses [1] v free velocity ofmyosin molecules 100 nm/s estimated based onnon-muscle myosinIIB [56, 58] E Young’s modulus 10000 N/ m varies ν Poisson’s ratio 0.45 - chosen τ substrate thickness 10 µ m [83] FA’s γ growth rate 0.05 /s estimated [41] N size initial adhesion 5000 - estimated based onnascent adhesions[82] N m maximum freebonds 8000000 - chosen N b maximum size focaladhesion 39062 - estimated based onnumber of integrinsthat fit in one lat-tice site [82] φ s slip tension 4.02 pN/ m [41] φ c catch tension 7.76 pN/ m [41] t FA focal adhesiongrowth time 10 s estimated based onprotrusion lifetimes[84]∆ t FA time steps 0.01 s chosenTable S1. Parameter values.33igure S1. The number of integrin bonds per cluster ( N ) in model M1 as afunction of distance from the cell center. All clusters at 2000 MCS from 25simulations were pooled. Shaded regions show standard deviations.34igure S2. Model sensitivity to actin-integrin strength p . (A) Cell area asa function of substrate stiffness, shaded regions: standard deviations over 25simulations; (B) Cell eccentricity as a function of substrate stiffness, shadedregions: standard deviations over 25 simulations.Figure S3. Model sensitivity to saturation value for actin-integrin strength σ h .(A) Cell area as a function of substrate stiffness, shaded regions: standarddeviations over 25 simulations; (B) Cell eccentricity as a function of substratestiffness, shaded regions: standard deviations over 25 simulations.35igure S4. Model sensitivity to focal adhesion growth time t FA . (A) Cell areaas a function of substrate stiffness, shaded regions: standard deviations over25 simulations; (B) Cell eccentricity as a function of substrate stiffness, shadedregions: standard deviations over 25 simulations.Figure S5. Model sensitivity to cellular temperature T . (A) Cell area as afunction of substrate stiffness, shaded regions: standard deviations over 25 sim-ulations; (B) Cell eccentricity as a function of substrate stiffness, shaded regions:standard deviations over 25 simulations.36igure S6. Model sensitivity to traction force magnitude µ . (A) Cell area asa function of substrate stiffness, shaded regions: standard deviations over 25simulations; (B) Cell eccentricity as a function of substrate stiffness, shadedregions: standard deviations over 25 simulations.37igure S7. Cells elongate on substrates of intermediate stiffness. Model M2 wasused where matrix stress reinforces traction force (cid:126)F s with p = 5. (A) Exampleconfigurations of cells at 2000 MCS on substrates of 1,50 and 50 kPa. Col-ors: hydrostatic stress; (B) Cell eccentricity as a function of substrate stiffness,shaded regions: standard deviations over 25 simulations; (C) distribution of N,the number of integrin bonds per cluster, all focal adhesion at 2000 MCS from25 simulations were pooled. We indicate the median. Color coding (C): Seelegend next to (C). 38igure S8. Durotaxis speed in µ m/h as a function of cell stiffness λ and cellulartemperature T . Values: mean ±±