## Abstract

When crawling on a flat substrate, living cells exert forces on it via adhesive contacts, enabling them to build up tension within their cytoskeleton and to change shape. The measurement of these forces has been made possible by traction force microscopy (TFM), a technique which has allowed us to obtain time-resolved traction force maps during cell migration. This cell ‘footprint’ is, however, not sufficient to understand the details of the mechanics of migration, that is how cytoskeletal elements (respectively, adhesion complexes) are put under tension and reinforce or deform (respectively, mature and/or unbind) as a result. In a recent paper, we have validated a rheological model of actomyosin linking tension, deformation and myosin activity. Here, we complement this model with tentative models of the mechanics of adhesion and explore how closely these models can predict the traction forces that we recover from experimental measurements during cell migration. The resulting mathematical problem is a PDE set on the experimentally observed domain, which we solve using a finite-element approach. The four parameters of the model can then be adjusted by comparison with experimental results on a single frame of an experiment, and then used to test the predictive power of the model for following frames and other experiments. It is found that the basic pattern of traction forces is robustly predicted by the model and fixed parameters as a function of current geometry only.

## 1. Introduction

During immunoresponse and cancer metastasis formation, cells crawl on the blood vessel wall [1]. This type of cell motion has been reproduced *in vitro* [2] and has been the subject of many modelling studies [3–5].

As inertial effects and body forces are vanishingly small in this process, all forces are instantaneously balanced in the system and, in particular, the resultant of the traction forces that the cell exerts on its environment has to be zero. The motion of the crawling cell is thus necessarily driven by its deformation, as the cell changes shape both by growth and shrinkage due to (de)polymerization at its leading edges [6,7] and simultaneously as actin cytoskeleton undergoes a persistent centripetal deformation, called retrograde flow [8–10]. To result in a net displacement of the cell with respect to its surroundings, forces need to be transmitted to it. Although fluid drag and non-specific interactions with the solid substrate are present, most of the stress is transmitted via specific adhesion interaction between ligands present on the surface of the substrate and transmembrane receptors which are bound to the actin cytoskeleton [11].

The mechanical models attempting to explain cell migration from the dynamics of its microstructure are thus focusing on the dynamics of actomyosin and adhesion complexes [12]. Important modelling efforts have been made to understand the initiation and maintenance of motility [13–17]. These works are using the simple and stable shape of keratocyte cells, or a one-dimensional (1D) simplification. While this allows for a fine understanding of possible detailed mechanisms of motility and is based on the same mechanisms of actin (de)polymerization and myosin-driven retrograde flow, the migration of keratocyte does not present the cycle of events observed in most other cell types during migration [18]. Moreover, although these models are shown to fit experimental results, in general, they have not yet been exploited in a systematic manner. Indeed, quantitative prediction of cell dynamics and exerted forces is only at its beginning [19–21]. Whereas fitting models remains necessary to acquire a minimal set of parameters from a subset of the experiments available in all cases, these recent papers are additionally predictive of other experimental conditions for which no further adjustment is done.

Here we combine the prediction of a simple yet quantitatively validated rheological model of actomyosin [19] with a nonlinear model of cell adhesion adapted from [16] and simulate it on the actual geometry of cells tracked while crawling. Monitoring the deformation of the substrate during the experiment [22] and a traction force microscopy (TFM) method [23–27] allow us to compute independently the traction forces that the cell exerts on the substrate, to which the predicted traction fields can be compared. The number of adjustable parameters is reduced to a minimum (two for the linear model, four for the nonlinear adhesion model) and the robustness of the parameter choice is assessed in a systematic way. The preditive capability of the model is tested over different cell migration events and cell types.

Experimental observations in the literature give consistent pictures of two different scales: the microscopic scale, at which the dynamics of the relevant molecules are well described (actin, actin-binding molecules and adhesion molecules), and the mesoscopic scale of the cell itself. Our approach is to write a mechanical model based on the microscale knowledge, and to investigate how these microscale dynamics yield the emergent mesoscale behaviours that are observed. In a previous paper [19], we have successfully used this type of approach and validated quantitatively at the mesoscale a rheological constitutive law based on a microscale model of actomyosin dynamics.

However, the set-up used in that work did not require a precise model of the mechanics of cell adhesion, whereas this is needed here in order to address cell crawling. Cell adhesion models of graded complexity have been introduced by many authors [16,28,29]. Our numerical resolution procedure and quantitative comparison with experimental data allow us to investigate which of these models match best the observations.

## 2. Mesoscale experimental observations

We observe cancer cells from two different cancer cell lines (T24 and RT112 cell lines) which are crawling on a relatively stiff gel, *E* = 10 kPa. Our experimental observations, traction force recovery and the relevant other observations in the literature are described briefly in appendix A(a) and A(b), and at length in [27]. Here we summarize only some salliant features that will need to be accounted for by the model predictions.

When plated on the gel, T24 cells assume an elongated and digitated shape, while RT112 cells have a rounded shape (figure 1). For both lines, as well as in all other observations in the literature, the traction forces they exert are minimal close to their geometric centre and increases distally (close to the cell edge). The tractions are oriented approximately along the normal to the cell edge, pointing inwards (figure 1*c*). The rate of increase of traction force along an imaginary line from cell centroid to cell edge is greater when the cell edge is close to centroid, but the intensity of traction goes generally to much larger values close to the part of cell edges which are more distal (farther away from centroid). In many instances, the maxima of the traction field are not situated right at the cell edge but somewhat proximal (inwards) from it, and tractions can be vanishingly small at the edge. In what follows, we will attempt to link these observations with both the phenomenology of adhesion complexes (as done, e.g. in [30]) and with the mechanics of the cytoskeleton.

## 3. Microscale-based mechanical model

It is well established experimentally that the mechanical properties of crawling cells are controlled by their actin structures and the proteins that bind to actin [2]. Although other cytoskeletal components have a smaller contribution in the mechanical balance, we will neglect them in what follows. Above a timescale between 0.1 and 10 s, the pressure in the cytosol equilibrates [31] and the poroelastic behaviour of the cell becomes negligible. Thus, the variable of interest is the stress tensor *σ*^{3D} in the actin meshwork (and its strain and rate-of-strain tensors *ɛ*^{3D} and , see table 1 for a list of model variables and parameters) at any position in the cell. In the absence of inertia and at sufficiently long timescale, the force balance is written as
where ** F** are bulk forces, discussed below. This internal stress of the actin needs to be balanced at the boundaries of the actin meshwork: thus the actin stress acting at boundaries where adhesion molecules are present is equal to the stress

**that these exert on the environment at its boundaries.**

*T*As the actin flows, its density ** ρ** will evolve. However, there is strong experimental evidence that the density of actin is tightly regulated by filament nucleators and various molecules favouring growth or shrinkage of actin filaments [32]. Denoting by the target density field of this regulation, and assuming that it has a characteristic time

*τ*, we can describe this with an advection–reaction equation: where

_{ρ}

*v*^{3D}is the velocity of actin. In what follows, we will assume , where

*D*is the cell diameter, which leads to the solution of the above equation , a uniform constant. This is supported by fluorescent speckle microscopy [33] and indeed, although there are visible local variations in the actin density on fluorescence images, the distribution of fluorescence intensity is well peaked and has a standard deviation of less than 50%.

When adhering on a substrate of sufficiently large stiffness [34], cells spread until their projected surface has a diameter *D* of the order 40 (RT112) to 100 µm (T24), and present a flat structure called the lamella [33] whose height is typically less than a few micrometres, and a dome-like structure referred to as the actin-cap, under which the nucleus is located, which has a height of 5–10 µm [35]. In this work, we will consider that this aspect ratio is sufficiently small to justify a two-dimensional (2D) approach in which the cell is treated as a thin layer of thickness *h* subject to tangential surface tractions ** T** and bulk forces

**only, but it should be borne in mind that this is only a first approximation. In this context, we will also consider that the variations of**

*F**h*are small. This geometric setting constrains some components of the 3D stress tensor

*σ*^{3D}: we have and for [36] (figure 2

*e*) which allows to rewrite the problem in terms of a 2D stress tensor, with components , and write the 2D force balance as See figure 2

*a,b*for a sketch of this mechanical balance. In practice,

**=**

*F***0**as bulk forces such as gravity are negligible and no external force is applied on the cell. At the cell edge (which is noted experimentally to correspond to the contact line of the cell with the substrate, ), there is no specific line force so the stress tensor

**along this boundary must vanish (figure 2**

*σ**b*) 3.1The stress tensor

**needs then to be related with the deformations by a constitutive law. As most of the intracellular stress is borne by the actin meshwork, it is the deformation of actin which is relevant to study in a first approximation. We have derived and validated such a rheology for the actomyosin cortex in [19], by taking into account the dynamics of cross-linkers that bind to actin and the input of mechanical energy by the myosin motor molecules. The dynamics of cross-linkers is modelled by a single constant residence time,**

*σ**τ*, which was found to be of the order of 10

_{α}^{3}s for two cell types (fibroblasts and myoblasts). The myosin molecular motors, which are among the cross-links that bind and unbind, are in addition responsible for a supplementary term in the constitutive relation, which reads as a pre-stress , where

*α*

_{myo}is the fraction of cross-linkers which are myosin filaments, and effectuate a power-stroke of step length at frequency 1/

*τ*

_{myo}.

*G*is the elastic modulus of the cross-linked actin network and

*β*is inversely proportional to its Kuhn length. In [37], we have shown that the fluorescence intensity of labelled myosin molecules could provide a proxy for the local variations of

*α*

_{myo}; however here, in the absence of measurements for most of individual terms in

*σ*

_{a}, we will treat it as a constant globally, as was done successfully in [19]. For the sake of simplicity, we will also assume that the actin filaments are isotropically distributed in the plane parallel to the substrate, and define the pre-stress tensor . With these assumptions, the constitutive equation writes 3.2where is the rate-of-strain tensor, the objective upper-convected time-derivative of the stress tensor

*and*

**σ***κ*is Lamé's first coefficient. The group has the dimension of a viscosity and this term corresponds to the energy dissipated in (slowly) deforming the actomyosin network [38] in an irreversible manner due to cross-linker unbinding [19]. This balance of stresses and the deformations linked with the viscous stresses are described in figure 2

*c,d*. This rheology is in line with early models of actomyosin [39] and active gels models [40]. We do not supplement this contractile liquid behaviour with an elastic resistance term of the cell, contrarily to what is done in mechanosensing [41,42] and migration models [17]. Other models have also considered the same rheology for cell migration, with an additional inertial term used for numerical stabilization [4].

Finally, a rheological law must be proposed for the relationship between the traction forces ** T** and the relative displacement between the substrate and the cell [3,36]. Here again, the relevant structure to define a displacement within the cell is the actin meshwork, which is mechanically bound to adhesion molecules. In a first approximation, we assume that the substrate displacement rate is small compared with the velocity of the actin within the cell, . This hypothesis will need to be questioned in future work, as experiments show that substrate displacement rates are of the order of 10

^{−3}µm s

^{−1}, comparable with the speed of retrograde flow. With this simplifying hypothesis, the relative speed of actin with respect to the substrate is the velocity

**of actin in the laboratory reference frame and . On the ground that adhesion complexes are very dynamic [11], we can expect a friction-like behaviour (figure 2**

*v**f*) 3.3where

*c*

_{f}may depend on the velocity |

**|. This is assumed in many modelling approaches, including [17,28] where**

*v**c*

_{f}is taken constant, and in [43] where position-dependent and dissymmetric adhesion is implemented.

However, experiments show that although local traction force and actin velocity are positively correlated on the whole, above some critical velocity, the traction forces drop by several folds [16,30], indicating that *c*_{f} is not uniformly constant. The causes of the drop in *c*_{f} are only speculated, as locations with a lower *c*_{f} share three related characteristics: the actin flow speed is above a threshold *v** which seems to be a cell-type specificity ( in fast-migrating keratocytes [16], in mammalian Ptk1 epithelial cells [30]), they are located more distally and adhesion complexes there are less mature [30]. Barnhart *et al*. [16] propose a differential equation that implements these different effects. Note that the precise role of the different terms in this model has not been experimentally tested so far, and Barnhart *et al*. [16] claim that a simple algebraic relation as exemplified in figure 2*g* captures the phenomenology. We choose to use the same algebraic relationship as they do, , in order to test whether it is able to produce a fair quantitative prediction of traction stresses.

The modelling discussed so far describes only a (dynamic) mechanical equilibrium, giving a snapshot of the force balance and rate of strain of the actin network. It is not in the scope of this paper to try to predict the subsequent dynamics of the cell, which would require to describe also processes such as actin polymerization-based protrusion [2], or to provide some other more or less explicit dissymetrization of the dynamics. Indeed, in order to obtain a persisting motility, models require a more or less explicit dissymetrization either of the actin treadmilling [28,44], of the myosin contractility *σ*_{a} [16,45] or of the curvature of a contractile structure [15]. In one dimension, the need for such an effect can be proved [46]. Here, we show that the traction forces observed in cells are not dominated by this dissymetrical component, since an entirely symmetrical model allows us to give good predictions of the observations. Further refinement of the model and comparison with the experiments, and new experiments tracking explicitely myosin and adhesion molecules will be needed in the future to address this question.

Two length scales appear in the problem: one, the cell diameter *D*, is directly observable but depends on cell type and fluctuates during migration, the other is a friction length . The speed of retrograde flow *v*_{t} measured at the leading edge in [47] and calculated from the modelling of cell-scale experiments in [19] is convenient to scale the rate of deformation of the cytoskeleton: as *v*_{t} is of the order 10^{−3}–10^{−2} µm s^{−1}, and the cell radius of the order of tens of micrometres, a characteristic time is *T* = 10^{3}–10^{4} s. The other timescale in the problem is the relaxation time *τ _{α}*, in [19], we find : thus the visco-elastic term can be expected to be of a lower magnitude than

*in the consitutive equation (3.2), although it is not*

**σ***a priori*negligible. If we choose to scale stresses with , the non-dimensional model is 3.4

*a*3.4

*b*where we introduce the friction non-dimensional number , based on a friction length for some reference friction coefficient and the Deborah number .

## 4. Results

### 4.1. One-dimensional model predictions

In order to reach a good understanding of the mechanical balance that the model describes, we first study it in 1D. The cell is assumed to occupy a segment of the real axis, and as we do not introduce explicitly a dissymetry, our model will predict a solution which is symmetrical with respect to the midpoint of this segment. This simplistic setting thus corresponds to cells which are not migrating or polarized. The non-dimensional equations for the permanent regime on the domain (−1,1) are
4.1*a*
4.1*b*
4.1*c*where we have slightly changed the non-dimensionalization, using instead of *G* when scaling stresses. Note the nonlinear coupling term , which arises from the upper-convected objective derivative of the stress tensor, in [36,43], an analogue 1D model with the co-rotational derivative is proposed.

Analytical solutions can be obtained in the case De = 0, and numerical approximations otherwise, the solution procedure is described in the electronic supplementary material, text A. An example of dynamical mechanical equilibrium is shown in figure 3*b*, for the case of a uniform friction number *ζ* = *ζ*_{0} and a vanishing relaxation time De = 0. A centripetal flow
corresponding to the retrograde flow, is observed [36]. It is maximal close to the cell edge, and its intensity is proportional to the myosin activity *σ*_{a}, consistent with experiments [33,47]. The traction force density *T* is by hypothesis directly proportional to the retrograde flow here, and thus centripetal and maximal close to cell edges too. The stress in the actomyosin vanishes at the cell edge, which corresponds to our imposed boundary condition, and is maximal at the cell centre in proportion with myosin activity *σ*_{a}:
It is thus myosin activity that puts the actin network under tension, a tension which is balanced at the cell edges by friction via the friction number *ζ*. If *ζ* is large, the tension is constant in most of the domain and balanced by traction forces in a narrow zone close to the edge (figure 3*b*).

In the case when De = 0.1, which corresponds to our estimates for experimental cases, the solution differs little from the De = 0 solution (electronic supplementary material, figure S3*a*). For De ≥ 1, a quantitative deviation can be appreciated (electronic supplementary material, figure S3*b*), but there is no strong qualitative difference. We will thus focus on the case De = 0 from this point, as this allows analytical 1D calculations and less involved numerical simulations in 2D (although the visco-elastic case De > 0 can be treated, see e.g. [48] using the approach in [49]).

We then turn to the case of a nonlinear friction number *ζ* = *ζ*(|** v**|), in order to allow analytical resolution we take a piecewise constant function (figure 2

*g*): 4.2with

*ζ*

_{1}<

*ζ*

_{0}consistently with observations [16,30]. In figure 3

*d*, we see that this results in a more uniform retrograde flow close to the cell edge, which is due to the fact that there is less of a gradient of tension there, due to a lower local friction. Because of the low friction number locally, traction is low close to the cell edge, which is the phenomenology that we wanted to reproduce.

Numerically, we can solve with (figure 2*g*) which is closer to the experimental observations [30], this gives a smoother behaviour but no essential qualitative difference.

### 4.2. Prediction of the traction field of a motile cell

Turning now to a 2D approach, we ask whether for a given cell shape, the above model can predict the traction pattern that we measure by TFM, as in figure 1. Starting from the data of *Ω*_{c} for a typical experimental result, we calculate using a finite-element approach (see appendix A(c)) the traction field for a linear friction law (*c*_{f} independent of ** v**) and a choice of the two parameters: friction number

*ζ*

_{0}and myosin contractility

*σ*

_{a}. As in 1D, the magnitude of the stress, retrograde flow and traction forces are in direct proportion to

*σ*

_{a}(see the electronic supplementary material, text B.2), while changes in

*ζ*

_{0}will modify the pattern of retrograde flow and traction forces. In consequence, we can find explicitly an optimal value for

*σ*

_{a}for a given value of

*ζ*

_{0}, and optimize for this parameter in order to get the best fitting approximation of the experimental traction field (figure 4

*a*). The relative error on the traction vectors is , that is, the error vectors have a magnitude 0.73 times the experimental vectors

*T*_{exp}in

*L*

^{2}norm. This can be considered a fair result for a two-parameter fit of a vector field with rich features. Indeed, the experimental traction field at this instant indicates that there is probably no mature adhesion under the protrusion visible on the left hand side, since the traction field decreases dramatically there. This feature cannot be described by this first model where the friction number is taken to be a homogeneous constant, thus limiting the possibility to approximate the experimental observations. The relative error on the intensity of traction is , thus the error is shared between a mismatch between experimental and predicted magnitude of traction field, and some misalignment of the experimental and predicted traction vectors. The optimal parameters are found to be , which implies a friction length of the order of 10 µm, and . Using the numerical value of parameters

*τ*and

_{α}*v*

_{t}that we have obtained in [19] for two other cell types, this would imply that the myosin pre-stress

*σ*

_{a}is about three times the zero-shear elastic modulus

*G*of the actomyosin meshwork, in [19] the same ratio was found to be 4.3.

During the experiments, cell position and shape and traction forces were acquired at several instants *t ^{i}* 2 min apart. We now ask whether our model and the parameters and can, for these different cell shapes , predict the traction forces without further parameter adjustment. We find that for the 21 frames of the experimental results, the relative error on traction vectors ranges from 0.53 to a maximum of 0.75, with a median of 0.60, which indicates that the parameters that where optimal for one instant give as good results for other frames (figure 4

*c,e*) even though the aspect ratio of the cell changes by a factor 2 depending on the stage of the migration process. The relative error on traction intensities ranges from 0.48 to 0.61, with a median of 0.57.

Thus, the parameters describing the mechanical behaviour of a migrating T24 cell are approximately conserved over more than half an hour during crawling. We then tested whether these parameters could also predict a good approximation of the traction stresses exerted by another cell of the same type in the same conditions. The comparison of the predictions of the model, using the same values and as for the first T24 cell, are shown in the electronic supplementary material, figure S4, the error of the predicted traction field varies in the same range as for the first cell (figure 4*e*).

Next, we considered the other cell type that was studied experimentally (RT112 cells). These cells spread to a much lower area on the substrate and maintain a rounded shape which evolves little in the course of migration. Although these characteristics are reminiscent of amoeboid migration, RT112 crawl at smaller speed and exert larger tractions than T24 [27]. All these characteristics make RT112 cells very different from T24, and we asked whether this is due to a different mechanical balance altogether, or different quantitative importance of several effects. However, when applying the same mechanical model as above with the parameters and , a fair approximation of the traction field obtained experimentally is recovered (figure 4*b,d*). This is equally true for the all 20 frames of this RT112 experiment (electronic supplementary material, figure S5) the median of the relative error on traction vectors is 0.63 and its range from 0.57 to 0.71, and traction intensities have a median error of 0.52 and a range from 0.43 to 0.57 (figure 4*e*).

For both RT112 and T24 cell types, experimental results show that the traction field maxima are generally not located quite at the cell edge, but somewhat proximal from it. As seen in the 1D calculations (figure 3*d*), this is compatible with the model provided that there is a switch from a large friction number *ζ*_{0} to a lower one *ζ*_{1} above some relative velocity *v**. For an appropriate choice of these parameters, the model prediction is improved (figure 5*a*). The median error on traction vectors drops from 0.73 to 0.61 and the one on the traction intensity from 0.59 to 0.48, the better agreement is not only in a large part due to a better match in magnitude of the traction in the protrusion at the cell front, but also in a better match of the directionality of tractions in this area.

Next, we asked whether this choice of parameters is robust in the same sense as the choice of and allowed the linear model to be predictive of the tractions at later migration stages and for other cells. We find that if *v** and *ζ*_{1} are assumed to remain constant across the different stages of migration, the nonlinear model does not significantly improve the match between predictions and experiments. However, if *v** is adjusted for each frame, we obtain a significant decrease of the error (figure 5*b*). In figure 5*c*, it is seen that the characteristic pattern of traction increase towards distal followed by an abrupt drop near the foremost areas of protrusions is recovered by this adhesion model.

## 5. Discussion

Cell migration is known to rely on a complex machinery subject to a large number of regulatory pathways [2]. Here, however, we show that cells which are migrating in a smooth way without shape changes (RT112) and cells which exhibit a multi-stage cycle of protrusion and retraction (T24) have a similar mechanical behaviour, whose baseline can be predicted with a simple model of actomyosin contraction and of adhesion. In particular, there is no obvious adjustment of the global mechanical balance corresponding to specific steps of mesenchymal migration, since all can be approximated with the same model parameters, which describe the myosin activity (pre-stress) and the effective friction due to the dynamics of adhesions.

This tends to imply that in a significant measure, the dynamics of actomyosin and of adhesions are not orchestrated at the time scale of the migration steps, but left to their self-organized assembly and disassembly pace. As shown in [19], this is not in contradiction with the ‘sensing’ properties of actomyosin, since the collective dynamics of actin and myosin give an emergent material which is intrinsically sensitive to the mechanical behaviour of the environment, even in the absence of a regulatory signal. This reliance of the mechanics of motility on self-organized processes is in agreement with the robustness of this behaviour—indeed even lamellar fragments of cells lacking nuclei, microtubules and most organelles can exhibit motility [13]. This is not to say that regulation via biochemical pathways would not be crucial to the migration process, but rather that they need to act only in order to tune an otherwise self-maintained system, just as adjusting the throttle control of an engine.

Beyond the fact that the global mechanical balance is well approximated, the traction fields observed contain many further details that cannot be captured by the linear model. The most prominent of these are the decreased traction forces at the leading edges, which are observed at most stages of mesenchymal migration in both T24 and RT112 cells, and had been noted in the literature before [16,30]. We find that these features can be reproduced by a nonlinear model which includes a velocity-dependent friction coefficient. In [30], they conclude from experimental data on local traction and actin velocity that the ageing and maturation of adhesion complexes are not correlated with their strength, and that the switch is based on a critical velocity *v**. Here we find, however, that the critical velocity *v**, contrarily to the myosin contractile pre-stress *σ*_{a}, varies at the scale of a few minutes during cell crawling. This suggests that *v** may be the target of a biochemical regulation that would be coupled with the migration steps.

## Competing interests

We declare we have no competing interests.

## Funding

J.E. thanks the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme *Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation* supported by EPSRC Grant Number EP/K032208/1. All authors thank Région Rhône-Alpes (CIBLE), ANR-12-BS09-0020-01 ‘Transmig’ and Tec21 (ANR-11-LABX-0030), and all except A.D. are member of GDR 3570 *MécaBio* and GDR 3070 *CellTiss* of CNRS.

## Appendix A. Methods

**(a) Cell crawling assays and image analysis**

The detailed protocol is described in [27]. Briefly, 10 kPa polyacrylamide gels were prepared with 0.2 µm fluorescent beads and coated with fibronection. Epithelial bladder cancer cell lines T24 and RT112 were seeded on the gel and left overnight to spread. Images were acquired with a time interval of 120 s for 2 h. The cell shape *Ω*_{c} was acquired either by transmission microscopy or by fluorescence reflection microscopy of actin-GFP transfected cells. Within a domain Ω (the field of view), the position of bead *j* in frame *i* is obtained with ‘ParticleTracker’ [50] in ImageJ software, the relaxed (also called initial) bead position is actually acquired at the end of the experiment by detaching cells using distilled water, allowing to construct the set of displacements (see [51] for details). *Ω* is chosen so that |*u*_{exp}| is vanishingly small close to . In general, we note that while the diameter of *Ω*_{c} is of the order of 50 µm.

**(b) Traction force inverse problem resolution**

Details can be found in [27,52,53]. The mechanical characterization of the gel is approximated as isotropic and elastic and the displacements are observed to be small enough so that linear elasticity applies. We assume that the gel can be approximated by a half-space {*z* ≤ 0}, the gel depth of 70 µm is sufficient to guarantee this for RT112 cells which have a diameter of 50 µm [54], but can introduce an error on some of the frames of the T24 cell migration, since it can transiently reach a length of the order of 100 µm. Following the study of Ambrosi [25], a reduced 2D problem is obtained by averaging in *z* over an effective thickness *w* (typically 1.5 µm) and the adjoint problem is written on in terms of an auxiliary unknown ** p**:
A 1

*a*and A 1

*b*where

*χ*

_{c}is the indicator function of

*Ω*

_{c}, , is the resultant of

**over**

*p**Ω*

_{c}and the reduced 2D Lamé coefficients are and . Problem (A 1) is solved for all frames

*i*using a finite-element method on a triangulation of

*Ω*, where are among the nodes and is a triangulation of a polygonal approximation

*Ω*

_{c,h}of order

*h*

^{2}of

*Ω*

_{c}. The calculated traction field is , which vanishes in

*Ω*\

*Ω*

_{c}and has zero resultant. For the same reason as the resultant is zero, the torque of the traction field also has to be zero. However, we do not enforce this currently in the method and the fact that the torque is small is only checked

*a posteriori*on the calculated traction field.

**(c) Finite-element simulations and parameter fitting**

In §3, we derive a tensorial visco-elastic model allowing to predict cell traction field *T*_{h} from the knowledge of cell shape *Ω*_{c} and some scalar parameters, namely *σ*_{a}, *ζ*_{0}, and in the nonlinear case *ζ*_{1} and *v**. The algorithm for the resolution of the full visco-elastic problem is given in [48]. Here we show in one dimension that within the range of experimentally relevant parameters, visco-elastic effects do not affect strongly the results. Thus simulations are shown only for the reduced viscous limit.

Briefly, we define a finite-element space *V*_{h} of piecewise quadratic functions based on the triangulation of at frame *i*, where is the cell domain *Ω*_{c} observed experimentally but with distances normalized by *D* = 50 µm. Using the finite-element software Rheolef, we solve the variational problem, electronic supplementary material, equation (9), in *V*_{h} for a given choice of and a uniform *ζ* = *ζ*_{0} for a solution . In the case of nonlinear friction, *ζ* depends on ** v**, a fixed-point algorithm is used to construct a sequence of solutions using the friction field , until convergence.

Next, the parameters can be optimized in order to best fit the experimental observations *T*_{exp}. Note that best fit is performed on one cell at the initial frame only throughout the paper to acquire parameters *σ*_{a} and *ζ*_{0}, no fitting is done for the other frames and the other cells. We normalize *T*_{exp} by a typical value 100 Pa, and aim to minimize by adjusting *ζ*_{0} and *σ*_{a}. Thanks to the linearity of operators and the scale-invariance of our choices for the function *ζ* (see the electronic supplementary material, text B.2), minimization for a given *ζ*_{0} with respect to *σ*_{a} can be done explicitly and the minimization writes:
In the rest of the article, we denote and simply by *Ω*_{c} and *T*_{exp}.

## Footnotes

One contribution of 12 to a theme issue ‘Coupling geometric partial differential equations with physics for cell morphology, motility and pattern formation’.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.