2.1 General overview of the model

We formulate a multiscale model of the mechanical interactions between cells and the extracellular matrix (ECM). The model comprises three building blocks. Firstly, an agent-based model captures cell shape changes caused by the mechanical stimuli from the ECM. Secondly, the ECM is modelled as an elastic material composed of elastic fibres that are interconnected by crosslinks. Finally, the cells and ECM dynamics are coupled via integrins. Integrins are cross-membrane receptors that cluster within FAs and bind ligands embedded within the ECM. Integrins are mechanosensitive and play a crucial role in transmitting mechanical stimuli from the ECM to cells. Fig 1A–1C illustrate the main components of our model and their interactions. We distinguish each fibre within the ECM, each cell and the FAs that mediate their interactions. Our model is force-based, i.e. the dynamics of the different components are driven by balancing the mechanical forces that act upon them. A schematic of these forces is presented in Fig 2.

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Fig 1. Schematic representation of the multiscale model.

(A) A cartoon illustrating the ECM network. Following [33], given the density of ECM fibres, we randomly distribute a set of points which are the seed of our ECM configuration. Then, the ECM network is generated by constructing a Voronoi tessellation on the seed points. The set of edges in this construction are the ECM fibres. We also illustrate different cell geometries: an elongated cell (purple), a round spread-out cell (green) and a fan-shaped cell (yellow). (B) Cells are represented as irregular polygons. For the elongated cells, where , we facilitate visualisation by plotting an ellipse whose major axis corresponds to the segment connecting the two cell vertices. Their vertices are given by the ends of viscoelastic segments that represent actin bundles that originate from the organising centre of the cell cytoskeleton (which does not necessarily coincide with the polygon centre of mass). These bundles are modelled as viscoelastic elements that deform when subject to forces exerted by the ECM which are transmitted by the integrins within the FAs. Cells can also contract under the action of myosin motors. Angular forces between these segments enable cells to maintain their geometry. The polygon vertices represent the sites where FAs are formed. (C) A cartoon showing our representation of FAs in which integrins, trans-membrane receptors which bind ECM fibres, are modelled as clusters of linear springs in parallel. (C) was created in BioRender [50].

https://doi.org/10.1371/journal.pcbi.1012698.g001

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Fig 2. A schematic representation of the forces included in the model.

(A) A zoom view of a randomly perturbed ECM region. The resultant force acting on the l-th crosslink is shown in red, and the elastic forces exerted by individual fibres at this crosslink are highlighted in purple. (B) A close-up view of the mechanical forces acting on an FA. In this illustration, the forces acting on the ECM are highlighted in red, while those acting on the cell are depicted in grey. In addition, we illustrate the cell-ECM interaction force, , acting between the ECM crosslink (in red) and the cell binding site (grey). (C) An illustration of the different structural and active forces incorporated into our cell model. From left to right: equilibrium cell configuration, elastic forces, angular forces and active contraction forces. Two different cell geometries are shown as representative examples: a round (at the top) and a fan-shaped cell (at the bottom).

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Model variables. Before detailing the various components of our model, we clarify the notation used, particularly we list the model’s elements and how we apply subindex and superindex notation to represent them. In our model simulations, we track the positions of ECM crosslinks, , positions at which individual matrix fibres are bound together, is the set of crosslinks. The number of cells in our model is given by , and each cell is represented by the index n. The cells are defined by their organising centre, whose position is given by , and a set of vertices, whose positions are denoted by . where the set of vertices of a certain cell n is . The number of vertices that define a cell is assumed constant and is given by . Each cell vertex represents a cell adhesion site (location where a FA is formed, i.e. interacting points). We also record the evolution of the number of bound integrins that form a FA and connect the cell vertex with a nearby ECM crosslink, . We list the variables of the system in Table 1.

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Table 1. Variables of the model.

https://doi.org/10.1371/journal.pcbi.1012698.t001

ECM model. We view the ECM as a network whose edges represent individual fibres and whose nodes represent crosslinks, i.e. sites at which individual matrix fibres are bound together. These bonds provide structural stability to the ECM network [51]. The position of each crosslink is referred to as where is the set of all crosslinks in the network, see Table 1. Following [33], we assume that the fibres are elastic and thus can be described by linear springs (see Fig 2A). In the absence of external forces, a crosslink evolves from an arbitrary initial position to an equilibrium configuration where all the elastic forces are in equilibrium. In addition to these internal forces, the ECM is also subjected to forces applied at specific points within the ECM (see Fig 2B). These points correspond to where cells have formed FAs that enable them to bind to the ECM. Active cellular behaviours, such as contraction driven by myosin motors, give rise to external forces which act upon the ECM and alter its (equilibrium) configuration.

In the overdamped limit, the evolution of crosslink positions is determined by balancing the (internal) elastic forces acting between neighbouring crosslinks and the (external) forces due to active cellular processes:

(1)

In (1), is the net elastic force acting on the l-th crosslink. This force includes an asymmetric response to fibre stretching and compression and acts as a source of non-linearity in the model (for more details see Sect S.2 in S1 File). represents the interaction force between this crosslink and i-th vertex of the n-th cell (see below, in the description of the mechanical coupling between the cells and the ECM). The parameter η denotes the damping coefficient.

This description, in which the ECM is represented as an elastic network, was previously employed by [33], where the authors examined how the structure of a random collagenous network influences the transmission of mechanical signals. They found that variations in the architecture of the ECM network significantly affect how mechanical signals are transmitted through the matrix. Specifically, the local arrangement of fibres influences how the ECM deforms, how fibres align in response to cellular forces, and how displacements propagate through the substrate. These effects were observed in simulations involving one or two contracting cells with circular shapes. Following their results, we generate a ECM network by constructing a Voronoi tessellation on the seed points. This configuration exhibits little local structure due to its random architecture with no consistent alignment or symmetry in the small-scale geometry of the network. Another important aspect is the low connectivity of the network, i.e. the number of fibres converging at a node is small, which is typical in fibrous networks [52,53].

Cell model. Cells are viewed as agents with polygonal shapes (see Fig 1A). In our model, we define different cells. Each cell , is defined by a set of points, , . We denote by the position of the cytoskeleton organising centre. The geometry of the cell is then modelled by a set of viscoelastic segments that emanate from the organising centre and terminate at the cell vertices, . Here, , with being the total number of structural segments in cell n (see Table 1 for details of the notation). can be varied to accommodate different cell shapes. Biologically, these segments can be understood as bundles of actin filaments or stress fibres that provide structural support to the cellular geometry [42,54,55]. They also correspond to the sites of cell-ECM interactions mediated by FAs [55]. In Fig 1A, we illustrate three possible cell geometries corresponding to an elongated (top-left), a round (top-right) and a polarised fan-shaped (bottom-right) cell. For the elongated cells, where , cells are not represented by polygonal shapes, but rather by two segments that emanate from the organising centre and terminate at two points, which for simplicity we term “cell vertices”. To facilitate visualisation of the cellular geometry, we plot an ellipse whose major axis corresponds to the segment connecting the two cell vertices (see Fig 1A).

The forces that drive the cellular dynamics can be divided into three classes (see Fig 2B–2C):

• Internal structural forces, . These forces regulate cell shape. We consider two types of structural forces. First we account for the viscoelastic properties of actin bundles. We describe the viscoelastic behaviour of the structural segments of the cell by the Kelvin-Voigt model [42,54]. We also account for the so-called angular forces, which reflect the tendency of the actin cortex to resist changes in cell shape. These angular forces set a target or natural angle between consecutive actin bundles, just as elastic springs have a natural length. This natural angle defines the equilibrium configuration of the cell in terms of its curvature, while the natural length defines the equilibrium cell size in the absence of forces. When the angle deviates from its target value, a restoring force acts to restore the angle to its target value, and prevents the cell from collapsing upon itself.
• Active contraction forces, . Active forces are generated within cells and usually driven by out-of-equilibrium ATP-dependent reactions. They are essential for most cell functions, including locomotion and adhesion. The present model includes only contraction forces. These forces are propagated along the actin bundles and are generated by myosin motors, which produce contractile forces on actin filaments [56]. Specifically, we represent active contraction forces as constant forces that contract the viscoelastic elements towards the cytoskeleton organising centre.
• Mechanical interactions with ECM, . These interactions are mediated by FAs, specialised anchoring points at which cells bind to the ECM. A detailed description is given below, in the paragraph describing the mechanical coupling between the cells and the ECM.

A more detailed description of the forces is given in Sect S.2 in S1 File. In our model, we assume that cells are sparsely seeded on the ECM and we only consider cell interactions transmitted through the ECM (direct cell-cell interactions are assumed negligible).

In the overdamped limit, where the inertial effects can be ignored relative to the viscous forces, the positions of the cell centre and vertices of the n-th cell are given by:

(2)

The solution of Eq (2) determines the evolution of the cell shape.

Mechanical coupling between the cells and the ECM. FAs are complex macromolecular clusters that transmit mechanical forces and regulatory signals between the ECM and an interacting adherent cell [11,12]. The formation of FAs is a complex process which involves inside-out cell mechanisms (signals originating from within the cell change the behaviour of surface receptors, affecting their ability to interact with external molecules), in conjunction with outside-in mechanisms (such as ECM biochemical composition and the biomechanics of the components that regulate the FA composition) [57,58]. In addition to anchoring cells to the ECM, FAs act as hubs for transmitting information from the ECM to the cells. This process involves hundreds of proteins, many of which associate and dissociate in a dynamic manner [13]. Integrins are the main component of FAs. They are the transmembrane mechanosensitive receptors that create mechanical bonds between the actin cytoskeleton and ECM fibres by binding to ligands embedded within the latter [13]. When subjected to mechanical stress, integrins deform and transmit mechanical stimuli from the ECM to the cell. In previous modelling studies [14,18,28,29], integrins were modelled as clusters of linear springs, i.e. they are assumed to stretch at a rate proportional to the applied force. Furthermore, the mechanical state of integrins also affects their unbinding kinetics [30,32]. This aspect of integrin mechanosensitivity has previously been modelled by assuming that the unbinding rate, , is a function of the force applied to the integrin [14,32].

In our model, cell vertices represent the ends of actin bundles, which correspond to the sites of FA assembly. Integrins within FAs transduce mechanical signals between cells and the ECM. They can be deformed due to the forces exerted by the ECM. This deformation, in turn, exerts forces on the actin bundles which, among other effects, can change the cell geometry. Similarly, forces exerted by the cell (e.g. myosin-motor mediated contraction) are transmitted to the ECM by stretching the integrins which, in turn, deforms the ECM fibres locally. For simplicity, we assume that integrins within FAs are organised as arrays of parallel linear springs (see Fig 1C). Therefore, all integrins undergo the same amount of deformation and the force exerted by the array of integrins is proportional to the number of bound receptors. This implies that to characterise the force exerted by the set of integrins, we must determine both their deformation (stretch) and the number of integrins bound to the ECM.

We assume that the cell-ECM mechanical linkages connect a cell vertex, , and a matrix crosslink, (see Fig 1C). We denote by the number of integrins bound to the ECM, denotes the number of unbound integrins at time t. Following [14], we assume that the total number of available integrins at a FA is conserved:

(3)

Based on our assumption that integrins behave as an assembly of parallel Hookean springs, the resulting force exerted by a cluster of bound integrins, , is determined by:

(4)

In Eq (4), denotes the integrins elastic constant and defines the extension of the integrins, which is given by:

where is the equilibrium length of an integrin.

The net force exerted by the bound integrins, , results from their deformation within a FA (see Fig 2B). This deformation is generated by the forces exerted by the ECM (elastic forces exerted by fibres converging at ) and the forces exerted by the cell (structural and active forces at ). By Newton’s third law, we know that the forces exerted at the cell adhesion site, , and the ECM crosslink, , have equivalent magnitude but pointing in opposite directions: .

Lastly, following [14,28], we assume that integrins within FAs unbind at rates which depend on the mechanical forces caused by their deformation, . Specifically, we use the so-called catch model [14,30] which has been proposed to more accurately describe the detachment of integrins. This model assumes that the maximal lifetime of a cell-ECM bond occurs at intermediate forces, whereas for large forces, it decays exponentially. This is captured by assuming that the integrin unbinding rate, , depends on the force which results in its deformation. The total number of available integrins in a FA, N_n;i,l, and their binding rate, , are assumed to be constant. Combining these effects, we arrive at the ODE for the number of bound integrins in a FA, which together with Eq (3) is:

(5)

To summarise, our model accounts for cell-ECM dynamics that span different spatial scales, which include ECM deformation, sub-cellular processes (such as myosin-driven contraction) and integrin binding and unbinding processes. In particular, the positions of ECM crosslinks are determined by Eq (1), and the deformation of each cell is governed by Eq (2), with cell-ECM coupling captured by the formation of the FAs. The number of bound integrins at each FA is given by Eq (5). A more detailed description of the model can be found in Sect S.2 in S1 File. The numerical methods used to solve the governing equations are described in Sect S.4 in S1 File. Unless otherwise stated, parameter values are selected to illustrate system behaviour while remaining within physiological ranges. A complete list of the parameter values used to generate the numerical simulations is presented in S1 Table.

We seek to quantify the effects of cell contraction on the ECM, i.e. to determine the deformation of the ECM caused by pulling forces due to cellular activity. We introduce several metrics to quantify fibre alignment and the transmission of mechanical stress between contracting cells. These metrics are based on local and intrinsic properties of the ECM network, which allow us to quantify its deformation and mechanical response to external forces. In our simulations, the mechanical perturbations of the ECM are caused by cellular activity.

2.2 Measuring fibre alignment

Local alignment metric.

To quantify the effects of active cell processes (such as myosin-driven contraction) on ECM structure, we introduce an alignment metric. Following [33], we consider a local orientation tensor which is defined at each ECM crosslink and accounts for the degree of alignment of the fibres converging to it. At crosslink l we have:

(6)

where is the tensor product, the sum is over the set of neighbouring nodes of a crosslink l, and

(7)

If the set of vectors that converge at crosslink l contains at least two non-parallel vectors, then the local orientation tensor, , is symmetric and positive definite with . As such it can be diagonalised and its eigenvalues lie within the interval (0,1). When the fibres are arranged isotropically, has two identical eigenvalues . By contrast, if the fibres are distributed anisotropically, then the eigenvalues are distinct . We note also that the eigenvector corresponding to the largest eigenvalue is parallel to the direction of the preferential orientation of the ECM fibres at this crosslink. The local orientation points in the direction associated with the largest eigenvalue. The anisotropy metric, , describes the local anisotropy of ECM fibre distribution at the l-th crosslink:

(8)

In near-to-isotropic scenarios, , while for anisotropic cases (i.e. fibres converging at crosslink l are strongly aligned in the same direction) . Fig 3A illustrates the principal alignment directions and the corresponding anisotropy parameter for three cases of increasing anisotropy. The left panel corresponds to the isotropic case, while the right panel corresponds to strong vertical alignment of ECM fibres.

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Fig 3. An illustration of the alignment measure.

(A) Examples of local alignment measures (see Eq (8)) for isotropic (left), partially anisotropic (middle) and strongly aligned (right) fibres. Here, ECM fibres are shown in light grey. The alignment vector corresponding to the principal eigenvalue of the tensor, , from Eq (6) is coloured according to the value of the anisotropy parameter, . Blue corresponds to isotropic fibre distribution , yellow indicates intermediate alignment , and orange corresponds to an anisotropic case . (B) ECM configurations at equilibrium (left) and after the contraction of two pairs of crosslinks (coloured in red and green), corresponding to two cells. The maximum stress transmission path after cellular contraction is highlighted in black. (C) Local alignment measures in the ECM fibres at an equilibrium (left) and after cellular contraction. ECM fibres within the region between the two cells (outlined in purple) are coloured according to the corresponding value of the local fibre anysotropy parameter . (D) Histograms of distributions of the principal eigenvector and the anisotropy parameter in the purple regions for the ECM configurations shown in (C). After cell contraction, ECM fibres realign along the contraction direction (), and the distribution of the anisotropy parameter skews towards an anisotropic scenario.

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2.3 Percolation: Measuring the spatial stress transmission

Another crucial aspect of mechanically-mediated cell-cell communication is the spatial propagation of stress through the ECM fibres. To quantify this effect, we propose to measure the stress percolation between ECM crosslinks. This technique involves determining a stress transmission path in the following way:

1. We calculate the tensional stress, , of the stretched fibres that comprise the ECM after their deformation due to cell contraction, i.e. those fibres whose length at time t is greater than their length at t = 0:
   for a pair of crosslinks l,m whose separation at time t is greater than their initial distance (). In the equation for the tensional stress, E_t represents the Young’s modulus of the stretched fibres (for more details, see Sect S.2 in S1 File).
2. We determine the set of fibres, , that compose the path connecting two crosslinks within the ECM as those that maximise the stress transmission. The crosslinks with positions and correspond to the two points of cell-ECM interaction (see Fig 3B). To find a path whose total stress is maximal (i.e. a path that maximises the sum of the individual stresses of the fibres which compose it), we utilise the Dijkstra algorithm [59], minimizing where refers to the pair of crosslinks that form a fibre. If there is no such connected path, i.e. a path of stretched fibres connecting both crosslinks, we conclude that there is no mechanical communication between the cells attached to them.
3. We then calculate the total stress in all fibres in . To quantify the degree of tortuosity of this path, we compare the Euclidean distance between the nodes with the length of the path: , where is the sum of lengths of the fibres that form the path. The tortuosity takes values between 0 and 1. As the tortuosity approaches 1, the connected path becomes increasingly similar to a straight line. Comparing the values of tortuosity in the initial configuration, a major increment observed after cell contraction indicates a major deformation of the ECM, which in turn leads to major transmission of mechanical cues.

In Fig 3B (right panel), we illustrate the percolation path between two ECM crosslinks (indicated by red points) after the contraction between the two pairs formed by a red and a green crosslink.

2.4 Focal adhesion dynamics for an isolated cell

To test the model’s ability to capture dynamic interactions between ECM crosslinks and FAs of a round contracting cell, we simulate an isolated cell attached to a randomly oriented (isotropic) ECM. Fig 4A–4B show the initial and final configurations of the cell-ECM system after 5 s of constant cell contraction. The initial configuration is obtained by letting the system relax to its equilibrium configuration, with the cell attached to the ECM and no contraction activity, i.e. we simulate the system of equations setting . Once the system reaches the latter state, we use this configuration as an initial condition for the model. Then the cell contracts under the action of active contractile forces, i.e. . As expected, the transmission of stress attenuates with distance: from the adhesion sites the stress emerges in radial paths, with compressed transversal fibres along these paths.

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Fig 4. Single cell contraction.

(A) The initial configuration of the cell-ECM system before cell contraction. A round cell is placed on top of a relaxed ECM network (as described in Sect S.2 in S1 File). The fibres in equilibrium are shown in grey. (B) ECM and cell configuration at after contraction. Stretched fibres are plotted in dark red, and the compressed fibres are in blue. The colour map indicates the stress of the ECM normalized by the maximum stress in the ECM. Parameters: Young’s modulus of the ECM fibres , number of integrins per FA N = 857, . The remaining parameters are listed in S1 Table. (C) The time evolution of the mean force exerted at the FA (solid blue line) and the elastic force exerted by the FA (dark red line) for the simulation from (A) and (B). The bifurcation force of the FA for this system is shown by a dash dotted blue line. The black dot indicates the peak tension force exerted by the FA. Shaded regions indicate the standard deviation calculated over all the cell FAs. (D) Evolution of the proportion of bounded integrins per FA (in blue) and the stretch of the integrins (in red) during cell contraction.

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A key feature of our model is that it seamlessly accommodates the different temporal scales involved in the cell-ECM system. In particular, the model captures the evolution of the different forces involved in the cell-ECM system. Integrin deformation and detachment is greatly affected by the forces applied to the FA due to cell contraction and ECM deformation, i.e. the contractile mechanisms and the viscoelastic properties of both the cell and the ECM determine the dynamics of their interaction (see Sect S.2 in S1 File). In the specific simulation shown in Fig 4, the model shows the tension undergone by the FA peaking early to a value below the detachment threshold. The FA tension then relaxes as the transmission of the deformation through the ECM progresses (see Fig 4C).

Finally, the evolution of the focal adhesion components, specifically the ratio of attached integrins and their stretch, are illustrated in Fig 4D. When analysing the temporal evolution of the focal adhesion components, we found that the dynamics of the interaction forces are mainly influenced by the stretching process of the integrins: while the interaction forces and stretch of the integrins in Fig 4B–4C have similar timescales, the changes in the number of bound integrins occur on a longer temporal scale. As predicted by the model analysis presented in Sect 3.1, the binding-unbinding dynamics are slower than those associated with integrin deformation.

3.1 Analysis of the FA model

To gain further insight into the FA dynamics, we performed a bifurcation analysis of the force-binding model by varying the strength of the force applied to the FA system, which we call external force in this section. For simplicity, we focus on the dynamics of a specific FA, and modify our notation as follows: N_b and refer to the number of bound integrins in the FA and their stretch respectively, N is the total number of available integrins and the force resulting from deforming a single integrin.

With this notation, the system of ODEs for the FA model is as follows:

(9)

where denotes the external force applied to the assembly of integrins, i.e. forces resulting from cell activity and ECM deformation. We performed a bifurcation analysis of Eq (9), viewing the force, , as a control parameter. Fig 5A shows that the system exhibits a saddle-node bifurcation. For values of smaller than a critical value, , the system exhibits monostability. In particular, when , a positive equilibrium, which corresponds to a stable cell-ECM bond, coexists with an unstable equilibrium. For , the system evolves to a state in which no bound receptors exist, which we interpret as detachment from the ECM. This is illustrated in Fig 5C, where we plot trajectories corresponding to attachment or detachment.

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Fig 5. Bifurcation analysis of the FA ODE system.

(A) A series of plots showing bifurcation diagrams of the system described by Eq (9) as the force parameter, , is varied. The bifurcation diagram is obtained numerically using BifurcationKit from Julia [66]. The four panels from left to right correspond to the increasing total number of available integrins . (B) Three plots showing the values of the bifurcation force, the ratio of attached integrins in the FA, and their stretch at the bifurcation point for different values of the total available integrins, N. Coloured circles represent the values obtained numerically as in (A); red curves are calculated from an analytical expression (see section Exact calculation of the critical value in S1 File). (C) Phase portrait diagrams for the case of N = 1500 for a value of the force lower than the bifurcation value (top) and a value higher than the bifurcation value (bottom). In grey, we show trajectories of the solution of Eq (9) starting from the initial condition of zero-stretch and different proportions of bound receptors, N_b/N. The nullclines of the system are shown in orange for and in blue for dN_b/dt = 0. The region shaded in blue represents the basin of attraction of the stable fixed point, while the trajectories that enter the copper region will lead to the detachment of the cell from the ECM since the number of bound receptors tends to zero.

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We also analysed the system behaviour as the number of receptors, N, varies. Fig 5A shows that as N increases, the size of the region of stability also increases ( increases as N grows), which means that ECM binding becomes more robust when integrins are more abundant. In Fig 5B, we show how the system variables at the bifurcation point depend on the total number of available integrins N. We note that our model predicts a linear relationship between the number of available integrins and the bifurcation force, . However, as the proportion of bound integrins and their stretch remain constant at the bifurcation point, regardless of the changes in the total number of available integrins N, we conclude that the model exhibits a scale invariance property as the number of available integrins N changes: the proportion of bound integrins and the ratio of the force exerted per available integrin do not depend on the number of available integrins (see Fig 5A, 5B). We analyse the system dynamics in terms of these ratios by dividing the system given by Eq. (9) by N. Although the equilibrium points are not altered, the dynamics of the stretch changes; for large values of N, the stretch dynamics occur much faster than in cases with low values of N, indicating that a quasi-equilibrium approximation may be appropriate in this scenario. For systems with a small number of available integrins, N, the mean-field approximation is not valid, and a stochastic approach should be used to describe the system dynamics.

In section “Exact calculation of the critical value ” in S1 File we derive an implicit expression for the bifurcation force , and related expressions for the proportion of bound integrins and their stretch :

(10)

The scale invariance property is reflected in the bifurcation values, as only depend on the binding-unbinding ratio.

These results show how forces influence cell-ECM interactions. In our model, the forces acting on each FA arise from a combination of viscoelastic forces, generated by both the cells and the ECM, and active cell contraction. For the interaction between the cell site and the ECM crosslink , the force is therefore given by

where refers to the inner product and

3.2 Reorientation and deformation of the ECM influence cell-cell interactions at a distance

Fig 6 shows simulation results for two elongated contracting cells located within the ECM at a distance from each other. These simulations illustrate how cell-cell communication can be established via mechanical cues. Specifically, following active cell contraction, ECM realignment and deformation propagate mechanical cues. Mechanical communication emerges because the deformation field induced by the traction forces exerted by a cell propagates through the ECM, reaching and mechanically stimulating the neighbouring cell. This occurs when the ECM forms a network capable of transmitting stress, effectively linking the mechanical environment between both cells. Fig 6A (left panel) shows a typical initial configuration, where the ECM has relaxed to a near-equilibrium configuration for which the fibres do not carry any initial load. This configuration minimises the impact of non-linearities caused by changes in the compressed and stretched regimes. Such an initial ECM configuration allows us to disentangle the contribution of cell-generated forces from passive ECM mechanics. Fig 6A (right panel) shows that cellular crosstalk is established via the formation of a continuous (percolative) path of maximal stress between both cells (see Sect 2.3). Furthermore, the stress distribution within the ECM shows higher stress levels aligned along the axis connecting the two cells.

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Fig 6. Contraction of two elongated cells.

(A) Left panel: initial configuration of our model simulations in which two elongated cells are placed on a relaxed ECM. Right panel: the configuration of the cell-ECM system after of contraction. ECM fibres at equilibrium are shown in grey, extended fibres highlighted in dark red and compressed fibres in blue. The colour map represents the stress within the ECM, normalised by the maximum stress value. In addition, the ECM path of maximum stress transmission is highlighted in black. Parameters: Young’s modulus of the ECM fibres , number of integrins per FA N = 3000, . The remaining parameters are as listed in S1 Table. (B) Two plots comparing the local alignment of ECM fibres before and after cell contraction. For each crosslink, we calculate the principle alignment direction and the corresponding value of the anisotropy metric given by Eq (8). Orange color represents strong fibre alignment, yellow represents an intermediate anisotropy and grey indicates isotropic cases. We then visualise fibre alignment at crosslinks in a high interaction region between the two cells (purple rhombus), and in a low cell-cell interaction region (green rhombus). (C) Distributions of alignment direction and alignment measure in the high and low interaction regions outlined in (B). Red error bars indicate the standard deviation calculated over 11 realizations of our model for the scenario of two contracting cells. Left panels show distributions of these metrics at equilibrium (), while right panels correspond to after contraction. Black solid lines in distributions of alignment angle indicate the direction of the segment that connects the cell centres (in purple region of high cell interaction). The t-test of the distribution of the anisotropy parameter in the region of high interaction before and after contraction indicates that the distributions are different ( × 10⁻⁰⁴). On the other hand, the t-test for the low interaction region before and after the contraction indicate that both samples could come from the same distribution with .

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Another crucial aspect that influences cell behaviour is the alignment of the ECM fibres, as cells migrating on a fibrous environment tend to polarise along the matrix fibres and their migration direction can be biased by the local orientation of the ECM (in [67], the authors show how fibroblasts align with the ECM fibres in vitro). Although migration is not included in our simulations, Fig 6B indicates that ECM realignment following active cell contraction is heterogeneous and significantly stronger between the active cells (outlined in purple) than in regions further apart from them (shown in green).

Using the quantifications of local alignment and local anisotropy defined in Sect 2.2, Eq (8), we quantify the ECM realignment in response to cell contraction. Fig 6C shows the distribution of metrics for ECM alignment anisotropy within the area between (far from) the contracting cells, represented by the purple rhombus (green triangles) in Fig 6B. Since the contraction-induced stress decays with distance, there is no significant change in the level of ECM anisotropy in the region of low cell interactions. By contrast, ECM fibres reorient towards the path of maximal stress that connects the two contracting cells. In the high interaction region (in purple), our model simulations predict the alignment of fibres in the direction of highest stress transmission (corresponding to in our simulation setup). Furthermore, the distribution of the local anisotropy, , skews towards 1, which indicates an increase in ECM anisotropy in this region.

3.3 Parameter sensitivity analysis

In most of our simulations, we consider a scenario in which two contracting elongated cells are placed in mutual proximity within the ECM. The cells are initially facing each other, which is the most favourable configuration for cell communication. Using this configuration, we conduct a parameter sensitivity analysis to measure variations in how the system responds to changes in contraction activity and ECM stiffness.

Active contraction forces increase the force and stress transmitted between cells, but do not affect the speed of force transmission. A key parameter that determines the qualitative behaviour of the system is the active contractile force. Regulation of active contraction, specifically myosin motor activity, is crucial for controlling cell-substrate adhesion, cell migration and tissue architecture [8,9,38,56,68]. In order to analyse the role of the contraction force in the deformation of the substrate, we perform simulations in which we increase the cell contraction force within a range that maintains cell-ECM attachment. Specifically, the contraction force varies from to ⋅ . The results of this sensitivity analysis are shown in Fig 7.

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Fig 7. Parameter sweep analysis. Active contraction.

We analyse the influence of active contractile forces on the mechanical communication between cells by measuring the alignment of ECM fibres, stress transmitted between cells and the peak of the interaction force at after contraction. In these simulations, the contraction force varies from to ⋅ . Parameters: number of integrins per FA N = 3000, . The remaining parameters are as listed in S1 Table. All results are averaged over 11 realisations of our model with random initial ECM network topologies. (A) Two boxplots showing the mean and skewness of the distribution of the anisotropy parameter for ECM fibres at after cell contraction. Green corresponds to the low interaction region while the results for the high interaction region are shown in purple (see Fig 6). (B) A plot showing the tortuosity of the path of maximum stress transmission (see Sect 2.3) calculated as a ratio of the Euclidean distance between the cells and the length of the percolation path. Orange indicates statistics calculated at after contraction, while the tortuosity of the same path at the initial time is shown in blue. (C) A plot showing the mean tensional stress in the maximum stress transmission path. (D), (E) Two plots demonstrating the time at which the cell-ECM interaction force, , is maximum (D) and the value of this peak force (E) (see peak force in Fig 4C). In addition, we include a linear regression fit where we can see that the maximum of the interaction force grows linearly with the increment of the contractile forces. The fit has R-square  = 0.9833.

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Fig 7A–7C show that the ECM becomes more aligned as the contractile forces increase. The behaviours of the mean (left panel) and skewness (right panel) of the local anisotropy parameter, Eq (8), are presented in Fig 7A, which shows that increasing the contraction force increases the anisotropy of the ECM mesh. In the region between the cells, elevating the contraction force increases both the mean anisotropy and the skewness of its distribution. Furthermore, in agreement with the fibre alignment results shown in Fig 7A, the path of maximal stress transmits more stress and becomes less tortuous as the contraction force increases.

Crucially, larger contraction forces enhance cell-to-cell communication. Fig 7E shows that the force exerted on the FAs increases as the contractility grows, augmenting the local deformation of the ECM. This local deformation is then propagated through the ECM, stretching and realigning the fibres between the cells, increasing the anisotropy of the fibres in the interaction region (purple, see Fig 6). While increasing the contractility enhances the transmission of forces through the ECM, the kinetics of the cell-to-cell communication are weakly affected. Fig 7D shows that the time at which the interaction force reaches its maximum, and its value (see Fig 6), are independent of the active contraction force.

ECM stiffness is crucial for determining the timing of mechanical communication between the cells. The effects of the mechanical properties of the ECM on cell behaviour are widely reported in the literature (see [3] for a review). In particular, some studies [22,33,39] show that ECM viscoelasticity plays an important role in cell migration and communication. In [22], the authors show that the alignment of the fibres and tether formation between two contracting cells may result from the mechanical forces exerted by those cells. In [33], the authors analyse how the stress transmission between two contracting cells depends on the ECM network architecture. Similarly, in [39], the authors investigate the transmission of stress and deformation between two contracting beads embedded within the ECM mesh. Despite these insights, the role of ECM stiffness—particularly how variations in the Young’s modulus of individual ECM fibres influence stress transmission and fibre realignment—remains less explored. Here, we address this gap by investigating how varying the stiffness of the ECM fibres affects mechanical communication between cells. Specifically, we conduct model simulations involving two contracting elongated cells in which we vary the Young’s modulus of the ECM fibres. In Fig 8, we present simulation results comparing the fibre alignment and stress transmission after active contraction of the two elongated cells.

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Fig 8. Parameter sweep analysis: Varying ECM stiffness.

We analyse the influence of the stiffness of the ECM fibres on the mechanical communication between cells by measuring the alignment of the fibres, and the stress transmitted between cells at after cell contraction. ECM stiffness is varied from to . Parameters: number of integrins per FA N = 3000, . The remaining parameters are listed in S1 Table. We perform 11 realizations with arbitrary initial configurations of the ECM for each set of parameter values. (A) We present the mean and skewness of the anisotropy parameter distribution of the ECM configuration at after cell contraction. Similar to results shown in Fig 6, green (purple) colour indicates the ECM region with low (high) cell interaction. (B) A boxplot showing the tortuosity of the path of maximum stress transmission (see Sect 2.3), calculated as a ratio of the Euclidean distance between the cells and the length of the percolation path. Orange indicates statistics calculated at after contraction, while the tortuosity of the same path at the initial time is shown in blue. For Pa, we do not observe the formation of a path between cells along stretched fibres. (C) A plot showing the mean tensional stress along the maximum stress transmission path at after cell contraction. Since no path of stretched fibres forms between the cells for the softest ECM, , the mean stress in these cases is . (D) Time at which the interaction force, , is maximum. (E) Maximum value of the interaction force. (D) and (E) show the time and the force value of the peak force represented in Fig 4C. In this case, the mean peak force is calculated from the mean of the interaction force of all the adhesion sites of both cells.

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Fig 8A shows that anisotropy of the ECM mesh (both mean (left panel) and skewness (right panel)) within the high interaction region between cells increases with the Young’s modulus of its fibres, i.e. fibre realignment is transmitted better for the stiffer fibres. The ECM deformation saturates as the stiffness increases.

In Fig 8B–8C we compare the statistics associated with the percolation path of maximum stress transmission described in Sect 2.3 as the Young’s modulus of the ECM fibres varies. For cases when a communication path along stretched fibres is formed, we calculate its tortuosity (Fig 8B) and the mean stress along its length (Fig 8C). Fig 8C shows that, for softer ECM, the mechanical deformation caused by cell contraction is not transmitted sufficiently far through the fibres to connect two cells separated apart. Although the ECM fibres in the vicinity of the cell adhesion are stretched, the elastic forces resulting from this deformation are small, leading to less effective transmission of mechanical signals. Softer ECM also leads to less fibre realignment, with negligible reorientation after cell contraction for . In the softest case, we performed a longer simulation, which showed that the mechanical percolation occurs at a slower rate, see Fig A in S1 File. On the other hand, fibre realignment and mechanical stress are better transmitted for stiffer ECM. Despite the saturation in the mean and skewness of the anisotropy parameter for stiffer ECM fibres (see and in Fig 8A), a higher Young’s modulus results in higher deformation and greater mean stress along the percolation path, see Fig 8B–8C. For stiffer ECM, when contracting cells pull on ECM crosslinks, as the fibres are more rigid, small fibre stretch implies transmission of larger forces within the network.

Finally, in order to quantify the dynamics of the interaction force, in Fig 8D, 8E we determine the time at which the force resulting from the integrin stretching is maximum, and the value of this force (maximum of the interaction force illustrated in Fig 4C). For stiffer ECM fibres, increasing values of the Young’s modulus reduce the time at which the peak force appears and increase the peak force. Small deformations of the ECM fibres lead increased tensile stresses (see in Fig 8C and 8E), and more rapid transmission of deformation through the ECM (see Fig 8D).

3.4 Local topology of the ECM network connecting two cells influences the effective ECM stiffness

For larger values of the Young’s modulus, cells frequently detach from the ECM during contraction. This phenomenon has been reported in the literature as ‘frictional slippage’: the unbinding rates become faster than binding rates, the number of simultaneously bound integrins drops drastically, and overall force transmission decreases [16]. The frictional slippage regime was reported, for instance, in neuronal growth cones [69] and at the trailing edge of migrating keratocytes [70]. As cells pull on stiffer substrates, the peak force experienced by the integrin cluster is reached more rapidly, leaving insufficient time for the integrin binding to adjust. Consequently, cells enter the detachment basin of attraction (see Fig 5), even though they are experiencing lower forces than the critical force described in Sect 3.1. We posit that the time required to reach peak force at a FA is closely related to the topology of the ECM network. If the initial ECM configuration allows for the formation of a communication path, then cells remain attached because mechanical stress can be rapidly transmitted along this path, with the energy from the initial force dissipated through ECM deformation. On the other hand, if the initial ECM configuration lacks a clear communication path, then the network topology induces rigidity, preventing transmission of the deformation caused by the cell’s initial pull. This topological effect is illustrated in Fig B in S1 File, where we present two model configurations for which a cell detaches from the ECM, and one model configuration for which the cells do not detach. In these cases, the stress is transmitted via several paths. In cases for which a cell detaches, the initial cell contraction stretches the integrins that form FAs rather than the ECM fibres, forcing the FA system to enter the basin of attraction of detachment (see Fig 5C, the detachment region).

To further investigate how the local structure of the ECM network affects the stress exerted on the FAs, we perform simulations in which two cells attached to a simplified ECM are pulled away from each other (see Fig 9). We consider two ECM topologies that connect the two cells: a zigzag ECM path (see Fig 9A) and a more complex scenario where the cells are connected to two long fibres with small transversal fibres between them (Fig 9B, 9C). We assume that cells are attached to the ECM fibres at one of their adhesion sites, while being pulled away from each other with a constant force. We assume further that at the initial configuration, there are a certain number of bound integrins at FA sites. We use this setup to investigate how the ECM topology affects cell-ECM interactions as the ECM stiffness varies.

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Fig 9. Two-cells simulation: analysis of topological effects of the ECM network.

Numerical simulations with two cells connected to simplified ECM networks. We present three snapshots of the time evolution of the cell-ECM system, and the evolution of the key variables that define the interaction force: the proportion of bound integrins and their stretch . The cells are pulled away from each other with a force . In light grey, we plot the fibres that are in their equilibrium state, blue indicates compressed and red corresponds to stretched fibres. Rightmost panels show the dynamics of the proportion of bound integrins (blue) and their stretch (red). (A) Two cells connected by a single zigzag path of ECM fibres. The crosslinks of the main zigzag path are connected to lateral fibres with fixed outer endpoints. These fibres are plotted in black. In this case, the system achieves an equilibrium, where the ECM fibres are stretched, and they form a communication path between cells that remain attached to it. The Young’s modulus of the ECM fibres is . (B) Simulation with the ECM configuration consisting of two paths connected by transversal fibres. The rigidity and architecture of this ECM configuration prevent the formation of a communication path between the cells (i.e. the system cannot effectively dissipate the active forces by deforming the network of fibres that connect the cells). As a result, the mechanical load becomes concentrated at the FAs, leading to rapid integrin stretching which causes cell detachment. The ECM fibres have the same Young’s modulus as in (A), . (C) The same ECM configuration as in (B) but with softer fibres, . In this case, the cells do not detach from the ECM.

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Fig 9A shows that, for the zig-zag scenario, the stress generated by the pulling force is dissipated via deformation of the communication path, even though the Young’s modulus is in the range for which ‘frictional slippage’ may occur. This deformation is rapidly transmitted, and the pulling energy dissipated by deforming the ECM. The FA system evolves until it reaches an equilibrium with a positive number of bound integrins (see right panel of Fig 9A). By contrast, the ECM configuration presented in Fig 9B introduces extra rigidity into the system, resulting in an entirely different behaviour. Since this ECM network is less deformable at the same ECM stiffness, its effective stiffness is greater and the pulling force leads to rapid stretching of the integrins that form the FAs, which causes cell detachment from the ECM (see Fig 9B, right panel). Additionally, in S1 Video, we observe that the FA system enters the detachment basin of attraction, even though the cell-ECM interaction force is below its critical value. There is no difference in behaviour when we randomly remove the transversal fibres (see Fig C in S1 File), which means that the topological effects result from the branching of the stress transmission in different paths.

Cell detachment due to the local structure of the ECM is closely linked to the effective stiffness of the ECM mesh. In Fig 9C, we present a simulation for a softer ECM where the cells remain anchored to the ECM. As the fibres are softer, the force exerted at FAs is transformed into substrate deformation, allowing part of the energy to be dissipated without over-stretching the integrins.

3.5 Variety of cell shapes and mechanical communication of multiple cells

Cell shape and multicellular coordination are tightly regulated by mechanical interactions with the ECM. Here, we show that our model is able to capture a variety of cell morphologies and extends naturally to simulate interactions among multiple cells through a shared ECM network.

The polygonal cell geometry used in our model enables us to study cells with a variety of shapes. For example, in Fig 10A, we observe the stress transmission path between two contracting fan-shaped cells.

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Fig 10. Illustration of the model’s flexibility to accommodate various cells and different cell shapes.

Two model simulations with contracting fan-shaped cells. ECM fibres at equilibrium are shown in grey, extended fibres are highlighted in dark red and compressed fibres in blue. The underlying colour map indicates the stress within the ECM, normalised by the maximum observed stress. Young’s modulus of the ECM fibres was set to , the number of integrins per FA N = 1000 and the contraction force . The remaining parameters are listed in S1 Table. (A) A numerical experiment similar to the one shown in Fig 6 performed for fan-shaped cells. Left panel: initial configuration in which two fan-shaped cells are placed on a relaxed ECM. Right panel: the configuration of the cell-ECM system after of contraction. The ECM path of maximum stress transmission between two cell interaction sites is highlighted in black. (B) Model simulation as in (A) but with three fan-shaped cells.

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Our model can also accommodate several cells in a straightforward manner (see Fig 10B), enabling us to study the feasibility of indirect cell-cell communication mediated mechanical cues in the ECM. After cell contraction, we can see how the stress is mainly distributed in the region between the three contracting cells, see Fig 10B, right panel.