Free-energy-based framework for early forecasting of stem cell differentiation

Commitment of stem cells to different lineages is inherently stochastic but regulated by a range of environmental bio/chemo/mechanical cues. Here, we develop an integrated stochastic modelling framework for predicting the differentiation of hMSCs in response to a range of environmental cues, including sizes of adhesive islands, stiffness of substrates and treatment with ROCK inhibitors in both growth and mixed media. The statistical framework analyses the fluctuations of cell morphologies over approximately a 24 h period after seeding the cells in the specific environment and uses the cytoskeletal free-energy distribution to forecast the lineage the hMSCs will commit to. The cytoskeletal free energy which succinctly parametrizes the biochemical state of the cell is shown to capture hMSC commitment over a range of environments while simple morphological factors such as cell shape, tractions on their own are unable to correlate with lineages hMSCs adopt.


The homeostatic mechanics framework
Here, we provide a brief overview of the homeostatic mechanics framework of Shishvan et al. [1]. The aim is to provide the reader the key aspects of the framework required for fully appreciating the computational results presented in the main text, and to also emphasise the differences in the present formulation compared to the work of Shishvan et al. [1]. Readers are referred to Shishvan et al. [1] for a more complete treatment including the derivations of the relevant equations.
Making the ansatz that living cells are entropic, Shishvan et al. [1] introduced the concept of the homeostatic ensemble with cellular homeostasis providing the additional constraints and mechanisms for entropy maximisation. This defined the notion of a (dynamic) homeostatic equilibrium state that intervenes to allow living cells to elude thermodynamic equilibrium. They thus developed a statistical mechanics framework for living cells using the notions of statistical inference applicable over a timescale from a few hours to a few days as long as the cell remains as a single undivided entity (i.e. the interphase period of the cell cycle). The key ideas behind the framework can be summarised as follows. A system comprising the cell and the extracellular matrix (ECM) is an open system with the cell exchanging nutrients with the surrounding bath. These nutrients fuel a large number of coupled biochemical reactions that include actin polymerisation, treadmilling and dendritic nucleation that effect changes to the cell morphology. These biochemical reactions change the morphology of the cell but are not precisely controlled, and this manifests via the observed morphological fluctuations of the cell. Shishvan et al. [1] made the ansatz that these biochemical reactions provide the mechanisms to maximise the morphological entropy of the cell, but constrained by the fact that the cell maintains a homeostatic state 1 over the interphase period. Showing that the homeostatic constraint translates to a constraint on the average Gibbs free-energy, Shishvan et al. [1] developed a statistical mechanics framework to analyse the fluctuating response of cells. It is this framework that we extend to analyse stem cell differentiation.
Controlling only macro variables (i.e. macrostate) such as the temperature, pressure and nutrient concentrations in the nutrient bath results in inherent uncertainty (referred to here as missing information) in micro variables (i.e. microstates) of the system. This includes a level of unpredictability in homeostatic process variables, such as the spatio-temporal distribution of chemical species, that is linked to Brownian motion and the complex feedback loops in the homeostatic processes. Thus, this system not only includes the usual lack of precise information on the positions and velocities of individual molecules associated with the thermodynamic temperature, but also an uncertainty in cell shape resulting from imprecise regulation of the homeostatic processes. The consequent entropy production forms the basis of this new statistical mechanics framework motivated by the following two levels of microstates: (i) Molecular microstates. Each molecular microstate has a specific configuration (position and momentum) of all molecules within the system.
(ii) Morphological microstates (Fig. 1 in the main text). Each morphological microstate is specified by the mapping (connection) of material points on the cell membrane to material points on the substrate and/or within the adhesive island geometry. In broad terms, a morphological microstate specifies the shape and size of the cell.
Shishvan et al. [1] identified the (dynamic) homeostatic or equilibrium state of the system by entropy maximisation. Subsequently, we shall simply refer to this state as an equilibrium state to emphasise that it is a stationary macrostate of the system inferred via entropy maximisation as in conventional equilibrium analysis. The total entropy of the system is written in terms of the conditional probability ( | ) of the molecular microstate ( ) given the morphological microstate ( ) and the probability ( ) of morphological microstate ( ) as and Γ ≡ − ∑ ( ) ln ( ) are the entropies of molecular microstates in morphological microstate ( ) and the morphological microstates, respectively. Equilibrium then corresponds to molecular and morphological macrostates that maximise T subject to appropriate constraints. The molecular macrostate evolves on the order of seconds, limited by processes such as the diffusion of unbound actin. By contrast, transformation of the morphological macrostate involves cell shape changes and therefore, the morphological macrostate evolves on the order of minutes, limited by co-operative cytoskeletal processes within the cell such as meshwork actin polymerisation and dendritic nucleation. The evolutions of the molecular and morphological macrostates are therefore temporally decoupled, and Shishvan et al. [1] showed that Eq. (1.1) can be maximised by independently maximising M ( ) at the smaller timescale to determine the equilibrium distribution of molecular microstates for a given morphological microstate, and then maximising Γ at the larger timescale to determine the equilibrium distribution of the morphological microstates.
Over the (short) timescale on the order of seconds, the only known constraint on the system is that it is maintained at a constant temperature, pressure and strain distribution. The equilibrium of a given morphological microstate ( ) obtained by maximising M ( ) (denoted by M ( ) ) corresponds to molecular arrangements that minimise the Gibbs free-energy with ( ) . Since the connection between the cell and the substrate is fixed for a given morphological microstate, the determination of ( ) is a standard boundary value problem as described in Section 1.2.
Over the (long) timescale on the order of several minutes to hours, the equilibrium distribution eq ( ) is determined by maximising Γ , but now with the additional constraint that the cell is maintained in its homeostatic state. For the case of a cell on an ECM in a constant temperature and pressure nutrient bath, the homeostatic constraint translates to the fact that the average Gibbs free-energy of the system over all the morphological microstates it assumes, is equal to the equilibrium Gibbs free-energy S of an isolated cell in suspension (free-standing cell), i.e. the homeostatic processes maintain the average biochemical state of the system equal to that of a cell in suspension. In deriving this result, Shishvan et al. [1] did not consider every individual homeostatic process, but rather used just the coarse-grained outcome of the homeostatic processes. The application of this coarse-grained constraint is the key element of the homeostatic mechanics framework, with the morphological entropy Γ parameterising the information lost by not modelling all variables associated with the homeostatic processes.
The maximisation of Γ while enforcing ∑ ( ) ( ) = S gives the homeostatic equilibrium state such that eq ( ) = 1 exp(− ( ) ), where ≡ ∑ exp(− ( ) ) is the partition function of the morphological microstates, and the distribution parameter follows from the homeostatic constraint The collection of all possible morphological microstates that the system assumes while maintaining its homeostatic equilibrium state is referred to as the homeostatic ensemble. The homeostatic ensemble can therefore be viewed as a large collection of copies of the system, each in one of the equilibrium morphological microstates. The copies ( ) are distributed in the ensemble such that the free-energies ( ) follow an exponential distribution eq ( ) with the distribution parameter .
The equilibrium morphological entropy Γ = − ∑ eq ( ) ln eq ( ) (i.e. the maximum value of Γ ) follows from (1.2) and (1.3) as where eq ( ) is substituted from Eq. (1.2). Thus, Γ is related to via the conjugate relation Γ / S = . Thus, analogous to 1/ that quantifies the increase in uncertainty of the molecular microstates (i.e. molecular entropy M ( ) ) with average enthalpy, specifies the increase in uncertainty of the morphological microstates (i.e. morphological entropy Γ ) with the average Gibbs free-energy. We therefore refer to 1/ as the homeostatic temperature with the understanding that it quantifies the fluctuations on a timescale much slower than that characterised by .

The equilibrium Gibbs free-energy of a morphological microstate
Similar to conventional statistical mechanics calculations that require a model for the energy of the system, the homeostatic statistical mechanics framework requires a model for the Gibbs free-energy ( ) of morphological microstate ( ). Here, we calculate ( ) using the free-energy model of Vigliotti et al. [2], as modified in [1] and [3], that includes contributions from cell elasticity and the actin/myosin stress-fibre cytoskeleton, with the cell modelled as a twodimensional (2D) body in the 1 − 2 plane adhered to an elastic substrate (Fig. 1b of the main text) such that the out-of-plane Cauchy stress 33 = 0. The state of the system changes as the cell moves, spreads and changes shape on the elastic substrate. Here, we shall give a prescription to calculate the Gibbs free-energy of the system when the cell is in a specific morphological microstate ( ), where the connections of material points on the cell membrane to the surface of the substrate are specified. Calculating the Gibbs free-energy of the system when the material points on the cell membrane is connected to the surface of the adhesive island requires a small modification to the prescription outlined below. Readers are urged to refer to Supplementary S1.3 (and also Supplementary S2 of Buskermolen et al [3]) for further information. In the following, for the sake of notational brevity, we shall drop the superscript ( ) that denotes the morphological microstate, as the entire discussion refers to a single morphological microstate.
The Vigliotti et al. [2] model assumes only two components within the cell: (i) a passive elastic contribution from subcellular structures such as the cell membrane, intermediate filaments and microtubules, and (ii) an active contribution from contractile acto-myosin stress-fibres that are modelled explicitly with the nucleus not explicitly modelled. This model was modified in [1] to incorporate a non-dilute concentration of stress-fibres, and further modified in [3] to include the nucleus in the analysis as a passive elastic body, in addition to the cytoplasm comprising the two above mentioned components. We shall first describe the modelling of the active actomyosin stress-fibres in the cytoplasm and then discuss the elastic model of both the nucleus and the cytoplasm.
Consider a two-dimensional (2D) cell of thickness 0 and volume 0 in its elastic resting state comprising a nucleus of volume N and cytoplasm of volume C such that 0 = N + C (Fig. 1b of the main text). The representative volume element (RVE) of the stress-fibres within the cytoplasm in this resting configuration is assumed to be a cylinder of volume R = 0 ( where ℓ 0 is the length of a stress-fibre functional unit in its ground-state, and R is the number of these ground-state functional units within this reference RVE. The total number of functional unit packets within the cell is 0 T , and we introduce 0 = 0 T R / C as the average number of functional unit packets available per RVE; 0 shall serve as a useful normalisation parameter. The state of stress-fibres at location within the cell is described by their angular concentration ( , ), and there are ( , ) functional units in series along the length of each stress-fibre in the RVE. Here, is the angle of the stress-fibre bundle in the undeformed configuration with respect to the 2 − direction of the elastic substrate (Fig. 1b of the main text). Vigliotti et al. [2] showed that, at steady-state, the number ss of functional units within the stress-fibres is given by where ̃n om ss is the strain at steady-state within a functional unit of the stress-fibres, and nom ( , ) is the nominal strain in direction . The chemical potential of the functional units within the stress-fibres in terms of the Boltzmann constant B is given by [ where the normalized concentration of the unbound stress-fibre proteins is ̂u ≡ u / 0 with ̂≡ R / 0 , while m ax is the maximum normalised value of ̂ corresponding to full occupancy of all available sites for stress-fibres (in a specific direction) and ̂L is the number of lattice sites available to unbound proteins. The enthalpy b of R bound functional units at steady-state is given in terms of the isometric stress-fibre stress max and the internal energy where Ω is the volume of R functional units. By contrast, the chemical potential of the unbound proteins is independent of stress and given in terms of the internal energy u as For a fixed configuration of the 2D cell (i.e. a fixed strain distribution nom ( , )), the contribution to the specific Helmholtz free-energy of the cell from the stress-fibre cytoskeleton follows as where 0 ≡ 0 / R is the number of protein packets per unit reference volume available to form functional units in the cell. However, we cannot yet evaluate cyto as ̂u ( ) and ( , ) are unknown. These will follow from the chemical equilibrium of the cell as will be discussed in Section 1.2.1.
The total stress within the cell includes contributions from the passive elasticity provided mainly by the intermediate filaments of the cytoskeleton attached to the nuclear and plasma membranes and the microtubules, as well as the active contractile stresses of the stress-fibres. The total Cauchy stress is written in an additive decomposition as where and p are the active and passive Cauchy stresses, respectively. In the 2D setting with the cell lying in the 1 − 2 plane, the active stress is given in terms of the volume fraction ℋ 0 of the stress-fibre proteins as where * is the angle of the stress-fibre measured with respect to 2 , and is related to its orientation in the undeformed configuration by the rotation with respect to the undeformed configuration. The passive elasticity in the 2D setting is given by a 2D specialization of the Ogden [4] hyperelastic model as derived in [1]. The strain energy density function of this 2D Ogden model is

Equilibrium of the morphological microstate
Shishvan et al. [1] have shown that equilibrium of a morphological microstate reduces to two conditions: (i) mechanical equilibrium with , = 0 throughout the system, and (ii) chemical equilibrium such that u ( ) = b ( , ) = constant, i.e. the chemical potentials of bound and unbound stress-fibre proteins are equal throughout the cell.
The condition u = b implies that ( , ) is given in terms of ̂u by , (1.14) and ̂u follows from the conservation of stress-fibre proteins throughout the cytoplasm, viz. with the total concentration of bound stress-fibre proteins denoted as ̂b ≡ 1 −̂u. Knowing ̂u and ( , ), the stress can now be evaluated and these stresses within the system (i.e. cell and substrate) need to satisfy mechanical equilibrium, i.e. , = 0. In this case, the mechanical equilibrium condition is readily satisfied as the stress field within the cell is equilibrated by a traction field T exerted by the substrate on the cell such that , = −T , where ( ) is the thickness of the cell in the current configuration.
The equilibrium value of the Gibbs free-energy is then given as = cell + sub where where ψ is the strain-energy density of the substrate material, and sub is the volume of the substrate. The substrate is modelled as a half-space made from a linear elastic material with Young's modulus sub and Poisson's ratio sub = 0.5. We assume that it is sufficient to model the deformation of the substrate using the small strain assumption (i.e. linear kinematics). The problem of calculating the equilibrium strain energy density ψ within the substrate then reduces to a linear elasticity problem that is readily solved using well-known Green's functions via the boundary element method (BEM): the advantage of using BEM to solve the elastic half-space problem is that we only need to mesh the surface of the substrate, while a finite element (FE) calculation would require a 3D meshing of the half-space. However, while it is reasonable to assume small deformations within the substrate, the cell undergoes large deformation and full nonlinear kinematics along with the nonlinear constitutive model is employed to analyse the cell. We thus analyse the cell and the substrate separately and connect the two analyses by ensuring displacement and traction continuity along the portion of the substrate surface connected to the cell.
We apply the traction distribution −T on the surface of the substrate (with the remainder of the substrate surface being traction free) and calculate the equilibrium strain distribution within the substrate using BEM. This gives the substrate free-energy sub . The BEM mesh for the substrate surface comprised ∼2600 elements. Mesh convergence studies revealed that increasing mesh density resulted in changes in the total free-energy of less than 3%, which translates to an error in the estimation of the probability of a morphological microstate of about 1% for typical values of the homeostatic temperature 1/ .
Here, u is given by Eq. (1.8) with the equilibrium value of ̂u obtained from Eq. (1.15). Details of numerical computation of sub is provided in [1]. For the purposes of further discussion, we label the equilibrium value cyto ≡ 0 C u as the cytoskeletal free-energy of the cell, passive ≡ ∫ Φ C C + ∫ Φ N N as the passive elastic energy of the cell, and sub as the substrate free-energy.
It is essential to note that the morphological macrostate of the cell arises from a complex interplay between cyto , passive and sub , such that the average free-energy is maintained at a constant value. In other words, the distribution of cyto indirectly influences the distributions of passive and sub via the homeostatic constraint. For example, an elastic substrate of stiffness sub = 30 kPa can equilibrate a wide range of traction distributions applied by the cell, resulting in high 1/ , which corresponds to large morphological fluctuations of the cell. This implies that the cell can sample both microstates with high levels of stress-fibre polymerisation (i.e. low cyto ), and microstates with large deformation (typically penalised by high passive and/or sub ), such that the average free-energy of the ensemble of microstates is S . However, in any specific microstate, the strain distribution in the cell determines cyto , passive and sub , with no correlation between the three free-energy components. This non-local effect of cyto on passive and sub is evidenced by the low correlation between measures of cellular deformation such as cell area, aspect ratio and average tractions (which correlate well with passive or sub ) and cyto in Fig. 8 of the main text.

Numerical methods
We employ Markov Chain Monte Carlo (MCMC) to construct a Markov chain that is representative of the homeostatic ensemble. This involves three steps: (i) a discretization scheme to represent morphological microstate ( ), (ii) calculation of ( ) for a given morphological microstate ( ), and (iii) construction of a Markov chain comprising these morphological microstates. Here, we briefly describe the procedure which was implemented in MATLAB.
In the general setting of a three-dimensional (3D) cell, a morphological microstate is defined by the connection of material points on the cell membrane to the surface of the substrate. In the 2D context of cells on micropatterned substrates, this reduces to specifying the connection of all material points of the cell to locations within the elastic substrate, i.e. a displacement field respectively. The NURBS employ fourth order base functions for both the 1 and 2 directions, and the knots vector included two nodes each with multiplicity four, located at the extrema of the interval. We emphasise here that this choice of representing the morphological microstates imposes restrictions on the morphological microstates that will be considered. Therefore, the choice of the discretization used to represent ( ) needs to be chosen so as to be able to represent the microstates we wish to sample, e.g. the choice can be based on the minimum width of a filopodium one expects for the given cell type. Given ( ) , we can calculate ( ) using the model outlined in the main paper with the cell discretised using constant strain triangles of size ≈ 0 /10, where 0 is the radius of the cell in its undeformed configuration.
We construct, via MCMC, a Markov chain that serves as a sample of the homeostatic ensemble for cells on wide elastic substrates and cells confined to square adhesive islands of area p . The algorithm closely follows the approach developed by Shishvan et al. [1]. However, here we additionally model cells constrained on adhesive islands with cell adhesion outside the islands prevented. Over the range of island sizes used in the experimental investigation, cells that were partially or entirely outside the islands were not observed. Thus, we construct a sample of the homeostatic ensemble comprising solely of morphological microstates that are fully within the adhesive islands. This is done using the Metropolis [5] algorithm in an iterative manner using the procedure explained in detail in [1] (see section 4.3 therein) but now with the following modification. In constructing the Markov chain, if any portion of the proposed new configuration of the cell lies outside the adhesive island, then the nodal boundary points outside the island were pushed back to the boundary of the island along a line joining their locations in the current proposed morphological microstate and their corresponding positions in previous accepted morphological microstatethis corrected morphological state was then used as the configuration on which to check the probability of acceptance. Typical Markov chains comprised in excess of ℒ = 2 × 10 6 samples.

Material parameters for hMSCs
All simulations are reported at a reference thermodynamic temperature = 0 , where 0 = 310 K. Most of the parameters of the model are related to the properties of the proteins that constitute stress-fibres. These parameters are thus expected to be independent of cell type. Notable exceptions to this are: (i) the stress-fibre protein volume fraction ℋ 0 ; and (ii) the passive elastic properties. Here, we use parameters calibrated for hMSCs that give good correspondence with the wide range of measurements reported in the study. The passive elastic parameters of the cytoplasm are taken to be C = 0.3 kPa, C = 50 kPa and C = 6, while the corresponding values for the nucleus are N = 1 kPa, N = 50 kPa and N = 10. The maximum contractile stress max = 240 kPa is consistent with a wide range of measurements on muscle fibres [6], and the density of stress-fibre proteins was taken as 0 = 3 × 10 6 μm −3 with the volume fraction of stress-fibre proteins ℋ 0 = 0.032. Following Vigliotti et al. [2], we assume that the steady-state functional unit strain ̃n om ss = 0.35 with b0 − u = 2.3 B 0 and Ω = 10 −7.1 μm 3 . The maximum angular concentration of stress-fibre proteins is set to m ax = 0.8. The cell in its undeformed state is a circle of radius 0 and thickness 0 , with a circular nucleus of radius N whose centre coincides with that of the cell. The radius of the hMSCs in their undeformed state was taken to be 0 = 15 μm, while their thickness was set at 0 / 0 = 0.12. The radius of the nucleus of hMSCs in their undeformed state was taken to be N = 6.75 μm, so that the nucleus occupies a volume fraction ̅ N = 0.21 of the cell.

Definitions of normalised quantities and observables
Following [1], the free-energy ( ) can be decomposed as ( ) = Υ ( ) +Υ 0 , where Υ 0 ≡ 0 0 [ u / R − B ln( ̂L )] is independent of the morphological microstate. It is thus natural to subtract out Υ 0 and define a normalised free-energy as We note in passing that we have normalised the Gibbs free-energy of each morphological microstate and the homeostatic temperature by the homeostatic value of the Gibbs free-energy that the cell attains over all the morphological microstates it samples. For the hMSC parameters listed in Supplementary S1.4, the cell in suspension has a radius of ≈ 15 μm with a Gibbs freeenergy S − Υ 0 ≈ −6.1 × 10 9 B 0 = −0.03 nJ (see Shishvan et al. [1] for details of the calculation of S − Υ 0 ). This implies that in units of Kelvin, a typical normalised temperature 1/̂= 0.2 of hMSCs corresponds to 1/ ≈ 10 11 K. Of course, this high temperature stems from the homeostatic framework inherently recognising that the morphological fluctuations of cells, rather than having a thermal origin, are biochemical in nature and arise from the imprecise regulation of the exchange of nutrients with the nutrient bath.
Analogously, we define the normalised passive and cytoskeletal free-energies of the cell as where the dummy symbol x denotes either ̂c yto or a morphometric variable (such as cell area, aspect ratio, and total tractions).

Tractions exerted by cells on substrates
An observable commonly reported in experiments, typically measured via traction-force microscopy, are spatial distributions of tractions exerted by cells on substrates and the associated so-called total traction force [7][8][9]. These measurements are nearly exclusively reported on relatively soft substrates where tractions exerted by the cell induce significant deformations. This makes measurements of surface displacements feasible and allows for a relatively low error estimate of the associated surface tractions [8,9]. It is well-established that the response of cells is mechano-sensitive in the sense that the tractions they exert depend on the substrate stiffness, with the tractions increasing with increasing substrate stiffness [9,10]. Mechanical equilibrium (Section 1.2.1) of a given morphological microstate ( ) specifies the traction distribution T ( ) ( ) between the cell and the substrate. We define a normalised resultant traction where 0 and 0 are the radius and thickness of the cell in its undeformed state, respectively, C is the shear modulus of the cytoplasm and is the traction at location in morphological state ( ). These distributions for selected morphological microstates on elastic substrates of two stiffness ( sub = 3 kPa and 30 kPa) are included in Fig. 3d (in the main text). In addition, we also define the normalised total traction as and the distributions of the normalised total traction are shown in Fig. 3c (in the main text) for cells on elastic substrates of three different stiffness, in Fig. 4b (in the main text) for cells on adhesive islands of three areas, and in Fig. 9c (in the main text) comparing cells untreated and treated with ROCK inhibitor.

Construction of immunofluorescence-like images
The predictions in Fig. 2a (in the main text) show immunofluorescence-like images comprising of three layers; the nucleus is coloured in purple and blue (in the top and bottom panels, respectively), while the focal adhesions and actin stress-fibres in shades of green and red, respectively. Focal adhesion distributions in the experiments by Engler et al [11] were observed by staining for paxillin (Fig. 2b of the main text). In the current set of simulations, the focal adhesions were not explicitly modelled but rather the cell was assumed to perfectly adhere to the substrate without directly accounting for the distribution of adhesion proteins. This results in the simulations predicting traction distributions Τ ( ) as discussed above. However, it is well-known that traction magnitudes scale with concentration of adhesive proteins [12] and thus, here we use predictions of traction magnitudes as a surrogate to visualise the predictions of focal adhesion distributions. Specifically, we include in Fig. 2a (in the main text) distributions of Τ ( ) for selected morphological microstates ( ) with the darker shades of green representing higher values of Τ ( ) and thereby higher paxillin concentrations.
The predictions of the actin stress-fibre structure within the cytoplasm are coloured in red. These images, are meant to give two key pieces of information: (i) the local orientation of the dominant stress-fibre bundle, and (ii) the concentration of the stress-fibre proteins. However, no discrete fibres are present in the model of the acto-myosin stress-fibres employed here with the fibres represented via continuum internal state variables. Here, we use this continuum information to generate a discrete depiction of the fibres using the following prescription.
Recall that for the purposes of calculation of ( ) , the cell was discretized into 800 constant strain triangles of approximately equal size in the undeformed configuration. Within each element , we defined an actin stress-fibre concentration in direction * as ̂b ( * ) ≡ ( * )̂s s ( * ), where * is the angle of the stress-fibre measured with respect to 2 direction and is related to its angle in the undeformed configuration by the rotation of element . Next, we construct a dummy mesh comprising approximately 200 triangular elements to discretize the deformed cell. Elements of the original computational mesh are assigned to the element of the dummy mesh in which their centroid lies. In this manner, approximately four elements of the original computational mesh are assigned to each element of the dummy mesh.
We then define an average actin stress-fibre concentration for each element of the dummy mesh as 〈̂b( * )〉 ≡ 1 ∑̂b ( * ) ∈ , (1.25) where is the volume of element of the computational mesh that is assigned to element of the dummy mesh and = ∑ ∈ . The dominant stress-fibre direction max * in dummy element is defined as the value of * corresponding to the maximum value of 〈̂b( * )〉 . Each element of the dummy mesh is hatched with green lines in the direction max * . We set the spacing of the hatches to scale with the local density of bound stress-fibre proteins in order to replicate the higher intensity of the fluorescent phalloidin observed under light microscopy in locations with a higher concentration of stress-fibres. The total concentration of bound stressfibre proteins in element of the computational mesh is with ̂b T ≡ ∫̂b C denoting the total concentration of bound stress-fibre proteins in the cell.
The spacing of the green hatches in element of the dummy mesh is set to scale inversely with the concentration of bound stress-fibre proteins within that element, i.e. ≡̂b / ∑̂b ∈ . This process results in the predictions of the actin cytoskeletal structure shown in Fig. 2a (in the main text), and we include in the top panel of Fig. 10b (in the main text) only predictions of the actin cytoskeletal structure (i.e. without nucleus or focal adhesions) for cells on adhesive islands of area p = 2725 2 to give more insight into the details of the predicted cytoskeletal structure at various locations within the cell. The predictions in Fig. 4a (in the main text) are generated by the same method, with the nucleus coloured in blue, while the focal adhesions (i.e. distributions of Τ ( ) ) and actin stress-fibres are coloured in shades of pink and green, respectively.