Sticking around: Optimal cell adhesion patterning for energy minimization and substrate mechanosensing

Cell mechanotransduction, in which cells sense and respond to the physical properties of their micro-environments, is proving fundamental to understanding cellular behaviours across biology. Tissue stiffness (Young’s modulus) is typically regarded as the key control parameter and bioengineered gels with defined mechanical properties have become an essential part of the toolkit for interrogating mechanotransduction. We here, however, show using a mechanical cell model that the effective substrate stiffness experienced by a cell depends not just on the engineered mechanical properties of the substrate but critically also on the particular arrangement of adhesions between cell and substrate. In particular, we find that cells with different adhesion patterns can experience two different gel stiffnesses as equivalent and will generate the same mean cell deformations. For small adhesive patches, which mimic experimentally observed focal adhesions, we demonstrate that the observed dynamics of adhesion growth and elongation can be explained by energy considerations. Significantly we show different focal adhesions dynamics for soft and stiff substrates with focal adhesion growth not preferred on soft substrates consistent with reported dynamics. Equally, fewer and larger adhesions are predicted to be preferred over more and smaller, an effect enhanced by random spot placing with the simulations predicting qualitatively realistic cell shapes in this case. The model is based on a continuum elasticity description of the cell and substrate system, with an active stress component capturing cellular contractility. This work demonstrates the necessity of considering the whole cell-substrate system, including the patterning of adhesion, when investigating cell stiffness sensing, with implications for mechanotransductive control in biophysics and tissue engineering. Author summary Cells are now known to sense the mechanical properties of their tissue micro-environments and use this as a signal to control a range of behaviours. Experimentally, such cell mechanotransduction is mostly investigated using carefully engineered gel substrates with defined stiffness. Here we show, using a model integrating active cellular contractility with continuum mechanics, that the way in which a cell senses its environment depends critically not just on the stiffness of the gel but also on the spatial patterning of adhesion sites. In this way, two gels of substantially different stiffnesses can be experienced by the cell as similar, if the adhesions are located differently. Exploiting this insight, we demonstrate that it is energetically favourable for small adhesions to grow and elongate on stiff substrates but that this is not the case on soft substrates. This is consistent with experimental observations that nascent adhesions only mature to stable focal adhesion (FA) sites on stiff substrates where they also grow and elongate. These focal adhesions (FAs) have been the focus of work on mechanotransduction. However, our paper demonstrates that there is a fundamental need to consider the combined cell and micro-environment system moving beyond a focus on individual FAs.

Introduction this signal transduction remains unclear. FAs have received most attention as mechanosensors [32][33][34], although there is an increasing awareness of the need to 40 account for mechanosensing across the cell including at the nuclear envelope [35,36]. 41 Several theoretical models have been developed to gain an insight into the cellular 42 force generation and its effects. Largely, these treat the main body of a cell as an elastic 43 solid which is being acted upon by an active component; this is coupled to a substrate 44 which offers further resistance to the force. The way in which the active cellular 45 contractility is represented broadly falls into two categories: simulations of cytoskeletal 46 dynamics and active continuum theories. Computational simulations of the cytoskeleton 47 tend to focus on the dynamics of subcellular constituents of the contractile mechanism 48 to investigate the cell-scale effects of their collective behaviour (e.g. [37][38][39]). In the 49 continuum approach, an adaptation of linear elasticity can lead to the introduction of 50 an active contractile term to the material constitutive relations and in this way either a 51 force balance equation [40,41] or an equivalent energy minimisation [42] problem can be 52 constructed for the cell deformations and stress. Such models can be adapted to 53 incorporate different adhesion dynamics, including FA clustering [43], to investigate 54 intracellular mechanics [16]. 55 In this work, we adopt an active stress approach to model the cellular contractility. 56 We focus on the significance of adhesion distribution and patterning on a cell's overall 57 ability to deform. Two distinct cases are considered. First, we consider that the cell is 58 adhered in a ring around its edge before considering the case that the cell is adhered at 59 several distinct spots mimicking FA complexes. In the case of spot adhesion we vary the 60 distribution, total area and shape of adhered regions and consider the effects on the 61 mean cellular deformation, relating this to the resistance the cell experiences from the 62 underlying substrate. These results show that the substrate may be experienced as 63 more or less stiff depending on how the cell is adhered. Specifically, we show that a cell 64 with a sparse distribution of adhered regions around its periphery, with large gaps |x| < r 1 0, |x| ∈ [r 1 , r 0 ], depicted in Fig 1A, and by symmetry the deformations are purely radial so that 96 u = u(r)e r . Consequently, the force balance equation (1) can be solved analytically to 97 give 98 u P 0 r 0 = where α 0 , α 1 and β 1 are constants determined from the boundary conditions (see 99 Methods) and I 1 and K 1 are modified Bessel functions. The contraction parameter P 0 100 linearly scales the deformation altering its magnitude only. As an illustration, Fig 1B   101 plots the analytical solution for different thicknesses of adhesive ring. We discover one 102 key nondimensional parameter,  To model localised spots of adhesion and in particular FAs, we take T (x) = 1 only in 106 small circular or elliptical regions in the cell (e.g. Fig 2A). In this case, an analytical  For cells adhered uniformly at their outer edge we observe in Fig 1B that  such measure adopted which is relatively easy to interpret is the mean deformation over 117 the cell area [48]. Where the mean deformation is lower the apparent substrate stiffness 118 by this measure would be higher, whereas a larger mean deformation would correspond 119 to a softer substrate. For the adhered ring the mean cellular deformation (scaled by cell 120 radius) can be explicitly calculated, see Methods. We see that throughout γ e < γ, so that the resistance experienced by the partially adhered cell is less than that for a completely adhered cell. Specifically, cells 138 with thinner rings (with r 1 /r 0 near 1) sense an effectively softer substrate as they 139 experience less resistance, however, this effect becomes less pronounced for wider 140 adhesive rings. For example when r 1 ≈ 0.66r 0 , γ e /γ is already close to 1 (γ e /γ = 0.9 on 141 γ = 5), showing that the cell is sensing almost the same resistance as it would when 142 completely adhered. This effect is enhanced on stiffer substrates, for instance when 143 γ = 15, γ e /γ = 0.9 is obtained at r 1 ≈ 0.78r 0 , see S1  with the same increase in substrate stiffness.  In Table 1, we present the mean cellular deformations corresponding to the Correspondingly, we suggest that the cell effectively experiences a stiffer substrate with 184 an increase in adhered area, in analogy to the results for a cell with an adhesive ring.

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However, this effect can in part be compensated for if the increase in area is not 186 uniform but is generated through an elongation of the spot. In Fig 3B, we show how a given spot size), we find that elongating the spots inwards may result in an increase in 189 the mean cellular deformation ( Fig 3B)  grow as the applied force increases, but that they additionally tend to elongate in the 197 direction of applied forces [19,20]. Our results suggest that that such an elongation may 198 be being used to at least partially compensate for the effect of adhesion growth. Orange line corresponds to elliptical spots with a fixed width but increasing length so the aspect ratio increases as adhered area increases, here W S is plotted against the equivalent radius of circular spots.
Finally, we consider the effects of an elongating spot on the substrate strain energy, 233 shown in Figs 4F and G. In Fig 4F, we see that on a soft substrate increasing the aspect 234 ratio from circular to elongated elliptic spots (aligned towards the cell centre) decreases 235 August 6, 2020 11/30 the work done to the substrate as the cell is more able to deform in the gaps between adhesions. Furthermore, our investigations suggest that on stiffer substrates there is an 237 optimal spot aspect ratio for these adhesions depending on spot size. In the case of an 238 adhered area ranging from 5% to 15% and γ = 15 this is approximately 2-3 times as 239 long as they are wide (Fig 4G). In Figs

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With a uniform ring of adhesion we show that the mean deformation, a measure 303 commonly experimentally measured [48], increases with a decrease in adhered area 304 (Fig 1). In this way, an entirely different gel stiffness can generate the same mean demonstrated that the effective stiffness is always less than the true gel stiffness Table 1. 311 The difference between the true stiffness and the effective stiffness was also shown to be 312 reduced on stiffer substrates or arrangements with greater adhered area. When 313 considering adhesive spots (FAs) we demonstrated that both increasing the number of 314 spots and total adhered area through increasing spot size effectively 'stiffens' the 315 substrate (Fig 2). Where the points of adhesion are randomly distributed the cell 316 shapes qualitatively look more realistic (Fig 5C-F) and the introduction of variation in 317 the inter-spot spacing further 'softens' the surface (Fig 5A).
We considered further the substrate strain energy, i.e. the work the cell does to the 319 underlying substrate, for both adhesive ring and spots. We showed that where adhesive 320 spots are distributed in a region around the cell edge this is energetically favourable 321 compared with maintaining adhesions directly at the cell edge (Fig 5B), while the mean 322 deformation is also increased, at least for the more regular arrangements (Fig 5A). We 323 note that it is observed experimentally that FAs form on stiff substrates with soft 324 substrates having no stable adhesions, and that FAs grow and elongate on these 325 substrates [15,20]. Significantly, we here show that starting with nascent adhesions of 326 small area, FA growth and elongation would be energetically favourable on stiff 327 substrates but not on soft substrates (Figs 4D and E). We additionally show that 328 although FA growth would by itself reduce the mean deformation and effectively 'stiffen' 329 the substrate the elongation of the adhesion spot can compensate for this effect (Fig 3). 330 On soft substrates spot elongation also reduces W s . However, on stiff substrates spot The mechanical model laid out above is based on an active stress model for cell 351 contractility. Thus the cell stress is considered to comprise of two components such that 352 where P is the target area reduction of cell elements [40]. We take P = −P 0 constant 356 throughout. In these constitutive relations, E c and ν are the cell's Young's modulus and 357 Poisson's ratio respectively, δ ij the Kronecker delta and ij the linear strain tensor.

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Throughout, we impose the condition that there is zero stress at the outer cell edge.

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For the case where the cell is assumed to adhere in a ring symmetry ensures that all deformations are radial, i.e. the deformation u = u(r)e r . On substituting the constitutive relations into the force balance (1) we obtain where the cell is adhered in r 1 < r < r 0 , which has exact solution (2) with the constants α 0 , α 1 , β 1 given by August 6, 2020 16/30 For conciseness we introduced functions: In the limit r1 r0 → 0, we recover the solution of a completely adhered disc [40] u The mean cellular deformation of a cell with an adhered ring can be calculated as where L 0 and L 1 are modified Struve functions [54].

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From the solution for u the strain energy W = W CA + W CP + W S can be calculated from the integral expressions With the approximation of the substrate as a linear array of springs there are no 365 non-local effects with no deformation outside of the cell spread area A so the integrals 366 can be restricted to A without loss of generality. We note that minimising the energy 367 functional W is equivalent to solving the force balance equation (1) [26,42,55] 368 (although [26,55] include an additional term to account for the deformation of the cell 369 membrane). We scale all energies by a factor of (1 − ν 2 )/hE c r 0 2 .