Substrate Rigidity Deforms and Polarizes Active Gels

We present a continuum model of the coupling between cells and substrate that accounts for some of the observed substrate-stiffness dependence of cell properties. The cell is modeled as an elastic active gel, adapting recently developed continuum theories of active viscoelastic fluids. The coupling to the substrate enters as a boundary condition that relates the cell’s deformation field to local stress gradients. In the presence of activity, the coupling to the substrate yields spatially inhomogeneous contractile stresses and deformations in the cell and can enhance polarization, breaking the cell’s front-rear symmetry. Introduction. – Many cell properties, including cell shape, migration and differentiation, are critically controlled by the strength and nature of the cell’s adhesion to a solid substrate and by the substrate’s mechanical properties [1]. For instance, it has been demonstrated that cell differentiation is optimized in a narrow range of matrix rigidity [2] and that the stiffness of the substrate can direct lineage specification of human mesenchymal stem cells [3]. In endothelial cells, adhesion to a substrate plays a crucial role in guiding cell migration and controlling a number of physiological processes, including vascular development, wound healing, and tumor spreading [4]. Fibroblasts and endothelial cells seem to generate more traction force and develop a broader and flatter morphology on stiff substrates than they do on soft but equally adhesive surfaces [5, 6]. They show an abrupt change in their spread area within a narrow range of substrate stiffnesses. This spreading also coincides with the appearance of stress fibers in the cytoskeleton, corresponding to the onset of a substantial amount of polarization within the cell [6]. Finally, such cells preferentially move from a soft to a hard surface and migrate faster on stiffer substrates [7]. The mechanical interaction of cells with a surrounding matrix is to a great extent controlled by contractile forces generated by interactions between filamentary actin and myosin proteins in the cytoskeleton. Such forces are then transmitted by cells to their surroundings through the action of focal adhesions that produce elastic stresses both in the cell and in the surrounding matrix. Cells in turn are capable of responding to the substrate stiffness by adjusting their own adhesion and elastic properties, with important implications for cell motility and shape [1, 8]. In this letter we present a simple model of the coupling between cells and substrate that accounts for some of the observed substrate-stiffness dependence of cell properties. The cell itself is modeled as an elastic active gel, adapting recently developed continuum theories of active viscoelastic fluids [9–11]. In these models the transduction of chemical energy from ATP hydrolysis into mechanical work by myosin motor proteins pulling on actin filaments yields active contractile contributions to the local stresses. The continuum theory of such active liquids has led to several predictions, including the onset of spontaneous deformation and flow in active films [12,13] and the retrograde flow of actin in the lamellipodium of crawling cells [11]. Active liquids cannot, however, support elastic stresses at long times, as required for the understanding of the crawling dynamics of the lamellipodium and of active contractions in living cells. Models of active elastic solids on the other hand have been shown to account for the contractility and stiffening of in-vitro actomyosin networks [14–16] and the spontaneous oscillations of muscle sarcomeres [17,18]. Very recently a continuum model of a one-dimensional polar, active elastic solid has also been used to describe the alternating polarity patterns observed in stress fibers [19]. In all these cases the elastic nature of the network at low frequency is crucial to provide the restoring forces needed to support deformations and oscillatory behavior. We model a cell as an elastic active film anchored to a solid substrate and study the static response of the film p-1 ar X iv :1 10 6. 09 29 v2 [ qbi o. C B ] 2 9 A ug 2 01 1 S. Banerjee1 M.C. Marchetti1,2 to variations in the strength of the anchoring. Although in the following we refer to our system as a cell, we stress that, on different length scales, the active elastic gel could also serve as a model for a confluent cell monolayer on a substrate. The coupling of the cell to the substrate enters via a boundary condition controlled by a “stiffness” parameter that depends on both the cell/substrate adhesion as well as the substrate rigidity. The description is macroscopic and applies on length scales large compared to the typical mesh size of the actin network in the cell lamellipodium (or large compared to the typical cell size in the case of a cell monolayer). By solving the elasticity and force balance equations in a simple one-dimensional geometry we obtain several experimentally relevant results. First, in an isotropic active gel substrate anchoring yields stresses and contractile deformations. The stress and deformation profiles for an isotropic active elastic gel are shown in the top frame of Fig. 1. The stress is largest at the center of the cell. Interestingly, a very similar profile of tensile stresses has been observed in confluent monolayers of migrating epithelial cells [20], where the stress increases as a function of the distance from the leading edge of the migrating layer and reaches its maximum at the center of the cell colony. Although our model considers stationary active elastic layers (and the resulting stresses are contractile as opposed to tensile), in both cases these stresses originate from active processes in the cell, driven by ATP consumption. The deformation of the active layer is largest at the cell boundaries (see Fig.1, top frame), as seen in experiments imaging traction forces exerted by cells on substrates [21] and its overall magnitude increases with cell activity. The density of the active gel layer is concentrated at the boundary, where the local contractile deformations are largest. The net deformation of the cell over its length is shown in the bottom frame of Fig. 1 and it increases monotonically with decreasing substrate stiffness, in qualitative agreement with experiments on fibroblasts showing that these cells are more extended on stiff substrates [6]. Finally, if the cell is polarized on average, the coupling to the substrate generates a spatially inhomogeneous polarization profile inside the cell. The mean polarization is enhanced over its value in the absence of substrate anchoring and it is a non-monotonic function of substrate stiffness (see Fig. 4). This result is in qualitative agreement with recent experiments that have demonstrated an intimate relation between the matrix rigidity and the alignment of cell fibers within the cell, suggesting that maximum alignment may be obtained for an optimal value of the substrate rigidity [22]. The active gel model. – The cell is modeled as an active gel described in terms of a density, ρ(r, t), and a displacement field, u(r, t), characterizing local deformations. In addition, to account for the possibility of cell polarization as may be induced by directed myosin motion and/or filament treadmilling, we introduce a polar orientational order parameter field, P(r, t). Although we are 0.0 0.2 0.4 0.6 0.8 1.0 -0.2 0.0 0.2 0.4 0.6 0.8