Rigidity sensing by stochastic sliding friction

The sliding friction force exerted by stochastic linkers interacting with a moving filament is calculated. The elastic properties of the substrate on which the linkers are anchored are shown to strongly influence the friction force. In some cases, the force is maximal for a finite substrate rigidity. Collective effects give rise to a dynamical instability resulting in a stick-slip behaviour, which is substrate-sensitive. The relevance of these results for the motility of crawling cells powered by an actin retrograde flow is discussed. Introduction. – Large-scale biological adhesion often involves a collection of ligand and receptor molecules undergoing stochastic binding and unbinding. The transient nature of cellular adhesion is crucial to such processes as motility, during which the cell must both exert a traction force and slide over a substrate. The cytoskeleton interacts with the extra-cellular matrix (ECM) through transmembrane receptors such as integrin binding to components of the ECM such as fibronectin [1]. Crawling cells often form broad and flat protrusions, called lamellipodia, where polymerisation of actin filaments against the cell membrane and contraction of the actin network by myosin motors result in an actin retrograde flow moving away from the cell leading edge [2]. Transient cytoskeleton adhesion to the ECM amounts to an effective friction on the actin retrograde flow that pushes the cell edge forward. Cells can sense various external cues, including the rigidity of their environment [3–5]. Rigidity sensing might be in part permitted by the stochastic nature of the friction force. Furthermore, the actin retrograde flow is widely reported to be irregular [6], sometimes displaying periodic oscillations [7]. This behaviour may be due to collective effect among the proteins linking the cytoskeleton to the extracellular medium. The pioneering work of Schallamach [8] showed that the friction force on an object sliding over a substrate covered with microscopic stochastic linkers can show a nonmonotonic behaviour with the sliding velocity. The friction force may decrease with increasing velocity in some range of parameters, a behaviour that is usually associated with dynamical instabilities and stick-slip [9, 10]. Combination of experimental and analytical works showed that stick-slip occurs between surfaces coated by surfactant layers [11] and between an actin filament and a substrate coated with myosin motors [12]. Numerical simulations have confirmed the relationships between macroscopic frictional phenomena (including stick-slip) and the dynamics of formation and rupture of microscopic bonds [13]. More recently, this behaviour has been put in the context of cell motility by Chan and Odde [14,15] and has been theoretically investigated in depth by several groups [16–18]. Several models exist for cellular rigidity sensing [19,20], but the role of substrate elasticity, and in particular the influence of substrate-mediated elastic interactions, on the friction force exerted by stochastic linkers has not yet received a synthetic analytical treatment. Based on computer simulations, Chan and Odde argued that the stochastic traction force could be optimal for intermediate substrate stiffness if the element driving the filament motion (myosin motors in their case) impose a forcedependent filament velocity [14, 15]. We show below that their treatment of the substrate compliance (one large spring connecting all the adhesion molecules together) is insufficient. Furthermore, considering the possible generic role sliding friction might play in rigidity sensing during cell migration, a more general conceptual model is needed. We present here a simple derivation of the force-velocity relationship that qualitatively reproduces the relevant features of the complete solution presented in [16–18], and has the advantage of being amenable to analytical treatment. This expression is then used to analytically derive the stability of a collection of stochastic linkers interacting p-1 ar X iv :1 31 0. 81 65 v1 [ co nd -m at .s of t] 3 0 O ct 2 01 3