Spontaneous oscillations from turnover of an elastic contractile material

Single and collective cellular oscillations involving the actomyosin cytoskeleton have been observed in numerous biological systems. We show here that a generic model of a contractile material, which is turning over and contracts against an elastic element, exhibits spontaneous oscillations. Such a model can thus account for shape oscillations observed in amnioserosa cells during dorsal closure of the Drosophila embryo. We investigate the collective dynamics of an ensemble of such oscillators and show that the relative contribution of viscous and friction losses yield different regimes of collective oscillations. Taking into account the diffusion of contractile elements, our theoretical framework predicts the appearance of traveling waves which might account for the propagation of actomyosin contractile waves observed during morphogenesis. In the cellular cytoskeleton, myosin molecular motors bind to networks of actin filaments and exert forces. Inhomogeneities in contractile stresses result in flows and deformations that can occur at the cellular or tissue level [1]. A striking example of such deformations are cell shape oscillations that have been shown to be involved in the morphogenesis of biological tissues during the development of living organisms [2]. In many examples, these oscillations occur jointly with cycles of accumulation and depolymerization of actin and myosin in the cell cortex, a thin layer of actomyosin located near the surface of the cell and concentrated on the apical side of tissues [1]. A hallmark of the observed oscillatory behaviour is that the actomyosin density oscillates approximates in phase opposition with the cell shape, in cells [3, 4] as well as in tissues [5, 6]. More specifically, higher concentrations reached when the cell surface area reaches a minimum [3, 7]. A fundamental property of the cell actomyosin cortex is its ability to turn over and to constantly renew its constituents at a certain rate. A recent study [3] highlighted that cortical turnover is essential to account for shape oscillations observed during cytokinesis. We generalize here the model of the Ref. [3] and show that a minimal, generic description of actomyosin contractility predicts spontaneous oscillations and different patterns of collective oscillations observed in different biological tissues. The minimal ingredients to generate oscillations are (1) a contractile element whose constituents are turning over, (2) an elastic element and (3) a viscous damper, i.e. a dissipative element. We think of the contractile element as representing a contractile network of concentration ar X iv :1 40 2. 53 90 v1 [ ph ys ic s. bi oph ] 2 1 Fe b 20 14 Spontaneous oscillations from turnover of an elastic contractile material 2 c exchanging its material with a reservoir (Fig. 1A), such that the concentration c follows the following dynamical equation: