Segregation, finite-time elastic singularities, and coarsening in renewable active matter

Material renewability in active living systems, such as in cells and tissues, can drive the large-scale patterning of forces, with distinctive phenotypic consequences. This is especially significant in the cell cytoskeleton, where multiple species of myosin bound to actin, apply differential contractile stresses and undergo differential turnover, giving rise to patterned force channeling. Here we study the dynamical patterning of stresses that emerge in a hydrodynamic description of a renewable active actomyosin elastomer comprising two myosin species. Our analytical framework also holds for an actomyosin elastomer with a single myosin species. We find that a uniform active contractile elastomer spontaneously segregates into spinodal stress patterns, followed by a finite-time collapse into tension carrying singular structures that display self-similar scaling and caustics. Our numerical analysis carried out in one dimension shows that these singular structures move and merge and gradually result in a slow coarsening dynamics. We discuss the implications of our findings to the emergence of stress fibers and the spatial patterning of actomyosin. Our study suggests that with state-dependent turnover of crosslinkers and myosin, the in vivo cytoskeleton can navigate through the space of material parameters to achieve a variety of functional phenotypes.