Stabilization of active tissue deformation by a dynamic signaling gradient
A key process during animal morphogenesis is oriented tissue deformation, which is often driven by internally generated active stresses. Yet, such active oriented materials are prone to well-known instabilities, raising the question of how oriented tissue deformation can be robust during morphogenesis. Here we study under which conditions active oriented deformation can be stabilized by the concentration pattern of a signaling molecule, which is secreted by a localized source region, diffuses across the tissue, and degrades. Consistent with earlier results, we find that oriented tissue deformation is always unstable in the gradient-contractile case, i.e. when active stresses act to contract the tissue along the direction of the signaling gradient, and we now show that this is true even in the limit of large diffusion. However, active deformation can be stabilized in the gradient-extensile case, i.e. when active stresses act to extend the tissue along the direction of the signaling gradient. Specifically, we show that gradient-extensile systems can be stable when the tissue is already elongated in the direction of the gradient. We moreover point out the existence of a formerly unknown, additional instability of the tissue shape change. This instability results from the interplay of active tissue shear and signal diffusion, and it indicates that some additional feedback mechanism may be required to control the target tissue shape. Taken together, our theoretical results provide quantitative criteria for robust active tissue deformation, and explain the lack of gradient-contractile systems in the biological literature, suggesting that the active matter instability acts as an evolutionary selection criterion.