  Inspired by active shape morphing in developing tissues and biomaterials, we
investigate two generic mechanochemical models where the deformations of a thin
elastic sheet are driven by, and in turn affect, the concentration gradients of
a chemical signal. We develop numerical methods to study the coupled elastic
deformations and chemical concentration kinetics, and illustrate with
computations the formation of different patterns depending on shell thickness,
strength of mechanochemical coupling and diffusivity. In the first model, the
sheet curvature governs the production of a contractility inhibitor and
depending on the threshold in the coupling, qualitatively different patterns
occur. The second model is based on the stress--dependent activity of myosin
motors, and demonstrates how the concentration distribution patterns of
molecular motors are affected by the long-range deformations generated by them.
Since the propagation of mechanical deformations is typically faster than
chemical kinetics (of molecular motors or signaling agents that affect motors),
we describe in detail and implement a numerical method based on separation of
timescales to effectively simulate such systems. We show that mechanochemical
coupling leads to long-range propagation of patterns in disparate systems
through elastic instabilities even without the diffusive or advective transport
of the chemicals.
