Modeling mechanochemical pattern formation in elastic sheets of biological matter

TL;DR

This paper develops numerical models to study how elastic sheets in biological tissues form patterns through coupled mechanical and chemical processes, revealing long-range pattern propagation driven by mechanochemical interactions.

Contribution

It introduces two generic mechanochemical models with numerical methods to analyze pattern formation in elastic sheets influenced by chemical signals.

Findings

Pattern formation depends on shell thickness, coupling strength, and diffusivity.

Mechanochemical coupling enables long-range pattern propagation without chemical diffusion.

Different patterns emerge based on thresholds and stress-dependent activities.

Abstract

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.