Mechanochemical models for calcium waves in embryonic epithelia

In embryogenesis, epithelial cells, acting as individual entities or as coordinated aggregates in a tissue, exhibit strong coupling between chemical signalling and mechanical responses to internally or externally applied stresses. Intercellular communication in combination with such coordination of morphogenetic movements can lead to drastic modifications in the calcium distribution in the cells. In this paper we extend the recent mechanochemical model in [K. Kaouri, P.K. Maini, P.A. Skourides, N. Christodoulou, S.J. Chapman. J. Math. Biol., 78 (2019) 2059--2092], for an epithelial continuum in one dimension, to a more realistic multi-dimensional case. The resulting parametrised governing equations consist of an advection-diffusion-reaction system for calcium signalling coupled with active-stress linear viscoelasticity and equipped with pure Neumann boundary conditions. We implement a mixed finite element method for the simulation of this complex multiphysics problem. Special care is taken in the treatment of the stress-free boundary conditions for the viscoelasticity in order to eliminate rigid motions from the space of admissible displacements. The stability and solvability of the continuous weak formulation is shown using fixed-point theory. We investigate numerically the solutions of this system and show that solitary waves and periodic wavetrains of calcium propagate through the embryonic epithelial sheet. We analyse the bifurcations of the system guided by the bifurcation analysis of the one-dimensional model. We also demonstrate the nucleation of calcium sparks into synchronous calcium waves coupled with contraction. This coupled model can be employed to gain insights into recent experimental observations in the context of embryogenesis, but also in other biological systems such as cancer cells, wound healing, keratinocytes, or white blood cells.