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Mechanochemical Models for Calcium Waves in Embryonic Epithelia

Abstract:

In embryogenesis, epithelial cells acting as individual entities or as coordinated aggregates in a tissue, exhibit strong coupling between mechanical responses to internally or externally applied stresses and chemical signalling. One of the most important chemical signals in this process is calcium. This mechanochemical coupling and intercellular communication drive the coordination of morphogenetic movements which are characterised by drastic changes in the concentration of calcium in the tissue. In this paper we extend the recent mechanochemical model in Kaouri et al. (J. Math. Biol. 78, 2059–2092, 2019), 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 finite element method in perturbed saddle-point form 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. Guided by the bifurcation analysis of the one-dimensional model, we analyse the behaviour of the system as two bifurcation parameters vary: the level of IP3 concentration and the strength of the mechanochemical coupling. We identify the parameter regions giving rise to solitary waves and periodic wavetrains of calcium. Furthermore, we demonstrate the nucleation of calcium sparks into synchronous calcium waves coupled with deformation. This 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.