A new mechanochemical model for apical constriction: Coupling calcium signalling and viscoelasticity

Embryonic epithelial cells exhibit strong coupling of mechanical responses to chemical signals and most notably to calcium. Recent experiments have shown that the disruption of calcium signals during neurulation strongly correlates with the appearance of neural tube defects. We, thus, develop a multi-dimensional mechanochemical model and use it to reproduce important experimental findings that describe anterior neural plate morphogenetic behaviour during neural tube closure. The governing equations consist of an advection-diffusion-reaction system for calcium concentration which is coupled to a force balance equation for the tissue. The tissue is modelled as a linear viscoelastic material that includes a calcium-dependent contraction stress. We implement a random distribution of calcium sparks that is compatible with experimental findings. A finite element method is employed to generate numerical solutions of the model for an appropriately chosen range of parameter values. We analyse the behaviour of the model as three parameters vary: the level of IP3 concentration, the strength of the stretch-sensitive activation and the maximum magnitude of the calcium-dependent contraction stress. Importantly, the simulations reproduce important experimental features, such as the spatio-temporal correlation between calcium transients and tissue deformation, the monotonic reduction of the apical surface area and the constant constriction rate, as time progresses. The model could also be employed to gain insights into other biological processes where the coupling of calcium signalling and mechanics is important, such as carcinogenesis and wound healing.