Pattern formation in discrete cell lattices

Abstract

In recent years, models for lattices of discrete cells have been attracting increased attention due to their greater flexibility to represent signalling and contact-dependent cell-cell interaction than conventional reaction-diffusion models. Using the almost forgotten method of Othmer and Scriven (1971) to calculate eigenvalues and eigenvectors for the Jacobian of the homogeneous state, a Turing-like linear stability analysis is carried out for diffusion-driven (DD) and signalling-driven (SD) discrete models. The method is a generalisation of the original method of Turing (1952). For two-species models it is found that there are profound differences between the two types of model when the size of the lattice increases. For DD models, the homogeneous state is typically either always stable, always unstable, or becomes unstable when the lattice gets suffficiently large. For SD models, the homogeneous state is typically unstable independent of lattice size, and stable only in a minor part of parameter space. Thus, SD models seem in general more pattern-prone than DD models. The conjecture that the linear analysis predicts the final pattern is investigated for a DD system with Thomas internal dynamics. Commonly the final pattern resembles the pattern of the initial perturbation of the homogeneous state, but this is by no means a general feature. When applied to a recent model for Delta-Notch lateral inhibition, linear analysis must be supplemented by various non-linear techniques to get a deeper insight into the patterning mechanisms. The overall conclusion is that a linear Turing analysis may be useful for predicting pattern, but when it comes to explaining patterns, non-linear analysis cannot be ignored.