Tissue interaction and spatial pattern formation
The development of spatial structure and form on vertebrate skin is a complex and poorly understood phenomenon. We consider here a new mechanochemical tissue interaction model for generating vertebrate skin patterns. Tissue interaction, which plays a crucial role in vertebrate skin morphogenesis, is modelled by reacting and diffusing signal morphogens. The model consists of seven coupled partial differential equations, one each for dermal and epidermal cell densities, four for the signal morphogen concentrations and one for describing epithelial mechanics. Because of its complexity, we reduce the full model to a small strain quasi-steady-state model, by making several simplifying assumptions. A steady state analysis demonstrates that our reduced system possesses stable time-independent steady state solutions on one-dimensional spatial domains. A linear analysis combined with a multiple time-scale perturbation procedure and numerical simulations are used to examine the range of patterns that the model can exhibit on both one- and two-dimensions domains. Spatial patterns, such as rolls, squares, rhombi and hexagons, which are remarkably similar to those observed on vertebrate skin, are obtained. Although much of the work on pattern formation is concerned with synchronous spatial patterning, many structures on vertebrate skin are laid down in a sequential fashion. Our tissue interaction model can account for such sequential pattern formation. A linear analysis and a regular perturbation analysis is used to examine propagating epithelial contraction waves coupled to dermal cell invasion waves. The results compare favourably with those obtained from numerical simulations of the model. Furthermore, sequential pattern formation on one-dimensional domains is analysed; first by an asymptotic technique, and then by a new method involving the envelopes of the spatio-temporal propagating solutions. Both methods provide analytical estimates for the speeds of the wave of propagating pattern which are in close agreement with those obtained numerically. Finally, by numerical simulations, we show that our tissue interaction model can account for two-dimensional sequential pattern formation. In particular, we show that complex two-dimensional patterns can be determined by simple quasi-one-dimensional patterns.