  Since its introduction in 1952, Turing's (pre-)pattern theory ("the chemical
basis of morphogenesis") has been widely applied to a number of areas in
developmental biology. The related pattern formation models normally comprise a
system of reaction-diffusion equations for interacting chemical species
("morphogens"), whose heterogeneous distribution in some spatial domain acts as
a template for cells to form some kind of pattern or structure through, for
example, differentiation or proliferation induced by the chemical pre-pattern.
Here we develop a hybrid discrete-continuum modelling framework for the
formation of cellular patterns via the Turing mechanism. In this framework, a
stochastic individual-based model of cell movement and proliferation is
combined with a reaction-diffusion system for the concentrations of some
morphogens. As an illustrative example, we focus on a model in which the
dynamics of the morphogens are governed by an activator-inhibitor system that
gives rise to Turing pre-patterns. The cells then interact with morphogens in
their local area through either of two forms of chemically-dependent cell
action: chemotaxis and chemically-controlled proliferation. We begin by
considering such a hybrid model posed on static spatial domains, and then turn
to the case of growing domains. In both cases, we formally derive the
corresponding deterministic continuum limit and show that that there is an
excellent quantitative match between the spatial patterns produced by the
stochastic individual-based model and its deterministic continuum counterpart,
when sufficiently large numbers of cells are considered. This paper is intended
to present a proof of concept for the ideas underlying the modelling framework,
with the aim to then apply the related methods to the study of specific
patterning and morphogenetic processes in the future.
