Genesis of two-dimensional patterns in cross-gradient fields

Abstract

Tissue morphogenesis is controlled by the two-dimensional patterning of gene expression in epithelial layers, that determines cell fates. The mechanisms of pattern formation involve intracellular regulatory networks controlled by paracrine and autocrine signaling. We develop a general logical scheme to deduce the morphology of two-dimensional patterns in the field of two crossed morphogen gradients enriched by the action of autocrine signaling that may subdivide expression domains in nontrivial ways. A variety of persistent patterns, either stationary or oscillatory, are generated using the various interaction schemes, some of which have been generated by a special algorithm including random inputs and selected according to suitable criteria. We give the full classification of a variety of stationary and oscillatory expression patterns in the presence of a single autocrine signal based on logical arguments. These results are further confirmed by numerical computations based on reaction-diffusion equations for morphogens and ligands and the discrete (cell-level) description of intracellular dynamics. Model simulations also elucidate transient processes, in particular interaction schemes. Different internal schemes may lead to identical persistent patterns, although relaxation may proceed in distinct ways. As an illustration of the general method, we test a particular scheme suggested by genetic studies of dynamic gene expression patterns in the follicular epithelium of the Drosophila eggshell.