Spatial pattern formation is one of the key issues in developmental biology. Some patterns arising in early development have a very small spatial scale and a natural explanation is that they arise by direct cell-cell signalling in epithelia. This necessitates the use of a spatially discrete model, in contrast to the continuum based approach of the widely studied Turing and mechanochemical models. In this work, we consider the pattern-forming potential of a model for juxtacrine communciation, in which signalling molecules anchored in the cell membrane bind to and activate receptors on the surface of immediately neighbouring cells. The key assumption is that ligand and receptor production are both upregulated by binding. By linear analysis, we show that conditions for pattern formation are dependent on the feedback functions of the model. We investigate the particular form of the pattern: specifically, we look at how the range of unstable wavenumbers varies with the parameter regime and find an estimate for the wavenumber associated with the fastest growing mode.
A previous juxtacrine model for Delta-Notch signalling studied by Collier et al.  only gives rise to patterning with a length scale of one or two cells, consistent with the fine-grained patterns seen in many developmental processes. However, there is evidence of longer range patterns in early development of the fruit fly Drosophila; in particular during neuroblast segregation in the embryo  and cell differentiation in the developing eye . The analysis we carry out predicts that patterns of a length scale longer than one or two cells are possible with our positive feedback mechanism, and numerical simulations confirm this.
Our work shows that juxtacrine signalling provides a novel and robust mechanism for the generation of spatial patterns.
 J R Collier, N A M Monk, P K Maini and J H Lewis. Pattern formation by lateral inhibition with feedback: a mathematical model of Delta-Notch intercellular signalling. J. Theor. Biol., 183:~429-446 (1996).
 J B Skeath and S B Carroll. Regulation of proneural gene expression and cell fate during neuroblast segregation in the Drosophila embryo. Development, 114:~939-946 (1992).
 S L Zipursky and G M Rubin. Determination of neuronal cell fate: lessons from the R7 neuron of Drosophila. Annu. Rev. Neurosci., 17:~373-397 (1994).