MODELING THE DROSOPHILA PAIR-RULE PATTERN BY REACTION DIFFUSION - GAP INPUT AND PATTERN CONTROL IN A 4-MORPHOGEN SYSTEM

Various reaction-diffusion models will produce striped patterns, but the most effective models so far devised to do this require two matched pairs of interacting morphogens, i.e. four substances in all. This paper examines the behavior of one such model, of a fairly generalized type, and its application to the process of pair-rule pattern formation during Drosophila embryogenesis. It is assumed that the two self-activating morphogens required by the model, expressed in complementary out-of-phase stripes, are products of early-acting pair-rule genes. Possible candidates include the primary pair-rule genes, hairy and runt. The conditions under which regular stripes are generated by the model are then examined, with emphasis on the way pre-existing patterns act to control stripe formation via their effect on rates of reaction within the pair-rule system, specifically rates of pair-rule gene transcription, to show how gap gene products may act during pattern formation. A fully symmetrical set of reactions, in which rates of formation, self- and cross-activation are exactly matched, gives unaligned stripes. Pronounced asymmetries in this regard, e.g. differential rates of formation or self-activation, destabilize stripes or produce local interruptions in the pattern like those seen in gap mutants. A limited degree of asymmetry, coupled with a gradient in the value of one or more parameters will give a correctly aligned, well-controled pattern of stripes. The experimental evidence indicates that gap genes could be responsible for both of these effects: they activate the pair-rule system asymmetrically and, when first expressed, generate a sufficiently complex landscape of concentration peaks and gradations to provide the local cues needed to correctly position and align the pair-rule stripes. In this respect, the pair-rule system can be viewed as having an intrinsic pattern-forming capability, but it depends on the input of gap genes for pattern control. Gradients are involved, but from this analysis, it is the graded distribution of gap products that is important, not the overall antero-posterior gradient. The uniform spacing of stripes, despite underlying peaks and troughs of gap gene expression, shows that pattern wavelength is relatively insensitive to parameter change, also a property of the model. The main features of the pair-rule system accord well in general terms with what is needed to control the striping capability of a 4-morphogen model, but a model of this type necessarily imposes constraints on the molecular properties of the morphogens. This may explain why the hairy and runt gene products differ in molecular terms from those of other pair-rule genes. Specifically, strong binding to DNA may be incompatible with morphogen function, and this may preclude the primary pattern-forming gene products being homeobox proteins. Additional complementary sets of pair-rule products that do bind strongly, e.g. even-skipped and ftz, would then be needed to amplify and stabilize the rudimentary pattern produced by the morphogens to give the final, highly non-linear on/off switching pattern. © 1990 Academic Press Limited.