Scaling dictates the decoder structure

Despite fluctuations in embryo size within a species, the spatial gene expression pattern and hence the embryonic structure develop in proportion with embryo size, known as the scaling phenomenon. For morphogen induced patterning of gene expression, the positional information encoded in the morphogen profile is decoded by the downstream genetic network (the decoder). In this paper, we show that the requirement of scaling sets severe constraints on the geometric structure of the decoder, which in turn enables deduction of mutants behavior and extraction of regulation information without going into any molecular details. We demonstrate that the Drosophila gap gene system achieves scaling in the way consistent with our theory - the decoder geometry required by scaling correctly accounts for the observed gap gene expression pattern in nearly all maternal morphogen mutants. Furthermore, the regulation logic and the coding/decoding strategy of the gap gene system can also be revealed from the decoder geometry. Our work provides a general theoretical framework on a large class of problems where scaling output is induced by non-scaling input, as well as a unified understanding of scaling, mutants behavior and gene regulation for the Drosophila gap gene system.