Matching Models to Data in Modelling Morphogen Diffusion

The mechanism by which spatial patterns are established during embryonic development is usually modelled as passive diffusion of morphogen proteins translated from maternally deposited messenger RNAs. Such diffusion models assume a constant supply of morphogens at the source throughout the establishment of the required profile at steady state. Working with the bicoid morphogen which establishes the anterior-posterior axis in the Drosophila embryo, we note that this constant source assumption is unrealistic since the maternal mRNA is known to decay after a certain time since egg laying. We numerically solve the reaction diffusion equation for one dimensional morphogen propagation and match the resulting solution to measured data. By minimising the squared error between model outputs and measurements published in the FlyEx database, we show how parameters of diffusion rate, mRNA and protein decay constants, and the onset of maternal mRNA decay can be assigned sensible values