Deterministic and stochastic models of bicoid protein gradient formation in Drosophila embryos: Modelling maternal mRNA degradation

Passive diffusion of a class of molecules known as morphogens as a mechanism that helps to establish spatial patterns of gene expression during embryonic development was proposed by Turing [1]. This mechanism 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 [2]. 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. In [3], we have incorporated a more realistic model of the morphogen source since the maternal mRNA should be expected to decay.We explicitly model the source as a constant supply followed by exponential decay and solve the reaction diffusion equation numerically for one dimensional morphogen propagation. 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. We also extend this work to further show how such a realistic source model may be combined with a recently published flow model [4] that takes into account advective transport. Moreover, a stochastic simulation based model [5] which includes Bicoid molecule reactions has also been implemented with new source model in our work.