Exact probabilistic solution of spatial-dependent stochastics and associated spatial potential landscape for the bicoid protein.

We investigated the spatial-dependent stochastic effects originating from the finite number of bicoid proteins in Drosophila melanogaster, which are crucial to cell development. We obtained an exact solution to the spatial-dependent stochastic chemical master equation and recovered the usual reaction-diffusion solution for the average of the bicoid concentration, valid in the bulk. We also used the steady state probability to get the spatial potential landscape. The stochastic effects are captured by the Poisson distribution; so, as the average of the bicoid concentration decreases from the anterior (A) to the posterior (P) of the embryo, the statistical fluctuations also decrease. An alternative way of interpreting this is that the shape of the spatial potential landscape shrinks from A to P. While the mathematical result is known, we offer a simple approach to understanding why it is what it is and give associated physical intuitions. The approach can be generalized and applied to any problem with a particle that diffuses, decays, and has a stochastic source.