Physically-inspired Gaussian processes for transcriptional regulation in Drosophila melanogaster

The regulatory process in Drosophila melanogaster is thoroughly studied for understanding several principles in systems biology. Since transcriptional regulation of the Drosophila depends on spatiotemporal interactions between mRNA expressions and gap-gene proteins, proper physically-inspired stochastic models are required to describe the existing link between both biological quantities. Many studies have shown that the use of Gaussian processes (GPs) and differential equations yields promising inference results when modelling regulatory processes. In order to exploit the benefits of GPs, two types of physically-inspired GPs based on the reaction-diffusion equation are further investigated in this paper. The main difference between both approaches lies on whether the GP prior is placed: either over mRNA expressions or protein concentrations. Contrarily to other stochastic frameworks, discretising the spatial space is not required here. Both GP models are tested under different conditions depending on the availability of biological data. Finally, their performances are assessed using a high-resolution dataset describing the blastoderm stage of the early embryo of Drosophila.