  The regulatory process of Drosophila is thoroughly studied for understanding
a great variety of biological principles. While pattern-forming gene networks
are analysed in the transcription step, post-transcriptional events (e.g.
translation, protein processing) play an important role in establishing protein
expression patterns and levels. Since the post-transcriptional regulation of
Drosophila depends on spatiotemporal interactions between mRNAs and gap
proteins, proper physically-inspired stochastic models are required to study
the link between both quantities. Previous research attempts have shown that
using Gaussian processes (GPs) and differential equations lead to promising
predictions when analysing regulatory networks. Here we aim at further
investigating two types of physically-inspired GP models based on a
reaction-diffusion equation where the main difference lies in where the prior
is placed. While one of them has been studied previously using protein data
only, the other is novel and yields a simple approach requiring only the
differentiation of kernel functions. In contrast 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 gap gene
mRNA expression data. Finally, their performances are assessed on a
high-resolution dataset describing the blastoderm stage of the early embryo of
Drosophila melanogaster
