A stochastic model for gene transcription on Drosophila melanogaster embryos

We examine immunostaining experimental data for the formation of the strip 2 of $even-skipped$ ($eve$) transcripts on $D.$ $melanogaster$ embryos. An estimate of the factor converting immunofluorescence intensity units into molecular numbers is given. The analysis of the $eve$ mRNA's dynamics at the region of the stripe 2 suggests that the promoter site of the gene has two distinct regimes: an earlier phase when it is predominantly activated until a critical time when it becomes mainly repressed. That suggests proposing a stochastic binary model for gene transcription on $D.$ $melanogaster$ embryos. Our model has two random variables: the transcripts number and the state of the source of mRNAs given as active or repressed. We are able to reproduce available experimental data for the average number of transcripts. An analysis of the random fluctuations on the number of $eve$ mRNA's and their consequences on the spatial precision of the stripe 2 is presented. We show that the position of the anterior/posterior borders fluctuate around their average position by $\sim 1 \%$ of the embryo length which is similar to what is found experimentally. The fitting of data by such a simple model suggests that it can be useful to understand the functions of randomness during developmental processes.