The effect of reactions on the formation and readout of the gradient of Bicoid

During early development, the establishment of gradients of transcriptional factors determines the patterning of cell fates. The case of Bicoid (Bcd) in {\it Drosophila melanogaster} embryos is well documented and studied. There are still controversies as to whether {\it SDD} models in which Bcd is {\it Synthesized} at one end, then {\it Diffuses} and is {\it Degraded} can explain the gradient formation within the timescale observed experimentally. The Bcd gradient is observed in embryos that express a Bicoid-eGFP fusion protein (Bcd-GFP) which cannot differentiate if Bcd is freely diffusing or bound to immobile sites. In this work we analyze an {\it SDID} model that includes the {\it Interaction} of Bcd with binding sites. Using previously determined biophysical parameters we find that this model can explain the gradient formation within the experimentally observed time. Analyzing the differences between the free and bound Bcd distributions we observe that the latter spans over a longer lengthscale. We conclude that deriving the lengthscale from the distribution of Bcd-GFP can lead to an overestimation of the gradient lengthscale and of the degree of cooperativity that explains the distribution of the protein Hunchback whose production is regulated by Bcd.