Parameter estimation for a model of gap gene circuits with time-variable external inputs in Drosophila

We study a model for spatio-temporal pattern formation of gap gene products in the early development of Drosophila. In contrast to previous studies of gap gene circuits, our model incorporates a number of proteins as time-variable external inputs, including a protein Huckebein which is necessary for setting up the correct posterior domain boundary and its shift in time for the gap gene hunchback. Unknown model parameters are inferred by fitting the model outputs to the gap gene data and statistical analysis is applied to investigate the quality of the parameter estimates. Our results, while being consistent with previous findings, at the same time provide a number of improvements. Firstly, it takes into account correct regulation of hunchback at the posterior part of the embryo. Secondly, confidence interval analysis shows that the regulatory topology of the gene network in our model which consists of parameters representing the regulation between genes is more consistent with the experimental evidences. Our results also reveal that for data fitting the Weighted Least Squares sum is a more suitable measure than the Ordinary Least Squares sum which has been used in all previous studies. This is confirmed by a better fit of the boundaries of the gap gene expression domains and an absence of patterning defects in the model outputs, as well as by a correct prediction of mutant phenotypes.