  A common task in experimental sciences is to fit mathematical models to
real-world measurements to improve understanding of natural phenomenon
(reverse-engineering or inverse modeling). When complex dynamical systems are
considered, such as partial differential equations, this task may become
challenging and ill-posed. In this work, a linear parabolic equation is
considered where the objective is to estimate both the differential operator
coefficients and the source term at once. The Bayesian methodology for inverse
problems provides a form of regularization while quantifying uncertainty as the
solution is a probability measure taking in account data. This posterior
distribution, which is non-Gaussian and infinite dimensional, is then
summarized through a mode and sampled using a state-of-the-art Markov-Chain
Monte-Carlo algorithm based on a geometric approach. After a rigorous analysis,
this methodology is applied on a dataset of the post-transcriptional regulation
of Kni gap gene in the early development of Drosophila Melanogaster where mRNA
concentration and both diffusion and depletion rates are inferred from noisy
measurement of the protein concentration
