Bayesian inversion of a diffusion model with application to biology.

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 modelling). When complex dynamical systems are considered, such as partial differential equations, this task may become challenging or ill-posed. In this work, a linear parabolic equation is considered as a model for protein transcription from MRNA. The objective is to estimate jointly the differential operator coefficients, namely the rates of diffusion and self-regulation, as well as a functional source. The recent Bayesian methodology for infinite dimensional inverse problems is applied, providing a unique posterior distribution on the parameter space continuous in the data. This posterior is then summarized using a Maximum a Posteriori estimator. Finally, the theoretical solution is illustrated using a state-of-the-art MCMC algorithm adapted to this non-Gaussian setting.

Physically-Inspired Gaussian Process Models for Post-Transcriptional Regulation in Drosophila.

The regulatory process of Drosophila is thoroughly studied for understanding a great variety of biological principles. While pattern-forming gene networks are analyzed 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 analyzing 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, discretizing 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.