Bayesian inversion of a diffusion model with application to biology

Abstract

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

“…From the physicallyinspired models' point of view, they provide accurate predictions even in regions where data are not available. Since they can account for different types of differential equations, physically-inspired GPs have been applied successfully in several fields such as human motion capture and robotics (Álvarez et al, 2013;Agudelo-España et al, 2017;Guarnizo and Álvarez, 2018), neuroscience (Alvarado et al, 2014), and molecular biology and genetics (Lawrence et al, 2007;Álvarez et al, 2013;Vásquez Jaramillo et al, 2014;López-Lopera and Álvarez, 2019;Croix et al, 2018;Gao et al, 2008). In (Álvarez et al, 2013;Vásquez Jaramillo et al, 2014;Croix et al, 2018), they have been applied to model the early embryo development of Drosophila melanogaster.…”