A Bayesian framework for inverse problems for quantitative biology

In this thesis, we present a Bayesian framework to solve inverse problems in the context of quantitative biology. We present a novel combination of the Bayesian approach to inverse problems, suitable for infinite-dimensional problems, with a parallel, scalable Markov Chain Monte Carlo algorithm to approximate the posterior distribution. Both the Bayesian framework and the parallelised MCMC were already known but they were not used in this context in the past. Our approach puts together existing results in order to provide a tool to easily solve inverse problems. We focus on models given by partial differential equations. Our methodology differs from previous results in its approach: it aims to be as transparent and independent of the model as possible, in order to make it flexible and applicable to a wide range of problems emerging from experimental and physical sciences. We illustrate our methodology with three of such applications in the areas of theoretical biology and cell biology. The first application deals with parameter and function identification within a Turing pattern formation model. To the best of our knowledge, our results are the first attempt to use Bayesian techniques to study the inverse problem for Turing patterns. In this example, we show how our implementation can deal with both finite- and infinite-dimensional parameters in the context of inverse problems for partial differential equations. The second example studies the spatio-temporal dynamics in cell biology. The study provides an example that seeks to best-fit a mathematical model to experimental data finding in the process optimal parameters and credible regimes and regions. We present a new derivation of the model, that corrects the short-comings of previous approaches. We provide all the details from techniques for data acquisition to the parameter identification, and we show in particular how the mathematical model can be used as a proxy to estimate parameters that are difficult to measure in the experiments, providing an novel alternative to more indirect estimates that also require more complex experiments. Finally, our third example illustrates the flexibility of our implementation of the methodology by using it to study traction force microscopy (TFM) data with a solver implemented independent of the Bayesian approach for parameter identification. We limit ourselves to the classical TFM setting, that we model as a two-dimensional linear elasticity problem. The results and methods generalise to more complex settings where quantitative modelling driven by biological observations is a requirement.