Well-posedness and asymptotic behavior of a multidimensional model of morphogen transport

Morphogen transport is a biological process, occurring in the tissue of living organisms, which is a determining step in cell differentiation. We present rigorous analysis of a simple model of this process, which is a system coupling parabolic PDE with ODE. We prove existence and uniqueness of solutions for both stationary and evolution problems. Moreover we show that the solution converges exponentially to the equilibrium in $C^1\times C^0$ topology. We prove all results for arbitrary dimension of the domain. Our results improve significantly previously known results for the same model in the case of one dimensional domain

Mathematical analysis of a model of chemotaxis arising from morphogenesis

We consider non-negative solution of a chemotaxis system with non constant chemotaxis sensitivity function X. This system appears as a limit case of a model formorphogenesis proposed by Bollenbach et al. (Phys. Rev. E. 75, 2007).Under suitable boundary conditions, modeling the presence of a morphogen source at x=0, we prove the existence of a global and bounded weak solution using an approximation by problems where diffusion is introduced in the ordinary differential equation. Moreover,we prove the convergence of the solution to the unique steady state provided that ? is small and ? is large enough. Numerical simulations both illustrate these results and give rise to further conjectures on the solution behavior that go beyond the rigorously proved statements

Mathematical Biology

Mathematical biology is a fast growing field of research, which on one hand side faces challenges resulting from the enormous amount of data provided by experimentalists in the recent years, on the other hand new mathematical methods may have to be developed to meet the demand for explanation and prediction on how specific biological systems function

On a nonlinear flux--limited equation arising in the transport of morphogens

Motivated by a mathematical model for the transport of morphogenes in biological systems, we study existence and uniqueness of entropy solutions for a mixed initial-boundary value problem associated with a nonlinear flux--limited diffusion system. From a mathematical point of view the problem behaves more as an hyperbolic system that a parabolic one

Local and non-local mathematical modelling of signalling during embryonic development

Embryonic development requires cells to communicate as they arrange into the adult organs and tissues. The ability of cells to sense their environment, respond to signals and self-organise is of crucial importance. Patterns of cells adopting distinct states of differentiation arise in early development, as a result of cell signalling. Furthermore, cells interact with each other in order to form aggregations or rearrange themselves via cell-cell adhesion. The distance over which cells can detect their surroundings plays an important role to the form of patterns to be developed, as well as the time necessary for developmental processes to complete. Cells achieve long range communication through the use of extensions such as filopodia. In this work we formulate and analyse various mathematical models incorporating long-range signalling. We first consider a spatially discrete model for juxtacrine signalling extended to include filopodial action. We show that a wide variety of patterns can arise through this mechanism, including single isolated cells within a large region or contiguous blocks of cells selected for a specific fate. Cell-cell adhesion modelling is addressed in this work. We propose a variety of discrete models from which continuous models are derived. We examine the models’ potential to describe cell-cell adhesion and the associated phenomena such as cell aggregation. By extending these models to consider long range cell interactions we were able to demonstrate their ability to reproduce biologically relevant patterns. Finally, we consider an application of cell adhesion modelling by attempting to reproduce a specific developmental event, the formation of sympathetic ganglia

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