Anisotropic nonlinear PDE models and dynamical systems in biology

This thesis deals with the analysis and numerical simulation of anisotropic nonlinear partial differential equations (PDEs) and dynamical systems in biology. It is divided into two parts, motivated by the simulation of fingerprint patterns and the modelling of biological transport networks. The first part of this thesis deals with a class of interacting particle models with anisotropic repulsive-attractive interaction forces and their continuum counterpart. These models are motivated by the simulation of fingerprint databases, which are required in forensic science and biometric applications. In existing interacting particle models, the forces are isotropic and the continuum limits of these particle models are given by nonlocal aggregation equations with radially symmetric potentials. The central novelty in the models we consider is an anisotropy induced by an underlying tensor field. This innovation does not only lead to the ability to describe real-world phenomena more accurately, but also renders their analysis significantly harder compared to their isotropic counterparts. We discuss the role of anisotropic interaction, study the steady states and present a stability analysis of line patterns. We also show numerical results for the simulation of fingerprints, based on discrete and continuum modelling approaches. The second part of this thesis focuses on a new dynamic modeling approach on a graph for biological transportation networks which are ubiquitous in living systems such as leaf venation in plants, blood circulatory systems, and neural networks. We study the existence of solutions to this model and propose an adaptation so that a macroscopic system can be obtained as its formal continuum limit. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. We also show the global existence of weak solutions of the macroscopic gradient flow. Results of numerical simulations of the discrete gradient flow illustrate the convergence to steady states, their non-uniqueness as well as their dependence on initial data and model parameters. Based on this model we propose an adapted model in the cellular context for leaf venation, investigate the model analytically and show numerically that it can produce branching vein patterns.This thesis was supported by the EPSRC, the MSCA-RISE projects CHiPS and NoMADS, the Cambridge Commonwealth, European & International Trust, the German Academic Scholarship Foundation, the Cambridge Centre for Analysis, the Cambridge Philosophical Society, the CCIMI, Murray Edwards College, SIAM and IMA

Analysis of nonlinear poroviscoelastic flows with discontinuous porosities

Existence and uniqueness of solutions is shown for a class of viscoelastic flows in porous media with particular attention to problems with nonsmooth porosities. The considered models are formulated in terms of the time-dependent nonlinear interaction between porosity and effective pressure, which in certain cases leads to porosity waves. In particular, conditions for well-posedness in the presence of initial porosities with jump discontinuities are identified.Comment: 34 page

Mean-field optimal control for biological pattern formation

We propose a mean-field optimal control problem for the parameter identification of a given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian approach on the mean-field level. Based on these conditions we propose a gradient descent method to identify relevant parameters such as angle of rotation and force scaling which may be spatially inhomogeneous. We discretize the first-order optimality conditions in order to employ the algorithm on the particle level. Moreover, we prove a rate for the convergence of the controls as the number of particles used for the discretization tends to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility of the approach

Detection of high codimensional bifurcations in variational PDEs

We derive bifurcation test equations for A-series singularities of nonlinear functionals and, based on these equations, we propose a numerical method for detecting high codimensional bifurcations in parameter-dependent PDEs such as parameter-dependent semilinear Poisson equations. As an example, we consider a Bratu-type problem and show how high codimensional bifurcations such as the swallowtail bifurcation can be found numerically. In particular, our original contributions are (1) the use of the Infinite-dimensional Splitting Lemma, (2) the unified and simplified treatment of all A-series bifurcations, (3) the presentation in Banach spaces, i.e. our results apply both to the PDE and its (variational) discretization, (4) further simplifications for parameter-dependent semilinear Poisson equations (both continuous and discrete), and (5) the unified treatment of the continuous problem and its discretisation

Auxin transport model for leaf venation.

The plant hormone auxin controls many aspects of the development of plants. One striking dynamical feature is the self-organization of leaf venation patterns which is driven by high levels of auxin within vein cells. The auxin transport is mediated by specialized membrane-localized proteins. Many venation models have been based on polarly localized efflux-mediator proteins of the PIN family. Here, we investigate a modelling framework for auxin transport with a positive feedback between auxin fluxes and transport capacities that are not necessarily polar, i.e. directional across a cell wall. Our approach is derived from a discrete graph-based model for biological transportation networks, where cells are represented by graph nodes and intercellular membranes by edges. The edges are not a priori oriented and the direction of auxin flow is determined by its concentration gradient along the edge. We prove global existence of solutions to the model and the validity of Murray's Law for its steady states. Moreover, we demonstrate with numerical simulations that the model is able connect an auxin source-sink pair with a mid-vein and that it can also produce branching vein patterns. A significant innovative aspect of our approach is that it allows the passage to a formal macroscopic limit which can be extended to include network growth. We perform mathematical analysis of the macroscopic formulation, showing the global existence of weak solutions for an appropriate parameter range.HJ is supported by the Gatsby Charitable Foundation (grant GAT3395-PR4). LMK is supported by the EPSRC grant EP/L016516/1 and the German National Academic Foundation

Autophosphorylation and the Dynamics of the Activation of Lck

Abstract: Lck (lymphocyte-specific protein tyrosine kinase) is an enzyme which plays a number of important roles in the function of immune cells. It belongs to the Src family of kinases which are known to undergo autophosphorylation. It turns out that this leads to a remarkable variety of dynamical behaviour which can occur during their activation. We prove that in the presence of autophosphorylation one phenomenon, bistability, already occurs in a mathematical model for a protein with a single phosphorylation site. We further show that a certain model of Lck exhibits oscillations. Finally, we discuss the relations of these results to models in the literature which involve Lck and describe specific biological processes, such as the early stages of T cell activation and the stimulation of T cell responses resulting from the suppression of PD-1 signalling which is important in immune checkpoint therapy for cancer

Closing the ODE-SDE gap in score-based diffusion models through the Fokker-Planck equation

Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling, due to their state-of-the art performance in many generation tasks while relying on mathematical foundations such as stochastic differential equations (SDEs) and ordinary differential equations (ODEs). Empirically, it has been reported that ODE based samples are inferior to SDE based samples. In this paper we rigorously describe the range of dynamics and approximations that arise when training score-based diffusion models, including the true SDE dynamics, the neural approximations, the various approximate particle dynamics that result, as well as their associated Fokker--Planck equations and the neural network approximations of these Fokker--Planck equations. We systematically analyse the difference between the ODE and SDE dynamics of score-based diffusion models, and link it to an associated Fokker--Planck equation. We derive a theoretical upper bound on the Wasserstein 2-distance between the ODE- and SDE-induced distributions in terms of a Fokker--Planck residual. We also show numerically that conventional score-based diffusion models can exhibit significant differences between ODE- and SDE-induced distributions which we demonstrate using explicit comparisons. Moreover, we show numerically that reducing the Fokker--Planck residual by adding it as an additional regularisation term leads to closing the gap between ODE- and SDE-induced distributions. Our experiments suggest that this regularisation can improve the distribution generated by the ODE, however that this can come at the cost of degraded SDE sample quality