# Titles and Abstracts

###### Nigel Burroughs: CO2 transport in plant cells - carbon assimilation and cellular geometry

Carbon assimilation has diversified over the course of evolution producing a range of cellular architectures and two key biochemical pathways called C3 and C4 (named after the carbon backbone of the key intermediary). C4 photosynthesis has evolved many times (convergent evolution) as a solution to a major bottleneck in carbon assimilation efficacy - the low efficacy of the primary carbon fixing enzyme RubisCO. RubisCO evolved when atmospheric CO2 concentration was at high; selection for faster enzymes able to function efficiently at low CO2 concentrations is limited by the associated loss of specificity for CO2 relative to O2. We develop a spatial model of gas transport in plant cells and examine the link between assimilation efficacy and cell architecture. This is being used to assess the potential for incorporating C4 biochemistry into C3 plants to achieve higher productivity.

###### Helen Byrne: PDE Models of Angiogenesis

Angiogenesis is the process by which new blood vessels form from pre-existing vessels in response to externally supplied growth factors. It is tightly regulated during wound healing and placental development and aberrant during macular degeneration. It heralds a tumour’s transition from a relatively harmless, localised mass to one that grows rapidly, spreads to other parts of the body and is potentially life threatening.

Angiogenesis is typically initiated by cells which, when deprived of vital nutrients, release chemicals that diffuse through the surrounding tissue matrix. These angiogenic factors elicit multiple responses when they bind to receptors on endothelial cells that line nearby blood vessels. For example, small capillary tips emerge from the existing vessels and migrate, via chemotaxis, towards the source of the angiogenic factors. At the same time, endothelial cells in the developing capillary sprouts proliferate, causing the sprouts to elongate. When two tips collide (or when a tip comes into contact with a vessel), they fuse or anastamose, forming a hollowed tube through which blood may flow, increasing the supply rate of vital nutrients to the tissue.

A large number of mathematical and computational models have been proposed to describe angiogenesis. These models range from deterministic, continuum models that focus on macroscopic variables such as capillary tip and vessel densities, to cell-based models which distinguish between individual capillary tips and account for the detailed morphology of the emerging vascular network.

In this talk, I will show how existing (and new) PDE models of angiogenesis can be derived by coarse-graining discrete, cell-based models. Of particular interest will be investigating how volume exclusion effects at the cell-scale influence the system dynamics at the macroscale.

References

[1] HM Byrne & MAJ Chaplain (1995). Mathematical models for tumour angiogenesis: numerical simulations and nonlinear wave solutions. Bull Math Biol 57(3), 461-486.

[2] F Spill, P Guerrero, T Alarcon, PK Maini & HM Byrne (2015). Mesoscopic and continuum modelling of angiogenesis. J Math Biol, 70(3), 485-532.

[3] S Pillay, HM Byrne & PK Maini (2017). Modeling angiogenesis: A discrete to continuum description. Phys Rev E, 95(1), 012410.

###### John King: Mathematical modelling of biological tissue growth

Some continuum (PDE) models will be described, with a focus on multiphase formulations.

###### Balázs Kovács: Convergence of finite elements on an evolving surface driven by diffusion on the surface

For a parabolic surface partial differential equation coupled to surface evolution, convergence of the spatial semidiscretization is studied. The velocity of the evolving surface is not given explicitly, but depends on the solution of the parabolic equation on the surface. Various velocity laws are considered: elliptic regularization of a direct pointwise coupling, a regularized mean curvature flow and a dynamic velocity law. We give a stability and convergence analysis or evolving surface finite elements for the coupled problem of surface diffusion and surface evolution is developed. The stability analysis works with the matrix-vector formulation of the method and does not use geometric arguments. The geometry enters only into the consistency estimates. We show some numerical experiments as well.

###### Kei Fong Lam: Diffuse interface models for tumour growth

There has been a recent focus in modelling tumour growth with diffuse interface models, due to their ability to capture topological transitions and the nature of the equations allows for further mathematical treatment. The simplest model variant is a two-phase model (for tumour and healthy cells) that couples a Cahn--Hilliard equation with a reaction-diffusion equation for a nutrient through source terms and fluxes. Through specific choices of the fluxes we capture the mechanisms of chemotaxis and active transport, and the latter is responsible for some unusual behaviour of the nutrient near the tumour interface. Then, we present a multiphase model variant incorporating necrotic tumour cells, and discuss the types of source terms used to model proliferation. Numerical simulations of the multiphase model show that the presence of a necrotic core seems to delay the onset of invasive growth. If time permits, we will discuss various analytic results on well-posedness, parameter identification and sharp interface limits.

###### John Mackenzie: An Adaptive Moving Mesh Method for Geometric Evolutions Laws and Bulk-Surface PDEs: Application to a Model of Cell Migration and Chemotaxis.

In this talk I will consider the adaptive numerical solution of curve-shortening flow with a driving force. An adaptive moving mesh approach is used to distribute the mesh points in the tangential direction. This ensures that the resulting meshes evolve smoothly in time and are well adjusted to resolve areas of high curvature. Experiments will be presented to highlight the improvement in accuracy obtained using the new method in comparison with uniform arc-length mesh distributions. We will also discuss the use of the evolving adaptive curve mesh in the adaptive generation of bulk meshes for the solution of bulk-surface PDEs in time dependent domains.

The main motivation for developing these computational tools is the modelling of single cell migration and chemotaxis. Chemoattractant gradients are usually considered in terms of sources and sinks that are independent of the chemotactic cell. However, recent interest has focused on “self-generated” gradients, in which cell populations create their own local gradients as they move. Here we consider the interplay between chemoattractants and single cells. To achieve this we model the breakdown of extracellular attractants by membrane-bound enzymes. Model equations are parameterised using published estimates from Dictyostelium cells chemotaxing towards cyclic AMP. We find that individual cells can substantially modulate their local attractant field under physiologically appropriate conditions of attractant and enzymes. This means the attractant concentration perceived by receptors can be a small fraction of the ambient concentration. This allows efficient chemotaxis in chemoattractant concentrations that would be saturating without local breakdown.

###### Anotida Madzvamuse: A robust and efficient adaptive multigrid solver for the optimal control of phase field formulations of geometric evolution laws

In this talk, I will present a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws. The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth and cell motility. Despite this, many open problems remain in the analysis and approximation of such problems. In the current work we focus on a phase field formulation of the optimal control problem, hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations. Approximation of the resulting optimal control problem is computationally challenging, requiring massive amounts of computational time and memory storage. The main focus of this work is to propose, derive, implement and test an efficient solution method for such problems. The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement. An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally. Furthermore, parallelisation as well as an adaptive-step gradient update

for the control are employed to further improve efficiency. Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency. A highlight of the present work is simulation results on the optimal control of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.

###### Robert Nürnberg: Coupling Navier--Stokes to Willmore: Numerical approximation of dynamic biomembranes

A parametric finite element approximation of a fluidic membrane, whose evolution is governed by a surface Navier--Stokes equation coupled to bulk Navier--Stokes equations, is presented. The elastic properties of the membrane are modelled with the help of curvature energies of Willmore and Helfrich type. Forces stemming from these energies act on the surface fluid, together with a forcing from the bulk fluid. We introduce a stable parametric finite element method to solve thiscomplex free boundary problem. Local inextensibility of the membrane is ensured by solving a tangential Navier--Stokes equations, taking surface viscosity effects of Boussinesq--Scriven type into account. In our approach the bulk and surface degrees of freedom are discretized independently, which leads to an unfitted finite element approximation of the underlying free boundary problem. Bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by Dziuk. The obtained numerical scheme can be shown to be stable and to have good mesh properties.

###### Andreas Rätz: Modelling and Simulation of Lipid Raft Formation in Biological Membranes

In this talk, we investigate a model for lipid raft formation and dynamics in biological membranes. The model describes the lipid composition of the membrane and an interaction with cholesterol. To account for cholesterol exchange between the interior and the membrane we couple a bulk-diffusion to a surface PDE on the membrane. The latter describes a relaxation dynamics for an energy taking lipid-phase separation and lipid-cholesterol interaction energy into account. It takes the form of an (extended) Cahn–Hilliard equation. For numerical simulations and investigation of the long-time behaviour of the model we choose a diffuse-interface method for the coupled bulk--surface system. Moreover, we consider the limit of infinitely fast bulk diffusion, where one obtains a reduced non-local surface PDE model. This reduced model is numerically treated by surface finite elements and its stationary points are analysed.

###### Dumitru Trucu: Structured Models of Cell Migration Incorporating Membrane Reactions

The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes enables a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this work we establish a general spatio-temporal-structural framework that enables the description of surface- bound reaction processes coupled with the cell population dynamics. We first provide a general theoretical description for this approach and then illustrate it with concrete examples arising from cancer invasion.

###### Chandrasekhar Venkataraman: Modelling phenotypic heterogeneity in solid tumours

A growing body of evidence suggests that the emergence of phenotypic heterogeneity and drug resistance in tumours is due to a process of adaption or evolution. Specifically, it is hypothesised that spatial variations in the concentration of abiotic factors within the tumour environment may create local niches driving natural selection of the tumour cells. In this talk we investigate the validity of such a hypothesis by developing a mathematical model linking the dynamics of a structured tumour cell population with the concentrations of abiotic factors in the microenvironment. For biological relevance we use parameters in the model that are obtained from data and work on realistic geometries. We compare numerical simulations of the full model using a finite element approximation with semi-analytical results on the asymptotically selected trait and observe good agreement between the two.