Phd projects in Mathematical Biology and Medicine

## Modelling general mechanisms driving evolution in natural populations

*Sandro Azaele and Mike Evans*

Understanding the mechanisms of evolution is vital, both for enhancing our knowledge of existing ecosystems, and for modelling how natural populations will adapt to environmental change. A better understanding of evolution requires modelling groups of interacting organisms. The mathematical tools for modelling large numbers of interacting objects already exist in theoretical physics, for studying the interactions of molecules within solids, liquids and gases. For instance, the concepts of entropy and critical points are central to our understanding of those systems. Applying those tools to evolving biological systems, from genes (Wagner A., 2008) to large ecosystems (Borile et al., 2012), can yield fascinating and important new insights. For instance, there are recent theoretical developments which show that living systems could operate in the vicinity of a critical point (Hidalgo et al., 2014), because this confers them more robustness and flexibility when facing environmental variability. This project will involve analysing some models of Darwinian evolution that are important when understanding evolving ecosystems. Those will be sufficiently simple to be both analytically tractable and suitable for agent-based simulation. Studying such models will allow a link to be established between the concept of genotypic entropy, which is well-defined in the stochastic numerical models, and a Hamiltonian formulation of the models, which enable one to develop a statistical mechanical approach to the theory. The development of such links would allow the investigation of general principles by means of powerful tools borrowed from statistical mechanics.

## Investigating the potential role of stochastic mechanisms in the sequential formation of repetitive structures in animals

*Sandro Azaele and Andrew D. Peel*

Vertebrates, arthropods and annelids all exhibit segmented body plans. Each of these groups is closely related to other animal phyla that are not segmented. However, Dr. Peel recently helped show that the mechanisms underpinning sequential segment formation in arthropods and vertebrates share striking similarities; segments form under the control of a network of molecular oscillators, called the 'segmentation clock'. Segmentation might have been a feature of an ancient common ancestor that has been retained by arthropods/vertebrates, but lost by other phyla during evolution. The alternative possibility is that there is a predisposition for animals to use molecular oscillators to pattern reiterated structures, and similarities between the arthropod and vertebrate clocks reflect convergent or parallel evolution. The extent to which genetic and organizational similarities between the arthropod and vertebrate clocks can be used as evidence for common ancestry depends on the number of developmental mechanisms that potentially could have been deployed. If many different mechanisms could 'do the job', then similarities are more likely to reflect common ancestry. If evolution is constrained to one, or a few, mechanisms, then similarities are more likely to reflect convergent or parallel evolution. Investigating potential mechanisms of sequential segmentation will therefore inform our understanding of the origin and evolution of animal segmentation.

## Modelling Biodiversity and Pattern Formation with Evolutionary Games

*Dr M Mobilia*

Understanding the maintenance of biodiversity and the emergence of cooperation are important topics in the Life and Behavioural Sciences. Evolutionary game theory, where the success of one species depends on what the others are doing, provides a promising mathematical framework to study the coexistence dynamics of interacting populations. As paradigmatic examples, the prisoners dilemma and the rock-paper-scissors games have emerged as a fruitful metaphor for cooperative and co-evolutionary dynamics (with applications in microbiology and ecology). While mathematical biology classically deals with deterministic (and often spatially homogeneous) models, it has been shown that the joint effect of noise and spatial degrees of freedom are important and realistic ingredients to be considered. In our research, we use tools of nonlinear dynamics, stochastic processes, differential equations and the theory of front propagation, as well as methods of statistical mechanics, to study the co-evolutionary dynamics of structured and unstructured populations in the presence of intrinsic noise. More information

## Social Dynamics and Emergence of Collective Behaviours

Dr M MobiliaApproaches relying on nonlinear dynamics and statistical mechanics have provided compelling models and crucial insights to understand interdisciplinary problems and emergent phenomena in complex systems. One paradigmatic example in the realm of social dynamics is the "voter model", where individuals in a population can be in one of two opinion states. The voter model is also closely related to evolutionary games used to model social and cooperation dilemmas. In this class of models, an individual is selected at random and adopts (with some probability) the state of its randomly-chosen neighbour; this update step is applied repeatedly. In this project, we propose to develop equally simple and paradigmatic individual-based models to investigate social behaviours like the emergence of cooperation, polarization and radicalization. For this, the dynamics will be implemented on various types of graphs and we will study a series of nonlinear (deterministic and stochastic) problems using a well-rounded combination of mathematical methods, notably the theory of dynamical systems and differential equations, stochastic processes and tools borrowed from statistical mechanics. More information

## Stochastic models of the adaptive immune system

*Grant Lythe and C. Molina-París*

The goal of this project is to develop mathematical and computational models to help understand how the immune system maintains its diversity of millions of lymphocyte populations, able to protect against pathogens while avoiding auto-immune diseases. The processes of positive and negative selection in the thymus will be studied with stochastic modelling techniques, including computational modelling and analysis of experimental data.

## Mathematical modelling of vascular endothelial growth factors and receptors

*C. Molina-París, Dr Vas Ponnambalam and Grant Lythe*

Multicellular organisms contain biological tubes such as blood vessels that are used to move molecules, cells and fluids to different tissues. Endothelial cells that line every blood vessel can bind growth factors which regulate new blood vessel formation and vascular function. The vascular endothelial growth factor A (VEGF-A) binds membrane receptor tyrosine kinases (VEGFR1, VEGFR2) and regulates many aspects of vascular function. Dysfunction of this VEGF-VEGFR system can lead to diseases such as cancer.

This student will use experimental data generated in the Endothelial Cell Biology Unit (Faculty of Biological Sciences, University of Leeds) and put it into a quantitative framework using mathematical modelling with spatial and temporal constraints. The PhD student will derive mathematical models of VEGFR synthesis, endocytosis and degradation. The systems biology approach integrates experimental and modelling-based data. The goal is cellular output prediction in silico and experimental testing to validate model robustness.