Mathematical Biology

The basement membrane (BM) is a nanoporous extracellular matrix that surrounds most tissues and blocks the passage of cells, primarily composed of covalently cross-linked collagen IV fibres and laminins. While BM breaching has traditionally been thought to be mediated by protease-mediated degradation, the failure of protease-targeting clinical trials to reduce metastasis suggests the existence of alternative protease-independent mechanisms. Recent studies indicate that invasive cells extend filopodia capable of remodelling plastic extracellular matrices. However, the covalent cross-links in collagen IV fibres are very strong, prompting the question of how filopodia might facilitate BM invasion. Collagen IV fibres undergo turnover---a dynamic process of protein renewal that may create transient weak spots in the BM. We hypothesise that filopodia exploit these weak spots during turnover to initiate and expand pores, enabling protease-independent invasion. We propose a mathematical biophysical model to test the plausibility of this mechanism using biologically relevant protruding and turnover conditions obtained from experimental observations in the literature. Invasive cells are represented as energetic biomembranes using geometric-surface partial differential equations, allowing the formation of filopodial protrusions, while the BM is modelled as a barrier with collagen IV cross-links that stochastically transition between active and inactive states. The results of the model contrasted with experimental observations identify two subpopulations of filopodia in invasive cells: thin, short-lived filopodia that contribute to global BM degradation, and long-lived, widening filopodia that locally stabilise and enlarge pores where invasion can eventually occur. Under suitable conditions, the model predicts that random turnover and filopodia can synchronise, leading to progressive pore enlargement. Further, pore enlargement arise from the collaboration of several filopodia entering and leaving the same region of the BM at different times. Although our results cannot demonstrate that this mechanism occurs in vivo, they place turnover as a plausible contributor to protease-independent invasion.

Host–parasite interactions often resemble an evolutionary arms race, where each side must continually adapt just to keep up. These dynamics can produce oscillations in allele frequencies—often called Red Queen dynamics—that are a hallmark of host–parasite coevolution. Despite their prominence, we still have an incomplete understanding of how they influence broader evolutionary outcomes. Much of the existing theory has focused on their role in the evolution of sex and recombination, leaving their consequences for other life-history traits largely unexplored. In this talk, I will explore the mechanisms that generate coevolutionary cycles and the role of eco-evolutionary feedbacks in shaping them. I will then discuss how these cycles influence the evolution of parasite virulence.

‘Reef halos’ are rings of sand, barren of vegetation, encircling reefs. However, the extent to which various biotic (e.g., herbivory) and abiotic (e.g., temperature, nutrients) factors drive changes in halo prevalence and size remains unclear. The objective of this study was to explore the effects of herbivore biomass, primary productivity, temperature, and nutrients on reef halo presence and width. First, we conducted a field study using artificial reef structures and their surrounding halos, finding that halos were more likely to be observed with high herbivorous fish biomass, and halos were larger under high temperatures. There was a distinct interaction between herbivorous fish biomass and temperature, where at high fish biomass, halos were more likely to be observed under low temperatures. Second, we incorporated environmental drivers into a consumer-resource model of halo dynamics. Certain formulations of temperature-dependent vegetation growth caused halo width and fish density to change from a fixed to an oscillating system, supporting the idea that environmental drivers can cause temporal fluctuations in halo width. Our unique combination of field-based and mechanistic modeling approaches has enhanced our understanding of the role of environmental drivers in grazing patterns, which will be particularly important as climate change causes shifts in marine systems worldwide.

Mathematical biology offers powerful tools to tackle pressing problems at the interface of health and public policy. In this talk, I will share two vignettes demonstrating how mathematical and simulation modelling can be applied to tobacco regulatory science. The first uses a Markov state transition framework to capture population-level dynamics of two tobacco products, each with a flavour option. This structure highlights the challenges of modelling high-dimensional systems, parameter inference from sparse data, and representing policy interventions as modifications to initiation, cessation, and product switching rates. The second vignette focuses on social network modelling, where adolescent tobacco use is primarily shaped by peer influence and network structure. In this setting, stochastic processes and graph-based models describe how behaviours propagate and stabilise within adolescent populations. Together, these examples illustrate how applied mathematics can bridge data and policy in public health.

Phylogenetic trees are mathematical objects that encode information about ancestry relationships and are often used in the interpretation of genomic data. They have proved especially useful for advancing our understanding of pathogen populations that evolve on observable timescales, and the construction of phylogenies and our interpretations of them rely on mathematical models at every step. In this talk, we will discuss ongoing projects that focus on the bacterium that causes Tuberculosis. In the first project, we connect compartmental models of disease transmission to pathogen phylogenies in order to understand how epidemiological processes affect tree shape. In the second project, we aim to reconstruct movement patterns on phylogenies to inform the likely efficacy of geographically-targeted public health interventions. In both of these projects, mathematical models play an essential role in the interpretation of phylogenies, and that seems likely to be the case for any statistical inferences we hope to draw from genomic data for the foreseeable future.