UK Mathematical Biology Conference

My talk will describe a state of the art image based modelling in several seemingly different areas of biology. I will show examples from biomedical (lymphatic, vascular and lung system) and agricultural problems of plant-soil interaction. I will describe the workflow from imaging (X-ray CT, XRF, SEM-EDX, histology), image reconstruction, image segmentation, computation and how to utilize this work stream to synthesise new scientific knowledge. In particular I will also outline several challenges and bottle necks in this process to hopefully encourage more mathematicians to get involved in the full pipeline.

Sleep is essential for healthy human function. Sleep may seem mysterious, but it is an area in which high level mathematical models have been pivotal in the development of our understanding – and is an area where considerable opportunities exist for further mathematical insight. In this talk I’ll introduce the nonsmooth coupled oscillator systems that form the basis of current mathematical models of sleep-wake regulation and discuss their dynamical behaviour. I will describe how we are using models to unravel environmental, societal and physiological factors that determine sleep timing and how we are combining models and data to develop digital twins for sleep-wake regulation, with the aim of creating personalised guidance to ameliorate delayed sleep timing disorders.

In this talk, I will present a hybrid individual cell-based mathematical and computational model, incorporating single-cell based intracellular dynamics, the cell microenvironment and cell-cell interactions to study the growth and progression of cancer cell mass. The modelling framework will then be used to study the effects of various anticancer therapies such as radiotherapy and chemotherapy, illustrating its adaptability and potential use. Furthermore, we will see how this modelling framework can be used to understand the role of known mechanisms in driving treatment effects and while in some cases, how it can provide testable hypotheses on poorly understood concepts such as “radiation-induced bystander effects”.

Building a mechanistic understanding of cell fate decisions remains a fundamental goal of developmental biology, with implications for stem cell therapies, regenerative medicine and understanding disease mechanisms. Single-cell transcriptomics should provide key information for their analysis, but building quantitative and predictive models from these data remains a challenge. I will present dynamic landscape analysis (DLA), a framework that applies dynamical systems theory to identify stable cell states, map transition pathways, and generate predictive cell fate models from single-cell data. Applying this framework to vertebrate neural tube development reveals that progenitor specification by Sonic Hedgehog can be captured in a landscape with an unexpected circular topology, where initially divergent lineages converge through multiple routes. This model accurately predicted cellular responses and cell fate allocation for unseen dynamic signalling regimes. By modelling the dynamic responses that drive cell fate decisions, the DLA framework provides a quantitative and generative framework for extracting mechanistic insights from high-dimensional single-cell data.

Extra-chromosomal DNA (ecDNA) is a genetic error found in more than 30% of tumour samples across various cancer types. It is a key driver of oncogene amplification promoting tumour progression and therapeutic resistance and is correlated to the worse clinical outcomes. Different from chromosomal DNA where genetic materials are on average equally divided to daughter cells controlled by centromeres during mitosis, the segregation of ecDNA copies is random partition and leads to a fast accumulation of cell-to-cell heterogeneity in copy numbers. I will present our analytical and computational modeling of ecDNA dynamics under random segregation, examining the impact of copy-number-dependent versus -independent fitness, as well as the maintenance and de-mixing of multiple ecDNA species or variants within single cells. By integrating experimental and clinical data, our results demonstrate that ecDNA is not merely a by-product but a driving force in tumor progression. Intra-tumor heterogeneity exists not only in copy number but also in genetic and phenotypic diversity. Furthermore, ecDNA fitness can be copy-number dependent, which has significant implications for treatment.

The spatiotemporal coordination and regulation of cell proliferation is fundamental in many aspects of development and tissue maintenance. Cells can adapt their division rates in response to mechanical checkpoints, yet we do not fully understand how cell proliferation regulation impacts collective cell migration phenomena. I will present a suite of continuum models of collective cell migration with cell cycle dynamics, which differ in their ability to describe mechanical constraints and hence cell proliferation regulation. By combining these mathematical models, Bayesian inference, and recent experimental data, I evaluate the level of model complexity that is consistent with the data and quantify the impact of mechanical constraints across different cell cycle stages in epithelial tissue expansion experiments.

Bone Morphogenic Protein 9 (Bmp9), whose signaling through Activin receptor-like kinase 1 (Alk1) is dysregulated in several diseases, has previously been shown to independently activate Notch target genes in an additive fashion with canonical Notch signaling. Here, by integrating predictive computational modeling validated with experiments, we uncover that Bmp9 upregulates Lunatic Fringe (LFng) in endothelial cells (ECs), and thereby also regulates Notch activity in an inter-dependent, synergistic fashion. Specifically, Bmp9-upregulated LFng enhances Notch receptor activity creating a much stronger effect when Dll4 ligands are also present. During sprouting, this LFng regulation alters vessel branching by modulating the timing of EC phenotype selection and rearrangement. Our results further indicate that LFng can play a role in Bmp9-related diseases and in pericyte-driven vessel stabilization, since we find LFng contributes to Jag1 upregulation in Bmp9-stimulated ECs; thus, Bmp9-upregulated LFng results in not only enhanced EC Dll4-Notch1 activation, but also Jag1-Notch3 activation in pericytes.

In this talk, I present a new mechanochemical framework for morphogenesis in regenerating epithelia, offering mechanistic insight into how physical forces and biochemical signalling interact to control pattern formation in living tissues. Focusing on Hydra morphogenesis, the model couples morphogen dynamics with tissue mechanics through a positive feedback loop: mechanical stretching enhances morphogen production, while morphogen concentration modulates tissue elasticity. We analyse the stability and bifurcation structure of steady states and, for exponential elasticity–morphogen coupling, establish a variational structure proving the existence of non-constant steady states for sufficiently small diffusion coefficients. Bifurcation analysis reveals a transition from subcritical to supercritical pitchfork bifurcations at a critical morphogen production rate, leading to bistability regions and robust single-peaked pattern formation without the need for a second diffusible inhibitor. The new mechanochemical model is further compared to classical Turing pattern formation, highlighting how mechanical feedback provides an alternative route to patterning. Theoretical predictions are supported by experimental validation.

Aerial views of intertidal mussel beds often reveal large scale striped patterns aligned perpendicular to the direction of the tide; dense bands of mussels alternate periodically with near bare sediment. I will describe a mathematical model for mussel beds, and show that it predicts this type of striped pattern for some parameter sets, with spotted patterns predicted in other cases. I will show that mathematical investigation of the stability of the stripes in 2-D explains when spots occur rather than stripes, and reveals a new type of hysteresis in pattern formation.

Most animals coordinate behaviour using neural computations. Yet, single-celled organisms also exhibit stimulus-responsive, even cognitive, actions. To understand how a single cell can coordinate and drive complex behaviours without any neural encoding, we study an algal protist – a motile cell with four extremely long cilia. The organism displays a surprisingly rich locomotor repertoire, emerging from the intricate dynamics of the cilia, which form a tight bundle when swimming. We leverage high-speed quantitative live imaging to extract the spectrum of possible ciliary beating patterns, and derive a dispersion relation coupling the temporal frequency and spatial wavelength of cilia oscillations. We further reconstruct the attractor manifold embedded in the behavioural space, showing that despite the range and complexity of ciliary beating modes, the underlying behavioural manifold is intrinsically low-dimensional with elaborate topological structure. Dynamic and excitable transitions in motility behaviour are encoded as trajectories in this space.