UK Mathematical Biology Conference

Melanoma is the most aggressive type of skin cancer, yet survival rates are excellent if it is diagnosed early. However, if metastasis occurs, five-year survival rates drop significantly. During the early stages of tumour initiation, melanoma cells form clusters within the primary tumour which promote metastasis. In the absence of biological tools to visualise cluster formation at primary tumour sites, we develop mathematical models to generate mechanistic insight into their formation. For this work we utilise in vitro data for two distinct melanoma cell phenotypes, one more proliferative and the other more invasive. This data consists of experiments for each phenotype individually, resulting in homogeneous clusters, as well as mixtures of the two phenotypes, resulting in heterogeneous clusters. We develop a series of differential-equation-based models using a coagulation-fragmentation-proliferation framework to describe the growth dynamics of homogeneous clusters, incorporating different functional forms for cell proliferation and cluster splitting. We then extend these models to describe the formation of heterogeneous cell clusters by considering both cluster size and phenotypic composition. We fit the models to experimental data, using a Bayesian framework to perform parameter inference and information criteria to perform model selection. In this way, we characterise and quantify differences in the clustering behaviour of two melanoma phenotypes in homogeneous and heterogeneous clusters, particularly the cluster coagulation, proliferation, and splitting rates. We find that the coagulation rate for the invasive phenotype is much larger than that for the proliferative phenotype, and evaluate how well different modelling assumptions fit the data in order to increase our understanding of the mechanisms driving metastasis. In future work, the models will be used to inform further experiments and, in particular, to suggest and test strategies for inhibiting metastasis.

Adaptive therapy (AT) protocols have been introduced to combat drug-resistance in cancer, and are characterized by breaks in maximum tolerated dose treatment (the current standard of care in most clinical settings). These breaks are scheduled to maintain tolerably high levels of tumor burden, employing competitive suppression of treatment-resistant sub-populations by treatment-sensitive sub-populations. Promising clinical results in prostate cancer indicate the potential of adaptive treatment protocols, but demonstrate broad heterogeneity in patient response. This naturally leads to the question: why does this heterogeneity occur, and is a `one-size-fits-all' protocol best for patients across this spectrum of responses? Using a Lotka-Volterra representation of drug-sensitive and -resistant tumor populations' dynamics, we obtain a predictive expression for the expected benefit from Adaptive Therapy and show that this can identify the best responders in a clinical dataset. Building on prior theoretical analyses, we obtain an analytic expression for the optimal treatment strategy, based on individual patient tumor dynamics. In contrast to prior approaches, this strategy also accounts for clinical limitations such as fixed time intervals between clinical appointments, to produce clinically realistic treatment strategies. We show that this strategy varies significantly between patients, and present a framework to generate individual treatment schedules from a patient's initial treatment data. Finally, we develop metrics (‘mathematical biomarkers’) to predict which patients have the greatest sensitivity to unplanned schedule changes, such as delayed appointments, allowing clinicians to identify high-risk patients that need to be monitored more closely and potentially more frequently. Overall, the proposed mathematical biomarkers illustrate the potential of analytic approaches to improve personalised cancer care.

Cancer invasion is a hallmark of malignancy, driven by the ability of cancer cells to degrade the extracellular matrix (ECM) and infiltrate surrounding tissues. This process is central to tumour progression and metastasis and involves complex interactions among cancer cells, ECM components, matrix-degrading enzymes, and the epithelial-to-mesenchymal transition (EMT)—a key phenotypic shift enabling cancer cells to transition from a proliferative epithelial state to an invasive mesenchymal phenotype. In this presentation, we introduce a suite of 3D mathematical models that capture the proliferation of epithelial-like cancer cells (ECs) and the invasion strategies of mesenchymal-like cancer cells (MCs). We begin with simplified models, discussing the existence and properties of classical solutions, before advancing to a multiscale hybrid stochastic-deterministic (SDE-PDE) framework with predictive capacity in realistic experimental settings. We conclude by presenting recent extensions of our model towards a multi-organ cancer invasion framework, outlining our initial steps in developing a mathematical foundation for a virtual cancer patient. This presentation is based on joint works with M. Chaplain, A. Madzvamuse, N. Harbour, T. Williams, L. Franssen, C. Verbeeck, Ch. Surulescu, N. Kolbe, and D. Katsaounis.

Tendons are connective tissues between muscle and bone. Tendinopathy is a common and painful pathology of the tendon affecting all ages, characterised by changes in its structure and composition. We aim to build a mathematical model to elucidate how altered tendon structure affects its function. Tendons are made up of 50-70% water and a collagen-heavy solid matrix, and undergo large deformations. We present a non-linear poroelastic model, assuming Darcy flow through the pores of the solid skeleton, to capture the bulk mechanics of the tissue. The solid skeleton is assumed to be formed primarily of crimped collagen fibrils that only bear load once fully taut, embedded in an isotropic background matrix. To model this, we derive a strain energy density for the solid skeleton assuming the fibrils are individually Hookean and the background matrix is neo-Hookean. We apply a uniaxial cyclic stress or displacement at the muscle boundary, and assume the bone is rigid and impermeable. We explore the effect of tendon damage by considering modifications to the solid constitutive law, such as altering collagen stiffness, fibril crimp angle and distribution, and determine how this affects the bulk poroelastic response of the tissue.

Many biological systems contain material components that are repaired and replaced over time by accompanying cells. One common example is the extra-cellular matrix, an interconnected network of proteins that provides chemical and mechanical protection and support in many systems, such as tissue basement membranes and bacterial biofilms. Questions remain as to how the bulk mechanical properties of the material depend on cell maintenance. I will present a mathematical model of a system of springs that aims to capture how the microscale replacement of elastic material can give rise to different observed bulk mechanical behaviours. Despite the model’s simplicity, it has a number of interesting characteristics that are biologically-relevant.

In this talk I shall survey recent work by myself and collaborators on an alternative view to the classical Turing-pattern view of the onset of pattern in biological systems. The chief context is that of cell polarity formation in which small GTPases that come in inactive cytosolic and active membrane-bound forms. We shall argue that for typical parameter values the symmetry-breaking bifurcation is typically subcritical. The onset of spatial pattern then occurs in a perfect system at the Maxwell point where homogeneous and spatial patterned states have equal values of some notion of energy. The existence of this Maxwell point not only makes spatially localised patterns more prevalent, but also adds additional complexity to the so-called Busse balloon of stable periodic patterns with different wavenumber. There are additional complexities if timescales are such that protein species are preserved. Then there is a propensity to form polarity through spatial fronts, whose location can be predicted by a novel theory of wave-pinning. Applications are discussed at very different spatial scales, namely plant cell polarity formation and the spatial ecology of desertification and savanna-rainforest transition. The overall conclusion is that biology is likely to favour subcritical bifurcation with its consequent hysteresis as this allows robust, sudden transitions to occur in the presence of slow and/or noisy parameter variation.

Many patterned ecosystems, such as dryland vegetation patterns and intertidal mussel beds can be described by PDEs admitting periodic travelling waves (PTWs). Under a changing environment that increases stress, such systems undergo a cascade of wavelength changes before an extinction event occurs. Classically, wavelength changes have been predicted by identifying the intersection of a PTW’s wavelength contour with a stability boundary in the system’s Busse balloon. In this talk, I highlight that this information is often insufficient because of a delayed loss of stability phenomenon. I show that PTWs can persist as transients for ecologically significant times after the crossing of a stability boundary in the Busse balloon. I present a method that can predict the order of magnitude of the time delay between the crossing of a stability boundary and the occurrence of a wavelength change by linking the delay to features of the essential spectra of the PTWs.

Despite its high prevalence and impact on the lives of those affected, a complete understanding of the cause of inflammatory bowel disease (IBD) is lacking. We investigate a novel mechanism which proposes that mutant epithelial cells are significant to the progression of IBD since they promote inflammation and are resistant to death. We develop a simple model encapsulating the propagation of mutant epithelial cells and immune cells which results from interactions with the intestinal barrier and bacteria. Motivated by the slow growth of mutant epithelial cells, and relatively slow response rate of the adaptive immune system, we combine geometric singular perturbation theory with matched asymptotic expansions to determine the one-dimensional slow invariant manifold that characterises the leading order dynamics at all times beyond a passive initial adjustment phase. The dynamics on this manifold are controlled by a bifurcation parameter, which depends on the ratio of growth to decay rates of all components except mutants and determines three distinct classes of permanent-form travelling waves that describe the propagation of mutant epithelial and immune cells. These are obtained from scalar reaction-diffusion equations with the reaction being (i) a bistable nonlinearity with a cut-off, (ii) a cubic Fisher nonlinearity and (iii) a KPP or Fisher nonlinearity. Our results suggest that mutant epithelial cells are critical to the progression of IBD. However, their effect on the speed of progression is subdominant.

Epidemiological models are crucial for informing policies aimed at reducing disease spread, both within indoor environments and across larger populations. At the population level (macroscale), network models can be used to understand and predict disease transmission. However, inferring key disease spread properties—such as time to infection, reproduction number—from data presents significant challenges. These include: (a) the evolving nature of network structures, (b) the quantity and quality of available data, and (c) the method by which data is extracted. A notable challenge is size bias, where highly connected individuals are more likely to be sampled, skewing the data. To investigate and mitigate this bias, we compared two data extraction methods: the Random Walk (RW) and the Metropolis-Hastings Random Walk (MHRW), quantifying the extent of size bias in each approach.

Salmonella is a significant zoonotic pathogen responsible for numerous foodborne diseases worldwide. Its transmission occurs through the consumption of contaminated animal- and plant-derived foods, as well as environmental and vector-mediated pathways. In this study, we develop a mathematical model based on a system of coupled nonlinear ordinary differential equations to describe the transmission dynamics of Salmonella across these pathways. The model accounts for direct human-to-human transmission, contamination via food and water sources, and vector-borne spread. We analyze the basic properties of the model, including equilibrium points, positivity, and boundedness of solutions. The basic reproduction number (R₀) is derived using the next-generation matrix method, providing insights into transmission dynamics. Stability analysis is conducted to determine the conditions for disease persistence and eradication. The model is calibrated using European Centre for Disease Prevention and Control (ECDC) data on Salmonellosis cases from 2007–2023, where some parameters are assumed while some are fitted using the package SciPy.optimize of the SciPy 1.15.1 library in Python 3.13.1. A sensitivity analysis is performed to identify the most influential parameters affecting disease transmission. Additionally, optimal control strategies are investigated, incorporating interventions such as improved sanitation, food safety regulations, and vector control. Numerical simulations demonstrate the effectiveness of various control measures, providing a quantitative assessment of their impact on reducing Salmonellosis prevalence. The study also addresses challenges in numerical implementation, ensuring stable and accurate solutions. Our findings offer valuable insights for policymakers and public health officials, aiding in the optimization of resource allocation for Salmonellosis prevention and control in the EU/EEA.

We set up and analyse a mathematical model, the Serious Crime Model, which describes the interaction of mild and serious offenders and potential criminals. However we get more complete results for a simpler version of this model, the Mild Crime Model, with no serious offenders. For the full Serious Crime Model there are two key parameters R_0^1 and R_0^2, corresponding to the basic reproduction number in the mathematics of infectious diseases, which determine the behaviour of the system. For the simpler Mild Crime Model there is one such parameter R_0^1. Both forward and backward bifurcation can occur for this second model with two subcritical non-trivial equilibria possible for R_0 < 1 in the backward case. For backwards bifurcation there is another threshold value R_0^* such that the upper non-trivial equilibrium is unstable for R_0^1 < R_0^* and stable for R_0^1 > R_0^*. For forwards bifurcation there are two threshold values R_0^* and R_0^** such that the unique endemic equilibrium is stable for R_0^** < R_0^1 < R_0^* and stable for R_0^1 > R_0^*. At the end we return to the Full Serious Crime Model and discuss the behaviour of this model. The results are meaningful and interesting because in all of the other epidemiological and sociological models of which we are aware, analogous thresholds to R_0^* and R_0^** do not exist. For forwards bifurcation the unique non-trivial equilibrium and for backwards bifurcation with two subcritical equilibria the higher nontrivial equilibrium are also usually always locally asymptotically stable. So our models exhibit interesting and unusual behaviour.

A pivotal aspect of developing effective immunotherapies for solid tumours is the robust testing of product efficacy inside in vitro platforms.Collaborating with an experimental team that developed a novel microfluidic device at Children’s National Hospital (CNH), we developed a mathematical model to investigate immune cell migration and cytotoxicity within the device. Specifically, we study Chimeric Antigen Receptor (CAR) T-cell migration inside the channels, treating the cell as a moving boundary driven by a chemoattractant concentration gradient. The chemoattractant concentration is governed by two partial differential equations (PDEs) that incorporate key geometric elements of the device. We examine the motion of the cell as a function of its occlusion of the channel and find that certain cell shapes allow for multiple cells to travel inside the channel simultaneously. Additionally, we identify parameter regimes under which cells clog the channel, impairing their movement. All our findings are validated against experimental data provided by CNH. We integrate our model results into a broader model of the device, which also examines the cytotoxicity of CAR T-cells. This provides a tool for distinguishing experimental artefacts from genuine CAR T-cell behaviour. This collaboration enabled the team at Children’s National Hospital to refine experimental conditions and uncover mechanisms enhancing CAR T-cell efficacy. [1] D Irimia, G Charras, N Agrawal, T Mitchison, M Toner, Polar stimulation and constrained cell migration in microfluidic channels,, Lab on a Chip 7 (12), 1783-1790

Glioblastoma (GBM) is the most aggressive and most common primary brain tumour in adults and is uniformly fatal, with a poor median survival time of 15 months. Standard of care for GBM consist of radiotherapy either alone or following surgical resection, despite this, radio-resistance almost always occurs making recurrence inevitable. Failure of the current standard of care has been partly attributed to a special sub-population, the glioma stem cells (GSCs), which initiate and drive tumour growth. Treatment cannot be successful unless all GSCs are eliminated. However, GSCs are known to be highly resistant to radiotherapy, and complete surgical removal is impossible in GBM. Therefore, new treatments that specifically target GSCs could have a potentially large benefit. BMP4 has been shown to induce differentiation of GSCs towards a less malignant, astrocytic-like (ALCs) lineage reversing the GSC state and reducing radio-resistance. We develop a data driven mechanistic mathematical model that accounts for the GSCs, tumour cells (TCs) and ALCs as well as their response to both radiotherapy and BMP4 induced differentiation therapy. We parameterise our model based on data collected from twelve GSC cell lines, that underwent various BMP4 and radiotherapy treatments. Through virtual clinical trials we determine an optimal dosing strategy for BMP4 in combination with radiotherapy. We identify several key parameters that impact the efficacy of BMP4 therapy including radiosensitivity and proliferation rate. These parameters can be used to strategically select candidates for real clinical trials that will likely have the largest benefit.

Pancreatic Ductal Adenocarcinoma (PDAC) is a highly aggressive malignancy with a tumour microenvironment (TME) that plays a crucial role in disease progression and therapeutic resistance. The architectural organisation of TMEs exhibits significant spatial and phenotypic heterogeneity, but how these structures emerge from dynamic intercellular interactions remains poorly understood. Mathematical modelling offers a powerful approach to explore the mechanistic basis of these emergent properties, providing insights beyond what can be directly observed through temporally sparse sampling and descriptive analyses of tumours. Here, we present a spatially explicit, multi-scale agent-based model that simulates the dynamic crosstalk between Pancreatic Cancer Cells (PCC) and Cancer-Associated Fibroblasts (CAF), incorporating key signalling pathways and phenotypic transitions. The model, parameterised using single-cell transcriptomics, captures the spatio-temporal evolution of PCC-CAF interactions under varying initial conditions of cell abundance and spatial organisation. Simulations reveal that both the proportions and spatial configurations of cell populations influence the emergent phenotypic composition and dynamic trajectories of co-evolution, highlighting the role of tissue architecture in shaping tumour progression and heterogeneity. In this presentation, we will discuss the mathematical formulation of the model, key simulation results, and ongoing efforts to extend the framework for exploring therapeutic strategies. Our approach demonstrates how mathematical modelling can be leveraged to predict and quantify the influence of hetero-cellular crosstalk and tissue architecture on shaping phenotypic heterogeneity within the tumour ecosystem in PDAC.

Acute lymphoblastic leukemia (ALL) is a type of cancer in which the bone marrow produces too many lymphocytes without further differentiation into mature blood cells. The primary treatment for most ALL cases is chemotherapy; however, after the intense treatment phase, relapse is observed in some patients. Early-stage relapses might be due to some remaining leukemia cells not being detectable by conventional cytomorphology. The molecular tests on minimal residual disease (MRD) can quantify disease burden at relatively low levels, thus helping track cancer remission and relapse. We investigated the dynamics of residual disease to describe the observed response and relapse kinetics heterogeneity. We developed stochastic, patient-specific models based on longitudinal MRD data to further the understanding of the driving factors of relapse, aiming to quantify better prediction of likelihood and timing of relapse in individual patients based on individual treatment choices.

The specification of discrete cell states in developing tissues is driven by biological signals that force the cells to modify their molecular identity and transition to a new state. This process is well illustrated by Waddington’s landscape metaphor, which portrays a cell on its developmental pathway as a ball rolling down a landscape of branching valleys. The local minima of the landscape represent “attractor states" whose existence depends on the signals that continuously reshape the landscape making it highly dynamical.  We present a systematic approach, grounded in dynamical systems and bifurcation theory, to construct dynamical landscape models from high dimensional transcriptomic data.  The approach uses computational methods to identify attractor states and the pathways connecting them thereby revealing the complex topology of the cells decision-making landscapes. We apply this framework to in vitro data from the developing neural tube, where distinct types of neural progenitors are specified in response to pulses of the signal Sonic Hedgehog. The resulting quantitative model successfully replicates cells responses when subjected to signal perturbations and can be used to predict in vitro protocols for generating specific cell types.

An important early step in mammalian embryonic development is the specification of the anteroposterior, dorsoventral, and mediolateral axes of the embryo. The axial midline is a central stripe-like pattern that runs along the anteroposterior axis, providing signals that structure the body plan and position developing internal organs. Despite the midline being a critical component of embryonic development, little is known about the mechanism of its specification. Gastruloids, which are embryo-like 3D stem cell aggregates, self-organise to create all 3 embryonic axes, are more amenable to experimentation than mouse embryos, and form a midline stripe at 120hAA (after aggregation). Our analysis of published scRNA-seq, spatial transcriptomics, and proteomics datasets on mouse gastruloids has identified 3 crucial signalling modules to be active from early symmetry breaking (72-96hAA) to midline specification (96h-120hAA). We hypothesise that a GRN comprising key genes from some or all these signalling modules govern both early symmetry breaking and axial midline specification. We model the gastruloid as a 2D sheet of nuclei and examine the parameter space of all possible GRN topologies, which when coupled with directed cell movements, successfully achieve symmetry breaking.

Embryo development is driven by precise gene expression timing, which varies across species. While molecular mechanisms controlling this timing are being intensely studied, the integration of tempo control into mathematical models remains underexplored. In this chalk talk, I will present a new mathematical framework linking gene expression orbits to developmental programs, allowing us to identify tempo changes as dynamical system perturbations that preserve the function of the system.

We study the positional information conferred by the morphogens Sonic Hedgehog and BMP in neural tube patterning. We use the mathematics of information theory to quantify the information that cells use to decide their fate. We study the encoding, recoding and decoding that take place as the morphogen gradient is formed, triggers a nuclear response and determines cell fates using a gene regulatory network.

Parkinson’s disease (PD) is the fastest growing neurological condition, currently affecting 10 million people worldwide, with this number set to double by 2050. Treatment for PD focuses predominantly on pharmacological interventions. When these become ineffective, an alternative can be provided by the deep brain stimulation (DBS). In DBS, electrical leads are implanted in a patient’s brain and a high frequency-low amplitude current is delivered to the brain, usually at the same setting throughout the day and night. Although the treatment can be life changing, its effectiveness is still limited and the mechanisms behind its positive effect remain unknown In this talk, we discuss the brain oscillations characteristic to Parkinson’s disease and present an approach to modelling the effect of stimulation using a Wilson-Cowan framework. Stimulation is applied as a forcing term, making the model non-autonomous. We find that depending on the dynamical regime, the model either behaves as a driven or an entrained system. In the entrained case, we consider a non-smooth limit, which brings insight on how entrainment occurs. Overall, our work contributes to understanding how brain stimulation can be used to alter brain rhythms, in particular suppressing the pathological oscillations and promoting healthy ones. Insights from studies on PD might further prove useful for other therapies such as epilepsy.

Connectivity is one signature of brain function. Connectivity may be structural, as measured by looking at images of the brain and measuring the white matter pathways (tracts). Connectivity may also be functional, measuring the extent to which different areas of the brain share information over time. Here, we consider functional connectivity (FC) as computed from pairwise correlations between recordings from different electroencephalogram (EEG) electrodes. We consider people living with mild Alzheimer’s disease (AD) as well as age-matched healthy controls and evaluate the extent to which FC evaluated from overnight recordings during sleep distinguish between the two groups. Previous work has focussed on FC during wake, but its variation during sleep remains underexplored. We used gold-standard polysomnographic recordings collected at the Surrey Sleep Research Centre based on the American Academy of Sleep Medicine standard low-density EEG montage with six electrodes for one full night of sleep. Data from 20 people living with mild AD, 12 of their partners and 42 healthy controls were used. FC was computed across vigilance states and frequency bands using two reproducible techniques: phase lag index and weighted phase lag index. Previous studies performed during wake find that FC is lower in AD than in healthy controls. We find the converse during sleep, i.e. that FC is consistently higher in AD than in controls. The theta (4-8Hz) and delta (0.5-4Hz) bands in non-rapid eye movement (NREM) sleep provide the highest discriminatory power. Notably, effect sizes in NREM sleep surpass those in wakefulness, highlighting the potential of sleep EEG features as biomarkers for AD.

Neurons often switch between two distinct membrane potentials, known as up and down states, which correspond to depolarization and hyperpolarization, respectively. These states are crucial for synaptic inputs integration and consequently neuronal signals generation and propagation. At the population level, the transition between up and down states occurs in a synchronized manner across large networks of neurons, making this transition critical for information processing and memory consolidation. The precise mechanisms that control transitions between up and down states are not fully understood, but they are likely due to neuronal network properties. Here, we propose an organizing center underlying the dynamics of up and down states within neuronal populations, applying bifurcation analysis and simulations of neuronal network models. Our analysis demonstrates that the dynamic structures we reveal are general across established macroscopic models of neural population activity, including the classical Wilson-Cowan and Tsodyks models, underscoring their universality. Our findings provide insight into how this organizing center characterizes distinct qualitative behaviors and its involvement in controlling the balance between excitation and inhibition, which is fundamental to large-scale brain dynamics. We show an example of this phenomenon in on our recently developed medial amygdala circuitry model and explore its broader implications for reproductive function.

GnRH pulsatility is a critical process for fertility and reproduction. Despite understanding the individual properties of KNDy neurons, the mechanism by which they synchronise to generate pulsatile GnRH release is unknown. To address this gap, we constructed a computational model of a KNDy neuron network incorporating both fast and slow communication. Our findings indicate that fast glutamate signalling may trigger synchronisation, while slower NKB and dynorphin signalling controls the burst duration. This model provides a framework for exploring how network architecture and neurotransmitter interactions influence the generation of GnRH pulses. This is joint work with Krasimira Tsaneva-Atanasova and Margaritis Voliotis at University of Exeter.

Flowers and pollinators have co-evolved over millennia to produce fascinating sensory properties. One such recent discovery was that bees and spiders can detect natural electrical fields. Thus, our attention turns to flowers. Acting as dielectrics, flowers inductively charge in electrical fields. Considering the source as that of charged pollinators or the Earth's background electrical field, we seek to answer: whether flowers use this to their advantage to become more detectable to pollinators? And how does floral geometry (petal shape and number) affect the perturbed field? To investigate this, the electric field interior and exterior to the flower is modelled numerically. A two-dimensional approximation is taken, and a AAA-least squares method used to find a rational approximation of the harmonic electric potential. Some time is spent examining this method; its application to two-domain problems appears to be new. The algorithm gives accurate and rapid results dependent on only three parameters: the relative permittivity of the flower – its ability to store electrical energy - the petal number and the location of the pollinator(s). The results show that flowers display distinct information about their morphology and pollen levels at distance through the perturbed electric field. Pollinators then detect these signals while also sensing other nearby pollinators. Some arthropods, such as the crab spider, may even use the flower to mask their own presence and draw in unsuspecting prey. Results in 3D are also produced using the COMSOL package and show that the 2D results are a good proxy for the overall 3D behaviour.

Multiscale models can help us to understand how organ-scale developmental patterns arise through processes at the cellular scale. Understanding such developmental patterning within growth tissues is a particular challenge. Here we present a multiscale model to investigate hormone gradients within the growth Arabidopsis plant root [1]. The model predicts how the distribution of the hormone gibberellin (GA) depends on the metabolism network, delivery of precursors, passive and active transport between adjacent cellular compartments, and cell growth. Using asymptotic analysis, we derived a continuum approximation of the initial cell-based model which revealed how the effective hormone velocity, diffusivity, and dilution rate depend on the parameters governing the cell-scale processes [2]. Furthermore, the continuum approximation reduced computational costs enabling us to estimate key unknown parameter values using biosensor data. The model suggested that delivery of a shoot-derived precursor within the elongating cells contributes to the hormone gradient. Model predictions suggested that enzyme inactivation within dividing cells could explain the steep hormone gradients observed in over-expression lines and improves agreement between predictions and data in both wild type and loss-of-function mutants. Furthermore, the model predicted that the hormone gradient also depends on a balance of diffusion through cell-to-cell channels (termed plasmodesmata) and degradation, which we validated via additional biosensor imaging. In conclusion, our results suggest that local synthesis combined with diffusion and degradation creates a spatial hormone gradient within the plant root that provides positional information and patterns cell elongation. References [1] Kiradjiev KB, Griffiths J, Jones AM, Band LR. 2025. A balance of metabolism and diffusion articulates a gibberellin hormone gradient in the Arabidopsis root. bioRxiv [2] Kiradjiev KB, Band LR. 2023. Multiscale asymptotic analysis reveals how cell growth and subcellular compartments affect tissue-scale hormone transport. Bull. Math. Biol. 85:101

As the climate warms the environment changes, as do the stresses plants experience. The SUMO (Small Ubiquitin-like Modifier) pathway is a key process that mediates plant responses to stress. As part of an interdisciplinary project, we are developing mathematical models to understand how SUMO transduces environmental signals into specific physiological responses. The model incorporates key processes in the SUMO cycle, which involves SUMO proteins performing post-translational modifications on a Target protein. We used the mathematical models composed of systems of non-linear ordinary differential equations to simulate observed changes in SUMO-cycle components after salt, osmotic and flagellin stress in different root tissues (integrating new experimental data generated as part of the SUMOcode project, www.sumocode.org). The model reveals how the different stresses affect the level of SUMOylation in different root tissues.

For a century and more, mathematicians (notably including Alan Turing) have been explaining Fibonacci numbers within plant forms; for most of this time developmental and molecular biologists have ignored them for their own mostly valid reasons. Here I address one objection, the lack of testable predictions, by presenting the first quantitative evaluation of the ability of Schwendener disk-stacking models to generate the spiral patterns seen in a large dataset of sunflower seedheads. I’ll explain the standard van Iterson theory of lattices on cylinders, which accounts for the observed predominance of Fibonacci counts. But I’ll also show the generalisation to disk- stacking models and how these account for other observed phenomena: a smaller but detectable frequency of Lucas numbers, a comparable frequency of Fibonacci numbers plus or minus one, occurrences of pairs of roughly equal but non-Fibonacci counts, and an occasional lack of rotational symmetry. Moreover the Schwendener model produces these phenomena in the region of parameter space just beyond where the Fibonacci structure breaks down, and also allows an interpretation of apparently noisy observations as a chaotic deterministic dynamical system. The rich dynamics of these conceptually simple models are a fertile area for mathematical study, allow quantitative comparison with existing morphological data, can be related to and generate new hypotheses in molecular biology, and above all provide explanations of complex biological form that cannot be reduced to expression of a single gene. Phyllotaxis (still) has the potential to become a persuasive example of the need for mathematics in developmental biology.

Melanoma cells can transition between cell states, contributing to therapy resistance and immune evasion. These state changes involve dynamic and reversible shifts in gene expression, making it essential to understand the underlying regulatory mechanisms for developing effective therapies. We present a mathematical model of a minimal gene regulatory network comprising key transcription factors associated with melanoma cell states. Using deterministic temporal and spatio-temporal differential equation models, we analyse gene expression dynamics and classify stable states in a biologically meaningful way. We exploit an approximation, based on cooperative binding of transcription factors, in which the models are piecewise smooth. At the population level, we use a naïve model of intercellular communication to explore how cells within a tumour can exhibit coordinated behaviour through travelling waves of gene expression. Additionally, we propose a method for deriving a condition that determines the final state of a population of communicating cells. This model provides a framework for better understanding some of the mechanisms driving gene expression dynamics and to inform and validate experimental hypotheses.

Cells inevitably encounter unpredictable changes in their microenvironment. To optimally survive, cells consequently adapt by orchestrating large alterations in their molecular states that often result in appreciable phenotypic changes. The timescale of molecular, and therefore, cellular adaptation depends on how quickly the memory of past environment encounters is lost (and therefore forgotten) by the cell (e.g. degradation rate of proteins unfavourable to the current environment). Here, we study the dynamical implications of two distinct memory mechanisms on cellular responses to changing environment. The two phenomenological models considered are – 1) Undated memory, wherein each prior environmental exposure has equal probability to dissipate in the next time step, and 2) Dated memory, wherein each experience is dated and erased according to their order of encounter. We find that the dated memory adapts faster than undated memory and confers higher growth benefit to the cell under stochastic changes or long-term abrupt shifts in the environment. Intriguingly, optimal memory for swift adaptation and higher growth benefit under periodic environment depends on the combination of cell memory size and environment period. We extend the results with these memory models to show how cost incurred during cellular adaptation (e.g. energetic cost of mRNA and protein production) improves cellular decision by delaying the phenotypic response until enough environmental change is experienced by the cell.

Gene regulatory networks (GRNs) govern processes such as cell fate, patterning, and adaptation. While multistability and oscillations are both common GRN dynamics in cell biology, they are typically studied and engineered in isolation. Here, we challenge this separation by using the AC-DC circuit, a minimal three-gene network that merges the classical toggle switch and repressilator [1,2]. Using a thermodynamic formalism, we show that even a simplified, single-inducer version of the circuit can exhibit diverse, multifunctional dynamics, including the coexistence of oscillations and multistability. To assess its synthetic viability, we use Bayesian parameter inference (ABC-SMC) to explore robustness, classify emergent behaviours, and analyse timing, excitability, and regime transitions. Remarkably, we find that the AC-DC circuit can produce hundreds of topologically distinct bifurcation diagrams, challenging the classical view that network topology rigidly constrains dynamical outcomes. This fascinating flexibility enables novel synthetic capabilities such as robust period control, coexistence of multiple oscillators, and finely tuned excitatory responses. By revealing the hidden potential of minimal circuits and providing design principles for their implementation, this work opens new directions for cell decision and computation. [1] Panovska-Griffiths, et al.,J. Roy. Soc. Interface, 2013. [2] Perez-Carrasco, et al., Cell systems, 2018.

Membranes regulate transport in a wide variety of applications, from industrial filtration and synthetic fabrics to biological cells and tissues. In bacteria, membrane channels control sensing and communication, and enable cells to filter antibiotics and resist treatment. In this talk we systematically upscale the transport across a bacterial membrane, deriving effective boundary conditions that explicitly account for the microscale channel structure, combining multiscale methodologies including asymptotic homogenisation and boundary layer theory. This allows us to treat the bacterial membrane as an effective interface, over which a significant concentration difference can be sustained. The effective conditions we derive preserve information about the microscale structure while reducing computational complexity, providing insight into how microscale properties affect membrane permeability and metabolite transport over much larger lengthscales. Incorporating these conditions into an additional population-level upscaling allows us to derive a colony-level model that explicitly and efficiently accounts for membrane channel microstructure. More broadly, because we consider a generic membrane geometry, our results hold for general (outer) problems away from the interface. Therefore, the results we derive have a wide scope of applications beyond bacterial membranes, for example, in modelling water vapour and heat loss through fabrics, as well as in industrial filtration processes.

The Golgi has an intricate spatial structure characterized by flattened vesicular compartments, known as cisternae. These cisternae expose the contents of the Golgi to membrane-bound enzymes that catalyse glycosylation, the addition of polymeric sugars to protein cargo, which is crucial to the successful secretion of many cellular products. The unusual and specific shape of Golgi cisternae is highly conserved across eukaryotic cells, suggesting significant influence in the correct functioning of the Golgi. To explore the relationship between cisternal shape and Golgi function, we develop and analyse a mathematical model of polymerisation in a cisterna that combines chemical kinetics, spatial diffusion and adsorption and desorption between lumen and membrane. Exploiting the slender geometry, we derive a non-local non-linear advection-diffusion equation that predicts secreted cargo mass distribution as a function of cisternal shape. The model predicts a maximum cisternal thickness for which successful glycosylation is possible, demonstrates the existence of an optimal thickness for most efficient glycosylation, and suggests how kinetic and geometric factors may combine to promote or disrupt polymer production. The model is supported by experimental evidence that disruption to Golgi morphology leads to observable changes in secreted cargo mass distribution.

Cells constantly adapt to changes in their environment, and a key part of this process involves dynamically regulating the localisation of transcription factors (TFs). In Saccharomyces cerevisiae, these TFs shuttle between the cytoplasm and nucleus in response to stress, forming complex time series that encode information about external conditions. However, these signals are noisy and often non-linear, posing challenges for conventional analytical approaches. To better understand how yeast cells represent environmental stress, we apply deep learning techniques—specifically Long Short-Term Memory (LSTM) networks and Variational Recurrent Autoencoders (VRAEs)—to model TF nuclear localisation dynamics under osmotic, oxidative, and nutrient stress. We first train these models on synthetic time series generated via the Stochastic Simulation Algorithm (SSA), capturing the stochastic nature of TF responses. We then fine-tune the models using experimental microfluidic datasets, employing transfer learning to bridge the gap between idealised and real biological systems. This strategy improves generalisability while reducing the need for large volumes of labelled experimental data. By estimating mutual information between TF dynamics and environmental conditions, we assess how well these models capture the informational content of the cell’s response. Compared to traditional classifiers, our deep learning approach more accurately decodes environmental states and reveals meaningful latent representations that reflect the roles of generalist and specialist TFs. Together, this work offers a robust, data-efficient framework for decoding cellular stress responses—highlighting how machine learning can uncover the hidden structure of dynamic biological systems and inform future applications in systems and synthetic biology.

Cell and tissue movement in development, cancer invasion, and immune response relies on chemical or mechanical guidance cues. In many systems, this movement is locally guided by self-generated signaling gradients rather than long-range, pre-patterned cues. However, how heterogeneous cell mixtures navigate through self-generated gradients remains largely unexplored. In this talk, I will first summarize our recent discovery that immune cells steer their long-range migration using self-generated chemotactic cues (Alanko, Ucar et al, Sci. Immun. 2023). I will then introduce a new theoretical model that describes migration and patterning strategies in heterogeneous cell populations (Ucar et al bioRxiv 2024). Our model predicts that the relative chemotactic sensitivities of different cell populations control their long-time coupling and co-migration dynamics, with boundary conditions such as external cell- and attractant reservoirs substantially influencing the migration patterns. I will show how this model quantitatively reproduces in vitro experiments on co-migrating immune cell mixtures. Interestingly, immune cell co-migration occurs near the optimal parameter regime predicted by theory for robust and colocalized migration. Finally, I will discuss the role of mechanical interactions, revealing a non-trivial interplay between chemotactic and mechanical non-reciprocity in driving collective migration.

This talk will argue that cells are not passive sensors but active participants that generate new information to guide their migration. Drawing on integrated experimental work with Dictyostelium and cancer cells, and supported by mathematical modelling, we show that cells create and use a variety of secondary signals to dramatically improve their navigational accuracy. A key mechanism we have identified is the creation of self-generated gradients. By either secreting chemoattractants or locally degrading ambient ones, cells produce sharp, reliable, local cues that are far more informative than the shallow background gradient.

Collective cell migration is ubiquitous amongst multicellular communities and contributes to many phenomena, e.g., morphogenesis and cancer metastasis. Nonetheless, it is still poorly understood how cells coordinate to control the emergent collective motion of cell groups (or swarms). Recent experimental data suggests that physical interactions between cells within the swarms can result in emergent fluid-like properties. In this work, we propose a continuum, coarse-grained, active fluid model to study how physical interactions affect the complex spatiotemporal dynamics of cell swarms' collective chemotaxis in response to self-generated chemical gradients. Our results reveal that the interplay between physical interactions, cell proliferation and chemotaxis can lead to a new mode of pattern formation via self-organised shedding: as the swarms move collectively, they can periodically shed groups of cells at the rear. As such, our work offers a new perspective to the study of chemotaxis of multicellular communities revealing the role of physical interactions in mediating their collective dynamics.

Advances in experimental techniques allow the collection of high-resolution spatio-temporal data that track individual motile entities over time and could be used to calibrate mathematical models of individual motility. However, experimental data is intrinsically discrete and noisy, and these characteristics complicate the effective calibration of models for individual motion. We consider individuals whose movement can be described by velocity-jump models in one spatial dimension, characterised by successive Markovian transitions between a network of n states, each with a specified velocity and a fixed rate of switching to every other state. We develop a Bayesian framework to calibrate these models to discrete and noisy data, which uses a likelihood consisting of approximations to the model solutions which we previously obtained. We apply the framework to recover the model parameters of simulated data, including the probabilities of switching to every other state. Moreover, we test the ability of the framework to select the most appropriate model to fit the data, including comparisons varying the number of states n.

Cell movement is critical to many biological processes, such as collective cell migration or the immune response to cancer . An important driver of cell movement are interactions between nearby cells, yet it can be difficult to infer and express the rules governing cell-cell interactions in a manner that is both biologically accurate and interpretable. Here we present a suite of tools, based on the theory of deep attention networks, that can accurately quantify cell-cell interaction dynamics directly from cell trajectory data, and enable the learned dynamics to be presented to the user in an easily interpretable manner. We showcase the power of these tools for interpreting cell trajectory data using data from two different experimental systems, and describe how to integrate the learned information into a mathematical modelling framework.

Collective cell migration plays a crucial role in numerous biological processes, including cancer growth, wound healing, and the immune response. Often, the migrating population consists of cells with various different phenotypes. This study derives a general mathematical framework for modelling cell migration into the micro-environment, which is coarse-grained from an underlying individual-based model that captures some of the dynamics of cell migration that are influenced by the phenotype of the cell, such as: random movement, proliferation, phenotypic transitions, and interactions with the external environment. The resulting model provides a continuum, macroscopic description of cell invasion, which represents the phenotype of the cell as a continuous variable and is much more amenable to simulation and analysis than its individual-based counterpart when considering a large number of phenotypes. The results highlight how phenotypic structuring impacts the spatial and temporal dynamics of cell populations, demonstrating that different environmental pressures and phenotypic transition mechanisms significantly influence invasion patterns.

To interpret observations on animals' movement and on their selection of habitat and resources in a coherent way, it is necessary to model their movement over all timescales in a way that is consistent with their long-term use of space. The work presented here takes advantage of developments in stochastic processes and statistical algorithms to develop a range of new models in which both the dynamics and the long-term behaviour are tractable and described parametrically, and which are flexible enough to represent a wide range of patterns of movement and space use encountered in reality. I extend the mathematical analogy between movement modelling and Markov chain Monte Carlo algorithms, first proposed by Michelot, Blackwell & Matthiopoulos (2019; Ecology 100, e02452), to a wide range of continuous-time stochastic processes, including both diffusion processes and velocity-jump models, that in different ways are motivated by the simple discrete-time step-and-turn models widely used in practice. Particular cases include a diffusion process where the dynamics are defined in terms of speed and bearing, and a velocity-jump process in d dimensions, generalising the ‘bouncy particle sampler’ used in Bayesian inference, in which the distribution of velocity after a so-called ‘bounce’ event has support over a region which itself has dimension d. This mathematical approach can be extended to models incorporating distinct behavioural states and to higher dimensional models representing the joint movement of interacting individuals.

Quantifying the dynamics of macroevolutionary trends, such as changes in body size and complexity, is vital for understanding the processes that have shaped patterns of extant and extinct biodiversity. Understanding the contribution from passive and driven trends in shaping this diversity is important as they are the macroscopic signature of the fundamental evolutionary mechanisms acting to cause change over deep time. In this talk I will discuss a non-phylogenetic approach to the study of these mechanisms which is based on the following dynamics: species, characterised by a trait, can be produced via speciation or can go extinct. They may also evolve their trait by two mechanisms: a passive mechanism and a driven mechanism. Crucially, the range of accessible traits is finite, as upper and lower boundaries will limit the possible trait values to simulate various constraints. The average dynamics of this system are described by a partial differential equation which can be solved analytically in the most simple cases. I will demonstrate how this model can be applied to various datasets with two examples: a large dataset of extant mammals' body mass and a dataset of brachiopod and bivalve maximum linear size through geological time. We find good support for bounds on these traits: the lower boundary on mammal mass allows the distribution to adhere to Cope's rule despite a driven process acting to reduce body mass; the upper boundary on size for bivalves and brachiopods allows our model to capture the slowing distribution peak and transition away from a symmetry towards the present day. We are also able to infer a dramatic change to the parameters controlling the evolution of the distribution of brachiopod size before and after the Permian-Triassic mass extinction event, demonstrating the utility of the model in studying shifts in evolutionary dynamics caused by environmental upheavals.

Evolutionary graph theory (EGT) considers evolutionary dynamics in a structured population that is represented by a graph. Fixation probability is a measure of probability that a mutation takes over the resident population. One of the major questions EGT tries to answer is how the graph structure impacts the fixation probability. It has been found that certain graphs can act as amplifiers or suppressors of selection. However, the type of update rule used in the model can impact this result. For example, the star graph is an amplifier for birth-death with fitness on birth (Bd) dynamics and is a suppressor for death-birth with fitness on birth (dB) dynamics. Typically, EGT has focused on discrete time models, which can be hard to link with realistic population dynamics. Recently, these discrete time models have been generalized to a continuous time Markov-process model based on eco-evolutionary dynamics, where the results for dB and Bd dynamics can be recreated by suppressing the ecological dynamics. This work shows that within this continuous time framework, there exists a continuous transition from Bd to dB results. Therefore, the interplay between the underlying biological parameters of birth rate, death rate, and competition, will drive the qualitative shift from amplifier (under Bd) to suppressor (under dB). Using the star graph as an example, we prove that the transition from Bd to dB depends on the magnitude of the natural death rate. By increasing the natural death rate of individuals, population structures that typically amplify selection under Bd dynamics can be driven to suppress fixation. Exploring the fundamental drivers behind this qualitative shift will provide further insights into whether population structures will amplify or suppress selection under realistic population dynamics.

The analysis of evolutionary tree shapes can provide insights into evolutionary processes as tree structures reflect the rates at which biological types arise, diversify, and become extinct. Phylogenetic imbalance has been studied for decades [1,2] but its causes and consequences remain poorly understood, partly due to a lack of general, robust methods for quantifying tree shape. We recently introduced new methods that solve this problem [3,4]. Here we use these methods to compare speciation trees for birds and Brassica – for which there is unusually extensive, high-quality data – and evolutionary trees of four human pathogens. We find two very distinct patterns. In the tree of bird species, tree balance is relatively high across taxonomic levels, whereas for pathogens and Brassica, balance decreases steeply as taxonomic level increases (that is, as we move up the tree). Both patterns differ from the predictions of simple mathematical models. Our findings provide a basis for using mathematical modelling to test alternative hypotheses regarding the generation of biological diversity. Similar methods can be applied to analysing the evolution of tumours and other systems. References: 1. Mooers & Heard. 1997. Q. Rev. Biol., 72:31-54. 2. Blum & François. 2006. Syst. Biol., 55:685-91. 3. Lemant et al. 2022. Syst. Biol., 71:1210-24. 4. Noble & Verity. bioRxiv. 2023.07.17.549219.

When two or more species compete, a typical problem is understanding which mechanisms lead to one prevailing over the others. While it is intuitive to posit that the prevailing species has a higher overall growth rate (e.g., through a higher replication rate), noise-induced selection mechanisms have attracted increasing attention in recent years. Models showing stochastic selection are often counterintuitive, as they can result in markedly fitness differences even with identical growth rates for all species. However, they are widely relevant, given the inherent randomness in biological systems. A crucial question is whether these two classes of mechanisms (noise-induced and growth-rate-based selection) can be distinguished experimentally. Here, we compare the frequency distributions of randomly occurring neutral mutations in a spatially extended system with two species, where one species is expanding in a wave-like fashion. We find qualitative and quantitative signatures of these frequency distributions that discriminate between selection based on a replicative advantage and noise-induced selection, driven by differences in carrying capacity or in baseline turnover rates. We find that standard statistical tests are able to detect these differences with practically feasible sample sizes. We also observe marked differences in the phylogenetic trees arising in the two scenarios. Our findings are applicable to current debates in the field of evolutionary biology. We recently applied noise-induced selection to a long-standing problem in mitochondrial biology, and this class of mechanisms has been repeatedly implied in the spread of altruistic traits.

The human placenta is a highly complex organ that fulfils dual functions of providing vital nutrients to, and removing waste products from, the developing fetus. Although it has been hypothesised that anatomical variations in the maternal placental domain (such as placental septal partitioning, and the distribution and size of decidual vasculature) are associated with adverse pregnancy outcomes, the functional implications of this structural variability remain an open question. To address this knowledge gap, we employ a high-fidelity computational model of fluid flow (Navier–Stokes–Darcy with non-uniform permeability) coupled to advection-diffusion-reaction dynamics in realistic three-dimensional placental geometries. We construct a reduced-order network model by partitioning the streamlines from the 3D simulations into a dyadic graph that connects the decidual vascular inlets and outlets. This representation enables efficient interpolation and extrapolation of the 3D simulations in a specific geometry for a broad range of physiological flow and solute exchange parameters, based entirely on mechanistic principles. We complement our computational approach by an analytical two-dimensional Darcy flow model. The model allows for a more comprehensive exploration of the anatomical parameter space, using a low-fidelity characterisation of the intervillous flow. We discuss the strengths and limitations of the two approaches, as well as the implications of gross variation in materno-placental anatomy on organ-scale function.

This formation of clogs in particle suspensions can have catastrophic consequences in biological systems, such as the flow of stiffened red blood cells in conditions like sickle cell disease (SCD). In particular, clogging can result in painful vaso-occlusive crises, the leading cause of hospitalisations among SCD patients. Predicting the locations of clogs and how they propagate across complex vascular networks is therefore of significant importance. Building on recent work that investigates clogging in a single constricted channel (Herale et al. 2025), we present a two-phase continuum model consisting of a suspended particle phase and Darcy seepage flow to examine the onset of clogging in branching vascular networks with varying channel geometries. By applying appropriate conditions for mass conservation and modelling the solid flux distribution at junctions, we explore the variation in particle volume fraction across the network. We demonstrate the geometric conditions under which vascular networks experience clogging and how these blockages propagate through the network. Additionally, we show how the diversion of solid flux away from clogs can lead to intermittent clogging in other regions of the branching structure.

Sight loss is one of the most feared health conditions, and in developed countries the most common cause is age-related macular degeneration, affecting 1 in 8 people over 60 years old. Some forms can be treated by injection of drugs into the eye. We seek to test a potential alternative delivery route. Aqueous humor flows within the eye to nourish the tissues, and its outflow has two pathways: conventional and unconventional. We build on our previous mathematical model of the unconventional flow route - an anterior-posterior fluid flow in the tissues of the outer eye. We develop a mathematical model to assess the feasibility of topical delivery of ranibizumab (as our exemplar), exploiting the unconventional outflow pathway. We build on our previous mathematical model of this flow, which predicts the fluid flow, pressure, and albumin concentration within the choroidal tissue. Adding drug transport to this model allows us to predict the fraction of the drug molecules that reach the posterior choroidal tissue, with the remainder being washed out through the sclera. Our mathematical model of the eye consists of several coupled components. We assume Darcy flow in the porous choroidal tissue, Stokes flow in the potential space between the choroid and the sclera (the suprachoroidal space or SCS) and boundary conditions that couple the domains. We also include transport equations for albumin and drug concentrations, which also influence the fluid flow via osmotic effects. The results suggest that topical application of a drug could result in drug molecules reaching the macula, potentially reducing the need for injections.

In this talk, we investigate how the rapid motion of 3D microswimmers affects their emergent trajectories in shear flow. This is an active version of the classic fluid mechanics result of Jeffery's orbits for inert spheroids, first explored by George Jeffery in the 1920s. We show that the rapid short-scale motion exhibited by many microswimmers can have a significant effect on longer-scale trajectories, despite the common neglect of this motion in some mathematical models. We further demonstrate that fast-scale yawing can generate emergent asymmetry and subsequent chaos, in stark contrast to constant fast-scale rotation.

Hydrodynamic interactions are a known mechanism for the synchronization of cilia and flagella. Typically, the fluid flow communicates information between the oscillating filaments, while some form of elastic compliance (either the flexibility of the filaments or that of their anchoring points) provides the adaptability necessary for synchronization. Previous studies have shown that rigid objects rotating on fixed trajectories can also synchronize if driven by phase-dependent forcing, making them suitable for describing the power and recovery strokes of eukaryotic cilia. However, such models do not apply to bacterial flagella which rotate continuously under a phase-independent driving torque that depends instead on the rotation speed of the motor. The distinctive torque-speed relationship of the bacterial flagellar motor arises from the dynamic remodeling of the motor in response to the load exerted by hydrodynamic forces on the rotating flagellum. In this talk, we will present a recent theoretical model which demonstrates that hydrodynamic interactions combined with a load-dependent driving torque can lead to synchronization, and we compare our findings to previous models of bacterial flagellar synchronization in the literature.

Information on the effect of fluid stresses on cells is needed for progress in regenerative stem cell therapies, wherein stem cells are injected into the bloodstream with the aim of reaching a target damaged organ and subsequently repairing the organ. This is hoped to avoid organ transplants. The fluid stresses experienced by the cell on its journey are known to directly impact its biomechanical response at the target site and hence the success of the therapy. However, this is not currently well-understood. In this presentation, we introduce a new mathematical model to capture the response of a biological cell to the stresses exerted on it by a Newtonian fluid. The cell is modelled as a sac of viscoelastic fluid, obeying the Upper Convected Maxwell constitutive law. The surrounding fluid is a viscous Newtonian fluid, obeying the Stokes equations. Exploiting the larger viscosity of the cell compared with the surrounding fluid enables the cell constitutive law to be linearized. New leading-order solutions for the intracellular flow and stress distributions over time for given initial conditions can then be obtained by primarily analytical means.

Understanding tumour dynamics under hypoxic conditions is critical for optimising cancer therapies, particularly with chemotherapeutic agents like Paclitaxel. This study presents a refined mathematical model of tumour growth that incorporates Paclitaxel effects and hypoxia-driven resistance using a system of nonlinear ordinary differential equations (ODEs). We employ the Metropolis-Hastings Markov Chain Monte Carlo (MH MCMC) algorithm for Bayesian inversion and parameter estimation, providing a probabilistic framework to capture uncertainties. Sensitivity analysis is conducted using the multiple shooting method, which enhances the stability and accuracy of local sensitivity estimates across time intervals. The simulation results demonstrate that cell viability is reduced under moderate hypoxia when treated with Paclitaxel, which is consistent with experimental data from HCC1806 breast cancer cell lines. This agreement between model predictions and experimental outcomes supports the model’s validity in capturing key biological mechanisms. Future work will extend the model using Physics-Informed Neural Networks (PINNs) to improve computational efficiency and explore advanced inverse problem-solving techniques for robust cancer treatment optimisation.

While cancer has traditionally been considered a genetic disease, mounting evidence indicates an important role for non-genetic (epigenetic) mechanisms. These mechanisms usually operate faster than genetic mutations and they can drive cancer evolution and the evolution of drug resistance in the absence of any genetic events. Moreover, common anti-cancer drugs have recently been observed to induce the adoption of non-genetic drug-tolerant cell states, thereby accelerating resistance evolution. This confounds conventional high-dose treatment strategies aimed at eradicating the tumor bulk, since high doses can simultaneously promote non-genetic resistance. In this work, we study optimal dosing of anti-cancer treatment under drug-induced cell plasticity. We show that the optimal strategy steers the tumor into a fixed equilibrium composition while precisely balancing the trade-off between cell kill and tolerance induction. The optimal equilibrium strategy ranges from applying a low constant dose to alternating between the maximum allowable dose and no dose, depending on the dynamics of tolerance induction. The directionality of tolerance induction, whether the drug elevates transitions from sensitivity to tolerance or inhibits transitions back, significantly affects optimal dosing, both in the short and long term. To demonstrate the applicability of our approach, we use it to identify an optimal low-dose strategy for colorectal cancer using publicly available in vitro data.

The recent revolution in spatial proteomics and transcriptomics technologies enables transcript and protein expression levels to be mapped across entire tissue sections, for thousands of markers spanning spatial scales from the subcellular to the tissue scale. Simultaneously, advances in computational performance mean that agent-based models, mathematical models which simulate emergent population level behaviour from the interactions of individuals, can be simulated at a larger scale than ever and under increasingly vast parameter regimes. These twin scientific advances, in both biological imaging and mathematical modelling, mean that the spatial complexity of both in vivo and in silico systems can be recorded and compared as never before. Unlocking the potential of ABMs in the context of these images presents a key challenge to the mathematical modelling community: how can we compare in silico models against data quantitatively, rigorously, and at scale, while ensuring that key emergent spatial behaviours of the system are adequately captured? Standard spatially aggregated statistics, such as cell counts or densities, do not capture key variations in geometry, topology, and intercellular interactions. A wide range of methods have been proposed to describe this spatial structure, from fields as diverse as topological data analysis, spatial statistics, and networks. In this talk, we show how MuSpAn, a new Python package for spatial analysis, allows users to easily integrate key mathematical methods from these fields and beyond into a robust set of metrics to comprehensively describe spatial data. We use MuSpAn to directly compare the spatial behaviours of ABMs against experimental and clinical data, unlocking new possibilities for model parameterization and data analysis. We demonstrate the power of this approach in a range of biological contexts, including pattern formation in developmental biology, morphological changes in models of tumour-vasculature interactions, and the formation of tertiary lymphoid structures in colorectal cancer.

Modeling biological dynamic processes with Stochastic or Ordinary Differential Equations (SDEs/ODEs) is often an iterative process where multiple models are constructed and evaluated. Consequently, for efficient modeling, software that enables rapid model evaluation is important. A common approach to evaluate a model is to fit the model to time-lapse data by estimating unknown model parameters. However, parameter estimation for ODEs and SDEs is often computationally demanding. Further, for ODEs, access to the gradient is beneficial, and although many toolboxes exist for gradient-based estimation, most do not support automatic differentiation. Therefore, gradient computations partly rely on analytically derived expressions which makes it hard to handle non-standard scenarios such as scientific machine learning (SciML) models. Lastly, most toolboxes lack support for parameter estimating SDEs. To address these factors, we have developed PEtab.jl, a Julia package for formulating parameter estimation problems for dynamic models. In addition to focusing on high performance and flexibility through automatic differentiation, PEtab.jl is extensively documented and designed to be user-friendly. Here, we demonstrate how PEtab.jl can be used for ODE-based automatic model selection to study histone acetylation in Drosophila melanogaster, as well as how it can be used to fit a stochastic SIR model to study disease spread in a population. We will also describe how the package is currently being expanded to support hybrid SciML models that combine data-driven neural networks with mechanistic models. Finally, we will address the question whether Julia should be used for parameter estimation in biology by presenting results from an extensive benchmark on real models with real data, in which PEtab.jl was evaluated against pyPESTO using the C++ AMICI simulation engine.

Quantifying transcriptional bursting from live-cell imaging data is critical for understanding stochastic gene regulation. We introduce an AI–based framework for inferring transcription bursting periods from fluorescence trajectories, achieving improved accuracy and robustness compared to widely used approaches such as hidden Markov models (HMMs) and threshold-based methods. Trained on synthetic data and validated against experimental data, our method shows robustness under experimental noise, reliably capturing complex transcriptional dynamics. By integrating AI with stochastic modeling, this approach provides a flexible and computationally efficient tool for the quantitative analysis of gene expression dynamics in single cells.

In this work, a compartmental mathematical model incorporating vital dynamics to analyze the spread and control of Hepatitis B Virus (HBV) is developed. The model consists of susceptible, exposed, infected, and recovered compartments, incorporating key epidemiological factors such as vertical transmission, disease progression, and recovery rates. The basic reproduction number, , which serves as a threshold parameter for disease persistence or eradication is derived. Stability analysis of the disease-free and endemic equilibrium points is conducted to assess the long-term behavior of the infection. Numerical simulations are performed to illustrate the impact of vaccination, treatment, and other control measures. The results provide insights into the effectiveness of various public health interventions and offer guidance for policymakers in designing strategies to mitigate the spread of HBV. Keywords: Hepatitis B, mathematical modeling, compartmental model, reproduction number, stability analysis, epidemiology

Tristetraprolin (TTP) is a messenger RNA (mRNA) binding protein that binds to pro-inflammatory mRNA and recruits protein complexes that promote mRNA degradation. As such, TTP is a key regulatory of the resolution of inflammation. TTP phosphorylation and activity is regulated by the mitogen-activated protein kinase (MAPK) p38 pathway, which is a cascade of phosphorylation events that promote thes expression of pro-inflammatory mRNAs. This pathway is complex with multiple protein interactions and a feedback loops, making it difficult to understand intuitively. To investigate this, a mathematical model of the pathway was developed using ordinary differential equations. The model can be used to make experimental predictions and, where biological results and mathematical results contradict, deepen our understanding of the pathway. In this way, the mathematical model informs experimental design and the experimental results refine the mathematical model to help identify new methods for the treatment of inflammatory disease.

Adult neurogenesis is the process through which mature neurons are generated from neural stem cells in the adult brain. A comprehensive understanding of the mechanisms regulating this process is essential for developing effective interventions aimed at decelerating the decline of adult neurogenesis associated with ageing. Mechanistic models provide a valuable tool for studying the dynamics of neural stem cells and their lineage, and have revealed alterations in these mechanisms during the ageing process. In my presentation I will describe how these processes are modulated, by investigating regulatory feedbacks among neural populations through the lens of nonlinear differential equations models. I will discuss our observations regarding the manner in which different neural populations govern the time-evolution of the nerual lineage, and the impact that specific perturbations have on the system. Finally, I will draw attention on the perils of inference from preliminary models based on insufficiently informative data, without an extensive and unbiased investigation.

Aggregation-diffusion equations are foundational tools for modelling biological aggregations. Their principal use is to link the collective movement mechanisms of organisms to their emergent space use patterns in a concrete mathematical way. However, most existing studies do not account for the effect of the underlying environment on organism movement. In reality, the environment is often a key determinant of emergent space use patterns, albeit in combination with collective aspects of motion. Here, I will present work towards this end, studying aggregation-diffusion equations in a heterogeneous environment in one spatial dimension. Under certain assumptions, it is possible to find exact analytic expressions for the steady-state solutions when diffusion is quadratic. Minimising the associated energy functional across these solutions provides a rapid way of determining the likely emergent space use pattern, which can be verified via numerical simulations. This energy-minimisation procedure is applied to a simple test case, where the environment consists of a single clump of attractive resources. Here, self-attraction and resource-attraction combine to shape the emergent aggregation. Two counter-intuitive findings emerge from these analytic results: (a) a non-monotonic dependence of clump width on the aggregation width, (b) a positive correlation between self-attraction strength and aggregation width when the resource attraction is strong.

Differential operators of integer order are known to capture local and isotropic effects, both in space and time. However, especially in biology, the increasing need to model complex phenomena with underlying properties such as spatial heterogeneity requires new modelling tools. The fractional calculus framework enables the development of more sophisticated models that capture the complex dynamics inherent to various biological systems. This talk will focus on how fractional reaction-diffusion equations naturally arise and can be derived from continuous-time random walks, highlighting the role of heavy-tailed distributions in the process. Both fractional partial differential equations, on the macroscopic level, as well as fractional stochastic differential equations, on the microscopic level, will be examined. For simple Riesz-fractional diffusion models, we will showcase comparative simulations, highlighting the key differences between fractional and classical diffusion. We propose a new numerical scheme that implements periodic boundary conditions, to control the loss of mass density.

Observing the decisions and actions of others provides social information that can inform your actions such as whether to follow. We consider a model where all agents simultaneously gather stochastic private information (weighted towards an unknown preference), coming to a decision once sufficiently confident. However, decisions (and indecisions) by agents are observed by all other agents and provide social information. In particular, following a decision by one agent, other agents incorporate this social information with their private information and may follow this decision if the agent becomes sufficiently confident; forming a wave of decisions. This wave (or lack thereof) provides further social information about the private information of other agents, leading to potentially more waves in response to the (in)decision. We explore small groups of agents to see if all this social information leads to quicker and/or better decisions as well as the number of waves of decisions. We also explore whether agents remain undecided after these waves and what happens to these undecided agents.