Kevin Burrage

Mathematical modelling is essential for understanding how complex biological systems respond to genetic, physiological, and environmental changes. Existing approaches, however, often require trade-offs between mechanistic detail, model size, parameter uncertainty, and interpretability. Ordinary differential equation (ODE) models capture biochemical processes with quantitative precision but can demand extensive parameterisation. In contrast, large statistical and machine-learning models rely on substantial datasets and frequently lack mechanistic transparency. Qualitative approaches such as Boolean networks improve scalability but may oversimplify biological behaviour. To address some of these limitations, we present PSoup, an R package that automatically converts knowledge graphs into transparent, parameter-free, qualitative models. PSoup uses algebraic update rules designed around a fixed, biologically interpretable baseline, enabling predictions of relative change across diverse perturbations without requiring kinetic parameters. This design allows PSoup to integrate information across biological scales and from heterogeneous experimental sources. We evaluated PSoup using the well-studied shoot branching network of Bertheloot et al. (2019), which incorporates hormonal (auxin, strigolactone, cytokinin) and metabolic (sucrose) regulation. Across 78 experimental conditions, PSoup correctly predicted 88.5%of perturbation outcomes, including 89.5%accuracy for unique, biologically consistent comparisons. We further demonstrate how PSoup can distinguish among alternative plausible network topologies, revealing how structural differences influence emergent system behaviour. PSoup offers an intuitive, accessible, and mathematically transparent framework for exploring biological networks. Its capacity to integrate diverse knowledge and test alternative hypotheses positions it as a powerful tool for biological discovery and a valuable complement to existing modelling approaches.

Despite its central role in development, cell homeostasis, and cancer, the mechanisms that govern canonical Wnt signaling, particularly how receptor internalization and membrane organization shape signalosome formation and downstream pathway activation, remain a topic of ongoing debate. To resolve this longstanding controversy, we developed a mechanistic model that integrates seemingly contradictory experimental observations into a coherent explanatory framework. Using an optimized multi-level rule-based modeling approach the model explicitly captures lateral membrane organization, receptor complex dynamics, endocytic routing and downstream signaling. Simulations reveal a previously unrecognized compensatory mechanism in which rapid recycling of partially immobilized receptor assemblies amplifies and stabilizes signalosome formation. The results further indicate that internalization and recycling are essential for stable long-term β -catenin activation, offering an explanation why these processes are required for excessive signaling in cancer cells. The model further predicts that ordered membrane domains enhance signaling not merely by colocalizing receptors and DVL but by increasing effective residence time through rebinding-driven cluster stabilization. Integrating diverse experimental measurements into a unified simulation framework quantitatively reconciles conflicting experimental reports on the necessity of receptor internalization for pathway activation. By resolving these debates and providing falsifiable predictions for how trafficking and membrane organization jointly tune pathway output, the study offers a mechanistic foundation for targeted modulation of Wnt signaling in developmental and disease contexts.

This paper introduces Magnus-based methods for solving stochastic delay-differential equations (SDDEs). We construct Magnus--Euler--Maruyama (MEM) and Magnus--Milstein (MM) schemes by combining stochastic Magnus integrators with Taylor methods for SDDEs. These schemes are applied incrementally between multiples of the delay times. We present proofs of their convergence orders and demonstrate these rates through numerical examples and error graphs. Among the examples, we apply the MEM and MM schemes to both linear and nonlinear problems. We also apply the MEM scheme to a stochastic partial delay-differential equation (SPDDE), comparing its performance with the traditional Euler--Maruyama (EM) method. Under fine spatial discretization, the MEM scheme remains numerically stable while the EM method becomes unstable, yielding a significant computational advantage.