In this talk, I motivate a bulk-surface reaction-diffusion model that captures the coupling between cytoplasmic and membrane-bound GTPase dynamics, and connect it to an evolution law that deforms the domain in response to those dynamics. By integrating the biochemical processes governing GTPase activity with the physical deformation of the cell boundary, this framework provides a unified approach to studying how intracellular signaling drives cellular shape changes and motility.

Using numerical simulations, we analyze how diffusion influences an ODE system exhibiting different dynamical behaviours; including stable steady state dynamics, bistable dynamics, limit cycle dynamics, and coexistence between uniform steady states and limit cycles.

In this talk, I will present a two-species reaction-diffusion model whose reaction kinetics are derived from first principles based on experimental observations. I will describe the model formulation and highlight key temporal dynamics of the model in the absence of diffusion. I will thereafter describe diffusion-driven transitions, such as spatial pattern formation and migration between steady states and migration of limit cycle in parameter regions far from classical Turing scenarios.

If the domain is sufficiently thin, the pattern consists of stripes which are nearly one-dimensional. We analyse patterns consisting of one, two or many stripes. We find that a single stripe can be located either at the thickest or thinnest part of the channel, depending on the choice of parameters. In the limit of many stripes, we derive an effective pattern density description of the equilibrium state. The effective density is easily computable as a solution of a first order ODE subject to an integral constraint. Depending on problem parameters, the resulting pattern can be either global spanning the entire domain, or can be clustered near either thickest or thinnest part of the domain. In addition, instability thresholds are derived from the continuum density limit of many stripes. Full two-dimensional numerical simulations are performed and are shown to be in agreement with the asymptotic results. Results are shown to be applicable to

We present a robust extension of the closest point method (CPM) for handling interior boundary conditions. The CPM reformulates the manifold PDE as a volumetric PDE in the Cartesian embedding space, requiring only the closest point representation of the manifold. This approach inherently supports open or closed manifolds, orientable or not, and of any codimension. To address interior boundary conditions, we derive a technique that implicitly partitions the embedding space across interior boundaries, modifying finite difference and interpolation stencils to respect these partitions while preserving second-order accuracy.

Our method includes an efficient sparse-grid implementation and scalable numerical solver capable of handling tens of millions of degrees of freedom, enabling solutions on complex manifolds. We demonstrate the convergence and accuracy of our approach using model PDEs and showcase applications to a range of geometry processing problems.

This is joint work with Nathan King (University of Waterloo), Haozhe Su (Lightspeed Studios), Mridul Aanjaneya (Rutgers University), and Christopher Batty (University of Waterloo).

In this talk, we analyze experimental data to investigate Mycobacterium smegmatis morphology over time, and model pattern dynamics using a reaction-diffusion system. This non pathogenic and fast growing bacterium is commonly studied as a model for harmful mycobacteria such as M. tuberculosis, since they share a similar cell wall structure. Upon using our pipeline to reduce the cell surface geometry to its center-line and measure height variation along it, we confirm the presence of peaks and troughs on the cell surface, as consistent features of the morphology. We also show how these features relate to bi-phasic and asymmetric polar growth dynamics, as well as division site selection. Finally, we show that a minimal reaction-diffusion model on a growing domain can reproduce and maintain similar feature over time, enabling a better understanding of yet unknown morphology controlling pathways.