Mathematical Biology

For over a century, scientists have studied striking spatiotemporal patterns during the continual tooth replacement of reptiles. Aside from the compelling aesthetics of this phenomenon, it is thought that understanding the underlying mechanisms may provide the insight required to trigger adult tooth replacement in humans. Theoretical frameworks have long been proposed to understand the rules behind the observed spatiotemporal order, but have only been analyzed mathematically more recently. Starting from Edmund's observations in crocodiles and proposed theory of replacement waves, we show how a simple model consisting of a row of non-interacting phase oscillators predicts several experimental observations. Next, inspired by the hypothesis put forth by Osborn, we consider a variation of the phase model with ODEs that account for mutual inhibition between tooth sites, and use continuation methods to thoroughly search parameter space for experimentally validated solutions. We then extend the model to a PDE that explicitly accounts for the diffusion of inhibitory signals between teeth, yielding some novel solution types. Using continuation methods once again, we delineate parameter regimes with solutions that closely resemble experimental observations in leopard geckos.

The questions of how healthy colonic crypts maintain their size under the rapid cell turnover in intestinal epithelium, and how homeostasis is disrupted by driver mutations, are central to understanding colorectal tumorigenesis. We propose a three-type stochastic branching process, which accounts for stem, transit-amplifying (TA) and fully differentiated (FD) cells, to model the dynamics of cell populations residing in colonic crypts. Our model is simple in its formulation, allowing us to estimate all but one of the model parameters from the literature. Fitting the single remaining parameter, we find that model results agree well with data from healthy human colonic crypts, capturing the considerable variance in population sizes observed experimentally. Importantly, our model predicts a steady-state population in healthy colonic crypts for relevant parameter values. We show that APC and KRAS mutations, the most significant early alterations leading to colorectal cancer, result in increased steady-state populations in mutated crypts, in agreement with experimental results. Finally, our model predicts a simple condition for unbounded growth of cells in a crypt, corresponding to colorectal malignancy. This is predicted to occur when the division rate of TA cells exceeds their differentiation rate, with implications for therapeutic cancer prevention strategies.

While mathematical models have classically been used in the study of physics and engineering, recently, they have become important tools in other fields such as biology, ecology, and sociology. In this talk I will discuss the use of partial differential equations and dynamical systems to shed light onto social and ecological phenomena. In the first part of this talk, we will focus on an Ecological application. For an efficient wildlife management plan, it is important that we understand (1) why animals move as they do and (2) what movement strategies are robust. I will discuss how reaction-advection-diffusion models can help us shed light into these two issues. The second part of the talk will focus on social applications. I will present a few models in the study of gentrification, urban crime, and protesting activity and discuss how theoretical and numerical analysis have provided intuition into these different social phenomena. Moreover, I will also point out the many benefits of utilizing a mathematical framework when data is not available.

One of the central questions in developmental biology is how organs form in the correct positions in order to create a functional mature organism. Plant leaves offer an easily observable example of organ positioning, with species-specific motifs for leaf arrangement (phyllotaxis). These patterns arise through a combination of chemical pattern formation, mechanical stresses and growth. Mathematical modelling in each of these areas (and their combinations) contributes to quantitative understanding of developmental mechanisms and morphogenesis in general. Conifer trees are some of the most characteristic plants of BC. They also display a type of ring patterning of their embryonic leaves (cotyledons), which I believe offers a unique route to understanding plant phyllotaxis in general. I will discuss how early work at UBC on similar patterning in algae led to application of reaction-diffusion models in conifer development. This framework has guided experiments at BCIT and recently led to a model that accounts for the natural variability in conifer cotyledon number. The model involves the kinetics of a highly conserved gene regulation module and therefore sheds light on the chemical pattern formation control of phyllotaxis across plants. Conifer patterning also demonstrates scaling of position to organism size, an active area of research in animal development: the model provides some mechanistic insight into how this can occur via chemical kinetics.

We analyze the existence, linear stability, and slow dynamics of localized 1D spike patterns for a Keller-Segel model of chemotaxis that includes the effect of logistic growth of the cellular population. Our analysis of localized patterns for this two-component reaction-diffusion (RD) model is based, not on the usual limit of a large chemotactic drift coefficient, but instead on the singular limit of an asymptotically small diffusivity of the chemoattractant concentration field. In the limit, steady-state and quasi-equilibrium 1D multi-spike patterns are constructed asymptotically. To determine the linear stability of steady-state N-spike patterns, we analyze the spectral properties associated with both the “large” O(1) and the “small” o(1) eigenvalues associated with the linearization of the Keller-Segel model. By analyzing a nonlocal eigenvalue problem characterizing the large eigenvalues, it is shown that N-spike equilibria can be destabilized by a zero-eigenvalue crossing leading to a competition instability if the cellular diffusion rate exceeds a threshold, or from a Hopf bifurcation if a relaxation time constant is too large. In addition, a matrix eigenvalue problem that governs the stability properties of an N-spike steady-state with respect to the small eigenvalues is derived. From an analysis of this matrix problem, an explicit range of cellular diffusion rate where the N-spike steady-state is stable to the small eigenvalues is identified. Finally, for quasi-equilibrium spike patterns that are stable on an O(1) time-scale, we derive a differential algebraic system (DAE) governing the slow dynamics of a collection of localized spikes. Unexpectedly, our analysis of the KS model with logistic growth in the small chemical diffusion rate regime is rather closely related to the analysis of spike patterns for the Gierer-Meinhardt RD system.
This paper is a joint work with Professor Michael J. Ward and Juncheng Wei.