IGTC Graduate Summer School in Mathematical Biology

The summer school will consist of a one month long graduate course, given by Leah Edelstein-Keshet, with three guest instructors: Profs. Karl Hadeler, Bard Ermentrout, and Alex Mogilner.

Descriptions of the course can be found below. The course will include lectures in the morning and tutorials/computer labs in the afternoons. Course credit will be available for graduate students in Western Canada through the Western Deans' Protocol.

Alex Mogilner

In the last decade, cell biology has been undergoing revolutionary change, in no small part due to rapid accumulation of quantitative data and development of mathematical and computational models that helped to make sense of these data. As a result, mechanistic understanding of molecular processes in the cell increased dramatically. In the course of five lectures, we will use recent research papers to examine a few cell biological phenomena (i.e. biochemical signaling, cell cycle, cell division, intracellular transport) to understand what was the logic of the controlled experiments, how the data was used to formulate and test the models, and how the models helped to make sense of the data. No prior biological knowledge will be needed. Basic applied mathematics tools will be used (ODEs, PDEs, elementary probability). The lectures will be followed by three problem solving sessions, during which we will work on an open modeling problem. Numerical simulations involved will be done using Matlab.

Leah Edelstein-Keshet

The IGTC Summer School in Mathematical Biology will consist of a set of introductory survey lectures, and a parallel set of three more advanced mini-courses that expose the students to advanced topics in mathematical biology research.

Cell biology offers a rich area where concepts and methods of applied mathematics can come into play. I plan to make this one central core of my own set of lectures in this course.

In my survey, I will first discuss chemical networks with specific dynamic properties (modeled by ODEs), focusing on recent applications to biochemical signalling in cells. I will review properties of several reaction-diffusion systems, illustrating both pattern and wave-based phenomena, with applications to the problem of gradient sensing and directional selection in a polarizing cell. Students will be exposed to papers in the literature that are interesting both from a mathematical and a biological perspective, where classical ideas of mathematical biology intersect with new biological applications. Student-led presentations of important papers, and hands-on experimentation with analysis and simulations is planned. This will form an introduction to the more advanced cutting-edge lectures by Alex Mogilner (see description above).

In a second part of my lectures, I will introduce a number of additional topics. These will include (1) models for interacting individuals (PDE's, systems of ODE's, and non-local models), (2) recent models for biomedical problems such as diabetes and Alzheimer's disease. Students will gain experience with stage-structured models, integro-PDE's, and both Lagrangian and Eulerian models for swarms and schools.

Additional guest lecturers invited to give occasional seminars will expose the students to other exciting current research in mathematical biology.

K.P. Hadeler

Mathematical approaches to biological modeling

Bacteria perform peculiar movements and control size and shape of their colonies by "quorum sensing". They "want" to optimize their environment, e.g. by "chemotaxis", and they "want" to spread. What are the mechanisms? Are movements and growth patterns arbitrary or governed by laws for the individual and the population? Can we cast these laws into equations with few parameters which can be identified in experiments? Similar questions can be posed for animal and plant species and for the spread of infectious diseases (West Nile virus is a recent example).

A suitable framework are reaction diffusion equations which exhibit biologically and mathematically challenging phenomena: patterns, localization (spikes), blow up, fronts and pulses. These occur also in refined models like damped wave equations, reaction transport equations, and Langevin equations. We demonstrate phenomena and establish connections between levels of complexity, e.g. by diffusion approximation. We ask whether a species is stable within an ecosystem (persistence) and how a species like the zebra mussel can invade (it took 150 years to get from England to North America).

We ask whether the dynamics of a system is determined by its components, when all components are similar (lattice dynamics) and when components represent quiescent or dormant phases. Examples are microbes (spores), cells ("most cells in the body are quiescent"), plants (seed banks) and mammals (hibernation).

Rather than modeling the size of fishes or rodent colonies by size classes one can use first order partial differential equations which allow dependence on individual growth rates and total population size, but mostly not stochastic variation. We incorporate stochastic behavior by diffusion (mathematically: viscosity approach to conservation laws).

Delays often represent unknown transport or reaction processes. Relations between delay differential equations and structured population models have been known since long, but their true nature became clear only recently. They help us to understand delay equations as models in their own right and to justify neutral delay equations.

Modeling infectious diseases is a most important application: vaccination, education and quarantine policies, effect of social structure, core groups and small reservoirs, demographic impact, and spread in space, as well as parameter identification from field data.

G. Bard Ermentrout

Oscillations in biology

In this set of lectures I will discuss the modeling and analysis of rhythmic phenomena in biology. I will start with models of oscillations including ODE models, PDE models and delay equations, laying out the necessary conditions that are required for oscillations. I will discuss the underlying geometry of mechanisms from which a system goes from a stable fixed point to a periodic orbit. I will then switch to questions concerning the interaction of intrinsic oscillators, that is, what kinds of behavior can we expect when oscillators are coupled. My main approach will be the reduction to "phase models" in which each oscillator is reduced to a one-dimensional variable on a circle. The interactions between the oscillators as well as the topology of the connections determines the stable patterns which are possible. I will show how both the intrinsic properties as well as the time scales of interactions combine to determine the stability of patterns in networks. In addition to synchronization behavior, I will provide conditions for various types of wave-like behavior including target waves, spiral waves, and complex dynamics. Applications to neural and cellular phenomena will be provided as well as many computer exercises in XPPAUT or if the students want to write their own code, Matlab.