# Mathematical Biology

## Alan Turing and the Patterns of Life

In 1952, Turing published his only paper spanning chemistry and biology: "The chemical basis of morphogenesis". In it, he proposed a hypothetical mechanism for the emergence of complex patterns in chemical reactions, called reaction-diffusion. He also predicted the use of computational models as a tool for understanding patterning. Sixty years later, reaction-diffusion is a key concept in the study of patterns and forms in nature. In particular, it provides a link between molecular genetics and developmental biology. The presentation will review the concept of reaction-diffusion, the tumultuous path towards its acceptance, and its current place in biology.

## 2012 IGTC Summit: Prof. Steve Krone (Part II)

Spontaneous pattern formation in spatial populations with cyclic dynamics

There are many examples in nature where a system goes through a succession of states that are cyclically related. Examples include ecological succession in a forest and SIRS models of epidemics. When such populations are spatially arranged (as are *all* populations to some degree), these cyclic dynamics can sometimes lead to the spontaneous formation of spatial patterns such as spiral waves. We will explore this phenomenon via interacting particle system models and related differential equations.

## 2012 IGTC Summit: Prof. Steve Krone (Part I)

Individual-based stochastic spatial models and population biology

These talks will provide an introduction to individual-based stochastic spatial models (sometimes called interacting particle systems or stochastic cellular automata). We will proceed from very simple basic models to more elaborate ones, illustrating the ideas with examples of spatial biological population dynamics. We will compare these models and results with analogous differential equations (ODE and PDE) and see how they are connected. Biological topics will include spatial population growth and spread, epidemics, evolution of pathogens, and antibiotic resistance plasmids. Throughout, we will point out situations in which spatial structure can dramatically influence the ecology and evolution of populations.

## IGTC 2012 Math Bio Summit

2012 IGTC Summit in Naramata

Lecture notes from Prof. Krone are available at http://www.mathtube.org/date/lecture_notes/2012.

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## Mathematical Cell Biology Summer Course Student Lecture 11

Cell-cell signaling between macrophages and mammary tumor cells

## Mechanical Simulations of Cell Motility

Here I survey a broad range of recent computational models for 2D and 3D cell motility. Some of these models depict chemical activation on the perimeter of a (static or deforming) domain. Others consider fluid and/or mechanical elements and/or biochemical signalling on the interior of a deforming 2D region

representing a cell. Examples of platforms include the immersed-boundary method and level set methods. I describe some of the computational challenges and how these have been addressed by various researchers.

## Polymer Size Distributions (continued)

We continue the discussion from last time, and solve the polymer size distribution equations, which are linear in the case of constant monomer level.

In a distinct case, when monomer is depleted, we show that the size distribution evolves in two phases, where in the first, the entire distribution appears to satisfy a transport equation, and then, later on, once monomer is at its critical level, the process of length adjustment appears to be governed by

an effective diffusion (in size-class). Next, I introduce the problem of determining features of polymer assembly from experimental

polymerization versus time data. (Based on work by Flybjerg et al, this leads to an extended homework exercise carried out by the students.) Finally, I revisit microtubule growth and shrinking by discussing the Dogterom-Leibler equations and their steady state exponential solutions. I illustrate the use of XPP software to solve several problems in this lecture.

## Mathematical Cell Biology Summer Course Student Lecture 10

Diffusion of receptors on a cell membrane

## Mathematical Cell Biology Summer Course Student Lecture 9

Size regulation in cell organelles

## Mathematical Cell Biology Summer Course Student Lecture 7

Transport of early endosomes on microtubules