July 30-August 2, 2008
Society for Mathematical Biology Conference

hosted by the Centre for Mathematical Medicine, Fields Institute
held at University of Toronto, Medical Sciences Bldg


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2) Pattern formation on continuously deforming domains: I and II (2 minisymposia)
Principal organiser: Dr. Anotida Madzvamuse, University of Sussex, UK

Many problems in biology and bio-medicine involve growth and shape changes. The growth and spread of tumors, the patterns formed on the skin of growing creatures, and the initiation of limb buds in the developing foetus are just some of the many examples. Mathematical modelling in patter formation, wound healing, cancer and angiogenesis gives rise to highly complicated nonlinear partial differential equations (e.g. reaction-diffusion systems, chemotaxis models etc).

Recent advances in scientific computing offer a unique opportunity to extend mathematical models derived on fixed domains to continuously changing environments. This minisymposium seeks to offer a platform for scientists to present innovative research in the area of mathematical models for pattern formation on continuously deforming domains and evolving surfaces. The aim is to bring together experimentalists and modellers to share scientific knowledge in recent advances in pattern selection and transition from an experimental point of view and for modellers to present methods that could possibly elucidate the mechanisms that could give rise to such patterns.

Patterns are observed to bifurcate as the domain or surface continues to evolve. The development of bifurcation theory on continuously deforming domains is still an unresolved issue: both from a mathematical and a developmental biology point of view. The question of the role of domain growth in biological systems is still an open question. Tumour growth in cancer biology and imaging are just two examples of other themes where domain growth plays a pivotal role.

Speakers and Title: Session I

1. Cheng Ming Chuong, University of Southern California
Changes of integument patterns on continuously deforming domains: a biological perspective.

Integument patterns are most striking because of their visibility. These patterns include the arrangement of feathers, hairs, or the number, spacing and of pigment dots or stripes. Adding on top of the complexity is that the field, i.e., the region where these patterns form, grow in size during development. This growth can be based on homogeneous expansion, uneven addition or localized growth. The pattern may be set up a particular time followed by growth in deforming domain; or the patterning elements may be continuously added and adjusted. We will discuss these possibilities in biological examples (chicken feather, mouse hair, zebra stripes, leopard spots, etc.) and evaluate the possible solution.

2. Shigeru Kondo, Department of Biological Science, Nagoya University
Zebras did not get the stripes, but lost the uniform color.

There are two major questions about the stripes of zebras. The first is the mechanism that generates the pattern in the skin. The second is the process of evolution that zebras got the conspicuous pattern. The answer to the first question has turn out to be "Turing pattern”. But how could such complex mechanism have evolved to give rise to a phenotype that seems to have no advantage for survival? We have recently identified the cellular interaction network among the pigment cells of zebrafish, and found that the network satisfies the necessary conditions of Turing pattern formation. Based on the identified network, we present a hypothesis that explains how the animal stripes evolved.

3. Anotida Madzvamuse, University of Sussex, Department of Mathematics
Development of bifurcation theory for reaction-diffusion systems on continuously deforming domains.

In this talk, I will present a detailed linear stability theory for reaction-diffusion systems (RDS) with constant coefficients on continuously deforming domains. By using the arbitrary Lagrangian-Eulerian formulation (ALE), the model equations on a continuously deforming domain are transformed to a fixed domain at each time, resulting in a conservative system. First, I prove that if the domain velocity is divergence free, then the linearised system of RDS reduces to one obtained for the RDS on a fixed domain. Secondly, I derive and show that the diffusion-driven instability conditions for an exponentially growing domain depend
on the domain growth rate. More important, the parameter space is a direct shift of the Turing space obtained on a fixed domain in the absence of domain growth. Alternatively, by looking at the eigenvalues, I show that the shifting of the Turing space is equivalent to the standard Turing space of the RDS on a fixed domain but with eigenvalues shifted to the left of the complex plane by a constant factor given by the divergence of the domain velocity.

Speakers and Title: Session II

1. Paul Kulesa, The Stowers Institute
Investigating Early Brain Formation in the Developing Embryo

The neural crest are an excellent model to study embryonic pattern formation and cancer since these cells arise along nearly the entire vertebrate axis and invade in a programmed manner of discrete migratory streams. How neural crest cells navigate through growing tissue microenvironments they encounter has typically been considered as pattern formation on a fixed domain. Here, we test our hypothesis that neural crest cell migration and proliferation are regulated within a dynamic migratory niche to compensate for growth of the migratory route, leading to proper target invasion. We measure differences in cell proliferation, growth of the migratory route, and cell-microenvironment interactions using a combination of molecular biology, photoactivation, embryo microsurgery, and high resolution imaging in the chick embryo. Our results suggest a model in which cell proliferation at the migratory front drives expansion of the niche into target sites, the invasion of which is regulated by specific receptor-ligand interactions.

2. Adela Comanici, Virginia Tech, Department of Mathematics
Patterns on Growing Square Domains via Mode Interactions.

In this talk, I will consider reaction-diffusion systems on growing square domains with Neumann boundary conditions (NBC). As suggested by numerical simulations, I will study a relevant mode interactions in steady-state bifurcation problems with both translational symmetry and square symmetry, combined with the symmetry constraint imposed by NBC. I will show that the transition between different types of squares can be generically continuous. Also, I will show that transitions between squares and stripes can occur generically via jump, or via steady-states and time-periodic states. I will point out the differences between the transitions from squares to stripes in the NBC problem and in the periodic boundary conditions problem.

3. Allen Sanderson, SCI Institute, University of Utah.
Rapidly Generating and Controlling Reaction-Diffusion Pattern Formation for use in Computer Visualization and Computer Graphics Applications.

In the course of numerically analyzing Reaction-Diffusion systems researchers often rely upon a one dimensional model. This analysis provides many clues as to the type of the pattern that will be formed. However, to fully understand the pattern formed it is typically necessary to generate it. This is especially true when the domain is expanded to two or three dimensions. We present a toolkit to explore and rapidly generate two and three dimensional patterns using GPU hardware acceleration. The hardware acceleration allows for interactive viewing the pattern formation process thus providing for more opportunity to explore affect
2 of different conditions within a Reaction-Diffusion system that are not easily predicted via the numerical analysis.

4. Ruth Baker, Centre for Mathematical Biology, Mathematical Institute, University of Oxford
Mathematical modelling of morphogen-controlled domain growth

Growth is a fundamental aspect of development: it results from the tight regulation of a combination of processes including cell differentiation, division and movement. Many of these phenomena have been shown to be controlled by gradients of chemical factors, known as morphogens, which regulate cell behaviour using threshold-specific responses. Here I will describe a mechanism via which growth can be regulated by morphogen concentration. Focussing on a paradigm developmental system, I will outline the biological detail, the mathematical model and present analytical and numerical results from our investigation.

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