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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
Abstract
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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