


Mathematical Biology / Biologie mathématique Org: Leah Keshet (UBC)
 FRED BRAUER, University of British Columbia, Department of Mathematics,
Vancouver, British Columbia V6T 1Z2
A discrete model for SARS transmission

We formulate and analyze a discrete model for the transmission of
Severe Acute Respiratory Syndrome (SARS), estimating parameters to fit
the data obtained on the spread of SARS in China. Simulations using
this model and these data indicate that early quarantine and a high
quarantine rate are crucial for the control of SARS.
(joint work with Zhou Yicang and Ma Zhien, Xi'an Jiaotong University)
 DANIEL COOMBS, UBC Mathematics, Department of Mathematics,
Vancouver, British Columbia V6T 1Z2
Equilibrium behaviour of cellcell synapses

In many situations, cellcell adhesion is mediated by multiple
ligandreceptor pairs. For example, the interaction between T cells
and antigenpresenting cells of the immune system is mediated not only
by T cell receptors and their ligands (peptidemajor
histocompatibility complex) but also by binding of intracellular
adhesion molecules. Interestingly, these binding pairs have different
resting lengths. Fluorescent labelling reveals segregation of the
longer adhesion molecules from the shorter T cell receptors in this
case. We explore the thermal equilibrium of a general cellcell
interaction mediated by two ligandreceptor pairs to examine
competition between the elasticity of the cell wall, nonspecific
intercellular repulsion and bond formation, leading to segregation at
equilibrium. We make detailed predictions concerning the relationship
between physical properties of the membrane and ligandreceptor pairs
and equilibrium pattern formation and suggest experiments to refine our
understanding of the system. We demonstrate our model by application to
the T cellantigenpresentingcell system and natural killer
celltarget cell adhesion. Our results underline the importance of
active, energyconsuming processes in this system.
 GERDA DE VRIES, Univeristy of Alberta, Department of Mathematics and Statistical
Sciences, Edmonton, Alberta T6G 2G1
Using mathematical models to deduce the spatiotemporal
dynamics of nuclear proteins from experimental fluorescence
recovery curves

Fluorescence Recovery After Photobleaching (FRAP) is an imaging
technique used to study the mobility of proteins in the cell nucleus.
In FRAP experiments, the protein being studied is tagged with Green
Fluorescent Protein (GFP). An intense laser beam is used to bleach the
fluorophore of the tagged proteins within a small region of the cell
nucleus. Due to diffusional exchange between the bleached and
unbleached proteins, fluorescence in the targeted area recovers. The
fluorescence recovery data can be used to quantify the mobility of the
proteins.
In this talk, we characterize the behaviour of the fluorescence
recovery curves for diffusing nuclear proteins undergoing binding
events with an approximate spatially homogeneous structure. We discuss
two mathematical models to interpret the data, namely a
reactiondiffusion model and a compartmental model. Perturbation
analysis leads to a clear explanation of two important limiting types
of behaviour exhibited by experimental recovery curves, namely 1) a
reduced diffusive recovery, and 2) a biphasic recovery characterized
by a fast phase and a slow phase. The results can be used to simplify
the task of parameter estimation. Application of the results is
demonstrated for nuclear actin and type H1 histone.
 RODERICK EDWARDS, University of Victoria, Mathematics and Statistics, Victoria,
British Columbia V8W 3P4
Central pattern generators for digging behaviour in sandcrabs

Electromyographic studies of sandcrabs during digging behaviour have
shown interesting phase changes between right and left limbs in
relation to the uropods (tail appendages). Initially both hindmost
limbs are antiphase with the uropods, but typically, one advances and
the other lags as the dig progresses, but never so far as to come in
phase with the uropods. Based on the better known neuroanatomy of the
closelyrelated crayfish, we propose models of the central pattern
generators coordinating each appendage, first via MorrisLecar
equations for individual neurons, and then by reduction to phase
equations. We show that the pattern of coupling produces the observed
behaviour, for any specific structure of the central pattern generators
as long as they are oscillators with an appropriate phase response
curve. (joint work with Alex Hodge, Pauline van den Driessche, Dorothy
Paul)
 MAREN FRIESEN, University of California Davis, USA
Sympatric ecological diversification due to frequencydependent
competition in Escherichia coli

We develop a model of the adaptive dynamics of diauxic growth to
describe adaptive diversification in experimental Esherichia coli
populations. The model is a modification of MichaelisMenton kinetics
on two resources that allows for sequential resource use, since E. coli
are known to preferentially use glucose. Twelve experimental lines
were started from the same genetically uniform ancestral strain and
grown in serial batch cultures on a mixture of glucose and acetate.
After 1000 generations, all populations were polymorphic for colony
size. Five populations were clearly dimorphic and thus serve as a
model for an adaptive lineage split. We analyzed the ecological basis
for this dimorphism by studying bacterial growth curves and found that
the two colony sizes differed in the pattern of diauxic growth. Using
invasion experiments, we show that the dimorphism of these two
ecological types is maintained by frequency dependent selection. Our
results support the hypothesis that in our experiments, adaptive
diversification from a genetically uniform ancestor occurred due to
frequency dependent ecological interactions. Our results have
implications for understanding the evolution of crossfeeding
polymorphism in microorganisms, as well as adaptive speciation due to
frequencydependent selection on phenotypic plasticity.
 DANIEL GRUNBAUM, University of Washington, School of Oceanography, Seattle,
Washington 981957940, USA
Extracting social behavior rules from group dynamics of
schooling fish

Social animal behaviors such as schooling, flocking and herding are
remarkable for how effectively such groups perform coordinated tasks
(predator detection and avoidance, food acquisition, migration, etc.)
while operating without centralized control and with biomechanical
constraints on locomotion and information exchange. The mechanics
underlying animal groups, for example, the relationships between
behaviors of individual fish and the characteristics of the schools
they collectively produce are poorly understood, in part because the
behaviors are difficult to observe experimentally. We used computerized
motion analysis to track the precise 3dimensional positions of fish in
small schools as they moved and interacted in experimental tanks.
Analyses of relative motions of neighboring fish suggest a reasonable
degree of consistency with some previously hypothesized social
behaviors. However, new inverse methods for mathematically deducing
behavioral rules from observed animal trajectories, as opposed to
simulating hypothetical rules to obtain trajectories, are needed to
resolve the underlying biological mechanisms. Coauthors: Steven
Viscido and Julia Parrish.
 YUEXIAN LI, University of British Columbia, Department of Mathematics,
Vancouver, British Columbia V6T 1Z2
A theory of forced pattern formation in excitable media

An excitable medium generally refers to a medium that is capable of
generating traveling waves. It has been widely encountered in biology,
chemistry, and physics. Many excitable media have been modeled by
systems of PDEs of reactiondiffusion type. Excitable neural media are
often modeled by integrodifferential equations (IDEs). In both PDE and
IDE models of excitable media, stationary spatial patterns of Turing's
type can occur under certain conditions. Such patterns have been used
to explain a variety of biological pattern formation processes
including morphogenesis and hallucination. In this talk, I'll discuss a
pattern formation mechanism that is different from Turing's, called
inhomogeneityinduced pattern formation. Such patterns occur in
an excitable medium due to the existence of an inhomogeneous but
stationary forcing. The interesting thing we found is: introducing a
stationary bump into such a medium does not always produce just a
simple bumpshaped output pattern. A complex bifurcation scenario can
occur giving rise to the coexistence of multiple patterns. Stability
analysis shows that instability of such patterns often occur through a
Hopf bifurcation giving rise to oscillatory pulse solutions. Such
oscillatory pulses can behave like a pulsegenerator that emits
traveling pulses periodically into the medium. Possible areas in
biology where this theory can be applied will be discussed. (joint work
with Alain Prat)
 GERALD LIM, Simon Fraser University, Department of Physics, Burnaby,
British Columbia V5A 1S6
A numerical study of the mechanics of red blood cell shapes and
shape transformations

A mature human red blood cell normally assumes the shape of a doubly
dimpled disc. However, it has been known for more than 50 years that,
under a variety of chemical or physical treatments in vitro, the cell
undergoes a quasiuniversal sequence of shape transformations. Unlike most
other cells, the red blood cell lacks internal stressbearing structure;
therefore, its shape can only be governed by its membrane. We describe the
membrane using a simple nonlinear mechanical model and use a numerical
minimisation technique to show that the entire sequence of shapes and
shape transformations can be reproduced by varying a single parameter of
the model.
 FRITHJOF LUTSCHER, University of Alberta, Centre for Mathematical Biology,
Edmonton, Alberta T6G 2G1
A solution of the drift paradox

The term "drift paradox" arose in the ecology of populations in
rivers and streams. It describes the surprising observation that
individuals such as aquatic insects, which are subject to downstream
advection, can persist in upper reaches of the stream.
In this talk, we present a general model for populations subject to
unidirectional flow. The model has the form of an integrodifferential
equation, i.e., movement of individuals is modeled by integration
with respect to a dispersal kernel. We derive an appropriate dispersal
kernel from a mechanistic movement model. We explore how the critical
domain size depends on the advection velocity and find two possible
explanations of the drift paradox. Then we determine the spread speed
of the population in the direction with and against the advection. We
show that the two ecologically relevant quantities "critical domain
size" and "spread speed", which have been studied separately so far,
are closely related in systems which unidirectional flow.
 NATHANIEL K. NEWLANDS, University of British Columbia, Department of Mathematics,
Vancouver, British Columbia V6T 1Z2
Multiscale modeling of tuna population dynamics: search
behaviour drives schooling, aggregation and dispersal

Movements of a wide variety of terrestrial and marine animals show
adaptation of search behavior to the environment. A movement analysis
of tuna reveals switching of search strategy. My talk will provide
relevant biological background and present results obtained from
simulation of mathematical models that describe movement behavior of
individuals, transitions between school formation structures and
population dispersal. Movement is modeled as a stochastic,
velocityjump process. Mathematical modeling of adaptive behavior may
better explain the use of space and resources by tuna.
 EIRIKUR PALSSON, Simon Fraser University, Department of Biology, Burnaby,
British Columbia V5A 1S6
Chemotaxis, cell adhesion, and cell sorting using Dictyostelium as a
model

The cellular slime mold Dictyostelium discoideum is a widely used model
system for studying a variety of basic processes in development,
including cellcell signaling, signal transduction, pattern formation
and cell motility.
In this talk I will discuss cell movements and signaling in
Dictyostelium and introduce a model that facilitates the simulation and
visualization of these processes. The building blocks of the model are
individual deformable ellipsoidal cells; each cell having certain given
properties, not necessarily the same for all cells. Since the model is
based on known processes, the parameters can be estimated or measured
experimentally. I will show simulations of the chemotactic behavior of
single cells, streaming during aggregation, and the collective motion
of an aggregate of cells driven by a small group of pacemakers. The
results are compared with experimental data and examples shown, that
highlight the interplay of chemotaxis and adhesion on cell sorting and
movements in Dictyostelium. The model predicts that the motion of
twodimensional slugs results from the same behavior that is exhibited
by individual cells; it is not necessary to invoke different mechanisms
or behaviors. I will also demonstrate how differences in adhesion
between prestalk and prespore cells, affect the sorting and
separation of those cell types, that occurs during the slug stage, and
I will suggest and explain why chemotaxis alone might not be sufficient
to achieve complete sorting. Finally I will discuss how different
models of the signaling system can influence the results.
 ALEXEI POTAPOV, Department of Mathematical and Statistical Sciences and Centre
for Mathematical Biology, University of Alberta, Edmonton,
Alberta T6G 2G1
Climate and competition: the effect of moving range boundaries
on habitat invasibility

Predictions for climate change include movement of temperature
isoclines up to 1000 meters per year, and this is supported by recent
empirical studies. This paper considers effects of a rapidly changing
environment on competitive outcomes between species. The model is
formulated as a system of nonlinear partial differential equations in a
moving domain. Terms in the equations decribe competition interactions
and random movement by individuals. Here the critical patch size and
travelling wave speed for each species, calculated in the absence of
competition and in a stationary habitat, play a role in determining the
outcome of the process with competition and in a moving habitat. We
demonstrate how habitat movement, coupled with edge effects, can open
up a new niche for invaders that would be otherwise excluded.
 PETER TAYLOR, Queen's University, Department of Mathematics and Statistics,
Kingston, Ontario K7L 3N6
Negotiation in evolutionary games of conflict

Allowing negotiation in an evolutionary game of conflict can alter the
outcome in an unexpected way. For example, contrary to some previous
findings, we show that if players are allowed to negotiate, cooperation
can be more likely to evolve. This is joint work with Troy Day and Ido
Pen.
 BRIAN G. TOPP, School of Kinesiology, Simon Fraser University,
Burnaby, British Columbia V5A 1S6
The pathogenesis of type 2 diabetes: a dynamical bifurcation?

Conditions such as obesity, ageing, and pregnancy are associated with
reduction in the ability of insulin to lower blood glucose levels
(insulin resistance). Most people are able to compensate for this
insulin resistance by increasing blood insulin levels (compensatory
hyperinsulinemia). However, in some people, the development of insulin
resistance leads to insufficient compensatory hyperinsulinemia
(insufficient adaptation) followed by progressive reductions in
insulinemia (bcell failure). These abnormal insulin dynamics lead
to the development and exacerbation of hyperglycemia that define type 2
diabetes. Recently, we developed a mathematical model to investigate
the mechanisms by which insulin resistance leads to compensatory
hyperinsulinemia as well as the mechanisms responsible for insufficient
adaptation and bcell failure. We incorporated into this model the
assumptions that 1) insulin resistance leads to compensatory
hyperinsulinemia by increasing the number of insulin secreting
pancreatic bcells, rather than increasing the function of existing
bcells, 2) that insulin resistance regulates bcell population
dynamics indirectly via changes in blood glucose levels, and
3) glucose has nonlinear effects on bcell dynamics (moderate
hyperglycemia is an expansion signal while extreme hyperglycemia causes
toxic reduction in the bcell population). We found that for normal
parameter values the model behaves like a typical negative feedback
loop. However, either by increasing the rate at which insulin
resistance develops or by inhibiting the maximal rate of bcell mass
expansion, we could generate a bifurcation in the models behaviour that
generates dynamics that are both qualitatively and quantitatively
similar to those observed in type 2 diabetes. We are presently
performing in vivo experiments to test the central assumptions of this
model.
 REBECCA TYSON, Okanagan University College, Department of Mathematics and Statistics,
Kelowna, British Columbia V1V 1V7
Dispersal of the codling moth

The codling moth is a major pest for apple and pear growers worldwide.
In recent years, orchardists have begun using massreared sterile
codling moth populations to control the wild population. The programme
has the advantage of eliminating the need for chemical sprays, but is
very expensive. Thus it is important to understand exactly how the
insects disperse in the field in order to make their release as
effective as possible. We present a diffusionbased model for codling
moth dispersal, along with field data being gathered in the Okanagan
Valley. Our main goal is to understand the effect of heterogeneous
landscapes on codling moth dispersal behaviour.
 PAULINE VAN DEN DRIESSCHE, University of Victoria, Department of Mathematics and Statistics,
Victoria, British Columbia V8W 3P4
The spatial spread of a multispecies disease

A general sspecies, npatch epidemic model with four disease
status compartments is formulated as a system of 4sn ordinary
differential equations with terms accounting for disease transmission,
demographics, and travel (migration) between patches. For each
species, the spatial component is represented as a directed graph with
patches as vertices and arcs determined by travel. Analysis of the
system includes determination of the local stability properties of the
disease free equilibrium by deriving the basic reproduction number,
R_{0}, as the spectral radius of a nonnegative matrix product. The
special case of 3species on 2patches, modeling bubonic plague,
with the species being fleas, rodents and humans, between an urban area
and its suburbs, is given as an example. Numerical simulations
demonstrate the effect of small migration on disease propagation for
one species in a 2patch system. (joint work with J. Arino and
R. Jordan)
 JAMES WATMOUGH, University of New Brunswick, Department of Mathematics and
Statistics, Fredericton, New Brunswick E3B 5A3
Spatial patterns of biological invasions

Much of the modelling work on biological invasions has focused on the
rate of spread of the invading species. However, much more effort is
needed to understand the spatial pattern of the invasion and how an
invading species can influence the distribution of native species. For
example, does the spatial arrangement of potato crops influence their
colonization by the potato beetle, and if so, what are the optimal
field size and spacing? How should shellfish aquaculture sites be
allocated, and does their proximity to natural foraging sites and
staging areas influence predation by sea ducks?

