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Mathematical Biology / Biologie mathématique
Org: Leah Keshet (UBC)

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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 cell-cell synapses

In many situations, cell-cell adhesion is mediated by multiple ligand-receptor pairs. For example, the interaction between T cells and antigen-presenting cells of the immune system is mediated not only by T cell receptors and their ligands (peptide-major 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 cell-cell interaction mediated by two ligand-receptor pairs to examine competition between the elasticity of the cell wall, non-specific intercellular repulsion and bond formation, leading to segregation at equilibrium. We make detailed predictions concerning the relationship between physical properties of the membrane and ligand-receptor pairs and equilibrium pattern formation and suggest experiments to refine our understanding of the system. We demonstrate our model by application to the T cell-antigen-presenting-cell system and natural killer cell-target cell adhesion. Our results underline the importance of active, energy-consuming 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 spatio-temporal 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 reaction-diffusion 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 closely-related crayfish, we propose models of the central pattern generators coordinating each appendage, first via Morris-Lecar 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 frequency-dependent 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 Michaelis-Menton 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 cross-feeding polymorphism in microorganisms, as well as adaptive speciation due to frequency-dependent selection on phenotypic plasticity.

DANIEL GRUNBAUM, University of Washington, School of Oceanography, Seattle, Washington  98195-7940, 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 3-dimensional 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. Co-authors: Steven Viscido and Julia Parrish.

YUE-XIAN 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 reaction-diffusion type. Excitable neural media are often modeled by integro-differential 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 inhomogeneity-induced 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 bump-shaped output pattern. A complex bifurcation scenario can occur giving rise to the co-existence 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 pulse-generator 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 quasi-universal sequence of shape transformations. Unlike most other cells, the red blood cell lacks internal stress-bearing 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 integro-differential 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
Multi-scale 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, velocity-jump 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 cell-cell 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 two-dimensional 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 pre-stalk and pre-spore 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 (b-cell 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 b-cell failure. We incorporated into this model the assumptions that 1)  insulin resistance leads to compensatory hyperinsulinemia by increasing the number of insulin secreting pancreatic b-cells, rather than increasing the function of existing b-cells, 2)  that insulin resistance regulates b-cell population dynamics indirectly via changes in blood glucose levels, and 3)  glucose has nonlinear effects on b-cell dynamics (moderate hyperglycemia is an expansion signal while extreme hyperglycemia causes toxic reduction in the b-cell 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 b-cell 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 mass-reared 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 diffusion-based 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 multi-species disease

A general s-species, n-patch 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, R₀, as the spectral radius of a nonnegative matrix product. The special case of 3-species on 2-patches, 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 2-patch 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?