The human spine must allow significant motion during such everyday activities as bending, lifting, looking overhead or emphatically shaking one's head no. In addition to allowing this motion, the spine must withstand significant mechanical loads, in its role as a component of the musculoskeletal structure, and protect a major component of the central nervous system, the spinal cord, from mechanical insult. The loads in the lumbar spine (low back) can range up to five thousand Newtons in some people for some activities. This approaches half the weight of some small cars!

The spine is anatomically complex and its size and structure changes from
one part of the body to the next. The corresponding kinematics (motion)
allowed by the spine varies greatly depending on the region of the body of
interest. Physicians, bioengineers and other basic scientists have long
studied the spine's kinematics for the investigation of natural
biomechanical processes, to characterize pathologic motion or instability
and to evaluate the efficacy of surgical devices and techniques. I will
present a review of the experimental techniques and underlying mathematical
principals that have been used to measure and communicate results regarding
human spine kinematics. One focus of the presentation will be to identify
and compare the fundamental differences between spine motion in the
cervical(neck), thoracic and lumbar regions of the spine. I will also review
the overall research questions and themes that have been addressed using
these techniques.

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Most mathematical models for signal generation in single neurons, such as the classic Hodgkin-Huxley model, assume that the single neuron is bathed in an infinite buffer solution. Thus the composition of the bath never changes. This assumption is appropriate for the comparison of model results to in vitro studies because in these studies the cell preparation is actually bathed in a relatively fixed environment. In their current state, such models are not able to take into account large changes in the external environment of a cell during ischemia. It is my goal to improve current neuron models so that the changing extracellular conditions can be taken into account in a single cell micro environment. The main challenges in this endeavor are due to the necessity of creating a finite extracellular compartment. This requires considering mass conservation and electroneutrality.

In this talk, I lay the foundation for a physically consistent model
based on a quasi-steady-state approximation. In the first part of the
presentation, an efficient numerical method for the solution of 1D
Poisson-Nernst-Planck (PNP) systems is developed. In the second part
of the talk, this numerical method is applied to solving the consecutive
steady-state dynamics of a two compartment system of ions. The results
of my approach are compared to the full PDE in order to confirm the
sensibility of the steady-state assumption. Finally, the quasi steady-
state approach is compared to a Hodgkin-Huxley type model for a cell
with intact gated channels but no ion pumps to maintain homeostasis.
In the future, I would like to incorporate active ion transport, applied
currents, and cell volume dynamics into my model.

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The recent SARS epidemic has drawn attention back to the classical Kermack - McKendrick epidemic model of 1927 (more precisely, to the very special case which is usually called the Kermack - McKendrick model). We study some natural extensions to include such aspects as an exposed period, quarantine, and the reduction of contacts by susceptibles in response to an epidemic.

[joint work with L. Berezansky and L. Idels] Population models with harvesting of various types are considered. The following question is in the centre of our discussion: when does the harvesting lead to the extinction of the population? When can we provide that there is a positive solution of the system? Two models are studied in detail: a delay logistic equation with continuous (proportional or nonlinear) harvesting which may also be delayed and a logistic equation with impulsive harvesting. In addition to non-extinction of the population, for the continuous model with linear harvesting the oscillation about a new equilibrium is studied; for a logistic equation without delay and with impulsive harvesting the asymptotics of solutions is described. It is demonstrated that the latter model can be reduced to some difference equation.

Kinesin is a motor protein that uses the energy of ATP hydrolysis to transport vesicles and organelles along microtubule tracks within cells. Biophysicists interested in understanding how such proteins can convert chemical energy into mechanical energy have studied kinesin and other motor proteins extensively. Recent experiments study the behavior of single kinesin motors using optical traps and tethered glass beads. Records of bead positions provide an indirect, noisy record of the progress of the motor protein. Hidden Markov models provide a tool to analyze these records statistically to infer model parameters for the motor's stepping cycle.

Scoliosis is an enigmatical disease (not so uncommon) which results in a three dimensional deformation of the spine. At present the only treatment for severe cases is spinal fusion, not a particularly desirable solution. The purpose of this study is to develop new tools for understanding this mystery and improving surgical procedures. We present preliminary results concerning a certain hypothesis about the three dimensional spinal configuration: namely, the static erect spine assumes a shape, which minimizes a hypothetical energy function. Ordinarily this type of minimization would be handled by Lagrange's method of undetermined multipliers. However due to the nature of the constraints, we obtain this minimum by an application of Newton methods on manifolds, which uses the group operations of orthogonal matrices to advantage.

If time permits, we shall discuss a connection of the scolitic deformity with the geometry of space curves. On one hand, the erect scoliotic spine is usually non-planar and involves twisted vertebrae. On the other hand, the normal spine is planar with no vertebral rotations, but rotation occurs upon bending. The medical literature gives a physiological explanation of this phenomena. We shall present a purely geometrical one.

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We consider the moment-closure approach to transport equations which arise in mathematical biology. We show that the negative L

(This talk is closely related to my math colloquium talk on Oct 10.)

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See Schedule for full details.

Childhood infections give us an unusually detailed empirical picture of natural enemy dynamics in space and time. I use wavelet analysis and models to explore the spatio-temrpoal dynamics and control by vaccination of measles and whooping cough in England and Wales. The results indicate marked hierarchical travelling waves of measles, driven by 'forest fire-like' dynamics and markedly different nonlinear behaviour of the measles and whooping cough attractors in the face of vaccination, despite the similarity of their natural history. A final study also explores waves. The talk also explores the impact of waves, sparks and vaccination in the control of foot and mouth disease.

Organizational Meeting

There will not be a formal seminar on this day, but we will meet at our usual venue at PIMS to introduce new students and faculty, and to chat over coffee, tea, and refreshments.

Neurons and other excitable cells often exhibit bursting oscillations; this behavior is characterized by alternations of silent phase of near steady state and active phase of rapid, spike-like oscillations. Mathematical models for bursting oscillations often display a rich structure of dynamic behavior. Besides periodic bursting oscillations, these systems may exhibit other types of periodic solutions, such as continuous spiking, as well as more exotic behavior including chaotic dynamics. The models contain multiple time scales and this often leads to very interesting issues related to the theory of singular perturbations. We introduce some different classes of bursting oscillations and discuss the underlying mathematical mechanisms responsible for these oscillations.

We consider an age-maturity structured model arising from a blood cell proliferation problem. This model is ``hybrid" i.e., continuous in time and age but the maturity variable is discrete. This is due to the fact that we include the cell division marker CarboxyFluorescein diacetate Succinimidyl Ester (CFSE). We use our mathematical analysis in conjunction with experimental data taken from the division analysis of primitive murine bone marrow cells to characterize the maturation/proliferation process. Cell cycle parameters such as proliferative rate, cell cycle duration, apoptosis rate and loss rate can be evaluated from CFSE+ cell tracking experiments.

Ryan is a visiting graduate student working with Nathaniel Newlands and Leah Keshet on a summer research project. He is modelling random motion and comparing data for tuna fish tracks with the simulated random walks. The goal is to learn more about how to characterise aspects of the individual behaviour given the record of the individual's motion. He will tell us about this project in this seminar.

This is intended to be a survey talk about the recent results in the theory of complex networks. Many real-world networks can be modelled using graphs, and their structure can be analyzed using graph theoretical results. Examples of such networks are: food webs, communication networks, social networks, protein interaction networks, etc. In the past, it was thought that random graphs, which are nearly regular, provide sufficient models for complex networks. However, in the late nineties, Barabási and Albert showed that complex networks are far from random. In fact, we now know that most networks have a degree distribution that follows a power law, i.e. many vertices of very low degree and few vertices of very large degree. So complex networks are best modelled using so called scale-free networks. Furthermore, Watts and Strogatz showed that real-world networks also exhibit high clustering and short characteristic path length, which defines a class of networks now known as small-world networks. In the past few years much research has focused on finding models that display both the scale-free and the small-world behavior to provide better ways to represent real-world networks. Attention has also been drawn to several particularly interesting properties of real-world networks such as path lengths, communities, and connectivity. Studying these properties promises to give insight into the structures underlying human interaction, gene interaction, and many other networks of significance to a wide variety of fields of research.

This talk will present an overview on recent results regarding models
and properties of small-world and scale-free networks using real-world
network examples. Furthermore, I will talk about the centrality of the
notion of clustering and some common misconceptions in the literature
about the definition of the clustering coefficient. The talk will also
give some insight about results from algebraic graph theory, in
particular, how the spectra of the Laplacian and normal matrices can
be used to study the structure of complex networks.

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There will be no seminar on the week of July 7-11 due to out-of town events.

Note: Prof Goldbeter is arriving from a trans-Atlantic flight at 2:30 pm on this same date. We plan to honor him with a reception around 4:00 pm followed by a seminar that will start between 4:30 and 5:00. In the event of flight delays that affect his arrival, we will hold the reception in any case.

We consider some simple models for constant yield harvesting of single species populations and of predators in a predator - prey system. There are some situations in which the critical harvest rate beyond which a system collapses must be obtained from the dynamics of the model and is much smaller than can be deduced from equilibrium analysis.

Here I present a modelling investigation of the factors that define the migratory route and behavior of sockeye salmon from the Fraser River. The model is fitness based and spatially explicit, and it is based on the hypothesis that fish behavior is constrained by the state of the individual and the environment. It also has a trade-off between foraging time and migration time. The environment here was defined as monthly fields of sea surface temperature, currents, prey density, and predation risk.

The model predicts the following characteristics of the sockeye salmon
migration: 1) Juveniles migrate along the coast and then move into the
Alaska Gyre where they stay for the rest of their oceanic residence.
Model predictions do not support the commonly held hypothesis of an
annual circuit around the Alaska Gyre. 2) The juvenile migration arises
as a response to high zooplankton density in the coast at the time of the
migration, although the high risk of mortality there creates a bottleneck
in their life cycle. 3) The model predicts a seasonal growth pattern as a
response to the seasonality of zooplankton density. 4) Juvenile fish
display higher swimming migration activity than adults. 5) Individuals
behaving optimally distribute below observed thermal limits, however
their distribution follows that of prey density.

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Coevolution is a broad topic connecting the study of individual traits within separate biological populations (e.g. separate species) to the study of ecosystems based on interactions among these populations. An early success in the analysis of stability in coevolutionary models introduced some twenty years ago was to separate evolutionary and ecological effects by assuming population densities track their equilibrium values (what I call the stationary density surface) for a given set of species' traits. This assumption is a more sophisticated version of the intuitively appealing idea that, since the evolution of population densities takes place on a much faster time scale (ecological time) than that of species' traits (evolutionary time), trait values can be taken as fixed parameters when studying the evolution of population densities (or vice versa). It has since been well documented that both assumptions have the potential to give misleading insights into the eventual outcome of general coevolutionary systems. The main purpose of this talk is to show that, notwithstanding the preceding statement, the stationary density surface remains of central importance. Moreover, there is no need for some artificial assumption of different coevolutionary time scales. This will initially be shown by considering coevolutionary Lotka-Volterra systems where individual fitness functions are assumed to be linear in the population state. If time permits, I will also connect the theory to more recent coevolutionary models based on adaptive dynamics where there continues to be a widespread conviction that the stationary density surface technique can be used to predict stable trait distributions.

I will then briefly survey some of the more recent biomedically related
projects, including modelling of Alzheimer's Disease (joint with A. Spiros)
and Type 1 Diabetes (with AFM Maree and Diane Finegood).
The common thread in the latter projects is the involvment of inflammation
and the immune system. My main goal will be to highlight the synergy between
the mathematical models and insight or understanding of underlying mechanisms
in the biological systems. (These project have been funded by MITACS.)

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This talk presents the work of the OFP's Tuna Ecology and Biology (TEB) section with particular emphasis on the development of individual-based models (IBM's) for the movement and population dynamics of Pacific tunas. The talk will briefly describe the work of the section and then focus on the development of an IBM for Pacific skipjack tuna,

The model has been developed in collaboration with colleagues in Bergen, Norway, and uses the Lagrangian IBM paradigm to track individual properties (e.g. movement, physiological state, feeding history, reproductive activity) of a population of model fish in an ocean simulated by prior numerical modelling of currents, temperature, primary production and tuna forage. Tuna larvae behave as passive drifters but juveniles and adults exhibit directed movement in relation to their environment, location, time and internal state. Movement and spawning behaviour is output from an artificial neural network with weights that are unique to each individual. A genetic algorithm is used to allow the weights to be passed on to offspring by sexual reproduction. Therefore life-cycle closure is achieved and both individual behaviour and large-scale spatial dynamics are emergent within the model without reference to a pre-determined fitness measure or a prescriptive rule-set.

The model is designed for eventual comparison with other population dynamics and stock assessment models and may be used to explore individual behaviour in comparison to data from archival and conventional tags.

On this day several students will present short summaries of work that they will take to the MITACS Annual General Meeting in Ottwa in May 2003.

Actin is a ubiquitous protein and is found in all eukaryotic cells. Pathogens such as Listeria hijack actin to propel themselves through infected cells. Listeria, as well as specially treated polystyrene beads, causes the growth of an actin "comet tail" through the process of polymerization where actin is built into thin filaments. As more actin is polymerized at the rear of the obstacle (bacterium or bead), it pushes the obstacle forward. We use a simple mathematical model to describe the dynamics of actin polymerization on a spatial grid to investigate this system. We will look at various parameters and whether they affect the formation of the actin comet tail and the velocity of the obstacle.

integrating tracking, tagging and aerial survey data

From 1993-1997, the New England Aquarium conducted fishery-linked aerial surveys with spotter pilots to document the surface abundance, distribution, and environmental associations of bluefin tuna schools in the Gulf of Maine. The long-term goal of this program is to develop fishery-independent estimates of abundance. Information obtained by direct monitoring of surface schools allowed us to conduct spatial analyses, modeling and simulations that are not usually available in CPUE based approaches. Our presentation will address measurement bias and uncertainty in population abundance estimation from bluefin tuna movement, spatial aggregation and distribution data. We used an integrated approach to analyze, calibrate, and correct the aerial survey data in order to obtain more reliable estimates of regional abundance. Bias and uncertainty in the size and aggregation of schools were directly estimated from the survey data, adjusted by additional data on movements and dispersal from electronic and hydro-acoustic tagging studies. Results of simulations will be presented for different survey designs, including random, systematic, stratified, adaptive and spotter-search aerial sampling.

Recommendations for achieving greater precision in abundance estimation in aerial surveys will be discussed. The work demonstrates how fishery-independent data can be integrated to provide more reliable estimates of bluefin tuna abundance, and a broader understanding of spatial heterogeneity in fish populations.

This talk will be presented at the
International Tuna Conference, Lake Arrowhead, CA,
U.S.A. May, 2003. (20 minutes). Feedback and comments
are welcomed.

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A biological swarm is a group of organisms undergoing large-scale coordinated movement. Typically, this movement is not due to centralized control, but rather to social interactions which occur on a length scale smaller than that of the global swarm formation. Swarms occur in populations of ants, locusts, fish, birds, wolves, and others, and are often observed to have sharp boundaries and a roughly spatially-constant population density. In this talk, I will discuss preliminary results for a simple continuum model for swarming in two dimensions. The population density $\rho$ satisfies an advection equation. The velocity depends nonlocally on $\rho$ by means of a convolution with a spatially decaying kernel $K$, which describes the social interaction between organisms. Using the Hodge decomposition theorem, the velocity field may be decomposed into a divergence-free component and a gradient component. This framework provides a convenient way to characterize the two-dimensional dynamics. The gradient component controls the expansion or contraction of the population, while the divergence-free component is responsible for its rotational motion. Numerical simulations of the model reveal vortex states similar to those observed in nature.

Classical evolutionary theories of aging make few assumptions about how many genes are involved in the aging process, or about the magnitude of their effects on rates of aging. I will discuss my recent work with gene networks. This work is an attempt to create a more genetically realistic model for the evolution of aging. In addition, I will discuss ways in which we might use networks to aid molecular biologists in predicting what types of genes might influence longevity.

We describe and analyze by elementary methods some simple models for disease transmission with vaccination. In particular, we give conditions for the existence of multiple endemic equilibria and backward bifurcations. We also extend the results to include the possibility of adaptive systems in which the parameter values may depend on the level of infection.

Actin filaments often form branched network structures which provide stable propulsion for cells or for intracellular parasites such as Listeria. We study the growth of such networks using a combination of stochastic simulation and kinetic rate equations. The stochastic simulations use a model of static, rigid filaments to which monomers and branches are added at random positions according to specified rates. They show that a simple model based on five basic unit processes produces three-dimensional network structures quite similar to those observed by electron microscopy. Surprisingly, the growth velocity of the network is independent of the opposing force, even though the velocity of a free filament depends exponentially on the force. To understand this effect, we study a rate-equation model based on the density of free filament tips near the cell membrane (or intracellular pathogen). The force-independence of the velocity results from the linearity of the rate equation, and holds under a broad range of plausible assumptions. Experiments testing the predictions are proposed.

Culmination of the morphogenesis of the cellular slime mould

We have simulated the culmination using a hybrid CA/PDE model. In the model, individual cells are represented as a group of connected automata, i.e. the basic scale of the model is subcellular. Initially we simulated in 2D, describing transverse sections through the culminant. Recently we have extended the model to 3D, in order to prove feasibility of the full model, and to open the road to describe the whole development as one continuous process.

With our model we are able to reproduce the main features that occur during culmination, namely the straight downward elongation of the stalk, its anchoring to the substratum and the formation of the long thin stalk topped by the spore head. We show that periodic upward movements, due to chemotactic motion, are essential for successful culmination, because the pressure waves they induce squeeze the stalk downwards through the cell mass, a mechanism which has a number of self-organising and self-correcting properties and can explain many experimental observations.

In this talk I will present some ways in which mathematical modeling has been helpful in studying T cell activation, and outline challenges for the future. No prior knowledge of the immune system will be assumed.

The activation of T cells is an essential part of the immune response to viruses and bacteria. Fragments ("antigens") of these enemies are presented to T cells on the surfaces of antigen-presenting-cells, but an individual T-cell carries receptors (TCR) that recognize only a few possible antigens. After the T cell becomes activated, it may kill the presenting cell (in the case of viral infection) or activate other components of the immune system (in the case of bacterial infection).

Activation appears to rely on the formation of a stable region of close apposition between the cells, termed the "immunological synapse". Within the synapse, each TCR may individually be activated and labeled for internalization by interaction with presented antigen. A key parameter controlling individual TCR activation and internalization is the lifetime of the bond between the TCR and a presented antigen.

We have developed a mathematical model consisting of reaction-diffusion equations describing spatial and temporal changes that take place within the synapse. From comparison of model predictions with experimental data, we draw conclusions about the requirements for T cell activation as well as the cellular internalization and degradation of TCR.

Many experimental investigations have shown that bacterial flagella (the long, whip-like structures that provide thrust during swimming) take on a variety of helical forms under differing mechanical and chemical conditions. During the 1980s a series of experiments examined the response of a single, detached flagellum to simple fluid stresses. In particular, when a flagellum is clamped at one end and placed in an axial external flow, it is observed that regions of the flagellum transform to the opposite chirality and travel as pulses down the length of the filament, the process repeating periodically.

We propose a theory for this phenomenon based on a treatment of the flagellum as an elastic object with multiple stable configurations. This theory is expressed in terms of coupled PDEs for the filament shape and twist configuration, and involves only physical, measurable properties of the flagellum. We generate simulations that quantitatively reproduce key features seen in experiment.

In lieu of nervous systems, single-celled organisms use complex networks of biochemical reactions to sense the world around them, make decisions, and take action. A wealth of quantitative biological data from bioinformatics to fluorescence microscopy has created the possibility of building biophysically realistic models of the information processing occurring inside cells, in analogy to models of biological neural networks. Biochemical networks present several unique mathematical challenges. Chemical reactants are localized within subcellular volumes, requiring PDE rather than ODE treatments of their behavior. Small numbers of interacting molecules make the typical biochemical network inherently noisy, leading us to consider approximations to stochastic PDEs. Finally, chemical systems typically occupy state spaces of large dimension, forcing us to look for effective means of "coarse-graining" the representation of chemical states. We have constructed a finite-element model for solving arbitrary boundary-coupled PDEs as a platform for studying spatially heterogeneous signal-transduction networks, and used it to develop a model for the orienting response of a eukaryotic cell during directed cell movement (chemotaxis)*. We are building on this finite-element framework to accommodate the effects of fluctuations as an approach to stochastic PDEs, and as a way of formalizing dimension-reduction of chemical state spaces.

*Eukaryotic chemotaxis will also be discussed in the context of Turing pattern formation in the 1/15 Wednesday Mathematics Colloquium.

In 1952 Alan Turing proposed a mechanism for the development of spatial patterns, such as animal coat patterns, from spatially homogeneous initial conditions, such as a presumably uniform embryo. Many systems have invited analogous treatments, from segmentation of the Drosophila embryo to Meinhardt's model for establishing direction in eukaryotic chemotaxis. The two essential elements underlying the Turing mechanism, a short-range activator and a long-range inhibitor, have not always been easily identified as the biology underlying pattern formation becomes better understood. In this talk I will explore two systems in which the biological details gave new insights into the possibilities of pattern formation. In the cerebral cortex, the local connectivity of nervous tissue gives an effective long-range inhibitory and short-range excitatory interaction that can lead to the creation of spontaneous patterned activity in the cortical sheet. In the visual cortex this spontaneous activity gives rise to a distinct set of geometric visual hallucinations. Careful analysis of the geometry of cortical connectivity allows classification of the observed patterns in terms of a particular class of subgroups of the Euclidean motions of the plane. As a second example, I will return to the problem of eukaryotic chemotaxis. An unbiased single-celled organism must respond to a weak gradient of a chemical attractant, organizing its internal chemistry in order to initiate movement in the direction of the gradient (chemotaxis). Using data from mutant cells showing anomalous chemotaxis, we have identified a rapidly diffusing intracellular inhibitory molecule that facilitates sharpening of the directional response. In addition to amplifying the weak spatial gradient signal, this variant of the Turing mechanism also exploits timing characteristics of the extracellular signal. (The Mathematical Biology Seminar on Thursday 1/16 will discuss biochemical networks in more detail.)

Investigators typically divide neurons of the primary visual cortex (V1) into two classes: simple and complex. Neurons with an approximately linear response to a visual stimulus are classified as simple; the remainder are classified as complex. Standard analysis methods lead to the view that V1 neurons fall neatly into these two categories. I develop a method to analyze neural response based on an explicit mathematical model that captures the nonlinear sign-independence of an idealized complex cell. When V1 neurons are analyzed by this method, one observes evidence of a broad continuum of nonlinear response without a division into two discrete classes. Discrete classes are observed with standard analysis methods because the methods do not account for the effect of a spike generating nonlinearity. I discuss how this modeling approach can be used to better understand the nonlinear response properties of visual and other stimulus-driven neurons.

Reconstructing the connectivity patterns of neural networks in higher organisms has been a formidable challenge. Most neurophysiology data consist only of spike times, and current analysis methods are unable to resolve the ambiguity in connectivity patterns that could lead to such data. I present a new method that can determine the presence of a connection between two neurons from the spike times of the neurons in response to spatiotemporal white noise. The method successfully distinguishes such a direct connection from common input originating from other, unmeasured neurons. Although the method is based on a highly idealized linear-nonlinear approximation of neural response, simulations demonstrate that the approach can work with a more realistic, integrate-and-fire neuron model. I propose that the approach exemplified by this analysis may yield viable tools for reconstructing neural networks from data gathered in neurophysiology experiments.

I study a process of self-organization that occurs inside a cell called a fish melanophore in which pigment particles are seen to aggregate. The process is mediated by subcellular components called microtubules, which form part of the cytoskeleton. The same components are at work in the centering of chromosomes during cell division, and therefore provide a good "warmup" problem for that more complicated but fundamentally important biological process. When a fragment of the cell is excised to eliminate the centrosome (the regular cytoskeletal organizer) and therefore the cytoskeletal structure, stimulating the cell with adrenaline somehow reintroduces cytoskeletal organization, leading to the formation of a microtubule aster and the aggregation of the cell's pigment particles at the center of the fragment. It is this centering behaviour that is analogous to chromosome alignment during cell division.

We derive a system of seven non-linear PDEs (1D) that describes the biological system. Numerical simulations of the equations demonstrate certain observed features (aggregation) but not others (centering). The system can be reduced so as to facilitate analysis which allow for an understanding of the successes and failures of the original model. Finally, we generalize the reduced model to 2D, incorporating a stochatic element, and present numerical results.

In this talk, I will also briefly mention some of my previous work on the phenomenon of ventricular fibrillation in the heart, and the analysis of wave phenomena in the Fitzhugh Nagumo equations that formed the focus of my PhD work.

Mitosis, the process by which a cell segregates two identical copies of its genome in preparation for division, is fundamental to cell replication and hence to life as we know it. In order to separate the two copies of the genome, a self-assembling protein machine, the mitotic spindle, employs several force generating molecules known as molecular motors. These motors, through interaction with spindle microtubules (semi-rigid protein filaments), aid in spindle assembly as well as chromosome segregation.

It has been proposed that mitosis occurs by progression through a sequence of steady states defined by a balance of motor forces. In close collaboration with experimentalists, we have developed a quantitative model describing this balance of forces during the process of spindle formation in Drosophila embryos. In particular, we describe how dynein and Ncd exert opposing forces on the spindle poles, moving them to opposite sides of the nucleus where they reach a steady-state separation distance. Complementing dynein is a polymerization force that explains the rapid initial separation that is seen in experimental measurements. The model compares favorably to data from both wildtype and mutant phenotypes.

In this talk, I will describe the details of the quantitative model as well as ongoing experimental work which was motivated by the quantitative study. Although more experimental work is required, I will also describe future plans for incorporating these new results into a "second generation" model.

Cooperative behavior among unrelated individuals is one of the fundamental problems in biology and social sciences. Reciprocal altruism fails to provide a solution if interactions are not repeated often enough or groups are too large. Punishment and reward can be very effective but require that defectors can be traced and identified. Here we present a simple but effective mechanism operating under full anonymity. Voluntary participation in public goods games can foil exploiters and overcome the social dilemma. This natural extension leads to rock-scissors-paper type cyclic dominance of the three strategies cooperate, defect and loner i.e. those unwilling to participate in the public enterprise. In voluntary public goods interactions the three strategies can co-exist under very diverse assumptions on population structure and adaptation mechanisms. In particular, spatially structured populations, where players interact only with their nearest neighbors, lead to interesting dynamical properties and intriguing spatio-temporal patterns. Variations of the value of the public good result in transitions between one-, two- and three-strategy states which are in the class of directed percolation. Although volunteering is incapable of stabilizing cooperation, it efficiently prevents successful spreading of selfish behavior and enables cooperators to persist at substantial levels.

Humans and animals display a high readiness to help their fellows in need - even if such assistance involves substantial costs in terms of money or fitness. Recently, economists became increasingly interested in this apparently irrational behavior. Extensive experiments suggest that fairness considerations, punishment and reputation play a crucial role. It now poses a challenging task to embed such mechanisms in an abstract game theoretical framework.

Various different games were designed to capture the essence and to highlight different aspects of such interactions between pairs or groups of individuals. The most prominent games are certainly the prisoner's dilemma, the public goods game as well as the ultimatum game. We show that from a mathematical point of view all three games are closely related and share a common core. In particular, the ultimatum game turns out to be a special case of a pairwise prisoner's dilemma interaction followed by punishment opportunities. This leads to a general class of two-stage games where an interaction is followed by a reaction. The replicator equation allows for a full analysis of the resulting game dynamics - even in the non-linear case of groups of interacting individuals.

The threat of punishment is efficient in creating incentives to cooperate, but punishment alone is not sufficient for persistence of cooperation. Additional mechanisms such as reputation or spatially structured populations are required to achieve highly social and fair outcomes related to the experimental evidence.