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Bacterial cells come in a wide variety of shapes and sizes, with the peptidoglycan cell wall as the primary stress-bearing structure that dictates cell shape. In recent years, cell shape has been shown to play a critical role in regulating many important biological functions including attachment, dispersal, motility, polar differentiation, predation, and cellular differentiation. How much control does a cell have over its shape, and can we tap into control mechanisms to synthetically engineer new morphologies? Though many molecular details of the composition and assembly of the cell wall components are known, how the peptidoglycan network organizes to give the cell shape during normal growth, and how it reorganizes in response to damage or environmental forces have been relatively unexplored. We have introduced a quantitative mechanical model of the bacterial cell wall that predicts the response of cell shape to peptidoglycan damage in the rod-shaped Gram-negative bacterium Escherichia coli. We have verified some of our predictions regarding morphological response to antibiotics using time-lapse imaging, suggesting that mechanical modelling of the cell wall can inform our understanding of cellular physiology. Our simulations based on our physical model also suggest a surprising robustness of cell shape to damage, allowing cells to grow and maintain their shape even under conditions that limit crosslinking. Our current research focuses on identifying the molecular factors responsible for cell shape determination and characterizing their phylogenetic diversity. In particular, we demonstrate that a small set of physical rules determines the cell's shape and the ability of the cell to maintain that shape, and we have used these rules to systematically alter the dimensions of rod-shaped bacteria. Our work has shown that many common bacterial cell shapes can be realized within both our model and in experiments via simple spatial patterning of the cytoplasm and cell wall, suggesting that subtle patterning changes could underlie the great diversity of shapes observed in the bacterial kingdom.

In numerous applications in the biological and engineering sciences, one encounters inverse problems where the uncertainty and/or variability in parameters and mechanisms to be modeled are a fundamental part of the problem formulation. This is in addition to the data-driven uncertainty that arises naturally in most inverse problems. We discuss a theoretical framework and an associated computational methodology for such problems. In statistical inverse problem formulations these problems are usually discussed in the context of "mixing distributions", but the mathematical foundations can be found in much earlier work on relaxed controls (sliding regimes, chattering controls) of Young, Filippov, Warga,...., the early treatment of two player non-cooperative differential games, and more recently in the treatment of Preisach hysteresis in smart materials. In this expository lecture we outline these connections and present results from our recent efforts using these ideas in several applications in biology.

The first step in RNA processing is the addition of an inverted GMP nucleotide, called a 'cap', onto the 5' end of the first transcribed nucleotide. In the nucleus this is bound by CBP80, which interacts with the processing machinery to coordinate intron splicing and addition of the poly(A) tail at the 3' end. The newly-processed mRNA is exported through the nuclear pore complex cap-end first, where CBP80 is replaced by eIF4E to begin the process of its translation into protein. The cap is removed as one of the first steps in the overall process of mRNA decay, and it was generally thought that loss of the cap results in rapid degradation by an exonuclease acting on the unprotected end.

In the dogma of molecular biology cap addition only occurs in the nucleus and its loss in the cytoplasm is irreversible. There are numerous reasons why this made sense, the most compelling of which is the concentration of the responsible protein (capping enzyme( in the nucleus and the biochemistry of cap addition, which requires a substrate with 2 phosphate groups, not the single phosphate that is left after the cap is removed. I will present work from my lab describing a new mechanism by which the cap can be restored onto cytoplasmic mRNAs after it has been removed by decapping or endonuclease cleavage. This work began with re-examination of results published in 1992 and never followed up describing a cap or cap-like structure on decay products of ß-globin mRNA in patients with ß-thalassemia (Cooley's anemia), a fatal disorder of hemoglobin production that is caused by inheriting two copies of this gene with a premature termination codon. I will describe how we validated those results, some of the basic biochemistry behind the re-capping process, and the identification and properties of a cytoplasmic complex that contains the enzymes that are responsible for mRNA re-capping. The loss of the cap is one of the key steps by which microRNAs repress translation and silence gene expression, and my talk will cover the cycle by which cytoplasmic re-capping may function in re-activating these silenced mRNAs. I will also touch on the possible links between cytoplasmic capping and the activation of neuronal or maternal mRNAs that must be kept in a silenced state until their translation is required. Although at this point it is highly speculative, cytoplasmic capping may also expand the proteome by enabling the translation of different forms of a protein from mRNAs that have lost the cap and sequences from their 5' ends, and the challenges the complexity of this process presents for bioinformatics, molecular and cell biology.

Selection is weak, so evolutionary change is slow. That's the classical picture. But over the last 25 years we've learned that it's often very wrong, and ecologically important traits can change markedly in just a few generations. I will talk about experimental work using aquatic predator-prey microcosms as a model system to study rapid evolution; general theory motivated by those experiments; and methods that we have been developing to apply these ideas to field data, to quantify the importance of evolutionary change and non-heritable trait changes for ecological dynamics. This is joint work with many colleagues, postdocs, and students.

At the cellular level, the detection, amplification, and processing of external chemical signals is affected by random fluctuations that arise within signaling pathways. In the case of the bacterial chemotaxis system we now have enough experimental data to go beyond ensemble averages. I will talk about our recent experimental and theoretical efforts to examine how the network design and spatial arrangement of this model signaling pathway shape the information processing and chemotactic capabilities of the single cell. An interesting result that emerges from this individual cell perspective is a molecular understanding of how cells resolve the compromise between the essential but likely competing behavioral modes of sensing and exploring.

Characterization of the intricate gene regulatory networks that govern organism behavior has led to the discovery a number of small and topologically distinct subnetworks known as 'network motifs'. Along with experimental efforts, dynamical systems modeling has provided new insights into the equilibrium states and transient dynamics of such subnetworks. In this talk I will give a brief overview of some of the more biologically-relevant motifs, detail the successes of the systems biological approach to motif identification, and discuss some of the challenges to the study of network motifs in complex organisms.

In this talk, we consider rigid properties of hyperbolic periodic solutions of dynamical systems of networks. Let X_0=(x^0_1, ... , x^0_n) be a hyperbolic periodic solution to an admissible dynamical system of a network G. Let T be the minimal period of X_0. Cells (nodes) i, j are phase-related on X_0 if there exists a real number theta ( 0 <= theta < 1), such that x^0_i(t)= x^0_j(t+ theta T). The phase relation is rigid if the same phase relation remains in the perturbed periodic solution of any sufficiently small admissible perturbation. When theta =0, the two cells are called rigid synchrony. Rigid phase relation is known to be a product of symmetry. However, Stewart and Parker found that rigid phase relation might occur on a network without symmetry, but one of its quotient network defined by collapsing all rigid synchrony cells to one cell has symmetry. Stewart and Parker conjectured that this is the only way to get rigid phase relations on a transitive network without symmetry.

Stewart et. al. reduced the proving of this conjecture to the proving of the following two other conjectures:

1. phase rigid property: Suppose phase relations on X_0 are rigid, then for each pair of phase-related cells, the signals they receive are also phase-related with the same phase-relation.
2. fully oscillatory property: In a transitive network, a hyperbolic periodic solution of an admissible vector field of the network is generically fully oscillatory (all cells are oscillatory on the periodic solution).

Prostate cancer (CaP) is the second most common cancer in American men. Although the majority of patients diagnosed with CaP are cured with primary treatment, it remains the second lead cause behind only lung cancer, of male cancer-related deaths in the western world. A few features set it apart from other cancers; it develops slowly over a period of years; CaP cells are dependent on male sex hormones for growth; treatment in the form of continuous androgen ablation fails due to the emergence of castrate-resistant CaP cells. Therefore, it has been proposed that intermittent androgen ablation therapy might be a better strategy for treating CaP. I present a model of prostate growth in humans, which can simulate the onset of CaP, as well as explain the emergence of resistance in response to therapy. Our model shall incorporate a variety of cell types such as healthy and CaP cells, as well as detailed biochemical pathways crucial to the growth of these cells. By being able to distinguish between various drug actions, and being fitted to individual patient data, we hope to develop a truly prescriptive tool to aid physicians in treatment choices for CaP patients.

In this talk, recent progress on the mathematical analysis of some reaction-cross diffusion models in population dynamics will be reported. In particular, some novel results concerning blowup of smooth solutions will be presented. This is a joint work with Yuan Lou and Dong Li.

Computational systems biology has brought many new insights to cancer biology through the quantitative analysis of molecular networks. Our goal is to apply a systems biology approach to the understanding of intracellular iron metabolism in normal breast epithelium and the changes the network undergoes as cells transition to malignancy. This talk will describe part of a complex cellular iron network that consists of multiple feedback loops as well as a mathematical model intended to help shed light on key regulatory nodes of iron metabolism dynamics.

This talk describes an inverse problem that arises in connection with cardiac surgery. Surgical correction of atrial fibrillation involves burning small scars on the cardiac wall, a process that in turn requires an accurate understanding of the geometry of the heart chamber. One approach for acquiring this geometry is to insert an electrode within the heart chamber and calculate its position via its readings of three induced orthogonal voltage potentials. While straightforward in principle, in practice this procedure is complicated by the fact that the unknown conductivity of the medium surrounding the heart chamber induces nonlinearities in the fields. If these distortions are modeled as having a low-order expansion in a harmonic basic, the problem can be reduced to solving an inverse problem in which the objective is to use electrical filed data to determine both the probe position and the expansion coefficients. This talk describes recent results illustrating how the success of this endeavor fundamentally depends on factors such as probe shape and basis choice. The talk will begin with a brief tutorial introduction to the field of inverse problems, highlighting important concepts and techniques. The second half will examine the position registration problem described above within the framework of these ideas.

Wound healing is a complicated orchestration of cells and biological signals that changes over the life of the wound. Chronic wounds, such as pressure ulcers or the foot sores of diabetics, are breaches in the skin that often refuse to heal. In this talk, we present a mathematical model of chronic wounds that incorporates the interactions of different type of cells, chemicals and the extracellular matrix (ECM) that are involved in the healing process.

The model consists of a coupled system of partial differential equations in the partially healed region, with the wound boundary as a free boundary. The ECM is assumed to be viscoelastic, and the free boundary moves with the velocity of the ECM at the boundary. The model variables include the concentration of oxygen, PDGF and VEGF, the densities of macrophages, fibroblasts, capillary tips and sprouts, and the density and velocity of the ECM. Simulations of the model demonstrate how oxygen deficiency may limit macrophage recruitment to the wound-site and impair wound closure. The results are in general agreement with experimental findings in an animal model.

During the 1990s the Gulf of Maine (GOM) underwent an ecosystem regime shift associated with an increase in freshwater inputs. This freshening has been linked to increased phytoplankton abundance, which in turn positively affected the growth of zooplankton and, consequently, many pelagic fish populations. Calanus finmarchicus is one of the most abundant species of zooplankton in the GOM and so is an important prey source for many species higher up the food chain such as herring and the North Atlantic right whale. While reproduction for C. finmarchicus was high during this period, abundance of the later stages of the surface population was paradoxically low. Adult herring preferentially feed on the later copepodid stages; it is therefore possible that increased herring presence exerted top-down control on C. finmarchicus. An alternative hypothesis is that the changes in phytoplankton abundance during the 1990s impacted recruitment of C. finmarchicus into the later stages. Specifically, phytoplankton variability may impact whether C. finmarchicus remain at the surface to reproduce or enter into a resting state until the following year, emerging to take advantage of the spring bloom. Using three simple differential equation models, we examined the interplay of top-down verses bottom-up processes on the observed changes in seasonal patterns of surface populations of late-stage C. finmarchicus.

Sleep and wake states are each maintained by activity in a corresponding neuronal network, with mutually inhibitory connections between the networks. In infant mammals, the durations of both states are exponentially distributed, whereas in adults, the wake states yield a heavy-tailed distribution. What drives this transformation of the wake distribution? Is it the altered network structure or a change in neuronal dynamics? What properties of the network are necessary for maintenance of neural activity on the network and what mechanisms are involved in transitioning between sleep and wake states? We explore these issues using random graph theory, specifically looking at stochastic processes occurring on random graphs, and also by investigating the accuracy of predictions made by deterministic approximations of stochastic processes on networks.

The problem of invasive species is thought to be second only to habitat destruction as a threat to biodiversity. Eradication strategies applied over spatial domains can take in many cases up to decades before achieving local extinction of the targeted invasive species. These practical efforts demand correct estimation of the outcome of the strategy before committing substantial economic and political resources over long periods of time. This talk discusses how the existence of global attractors in an infinite dimensional dynamical system, representing genetic control of an invasive species, can be used in spatial ecology to determine a state of local extinction. It is shown that in some cases it is possible to determine for a finite time the existence of a state of local extinction, and the conditions under which this happens. The Trojan Y Chromosome and the Daughterless Male eradication methods are presented and compared.

Synchronization is the essential function of many biological networks. The synchronization of pace-maker cells in the heart creates a pulse that drives blood throughout the body. The synchronization of specialized neurons in the brain creates a circadian clock that keeps the body in time with the day. It is difficult to determine the origins of this synchrony, in part, because biological networks are typically complex and impossible to observe. In particular, the topological structure of many biological networks remains unknown. Furthermore, it is costly to conduct experiments capable of determining the function of observed network features. As a result, theoretical explorations of emergent synchrony are called for. We use optimal control theory to build networks that maximize synchrony. Analysis of these optimal networks allows us to identify topological features that promote synchrony.

Periodic reversals of the direction of motion in systems of self-propelled rod shaped bacteria enable them to effectively resolve traffic jams formed during swarming and maximize their swarming rate. In this talk, a connection is shown between a microscopic one dimensional cell-based stochastic model of reversing non-overlapping bacteria and a macroscopic non-linear diffusion equation for the dynamics of cellular density. Boltzmann-Matano analysis is used to determine the nonlinear diffusion equation corresponding to the specific reversal frequency. A combination of microscopic and macroscopic models are used for studying swarming rates of populations of bacteria reversing at different frequencies. Cell populations with high reversal frequencies are able to spread out effectively at high densities. If the cells rarely reverse, then they are able to spread out at lower densities but are less efficient at spreading out at higher densities.

Neurons in the suprachiasmatic nucleus (SCN) of the hypothalamus are thought to communicate time of day information through circadian (~24-hour) variation of their firing frequency, with low rates during the night and higher rates during the day. Based on a mathematical model of the ionic currents within SCN neurons, we predict that the neural code of the SCN is more complex and that throughout the day clock-containing SCN neurons can transition between firing and quiescent states, including an unusual depolarized rest state. We also simulate networks of SCN neurons at a set circadian phase with GABAergic coupling, and observe the formation of clusters of neurons with near synchronous firing. We find that the clustering depends on network properties such as synaptic strength and density. Experimental data supporting these modeling results will be discussed.

Fiddler crabs form loose foraging groups, but under threat of predation, these groups (flocks) tend to clump up. We report on fieldwork of Viscido and Wethey testing Hamilton's "selfish herd" hypothesis, and then joint work we did attempting to model the observed behavior. The novelty was to take Hamilton's "domains of danger" literally; in mathematical terms these are the Voronoi polygons surrounding each individual in the aggregation. We took an individually based modeling approach, running simulations of various movement rules, each of which leads to some clustering of organisms. In terms of reduction of domains of danger, the "local crowded horizon" rule appeared to work the best. This leads to a number of questions about how to recognize groups and model their behavior.

Classical vaccines rely on the induction of serum antibodies that are maintained at high levels for prolonged periods after vaccination, resulting in practically sterilizing immunity. Although this ideal scenario is not feasible for most pathogens, vaccine researchers are progressively presenting partial solutions. There are candidate vaccines against various pathogens that offer modest protection, and there is growing conviction that new efficacy measures and new experimental designs are needed to characterize the effects of immunity, both at individual and population scales. Recent developments - conceptual and methodological - will be presented for addressing this timely problem that is both of theoretical and practical interest.

In this talk I will discuss how a neural network can learn to approximate optimal decision making in cases where there is overlap in the possible signals and an arbitrary number of alternatives. I will begin with a general discussion of decision making and introduce the drift-diffusion model which has been very successful in capturing much of observed experimental behavior in monkeys and humans when there are two alternatives. I will then discuss how these models for decisions between two alternatives may be extended to the general case of an arbitrary number of choices, and how optimality considerations place conditions on networks implementing such models. Finally, I will discuss how such networks can adapt (learn) over trials to approximate optimality.

NIH and FDA's vision of personalized medicine involves a drug, and a companion biomarker test identifying the patient subgroup that the drug targets. Examples of personalized medicine approved by the FDA are surprisingly few. (Can you name the only 3 microarray devices approved by the FDA?) Yet, personalized medicine is beginning to take shape. After the FDA issued its VGDS (Voluntary Genomic Data Submission) draft guidance in 2005, drug companies have been routinely banking biological samples from clinical trials. In June 2010, FDA held public hearing on regulation of genetic prognostic/diagnostic tests. This seminar will indicate what the hundreds of PhDs with quantitative training working in the pharmaceutical industry can expect to be the skills that will be needed as medicine transitions from "on average" to "for individuals".

I will first give a solid mathematical foundation of multiple testing that is difficult to gather from literature. Then I will focus on two typically overlooked issues in testing of biomarkers. One issue is interpretation of an unconditional expectation error rate such as FDR, however it is controlled. The other issue, which has only recently come to light, is the ever popular permutation testing requires a strong assumption on the (unknown) joint distribution of biomarkers to control its error rate. These and other issues will be illustrated in the Genome-wide Association Studies (GWAS) setting.

Cholera is a waterborne intestinal infection which causes profuse, watery diarrhea, vomiting, and dehydration. It can be transmitted via contaminated water as well as person to person, with 3-5 million cases/ year and over 100,000 deaths/year. A major public health question involves understanding the modes of cholera transmission in order to improve control and prevention strategies. One issue of interest is: given data for an outbreak, can we determine the role and relative importance of waterborne vs. person-to-person routes of transmission? To examine this issue, we explored the identifiability and parameter estimation of a differential equation model of cholera transmission dynamics. We used a computational algebra approach to establish whether it is possible to determine the transmission rates from outbreak case data (i.e. whether the transmission rates are identifiable), and then applied the model to a recent cholera outbreak in Angola which resulted in over 80,000 cases and over 3000 deaths. Our results show that both water and person-to-person transmission routes are identifiable, although they become practically unidentifiable with fast water dynamics. Using these results, parameter estimation for the Angola outbreak suggests that both water and person-to-person transmission are needed to explain the observed cholera dynamics. I will also discuss some ongoing work using this model, including modeling the spatial spread of outbreaks, public health interventions and control strategies, and applications to the ongoing cholera outbreak in Haiti.

Large-scale shifts in habitat during evolution require lineages to respond to new selective pressures, often resulting in a cavalcade of novel morphologies. In cases where distantly related taxa occupy similar or sympatric habitats, similar characteristics often arise independently, resulting in convergence. Cephalopods are a morphologically diverse group marine molluscs whose members have undergone several transitions between pelagic (free-swimming) and benthic (associated with the bottom) lifestyles, which has lead to the evolution of apparent convergences such as light-producing organs and a cornea covering the eye. To uncover the molecular mechanisms influencing convergent evolution in cephalopods, I utilize next-generation sequencing techniques to analyze gene expression patterns in the cephalopod cornea, a structure that has arisen independently in the squid and octopus lineages. Results from this study provide new insight into the origins of complexity, and into the impact molecular mechanisms such as gene sharing and duplication have on a macro-evolutionary scale.

In traditional maximum likelihood phylogenetic tree inference, only the mutational process is considered for explaining the variation seen in the sequences, and the history of each gene analyzed is assumed to reflect the history of the species.

However, there are other mechanisms responsible for genetic variation between species, and the most influential of them is the coalescent process, which explains how sequence variation can be retained in a population, and how each gene tree does not necessarily reflect the history of the species.

With next-generation sequencing becoming less expensive, there will be a massive influx of sequence data in the near future, and with multi-gene datasets, the effect of the coalescent process will be more important to take into consideration when estimating the species tree. A set of sequenced transcriptomes will have genes sampled randomly, with a high frequency of missing data for each gene when considered across all sampled species.

Here I present a simulation study on the effects of missing data on estimating the species tree from a set of gene trees when taking the coalescent process into consideration. We have examined the effects on species tree estimation from sampling several lineages per species, different degrees and patterns of missing data and recent and older speciations.

Infectious diseases have a long history of study within many subdisciplines of biology and mathematics. This talk will be an overview of two projects where dynamical systems theory is used to study two infectious disease systems. In each case, the multiple time scales naturally present in these system help simplify the mathematical analyses necessary for answering our motivating biological questions.

Within-host dynamics of Mycoplasma infections. The goal of this work is to clarify the role of the host immune response in shaping the the dynamics of Mycoplasma gallisepticum infections in the house finch (Carpodacus mexicanus) and other wild birds. I will present a pathogen-immune model of a localized Mycoplasma infection along with analytical and computational results describing the dynamics of this model. These results will be used to address the role of the host immune system in shaping individual and species level variation in disease progression, and to suggest directions for future empirical study of this system.

Infectious disease in predator populations: Consequences of prey-mediated transmission and infectiousness. The second half of my talk is motivated by recent field and laboratory study of Daphnia sp. (the "predators") and their parasites, which find that Daphnia's algal food source can directly drive rates of disease transmission and infectiousness. I will introduce a single model framework that includes predator-prey (a.k.a. consumer-resource) and disease processes, then I'll present both analytical and computational results that show how predator-prey and disease dynamics are affected when trophic interactions directly affect disease transmission and infectiousness among predators.

Noise in cardiac pacing cycles, for instance, the heart rate variability, has been observed and researched for decades. Contemporarily, various cardiac models have been constructed to investigate the electric activity of the cardiac cells. Yet there has not been a study on extracting information of the underlying dynamics if some noisy data are given. In this talk we will show a method to determine the cardiac restitution approximately in the range of the pacing cycles provided a series of noisy data. We assume the data are generated through some unknown mapping model with memory, and the memory is supposed to be hidden and not able to be detected.

Heterogeneity is a fundamental issue in mathematical epidemiology. We expect many factors influencing disease transmission to vary across populations and different spatial scales. Many results exist for the effect of heterogeneity on the spread of disease for SIR type models, where transmission occurs as a result of direct contact with infected individuals. Waterborne disease, such as cholera, may be spread through contact with a contaminated water source as well as through direct person-person transmission. We investigate the effect of heterogeneity in both transmission pathways on the value of the basic reproductive number R0 in multi-patch SIWR models, specifically a system of N patches sharing a common water source.

Several methods have been proposed for correlating genomic sequence patterns directly with phenotypes of similar organisms. However, the evolutionary relationships between organisms lead to non-independence among the sequences. A phylogenetic tree reconstruction uncovers sibling lineages where the phenotypes first start to differentiate, and, conditional on this tree, PhyloPTE adopts an additive hazard model to identify likely mutational paths along the tree as the phenotypes fully develop.

For example, the HIV-1 virus has a population structure reflecting both transmission between individuals and evolution of the HIV-1 quasispecies within each patient. Non-independence can introduce spuriously strong correlation between unrelated mutations giving a false appearance of causation. These evolutionary relationships are an issue even in HIV-1 where recombination is rapid, and they are pervasive in humans, where linkage disequilibrium is extensive. In human disease studies, spurious correlation can sometimes be overcome by pedigree analysis or simple sibling studies: alleles common only in "sick" siblings are likely true causative alleles.

PHYLOPTE's advantages include: incorporating information about branch lengths to infer mutational rates; computational speed practical for high-throughput (next generation)sequence data; estimates of relative influence of different effects; and improved precision even versus other tree-based methods: 50%-300% improvement in precision at same recall, either to predict experimental correlations (obtained from STRING:http://string-db.org/) or in simulations under biologically reasonable parameters on HIV quasispecies sequence trees.

This is joint work with Joseph Verducci, Daniel Janies, and Jesse Kwiek of The Ohio State University.

Overfishing, pollution and other environmental factors have greatly reduced commercially valuable stocks of fish. In a 2006 Science article, a group of ecologists and economists warned that the world may run out of seafood from natural stocks if overfishing continues at current rates. In this talk, we will explore the interaction between constant and periodic proportion harvest policies and recruitment dynamics. In case studies, we analyze these policies and illustrate how they might be applied to Gulf of Alaska Pacific halibut fishery and the Georges Bank Atlantic cod fishery based on harvest rates from 1975 to 2007.

In this talk, we use a generalized age-structured model to discuss the role that both compensatory (non-oscillatory) and overcompensatory (oscillatory) dynamics play in the long term dynamics of exploited fisheries. When each species is governed by compensatory dynamics via the Beverton-Holt model and the predator's response to species interaction is modeled using a linear function, we show that the predator-prey model exhibits a globally stable positive fixed point. In stark contrast, we show that when each species is governed by compensatory dynamics via the Beverton-Holt model and the predator response function is exponential, then the predator-prey model exhibits population oscillations.

The Ricker model is used to explore the impact of overcompensatory dynamics on the long-term viability of exploited fish populations. In particular, we investigated the bifurcations of the positive fixed point and give specific parameter values for when the predator-prey system undergoes flip bifurcation (stability shifts from the interior fixed point to a 2-cycle) and Neimark-Sacker bifurcation (stability shift from the interior fixed point to an invariant loop). We also illustrate the presence of alternate life-history outcomes (multiple attractors).

In addition, we use the predator-prey model to exam the state of several fish populations in a Georges Bank food chain. We consider Ricker recruitment functions and Beverton-Holt recruitment functions to determine the best model fit for interacting Cod-Silver hake populations, Cod-Herring populations and Silver hake-Herring populations and use these results to make predictions on the sustainability of the exploited populations.

This will be an introductory lecture on the auditory system for mathematicians. I'll start with sound waves, discuss the outer, middle, and inner ears, the VIIIth nerve, and up into the brain stem. Then I'll outline work on sound localization, stochasticity, hyperacuity, and the ocular-vestibular reflex. The emphasis will be on mathematical issues and the need for new conceptual constructs.

The talk is based on three chapters from a popular science book that I've just delivered to the publisher. The main aim of the book - and the talk - is to illustrate the broad range of mathematical ideas and methods now being used to provide insight into biological questions. The samples in the talk involve knot theory, multidimensional geometry, group theory (symmetry), and networks. As befits a popular science book, the mathematical technicalities will be omitted.

1. Virus from the Fourth Dimension (work of Reidun Twarock and others): Use of multidimensional geometry to understand the structure of virus protein coats.
2. Networking Opportunities (work of Atsushi Tero and others): Biological networks, including how to use slime mould to design the Tokyo railroad system.
3. Knots in DNA (work of Nicholas Cozzarelli and others): How the twisted topology of the DNA double helix can be used to work out how topoisomerases act, in particular Tn3 resolvase.

Identifying mechanisms for the onset of cardiac arrhythmias is an important component of ongoing research in electrophysiology. Mathematically, abnormal rhythms such as ventricular tachycardia and fibrillation can be identified with spiral waves and spatiotemporal chaos, respectively. Understanding the precursors of such arrhythmias is possible even if we restrict ourselves to an idealized one-dimensional fiber of cardiac cells. In this presentation, I will use asymptotic methods to reduce a standard Hodgkin-Huxley type PDE model of a cardiac fiber to a system of ODEs which is amenable to mathematical analysis. My calculations exploit a particular feature of cardiac tissue known as electrical restitution: the speed and duration of cardiac action potentials depends upon how [locally] well-rested the tissue is. The kinematic model that I will introduce is far less computationally expensive than standard PDE models, making it feasible to run repeated numerical experiments. I will discuss one such experiment: the use of far-field pacing and feedback control to terminate chaos.