
Thomas Kurtz
For chemical reaction networks in biological cells, reaction rates and chemical species numbers may vary over several orders of magnitude. Combined, these large variations can lead to subnetworks operating on very different time scales. Separation of time scales has been exploited in many contexts as a basis for reducing the complexity of dynamic models, but the interaction of the rate constants and the species numbers makes identifying the appropriate time scales tricky at best. Some systematic approaches to this identification will be discussed and illustrated by application to one or more complex reaction network models.

Steve Krone
Bacterial plasmids are circular extrachromosomal genetic elements that code for simultaneous resistance to multiple antibiotics and are thought to be one of the most important factors in the alarmingly rapid loss of our arsenal of antimicrobial drugs. Plasmids propagate horizontally by infectious transfer, as well as vertically during cell division. Horizontal transfer requires contact between donor and recipient cells, and so spatial structure can play a key role in mediating the spread of antibiotic resistance genes. We will discuss ODE and stochastic spatial models of plasmid population dynamics, as well as empirical results. As an example of the effects of spatial structure, we will use the spatial model to evaluate the effectiveness of a commonly used estimate of plasmid transfer efficiency when applied to surfaceassociated populations.

Priscilla Greenwood
How does a stochastic process move between the domains of attraction of locally stable points or cycles of an associated deterministic system, and cross unstable cycles? This question arises when we try to quantify the behavior of a neuron in terms of a stochastic neuron model. In the Morris Lecar model, for instance, the muchstudied interspikeinterval distribution depends on a process exiting from a quasistationary state near a fixed point and crossing an unstable limit cycle. When a process encounters an unstable cycle it tends to follow along a bit. But we need to do better than that.
References:
* P Baxendale and P E Greenwood, Sustained oscillations for density dependent Markov processes. J. Math Biology, Sept 2011.
* S Ditlevsen and P E Greenwood, (2011) The MorrisLecar neuron model embeds a leaky integrateandfire model, arXiv 1108.0073.
* P F Rowat and P E Greenwood, Identification and continuity of the distributions of burst length and interspike intervals in the stochastic Morris Lecar neuron. Neural Computation, to appear.

Jason Schweinsberg
Consider a population of fixed size that evolves over time. At each time, the genealogical structure of the population can be described by a coalescent tree whose branches are traced back to the most recent common ancestor of the population. This gives rise to a treevalued stochastic process. We will study this process in the case of populations whose genealogy is given by the BolthausenSznitman coalescent. We will focus on the evolution of the time back to the most recent common ancestor and the total length of branches in the tree.

Steven Evans
Metagenomics attempts to sample and study all the genetic material present in a community of microorganisms in environments that range from the human gut to the open ocean. This enterprise is made possible by highthroughput pyrosequencing technologies that produce a "soup" of DNA fragments which are not a priori associated with particular organisms or with particular locations on the genome. Statistical methods can be used to assign these fragments to locations on a reference phylogenetic tree using preexisting information about the genomes of previously identified species. Each metagenomic sample thus results in a cloud of points on the reference tree. In seeking to answer questions such as what distinguishes the vaginal microbiomes of women with bacterial vaginosis from those of woment who don't, one is led to consider statistical methods for distinguishing between such clouds. I will discuss joint work with Erick Matsen from the Fred Hutchinson Cancer Research Center in which we use ideas based on distances between probability measures that go back to Gaspard Monge's 1781 "M'emoir sur la th'eorie des d'eblais et des remblais" as well as some familiar objects (e.g. reproducing kernel Hilbert spaces) from the world of Gaussian processes.

Anton Wakolbinger
We discuss a FlemingViot model whose mutation process is a birth and death process on the nonnegative integers. In this model, new deleterious mutations accumulate at a constant rate per generation, and each mutation decreases the individual fitness by a constant amount. Other than in the classical case of Muller's ratchet, each of the present mutations has a small probablity per generation to disappear. In the infinite population limit we obtain the solution in a closed form by analyzing a probabilistic particle system that represents this solution. We will also discuss recent ideas to approach (yet unsolved) questions on the rate of Muller's ratchet. The talk is based on joint work with Peter Pfaffelhuber and Paul Staab.

Paul Joyce
In relating genotypes to fitness, models of adaptation need to be both computation ally tractable and to qualitatively match observed data. One reason tractability is not a trivial problem comes from a combinatoric problem whereby no matter in what order a set of mutations occurs, it must yield the same fitness. We refer to this problem as the bookkeeping problem. Because of their commutative property, the simple additive and multiplicative models naturally solve the bookkeeping problem. However, the fitness trajectories and epistatic patterns they predict are inconsistent with the patterns commonly observed in experimental evolution. This motivates us to propose a new and equally simple model that we call stickbreaking. Under the stickbreaking model, the intrinsic fittness effects of mutations scale by the distance of the current background to a hypothesized boundary. We use simulations and theoretical analyses to explore the basic properties of the stickbreaking model such as the distribution of fitness effects, fitness trajectories, and epistasis. Stickbreaking is compared to the additive and multiplicative models using a number of novel likelihood based approaches to account for error in the predictions. We apply or statistical methodology to a number recently published data sets and conclude the stickbreaking model is consistent with several commonly observed patterns of adaptive evolution.

Alison Etheridge
Classical models for gene flow fail in (at least) three ways. First, they cannot explain patterns in data observed over large scales; second, they predict much more genetic diversity than is observed; and third, they asssume that genetic loci evolve independently. I shall describe, as time permits, results of joint projects with Nick Barton, Nathanael Berestycki, Jerome Kelleher and Amandine Veber in which we have proposed a framework for modelling populations that are distributed across a spatial continuum and analysed aspects of a particular model that arises in this framework that we have called the spatial LambdaFlemingViot process.

Rick Durrett
It is common to use a multitype branching process to model the accumulation of mutations that leads to cancer progression, metastasis, and resistance to treatment. In this talk I will describe results about the time until the first type k (cell with k mutations) and the growth of the type k population obtained in joint work with Stephen Moseley, and their use in evaluating possible screening strategies for ovarian cancer, work in progress with Duke undergraduate Kaveh Danesh. The point process representation of the limit, which is a onesided stable law, together with results from 1060 years ago leads to remarkable explicit formulas for Simpson's index and the size of the largest clone. These results are important in understanding tumor diversity which can present serious obstacles to treatment. The last topic is joint work with Jasmine Foo, Kevin Leder, John Mayberry, and Franziska Michor.

Nick Barton
An individual passes on random segments of her genome to future generations: typically, most of the genome is lost, but a small fraction survives, in many copies. This distribution of surviving blocks can be calculated using a branching process argument. Remarkably, after a few tens of generations it has the same form for every individual, with variation in reproductive value between individuals only affecting the probability of survival. These results follow the descent of genomes forwards in time. The converse problem is to ask how far back we can reconstruct the pedigree, given a sample of complete genomes.

John White
One of the principal tasks of the nervous system is to generate internal representations of the world, in order that we might best interpret the present, predict the future, and thus pass our genes to the next generation. For this reason, it seems quite surprising that the nervous system is so noisy. This noise is reasonably well characterized at the level of ion channels and individual cells, but it is not clear why such a level of noise is tolerated. In this talk, I will describe some of the major sources of noise in the nervous system, and will discuss some of the current puzzles regarding how the effects of noise scale in the largeN limit.

John White
One of the principal tasks of the nervous system is to generate internal representations of the world, in order that we might best interpret the present, predict the future, and thus pass our genes to the next generation. For this reason, it seems quite surprising that the nervous system is so noisy. This noise is reasonably well characterized at the level of ion channels and individual cells, but it is not clear why such a level of noise is tolerated. In this talk, I will describe some of the major sources of noise in the nervous system, and will discuss some of the current puzzles regarding how the effects of noise scale in the largeN limit.

Nick Barton
An individual passes on random segments of her genome to future generations: typically, most of the genome is lost, but a small fraction survives, in many copies. This distribution of surviving blocks can be calculated using a branching process argument. Remarkably, after a few tens of generations it has the same form for every individual, with variation in reproductive value between individuals only affecting the probability of survival. These results follow the descent of genomes forwards in time. The converse problem is to ask how far back we can reconstruct the pedigree, given a sample of complete genomes.