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.
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 one-sided stable law, together with results from 10-60 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.
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 Lambda-Fleming-Viot process.
Metagenomics attempts to sample and study all the genetic material present in a community of micro-organisms in environments that range from the human gut to the open ocean. This enterprise is made possible by high-throughput 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 pre-existing 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.
A new generation of anti-cancer drugs targeted to specific oncogenic pathways has emerged in recent years as a promising alternative to chemotherapy and radiation. However, the clinical success of such drugs has been limited by the evolution of acquired resistance, which leads to a relapse after initial response to therapy. In the first part of this talk I will discuss the use of a simple two-type branching process model to represent sensitive and resistant cells in a population. Using this validated model we propose novel scheduling strategies in the treatment of non-small cell lung cancer with targeted therapy, which will delay resistance. Then, we investigate the large population limit of these processes in the time scale of extinction of the sensitive cell population. Using these limits we can study the turnaround point when the total tumor size goes from subcritical to supercritical (corresponding to the clinical time of disease progression), as well as the population levels of sensitive and resistant cells at various clinically relevant times.
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 much-studied interspike-interval distribution depends on a process exiting from a quasi-stationary 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.
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.
Bacterial plasmids are circular extra-chromosomal 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 surface-associated populations.
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.
We study models describing the evolution of a sexual (diploid) population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. We take into account the genetics of the reproduction. Each individual is characterized by two allelic traits and the associated phenotypic trait. The population is described as a point measure valued process with support on the genotype space. Its dynamics is a birth and death dynamics with selection and Mendelian rule in the reproduction and competition between individuals. Allelic mutations may occur during the reproduction events. The population size is assumed to be large and the mutation rate small. We prove that under a good combination of the scales, and if the mutation steps are small, the population process converges in a long time scale by a jump process jumping from a monomorphic homozygote equilibrium to another one. This study involves a three-types diploid nonlinear dynamical system, which is studied using small perturbations of a neutral case. The behavior of the process is thus studied near the evolutionary singularities.
This work is a joint work with Pierre Collet (Ecole Polytechnique) and Hans Metz (Leiden University).
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 tree-valued stochastic process. We will study this process in the case of populations whose genealogy is given by the Bolthausen-Sznitman 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.
We discuss a Fleming-Viot model whose mutation process is a birth- and death process on the non-negative 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.
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 large-N limit.
Deterministic dynamic models with delayed feedback and state constraints arise in a variety of applications in science and engineering, including biology. There is interest in understanding what effect noise has on the behavior of such models. Here we consider a multidimensional stochastic delay differential equation with normal reflection as a noisy analogue of a deterministic system with delayed feedback and positivity constraints. We obtain sufficient conditions for existence and uniqueness of stationary distributions for such equations. The results are applied to an example of a simple biochemical reaction system.
G. Edelman, O. Sporns and G. Tononi have introduced in theoretical biology the neural complexity of a family of random variables, defining it as an average of mutual information over subsystems, with the aim of quantifying the complexity of the brain. We provide a mathematical framework for this concept, studying in particular the problem of maximization of such functional for fixed system size and the asymptotic properties of maximizers as the system size goes to infinity. We shall also discuss some possible developments and applications of our work. (Joint work with Jerome Buzzi)
Even if Super Brownian Motion (SBM) in 2 dimensions is a measure valued process, the theory of historical processes allows to track the "path" that each particle takes, in the interpretation of SBM as a cloud of particles. This allows to define when two sets on the plane are connected through SBM, which we define as the event that a particle, or one of its ancestors, spent a positive amount of time on both sets before it died. I showed that the probability of this event is invariant under a bijective conformal transformation.