I will start with an overview over various models for the dynamics of the genetic composition of populations evolving under recombination. For the deterministic treatment that applies in the infinite-population limit, one has large, nonlinear dynamical systems; for the stochastic treatment required for finite populations, the Moran, or Wright-Fisher model is appropriate.
Recent years have seen increasing attention to the subtle effects on intracellular transport caused when multiple molecular motors bind to a common cargo. We develop and examine a coarse-grained model which resolves the spatial configuration as well as the thermal fluctuations of the molecular motors and the cargo. This intermediate model can accept as inputs either common...
A stochastic model for variable-length stepping of kinesins engineered with extended neck linkers is developed. This requires consideration of the separation in microtubule binding sites between the heads of the motor at the beginning of a step. It can be shown that the separation is a stationary Markov process and can be included in the calculation of standard experimenta...
The growing field of synthetic biology aims to forward engineer biology both for applications such as energy production, drug production, and bioremediation, as well as for the purpose of furthering the fundamental understanding of natural systems. However, engineering living cells is notoriously difficult due to issues such as mutation, epigenetic variation, fitness effec...
Populations, whether they be viral particles, bio-chemicals, plants or animals, are subject to intrinsic and extrinsic sources of stochasticity. This stochasticity in conjunction with nonlinear interactions between individuals determines to what extinct populations are able to persist in the long-term. Understanding the precise nature of these interactive effects is a cent...
The SLFV process is a population model in which individuals live in a continuous space. Each of them also carries some heritable type or allele. We shall describe the long-term behaviour of this measure-valued process and that of the corresponding genealogical process of a sample of individuals in two cases : one that mimics the evolution of nearest-neighbour voter model (...
The dynamics of a virus infection within its host is governed at its earliest stages by processes at the molecular and cellular scale. We are developing cell-culture measurements and computational models to better understand how these and other processes contribute to the early dynamics of virus growth and infection spread. As a model system we study vesicular stomatitis v...
Deterministic population dynamical models connect to reality through their interpretation as limits for systems size going to infinity of stochastic processes in which individuals are represented as discrete entities. In structured population models individuals may be born in different states (e.g. locations in space) after which they proceed through their h(eterogeneity)-...
In deterministic epidemic models, pathogen extinction in a population is determined by the magnitude of the basic reproduction number R0. In stochastic epidemic models, the probability of pathogen extinction depends on R0, the size of the population and the number of infectious individuals. For example, in the SIS Markov chain epidemic model, if the basic reproduction numb...
We derive kinetic differential or integrodifference equations for the mean and variance or of a quantitative trait as a function of space and time, in some cases recovering known equations and in some cases obtaining new ones that capture effects, such as nonmonotonicity of traveling waves, that can be seen in stochastic simulations. We then reanalyze kinetic equations due...
In a toy model of binary splitting branching processes with population size dependence (supercritical below and subcritical above a threshhold, the carrying capacity) the chance of a little population establishing itself in the sense of reaching a band around the carrying capacity is determined, and so is the persistence time of the population. Mutations and competition be...
I will focus on computational methods for stochastically modeled biochemical reaction networks. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain. I will show how different computational methods can be...
Stochastic biochemical systems and population genetics models are described by similar mathematical equations, and hence similar phenomena should be observed in both systems. Here we focus on stochastic kinetics with time scale separation. We show how to integrate out the fast degrees of freedom, while rigorously preserving their effects on the fluctuations of slower varia...