Workshop 2: Stochastic Processes in Cell and Population Biology

Organizers

Timothy Elston
Department of Pharmacology, University of North Carolina, Chapel Hill
Thomas Kurtz
Mathematics and Statistics, University of Wisconsin
Johan Paulsson
Dept of Systems Biology, Harvard University
Michael Simpson
CNMS, Oak Ridge National Laboratory

Many stochastic features of intracellular processes have close counterparts in population biology. Intrinsic and extrinsic noise in gene expression are similar to demographic and environmental noise in the sizes of metapopulations, with expression bursts corresponding to litter sizes. The near-critical dynamics when two subunits form hetero-dimers are in turn similar to gender-balance fluctuations for organisms that form couples, while intracellular incompatibility is similar to the mutual exclusion principle in ecology. The effects of the cell cycle on chemical abundances resemble seasonality effects on populations, while partitioning errors at cell division are similar to the combination of migration and death before returning to breeding grounds. Both fields also study density-dependent negative and positive feedback loops formed through mutual interactions between agents.

Though the selective pressures are different, there are very strong synergies between these two fields, both in terms of the effects studied and the mathematical methods used. This workshop will exploit those synergies by bringing together researchers who study stochastic aspects of cell and population biology.

Accepted Speakers

Linda Allen
Department of Mathematics and Statistics, Texas Tech University
David Anderson
Mathematics, University of Wisconsin - Madison
Ellen Baake
Faculty of Technology,
Gabor Balazsi
Systems Biology, Unit 950, University of Texas M. D. Anderson Cancer Center
Timothy Elston
Department of Pharmacology, University of North Carolina, Chapel Hill
John Fricks
Statistics, Pennsylvania State University
Peter Jagers
Mathematical Sciences,
David Karig
Center for Nanophase Materials Sciences, Oak Ridge National Laboratory
Peter Kramer
Mathematical Sciences, Rensselaer Polytechnic Institute
J. A. J. (Hans) Metz
Plant Ecology and Phytochemistry, Analysis and Dynamical Systems, Institute of Biology, Mathematical Institute
Judith Miller
Mathematics and Statistics, Georgetown University
Ilya Nemenman
Physics and Biology, Emory University
Hans Othmer
School of Mathematics, University of Minnesota
Garegin Papoian
Chemistry and Biochemistry, University of Maryland
Linda Petzold
Mechanical Engineering, University of California, Santa Barbara
Sebastian Schreiber
Department of Evolution and Ecology, University of California, Davis
Michael Simpson
CNMS, Oak Ridge National Laboratory
Gurol Suel
Pharmacology, UT Southwestern Medical Center
Lev Tsimring
BioCircuits Institute, University of California, San Diego
Amandine Veber
CMAP,
Leor Weinberger
Gladstone Institute For Virology/Department of Biochemistry and Biophysics, UCSF
John Yin
Chemical & Biological Engineering, University of Wisconsin
Monday, October 24, 2011
Time Session
09:00 AM
09:50 AM
Johan Paulsson - Fluctuations in cells and ecosystems
(no description available)
10:30 AM
11:20 AM
Linda Allen - Pathogen Extinction in Stochastic Models of Epidemics and Viral Dynamics
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 number R0>1, the population size is large and I(0)=a is small, then a classic result of Whittle (1955) gives an approximation to the probability of pathogen extinction: (1/R0)a. This classic result can be derived from branching process theory. We apply results from multitype Markov branching process theory to generalize this approximation for probability of pathogen extinction to more complex epidemic models with multiple stages, treatment , or multiple populations and to within host models of virus and cell dynamics.

Work done in collaboration with Yuan Yuan and Glenn Lahodny.
11:30 AM
12:20 PM
Lev Tsimring - Accelerated stochastic simulation algorithm for modeling evolutionary population dynamics
Evolution and co-evolution of ecological communities are stochastic processes often characterized by vastly different rates of reproduction and mutation and a coexistence of very large and very small sub-populations of competing species. This creates serious difficulties for accurate statistical modeling of evolutionary dynamics. In this talk, we review recent progress in this area and introduce a new exact algorithm for fast fully stochastic simulations of birth/death/mutation processes. It produces a significant speedup compared to the direct stochastic simulation algorithm in a typical case when the total population size is large and the mutation rates are much smaller than birth/death rates. We illustrate the performance of the algorithm on several representative examples: evolution on a smooth fitness landscape, Kauffmann's NK model, directed evolution of a regulatory gene network, and stochastic predator-prey system. (joint work with William Mather)
02:30 PM
03:20 PM
J. A. J. (Hans) Metz - Effective population sizes and the canonical equation of adaptive dynamics
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)-state space, e.g. spanned by their i(dividual)-state and location. On such models one can graft evolutionary processes like random genetic drift or adaptive evolution by rare repeated substitutions of mutants in heritable traits affecting the state transition and reproduction processes of individuals. From this general perspective I will derive the so-called Canonical Equation of adaptive dynamics, a differential equation for evolutionary trait change derived under the additional assumption that mutations have small effect. In the CE approximation the rate of evolution is found to correspond to the product of a parameter $n_{e,A}$, equal to the population size times a dimensionless product of life history parameters (including spatial movements), times the gradient of the invasion fitness of potential mutants with respect to their trait vector. From a heuristic connection with the diffusion approximation for genetic drift it follows that $n_{e,A} = n_{e,D}$, the effective population size from population genetics.
04:00 PM
04:50 PM
Gurol Suel - Cellular decision-making in the context of population dynamics
How do cells execute decisions to cope with and survive under environmental conditions? My laboratory focuses on understanding how the dynamics of genetic circuits comprised of interactions between genes and proteins allow cells to govern decision-making. Interestingly, we find that stochastic fluctuations that are inherent to the biochemical reactions within genetic circuits can allow cells to cope with unpredictable environmental conditions. In addition, since cells have the ability to alter their own environment, the decisions at the single-cell level can depend on the context of the population. I will be presenting our attempts to understand these problems.
Tuesday, October 25, 2011
Time Session
09:00 AM
09:50 AM
Linda Petzold - Stochasticity in Circadian Clocks
In mammals, the suprachiasmatic nucleus (SCN), a brain region of about 20,000 neurons, serves as the master circadian clock, coordinating timing throughout the body and entraining the body to daily light cycles. Experiments in which cell-to-cell signaling between SCN neurons is disrupted by physical separation of the cells, or by blocking vasoactive intestinal polypeptide (VIP) mediated signaling, show that the remarkable precision of the circadian clock at the level of the organism relies on this intercellular signaling. In the absence of cell-to-cell signaling, each SCN neuron and the SCN as a whole exhibits a high degree of stochasticity, with significantly less stable oscillations. We describe several novel findings that were obtained via a combination of experiment and discrete stochastic models, explored through wavelet analysis.
10:30 AM
11:20 AM
Ilya Nemenman - Stochastic processes in the adiabatic limit: applications to biochemistry and population genetics
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 variables. This procedure allows to speed up simulation of kinetic networks and reveals a number of interesting phenomena, previously unobserved in the context of classical stochastic kinetics. One of the most interesting is the emergence of geometric phases, which we show may have substantial effects on, in particular, the frequency of fixation of new mutations in slowly variable environments.
11:30 AM
12:20 PM
David Anderson - Computational methods for stochastically modeled biochemical reaction networks
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 understood and analyzed by using different representations for the processes. Topics discussed will be a subset of: approximation techniques, variance reduction methods, parameter sensitivities.
02:30 PM
03:20 PM
Peter Jagers - Simple, very simple, and not so simple models of populations lingering around a carrying capacity, and allowing evolutionary branching
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 between morphs are introduced, and it turns out that the resulting processes exhibit evolutionary branchings which occur in a manner slightly different from that predicted by established deterministic theory. The validity of conclusions is discussed in terms of more general branching processes. (Joint work with Serik Sagitov, Fima Klebaner et al.)
04:00 PM
04:50 PM
Judith Miller - Kinetic equations in spatial quantitative genetics
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 to Kirkpatrick and Barton for population range limits, showing that they exhibit bistability and hysteresis. This suggests a possible mechanism for lag times between establishment and subsequent explosive growth and range expansion in the absence of an Allee effect.
Wednesday, October 26, 2011
Time Session
09:00 AM
09:50 AM
John Yin - Impacts of genetics, environment and noise on virus growth
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 virus (VSV), a rabies-like RNA virus, growing on BHK cells. Established single-cycle measures of virus growth within infected cells provide population averages, which mask potential cell-to-cell variation. We used fluorescence-activated cell sorting to isolate single cells infected by single particles of a recombinant VSV expressing green fluorescent protein. Measured virus yields spanned a broad range from 8000 to below the detection limit of 10 infectious virus particles per cell. Viral genetic variation and host-cell cycle differences were unable to fully account for the observed yield differences. Computer simulations of the VSV dynamics within an infected cell suggest a potential role for stochastic gene expression to the observed yield variation. These studies are currently being extended to study the kinetics of virus production from individual infected cells.
10:30 AM
11:20 AM
Amandine Veber - Large-scale behaviour of the spatial Lambda-Fleming-Viot process
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 (but in a spatial continuum), and one that allows some individuals to send offspring at very large distances. This is a joint work with Nathana�l Berestycki and Alison Etheridge.
04:00 PM
04:50 PM
Sebastian Schreiber - Population persistence in the face of demographic and environmental uncertainty
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 central issue in population biology from theoretical, empirical, and applied perspectives.

For the first part of this talk, I will discuss, briefly, the relationship between attractors of deterministic models and quasi-stationary distributions of their stochastic, finite population counterpoints i.e. models accounting for demographic stochasticity. These results shed some insight into when persistence should be observed over long time frames despite extinction being inevitable.

For the second part of the talk, I will discuss results on stochastic persistence and boundedness for stochastic models accounting for environmental (but not demographic) noise. Stochastic boundedness asserts that asymptotically the population process tends to remain in compact sets. In contrast, stochastic persistence requires that the population process tends to be "repelled" by some "extinction set." Using these results, I will illustrate how environmental noise can facilitate coexistence of competing species and how dispersal in stochastic environments can rescue locally extinction prone populations. Empirical demonstrations from Kansas prairies, acorn woodpecker populations, and microcosm experiments will be discussed.
Thursday, October 27, 2011
Time Session
09:00 AM
09:50 AM
Ellen Baake - Recombination dynamics and ancestral recombination trees
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.

I will focus on models involving only single crossovers in every generation and contrast the situations in continuous and in discrete time. In continuous time, the deterministic model has a simple closed solution, which is due to the independence of the individual recombination events. In contrast, discrete time introduces dependencies between the links and leads to a much more complex situation. Nevertheless, the situation becomes tractable by looking backwards in time, starting from single individuals at present in a Wright-Fisher population with recombination and tracing back the ancestry of the various gene segments that result from recombination. In the limit of population size to infinity, these segments become independent. We identify the process that describes their history, together with the tree structures they define, which we like to call ancestral recombination trees. It turns out that the corresponding tree _topologies_ play a special role: Surprisingly, explicit probabilities may be assigned to them, which then leads to an explicit solution of the recombination dynamics.

This is joint work with Ute von Wangenheim.

[1] E. Baake, Deterministic and stochastic aspects of single-crossover recombination, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, Vol. VI, 3037-3053
10:30 AM
11:20 AM
David Karig - Cell-free synthetic biology in nanofabricated reaction devices
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 effects, and the interaction of synthetic components with host cell processes. Thus, simpler contexts such as cell-free expression systems offer great promise to engineering complex biological behavior in a quantitative fashion. Furthermore, the confinement of cell-free gene circuit reactions in nanofabricated reaction devices offers a flexible approach to investigating fundamental aspects of gene circuit function. Currently, we are using such devices to study noise in simple gene circuits. Cell-free reactions confined in different volume wells are imaged over time using fluorescent microscopy. The noise characteristics of the resulting gene expression trajectories are analyzed and compared for different gene circuits.
11:30 AM
12:20 PM
John Fricks - Bridging Scales in Molecular Motor Models: From Diffusing Heads to Multiple Steps
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 experimental quantities, such as asymptotic velocity and effective diffusion, through the appropriate limits of a semi-Markov process. Using this framework, asymptotic results for randomly detached motors are also obtained and linked to the statistical analysis of velocity data from motor assays. In addition, we will discuss how the framework developed here could be used as one component of a larger scale model for motor-cargo systems of the type to be presented in Kramer's talk.
02:30 PM
03:20 PM
Garegin Papoian - Physico-Chemical Simulations of Eukaryotic Cell Motility
Actin polymerization in vivo is regulated spatially and temporally by a web of signaling proteins. We developed a three-dimensional, physico-chemical, stochastic model of sheet-like lamellipodia, which are projected by eukaryotic cells during cell migration, and contain a dynamically remodeling three-dimensional actin mesh. A number of regulatory proteins and subtle mechano-chemical couplings determine the lamellipodial protrusion dynamics. Our work sheds light on how lamellipodial protrusion dynamics is affected by the concentrations of actin and actin-binding proteins. Overall, our work emphasizes that elongation and nucleation processes work highly cooperatively in determining the optimal protrusion speed for the actin mesh in lamellipodia. We also studied molecular mechanisms of growth retraction cycles in filopodia, finger-like protrusions based on bundles of actin filaments. In particular, we found that capping proteins and molecular motors may have a profound effect on filopodial dynamics. We also uncovered the rules of active transport in filopodia, mediated by molecular motors, allowing for highly efficient delivery of cytosolic proteins to the filopodial tip. We studies the concentration profile of motors and actin along the filopodial tube, and the way motor transport couples to filopodial growth dynamics.
04:00 PM
04:50 PM
Peter Kramer - Bridging Scales in Molecular Motor Models: From Single to Multiple Motor Systems
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 experimental quantities or the effective single-motor transport characterizations obtained through the kind of systematic analysis of detailed molecular motor models described in Fricks' presentation. Through stochastic asymptotic reductions, we derive the effective transport properties of the multiple-motor-cargo complex, and provide analytical explanations for why a cargo bound to two molecular motors moves more slowly at low applied forces but more rapidly at high applied forces than a cargo bound to a single molecular motor.
Friday, October 28, 2011
Time Session
09:00 AM
09:50 AM
Michael Simpson - (no title available)
(no description available)
10:30 AM
11:20 AM
Hans Othmer - Stochastic Problems in Pattern Formation and Development
Pattern formation in a developing tissue frequently involves the proper spatial localization of the boundary between different cell types. A standard mechanism to accomplish this uses one or more diffusible chemical signals called morphogens that are produced at boundaries of the tissue, and whose concentration determines gene expression and phenotypic characteristics in the tissue. How such boundaries, which lead to distinguished fates between adjacent cells, are set is an important issue in developmental biology. Since the concentration of morphogens or downstream components may be small, stochastic fluctuations make the reliable determination of the boundary more difficult. Models of this process are frequently based on reaction-diffusion equations, and in this talk we will address several questions related to the simulation of such systems. Firstly, how does one chose a computational cell size for a complex reaction-diffusion network, secondly, how does does one eliminate fast reactions in a stochastic reaction network, and thirdly, how does the network structure affect the resilience of boundary location determination when stochastic effects are important.
11:30 AM
12:20 PM
Timothy Elston - Stochastic models of cell movement
(no description available)
Name Email Affiliation
Aggarwal, Nitish aggarwal.nitish@gmail.com Mathematics, The Ohio State University
Agnihotri, Mithila agnihotri.2@buckeyemail.osu.edu Biophysics, The Ohio State University
Allen, Linda linda.j.allen@ttu.edu Department of Mathematics and Statistics, Texas Tech University
Anderson, David anderson@math.wisc.edu Mathematics, University of Wisconsin - Madison
Baake, Ellen ebaake@techfak.uni-bielefeld.de Faculty of Technology,
Balazsi, Gabor gbalazsi@mdanderson.org Systems Biology, Unit 950, University of Texas M. D. Anderson Cancer Center
Bewick, Sharon sharon_bewick@hotmail.com NIMBioS,
Bishop, Lisa lbishop@amath.washington.edu Gladstone Institute, University of San Francisco
Cao, Yang ycao@cs.vt.edu Computer Science, Virginia Tech
Conlisk, Terry conlisk.1@osu.edu Mechanical Engineering, The Ohio State University
Dar, Roy roy.dar@gladstone.ucsf.edu Gladstone Institute (GIVI), California State University, San Francisco
Elston, Timothy telston@amath.unc.edu Department of Pharmacology, University of North Carolina, Chapel Hill
Erban, Radek erban@maths.ox.ac.uk Mathematical Institute, University of Oxford
Fricks, John fricks@stat.psu.edu Statistics, Pennsylvania State University
G. T. Zanudo, Jorge jgtz@psu.edu Physics, Pennsylvania State University
Galante, Amanda agalante@cscamm.umd.edu Applied Mathematics & Statistics and Scientific Computation Program, University of Maryland
Gupta, Ankit gupta@cmap.polytechnique.fr Centre de Mathematiques Appliquees,
Higham, Des d.j.higham@strath.ac.uk Mathematics and Statistics,
Isaacson, Samuel isaacson@math.bu.edu mathematics and statistics, Boston University
Jagers, Peter jagers@chalmers.se Mathematical Sciences,
Ji, Lin ji.lin.ginny@gmail.com Department of Chemistry, Capital Normal University
Joo, Jaewook jjoo1@utk.edu Physics, University of Tennessee
Kang, Hye-Won hkang@math.umn.edu Department of Mathematics, University of Maryland Baltimore County
Karig, David karigdk@ornl.gov Center for Nanophase Materials Sciences, Oak Ridge National Laboratory
Kramer, Peter kramep@rpi.edu Mathematical Sciences, Rensselaer Polytechnic Institute
Kulkarni, Rahul kulkarni@vt.edu Physics, Virginia Polytechnic Institute and State University
Kurt, Noemi kurt@math.tu-berlin.de Mathematics, TU Berlin
Kurtz, Thomas kurtz@math.wisc.edu Mathematics and Statistics, University of Wisconsin
Li, Yao yli@math.gatech.edu Mathematics , Georgia Institute of Technology
Loewe, Laurence Loewe@wisc.edu Laboratory of Genetics, University of Wisconsin
Maheshri, Narendra narendra@mit.edu Chemical Engineering, Massachusetts Institute of Technology
McKinley, Scott scott.mckinley@ufl.edu Mathematics, University of Florida
Mejia-Guerra, Katherine mejia-guerra.1@osu.edu Molecular Genetics, The Ohio State University
Metz, Johan (=Hans) j.a.j.metz@biology.leidenuniv.nl Plant Ecology and Phytochemistry, Analysis and Dynamical Systems, Institute of Biology, Mathematical Institute
Miller, Judith jrm32@georgetown.edu Mathematics and Statistics, Georgetown University
Narula, Jatin jn3@rice.edu Bioengineering, Rice University
Nemenman, Ilya ilya.nemenman@emory.edu Physics and Biology, Emory University
Newby, Jay newby@maths.ox.ac.uk Mathematics, University of Oxford
Olofsson, Peter polofsso@trinity.edu Mathematics, Trinity University
Othmer, Hans othmer@math.umn.edu School of Mathematics, University of Minnesota
Papoian, Garegin gpapoian@umd.edu Chemistry and Biochemistry, University of Maryland
Paulsson, Johan Johan_Paulsson@hms.harvard.edu Dept of Systems Biology, Harvard University
Petzold, Linda petzold@cs.ucsb.edu Mechanical Engineering, University of California, Santa Barbara
Popovic, Lea lpopovic@mathstat.concordia.ca Dept of Mathematics and Statistics, Concordia University
Razooky, Brandon brazooky@ucsd.edu Biophysics, University of California, San Francisco
Reinhold, Dominik reinhold@email.unc.edu Mathematics and Computer Science, Clark University
Sardanyés Cayuela, Josep josep.sardanes@upf.edu Gladstone Institute of Virology and Immunology,
Schreiber, Sebastian sschreiber@ucdavis.edu Department of Evolution and Ecology, University of California, Davis
Seweryn, Michal mseweryn@math.uni.lodz.pl Biostatistics, Georgia Health Sciences University
Shih, Yu-Keng shihy@cse.ohio-state.edu Mathematics Department, Duke University
Simpson, Michael simpsonml1@ornl.gov CNMS, Oak Ridge National Laboratory
Singh, Abhyudai a2singh@ucsd.edu Electrical Engineering, University of Delaware
Smith, Daniel das92@pitt.edu Mathematics, University of Pittsburgh
Suel, Gurol Gurol.Suel@UTSouthwestern.edu Pharmacology, UT Southwestern Medical Center
Tosun, Kursad ktosun@siu.edu Mathematics, Southern Illinois University
Trousdale, James jamest212@gmail.com Mathematics, University of Houston
Tsimring, Lev ltsimring@ucsd.edu BioCircuits Institute, University of California, San Diego
Veber, Amandine amandine.veber@cmap.polytechnique.fr CMAP,
Walczak, Aleksandra awalczak@lpt.ens.fr Laboratoire de Physique Theorique,
Wang, Xiao xiaowang@asu.edu Bioengineering, Arizona State University
Weinberger, Leor lsw@ucsd.edu Gladstone Institute For Virology/Department of Biochemistry and Biophysics, UCSF
Yin, John yin@engr.wisc.edu Chemical & Biological Engineering, University of Wisconsin
Zheng, Likun zhen0107@math.umn.edu Department of Mathematics, University of California, Irvine
Zygalakis, Konstantinos zygalakis@maths.ox.ac.uk Oxford Centre for Collaborative Applied Mathematics,
Pathogen Extinction in Stochastic Models of Epidemics and Viral Dynamics
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 number R0>1, the population size is large and I(0)=a is small, then a classic result of Whittle (1955) gives an approximation to the probability of pathogen extinction: (1/R0)a. This classic result can be derived from branching process theory. We apply results from multitype Markov branching process theory to generalize this approximation for probability of pathogen extinction to more complex epidemic models with multiple stages, treatment , or multiple populations and to within host models of virus and cell dynamics.

Work done in collaboration with Yuan Yuan and Glenn Lahodny.
Computational methods for stochastically modeled biochemical reaction networks
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 understood and analyzed by using different representations for the processes. Topics discussed will be a subset of: approximation techniques, variance reduction methods, parameter sensitivities.
Recombination dynamics and ancestral recombination trees
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.

I will focus on models involving only single crossovers in every generation and contrast the situations in continuous and in discrete time. In continuous time, the deterministic model has a simple closed solution, which is due to the independence of the individual recombination events. In contrast, discrete time introduces dependencies between the links and leads to a much more complex situation. Nevertheless, the situation becomes tractable by looking backwards in time, starting from single individuals at present in a Wright-Fisher population with recombination and tracing back the ancestry of the various gene segments that result from recombination. In the limit of population size to infinity, these segments become independent. We identify the process that describes their history, together with the tree structures they define, which we like to call ancestral recombination trees. It turns out that the corresponding tree _topologies_ play a special role: Surprisingly, explicit probabilities may be assigned to them, which then leads to an explicit solution of the recombination dynamics.

This is joint work with Ute von Wangenheim.

[1] E. Baake, Deterministic and stochastic aspects of single-crossover recombination, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, Vol. VI, 3037-3053
Stochastic models of cell movement
(no description available)
Bridging Scales in Molecular Motor Models: From Diffusing Heads to Multiple Steps
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 experimental quantities, such as asymptotic velocity and effective diffusion, through the appropriate limits of a semi-Markov process. Using this framework, asymptotic results for randomly detached motors are also obtained and linked to the statistical analysis of velocity data from motor assays. In addition, we will discuss how the framework developed here could be used as one component of a larger scale model for motor-cargo systems of the type to be presented in Kramer's talk.
Simple, very simple, and not so simple models of populations lingering around a carrying capacity, and allowing evolutionary branching
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 between morphs are introduced, and it turns out that the resulting processes exhibit evolutionary branchings which occur in a manner slightly different from that predicted by established deterministic theory. The validity of conclusions is discussed in terms of more general branching processes. (Joint work with Serik Sagitov, Fima Klebaner et al.)
Cell-free synthetic biology in nanofabricated reaction devices
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 effects, and the interaction of synthetic components with host cell processes. Thus, simpler contexts such as cell-free expression systems offer great promise to engineering complex biological behavior in a quantitative fashion. Furthermore, the confinement of cell-free gene circuit reactions in nanofabricated reaction devices offers a flexible approach to investigating fundamental aspects of gene circuit function. Currently, we are using such devices to study noise in simple gene circuits. Cell-free reactions confined in different volume wells are imaged over time using fluorescent microscopy. The noise characteristics of the resulting gene expression trajectories are analyzed and compared for different gene circuits.
Bridging Scales in Molecular Motor Models: From Single to Multiple Motor Systems
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 experimental quantities or the effective single-motor transport characterizations obtained through the kind of systematic analysis of detailed molecular motor models described in Fricks' presentation. Through stochastic asymptotic reductions, we derive the effective transport properties of the multiple-motor-cargo complex, and provide analytical explanations for why a cargo bound to two molecular motors moves more slowly at low applied forces but more rapidly at high applied forces than a cargo bound to a single molecular motor.
Effective population sizes and the canonical equation of adaptive dynamics
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)-state space, e.g. spanned by their i(dividual)-state and location. On such models one can graft evolutionary processes like random genetic drift or adaptive evolution by rare repeated substitutions of mutants in heritable traits affecting the state transition and reproduction processes of individuals. From this general perspective I will derive the so-called Canonical Equation of adaptive dynamics, a differential equation for evolutionary trait change derived under the additional assumption that mutations have small effect. In the CE approximation the rate of evolution is found to correspond to the product of a parameter $n_{e,A}$, equal to the population size times a dimensionless product of life history parameters (including spatial movements), times the gradient of the invasion fitness of potential mutants with respect to their trait vector. From a heuristic connection with the diffusion approximation for genetic drift it follows that $n_{e,A} = n_{e,D}$, the effective population size from population genetics.
Kinetic equations in spatial quantitative genetics
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 to Kirkpatrick and Barton for population range limits, showing that they exhibit bistability and hysteresis. This suggests a possible mechanism for lag times between establishment and subsequent explosive growth and range expansion in the absence of an Allee effect.
Stochastic processes in the adiabatic limit: applications to biochemistry and population genetics
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 variables. This procedure allows to speed up simulation of kinetic networks and reveals a number of interesting phenomena, previously unobserved in the context of classical stochastic kinetics. One of the most interesting is the emergence of geometric phases, which we show may have substantial effects on, in particular, the frequency of fixation of new mutations in slowly variable environments.
Stochastic Problems in Pattern Formation and Development
Pattern formation in a developing tissue frequently involves the proper spatial localization of the boundary between different cell types. A standard mechanism to accomplish this uses one or more diffusible chemical signals called morphogens that are produced at boundaries of the tissue, and whose concentration determines gene expression and phenotypic characteristics in the tissue. How such boundaries, which lead to distinguished fates between adjacent cells, are set is an important issue in developmental biology. Since the concentration of morphogens or downstream components may be small, stochastic fluctuations make the reliable determination of the boundary more difficult. Models of this process are frequently based on reaction-diffusion equations, and in this talk we will address several questions related to the simulation of such systems. Firstly, how does one chose a computational cell size for a complex reaction-diffusion network, secondly, how does does one eliminate fast reactions in a stochastic reaction network, and thirdly, how does the network structure affect the resilience of boundary location determination when stochastic effects are important.
Physico-Chemical Simulations of Eukaryotic Cell Motility
Actin polymerization in vivo is regulated spatially and temporally by a web of signaling proteins. We developed a three-dimensional, physico-chemical, stochastic model of sheet-like lamellipodia, which are projected by eukaryotic cells during cell migration, and contain a dynamically remodeling three-dimensional actin mesh. A number of regulatory proteins and subtle mechano-chemical couplings determine the lamellipodial protrusion dynamics. Our work sheds light on how lamellipodial protrusion dynamics is affected by the concentrations of actin and actin-binding proteins. Overall, our work emphasizes that elongation and nucleation processes work highly cooperatively in determining the optimal protrusion speed for the actin mesh in lamellipodia. We also studied molecular mechanisms of growth retraction cycles in filopodia, finger-like protrusions based on bundles of actin filaments. In particular, we found that capping proteins and molecular motors may have a profound effect on filopodial dynamics. We also uncovered the rules of active transport in filopodia, mediated by molecular motors, allowing for highly efficient delivery of cytosolic proteins to the filopodial tip. We studies the concentration profile of motors and actin along the filopodial tube, and the way motor transport couples to filopodial growth dynamics.
Fluctuations in cells and ecosystems
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Stochasticity in Circadian Clocks
In mammals, the suprachiasmatic nucleus (SCN), a brain region of about 20,000 neurons, serves as the master circadian clock, coordinating timing throughout the body and entraining the body to daily light cycles. Experiments in which cell-to-cell signaling between SCN neurons is disrupted by physical separation of the cells, or by blocking vasoactive intestinal polypeptide (VIP) mediated signaling, show that the remarkable precision of the circadian clock at the level of the organism relies on this intercellular signaling. In the absence of cell-to-cell signaling, each SCN neuron and the SCN as a whole exhibits a high degree of stochasticity, with significantly less stable oscillations. We describe several novel findings that were obtained via a combination of experiment and discrete stochastic models, explored through wavelet analysis.
Population persistence in the face of demographic and environmental uncertainty
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 central issue in population biology from theoretical, empirical, and applied perspectives.

For the first part of this talk, I will discuss, briefly, the relationship between attractors of deterministic models and quasi-stationary distributions of their stochastic, finite population counterpoints i.e. models accounting for demographic stochasticity. These results shed some insight into when persistence should be observed over long time frames despite extinction being inevitable.

For the second part of the talk, I will discuss results on stochastic persistence and boundedness for stochastic models accounting for environmental (but not demographic) noise. Stochastic boundedness asserts that asymptotically the population process tends to remain in compact sets. In contrast, stochastic persistence requires that the population process tends to be "repelled" by some "extinction set." Using these results, I will illustrate how environmental noise can facilitate coexistence of competing species and how dispersal in stochastic environments can rescue locally extinction prone populations. Empirical demonstrations from Kansas prairies, acorn woodpecker populations, and microcosm experiments will be discussed.
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Cellular decision-making in the context of population dynamics
How do cells execute decisions to cope with and survive under environmental conditions? My laboratory focuses on understanding how the dynamics of genetic circuits comprised of interactions between genes and proteins allow cells to govern decision-making. Interestingly, we find that stochastic fluctuations that are inherent to the biochemical reactions within genetic circuits can allow cells to cope with unpredictable environmental conditions. In addition, since cells have the ability to alter their own environment, the decisions at the single-cell level can depend on the context of the population. I will be presenting our attempts to understand these problems.
Accelerated stochastic simulation algorithm for modeling evolutionary population dynamics
Evolution and co-evolution of ecological communities are stochastic processes often characterized by vastly different rates of reproduction and mutation and a coexistence of very large and very small sub-populations of competing species. This creates serious difficulties for accurate statistical modeling of evolutionary dynamics. In this talk, we review recent progress in this area and introduce a new exact algorithm for fast fully stochastic simulations of birth/death/mutation processes. It produces a significant speedup compared to the direct stochastic simulation algorithm in a typical case when the total population size is large and the mutation rates are much smaller than birth/death rates. We illustrate the performance of the algorithm on several representative examples: evolution on a smooth fitness landscape, Kauffmann's NK model, directed evolution of a regulatory gene network, and stochastic predator-prey system. (joint work with William Mather)
Large-scale behaviour of the spatial Lambda-Fleming-Viot process
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 (but in a spatial continuum), and one that allows some individuals to send offspring at very large distances. This is a joint work with Nathanaï¿½l Berestycki and Alison Etheridge.
Impacts of genetics, environment and noise on virus growth
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 virus (VSV), a rabies-like RNA virus, growing on BHK cells. Established single-cycle measures of virus growth within infected cells provide population averages, which mask potential cell-to-cell variation. We used fluorescence-activated cell sorting to isolate single cells infected by single particles of a recombinant VSV expressing green fluorescent protein. Measured virus yields spanned a broad range from 8000 to below the detection limit of 10 infectious virus particles per cell. Viral genetic variation and host-cell cycle differences were unable to fully account for the observed yield differences. Computer simulations of the VSV dynamics within an infected cell suggest a potential role for stochastic gene expression to the observed yield variation. These studies are currently being extended to study the kinetics of virus production from individual infected cells.

Recombination dynamics and ancestral recombination trees
Ellen Baake 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 dynamic

Bridging Scales in Molecular Motor Models: From Single to Multiple Motor Systems
Peter Kramer 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

Bridging Scales in Molecular Motor Models: From Diffusing Heads to Multiple Steps
John Fricks 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 c

Cell-free synthetic biology in nanofabricated reaction devices
David Karig 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. H

Population persistence in the face of demographic and environmental uncertainty
Sebastian Schreiber 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 exti

Large-scale behaviour of the spatial Lambda-Fleming-Viot process
Amandine Veber 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 ge

Impacts of genetics, environment and noise on virus growth
John Yin 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 proce

Effective population sizes and the canonical equation of adaptive dynamics
Johan (=Hans) Metz 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

Pathogen Extinction in Stochastic Models of Epidemics and Viral Dynamics
Linda Allen 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

Kinetic equations in spatial quantitative genetics
Judith Miller 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, su

Simple, very simple, and not so simple models of populations lingering around a carrying capacity, and allowing evolutionary branching
Peter Jagers 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 ba

Computational methods for stochastically modeled biochemical reaction networks
David Anderson 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 sp

Stochastic processes in the adiabatic limit: applications to biochemistry and population genetics
Ilya Nemenman 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 ho