Workshop: Stochastic Models in Cell Biology

April 9-11 2006, Cornell University


 Sunday, Apr 9 2006 : Malott Hall 253
10:00 - 11:00Dan T. Gillespie
11:30 - 12:30Linda Petzold, UC Santa Barbara
02:00 - 03:00Greg Rempala, University of Louisville
03:30 - 04:30Gheorge Craciun, University of Wisconsin
 (6:00)  dinner at a local restaurant
 
 Monday, Apr 10 2006 : Rhodes Hall 655
09:30 - 10:30Jonathan Mattingly, Duke University
11:00 - 12:00Ruth Williams, UC San Diego
02:00 - 03:30  open discussion
04:00 - 05:00
Mike Reed, Duke University (prob. sem.)
 (5:30)  reception in the Malott Hall lounge
 
 Tuesday, Apr 11 2006 : Rhodes Hall 655
09:30 - 10:30Thomas G. Kurtz, University of Wisconsin
11:00 - 12:00Lea Popovic, Cornell University
 
 
 

   Abstracts : 
Dan T. Gillespie : "Stochastic Chemical Kinetics"
The time evolution of a well-stirred chemically reacting system is traditionally modeled by a set of coupled ordinary differential equations called the reaction rate equation (RRE). The resulting picture of continuous deterministic evolution is, however, valid only for infinitely large systems. That condition is usually well approximated in laboratory test tube systems. But in biological systems formed by single living cells, the small population numbers of some reactant species can result in dynamical behavior that is noticeably discrete rather than continuous, and stochastic rather than deterministic. In that case, a more physically accurate mathematical modeling is obtained by using the machinery of Markov process theory, specifically, the chemical master equation (CME) and the stochastic simulation algorithm (SSA). This talk will review the theoretical foundations of stochastic chemical kinetics, and then discuss some recent efforts to (1) approximate the SSA by a faster simulation procedure, and (2) establish the formal connection between the CME/SSA description and the RRE description.
Linda Petzold : "Computational Tools for Systems Biology"
The discrete stochastic systems from chemical kinetics are inherently multiscale, involving both fast and slow reactions as well as a wide range of species populations. In this talk we will focus on the development of algorithms and software that exploit the multiscale nature of these problems.
Greg Rempala : "Estimating reaction constants in discrete stochastic kinetic systems"
One of the key issues of interest in analyzing a stochastic kinetic model of a biological system is how to infer the values of reaction constants. Under mass action kinetics assumption this is relatively straightforward when the system trajectories are fully observed, however, this is rarely the case in practice. The talk shall summarize some recent developments in the area of Bayesian inference for reaction constants using MCMC methodology in "data- poor" settings. We shall illustrate the issues and challenges of the approach via some examples of inferences for well-known biochemical networks models like e.g., gene transcription and auto- regulation.
Gheorghe Craciun : "Multiple equilibria in biochemical reaction networks"
In nature there are millions of distinct networks of biochemical reactions that might present themselves for study at one time or another. Each reaction network gives rise to its own system of differential equations. These are usually high dimensional, nonlinear, and have many unknown parameters. Nevertheless, each reaction network induces its mass-action differential equations (up to parameter values) in a precise way. This raises the possibility that qualitative properties of the induced differential equations might be tied directly to reaction network structure. We will show that reaction diagrams, similar to those that biochemists usually draw, carry subtle information about a reaction network's capacity to exhibit multiple equilibria. We will also discuss implications for the interpretation of experiments in cell biology. This is joint work with Martin Feinberg.
Ruth Williams : "Stochastic networks with non-exponential inter-event times"
Statistical fluctuations can play a significant role in biochemical reactions where some of the reactant species are only present in small numbers. Markovian stochastic models, using the chemical master equation for theoretical analysis and Gillespie-type stochastic simulation for computation, are useful for analysing such systems when inter-event times are (approximately) exponential. However, for reactions involving assembly processes such as transcription or translation, models allowing for inter-event times with more general distributions are desirable. Such models have elements in common with non-Markovian state-dependent queueing networks. This talk will present some preliminary work indicating some connections between such queueing network models and biochemical reaction networks.
Mike Reed : "Why Cell Biology Needs Mathematicians"
Epidemiology provides correlations that link gene polymorphisms and dietary deficiencies to heart disease, cancer, and other human health problems. Typically, the correlations are small and it is difficult to understand what they mean without understanding how the metabolism of specific cells changes in the presence of polymorphisms or specific diets. Further, one must understand how these changes lead to permanent alterations in cell function and to a cascade of events that lead to the disease is question. The dynamics of cells is highly complex involving mechanisms, both deterministic and stochastic, at many different time scales and includes a myriad of special features that have evolved to accomplish specific tasks. Cells from different animals or different tissues are different, and indeed the same cell is often different at different times, all of which makes the drawing of causal conclusions from the biolgical data difficult. Examples that illustrate these ideas will be discussed as well the need for new methods in the qualitative theory of ordinary differential equations.
Tom Kurtz : "Analysis of stochastic models for reaction network"
Standard methods for characterizing Markov models will be reviewed. Martingale and stochastic analytic methods will be applied to derive approximations and reduced models for reaction networks. In particular, averaging methods will be emphasized.
Lea Popovic : "Analysis of a stochastic model for viral infection"
Stochastic models of reaction networks treat the system as a continuous time Markov chain on the number of molecules of each species with reactions as possible transitions of the chain. We consider approaches to approximation of such models that take the multiscale nature of the system into account. Our primary example is a model of a cell's viral infection for which we apply a combination of averaging and law of large number arguments to show that the ``slow'' component of the model can be approximated by a deterministic equation and to characterize the asymptotic distribution of the ``fast'' components. The main goal is to illustrate techniques that can be used to reduce the dimensionality of much more complex models.