Tuesday, Apr 11
2006 : Rhodes Hall 655
09:30 - 10:30 | Thomas G. Kurtz, University of Wisconsin |
11:00 - 12:00 | Lea 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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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