David Anderson
Duke University

Propagation of Fluctuations in Biochemical Systems

We consider biochemical reaction systems equipped with an input subjected to (possibly large) ongoing stochastic fluctuations and study how the input fluctuations propagate through the system.  In particular, we consider how different geometric properties of reaction diagrams and different asymptotic limits in the kinetics of systems suppress, magnify, or otherwise shape the fluctuations.

By adding fluctuations to a biochemical system one is confronted with a complicated set of nonlinear stochastic differential equations.  A natural first step is to prove that there is a unique stationary measure on Euclidean space to which the distribution of any solution of the SDEs will converge as time goes to infinity.  This puts a restriction on the type of system we consider because it is an open question as to how to categorize all systems for which a unique stationary measure exists.  If such a measure does exist, however, the solution to the SDEs with distribution given by that measure becomes the natural object to consider when asking questions about how the variances of different concentrations and fluxes vary based on the different geometric and asymptotic properties of the system.

The work presented represents the beginning stages of an avenue possibly rich with both pure mathematical questions (for example, questions of convergence to a unique stationary measure) and serious applications.  One such application, numerical calculations of fluctuations used to study long range control mechanisms in methionine metabolism, will be briefly discussed.



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