Master's Thesis
Capturing Large Fluctuations in the Dynamics of Biochemical Reaction Networks using Path Integrals
October 2019 – December 2020
Abstract
Biological systems such as gene expression and metabolic networks are often
described by chemical reaction networks. Their stochastic dynamics is governed
by a chemical master equation which cannot be, in general, solved analytically,
so approximations are necessary. Standard approaches such as the linear noise
approximation tend to predict Gaussian fluctuations into the unphysical regime
of negative concentrations, particularly when small mean molecule numbers lead
to large fluctuations.
We present an alternative approach that works with Poisson fluctuations as its
baseline and so avoids unphysical negative concentrations. The method is based
on a Doi-Peliti coherent state path integral representation of the dynamics.
To this we then apply a Plefka expansion, which treats interactions as
perturbations to an effective non-interacting system. Up to first order in
the expansion, the standard mass action kinetics are recovered and an accurate
description in the large fluctuation regime of low copy numbers is obtained by
expanding the Plefka free energy up to second order.
We demonstrate the usefulness of this approach on simple but paradigmatic
reaction networks, comparing with the results of the linear noise and
moment closure approximation and the exact solution of the chemical
master equation.
Source code: here.
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