On the reduction of stochastic chemical reaction networks

The linear noise approximation (LNA) describes the random fluctuations from the mean-field concentrations of a chemical reaction network due to intrinsic noise. It is also used as a test probe to determine the accuracy of reduced formulations of the chemical master equation and to understand the relationship between timescale disparity and model reduction in stochastic environments. Although several reduced LNAs have been proposed, they have not been placed into a general theory concerning the accuracy of reduced LNAs derived from center manifold and singular perturbation theory. This has made it difficult to understand why certain reductions of the master or Langevin equations fail or succeed. In this work, we develop a deeper understanding of slow manifold projection in the linear noise regime by answering a straightforward but open question: In the presence of eigenvalue disparity, does the appropriate oblique projection of the LNA onto the slow eigenspace accurately approximate the first and second moments of complete LNA, and if not, why? Although most studies concentrate on the role of eigenvalue disparity arising from the drift matrix, we go further and examine the interplay between disparate 'drift" eigenvalues and the eigenvalues of the diffusion matrix, the latter of which may or may not be disparate. Furthermore, we place the previously established reductions of the LNA into a more general framework and formulate the necessary and sufficient conditions for the projected LNA to accurately approximate the first and second moments of the complete LNA.