Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators

The linear noise approximation (LNA) offers a simple means by which one can study intrinsic noise in monostable biochemical networks. Using simple physical arguments, we have recently introduced the slow-scale LNA (ssLNA), which is a reduced version of the LNA under conditions of timescale separation. In this paper we present the first rigorous derivation of the ssLNA using the projection operator technique and show that the ssLNA follows uniquely from the standard LNA under the same conditions of timescale separation as those required for the deterministic quasi-steady-state approximation. We also show that the large molecule number limit of several common stochastic model reduction techniques under timescale separation conditions constitutes a special case of the ssLNA.