The slow-scale linear noise approximation: an accurate, reduced stochastic description of biochemical networks under timescale separation conditions

Abstract

BackgroundIt is well known that the deterministic dynamics of biochemical reaction networks can be more easily studied if timescale separation conditions are invoked (the quasi-steady-state assumption). In this case the deterministic dynamics of a large network of elementary reactions are well described by the dynamics of a smaller network of effective reactions. Each of the latter represents a group of elementary reactions in the large network and has associated with it an effective macroscopic rate law. A popular method to achieve model reduction in the presence of intrinsic noise consists of using the effective macroscopic rate laws to heuristically deduce effective probabilities for the effective reactions which then enables simulation via the stochastic simulation algorithm (SSA). The validity of this heuristic SSA method is a priori doubtful because the reaction probabilities for the SSA have only been rigorously derived from microscopic physics arguments for elementary reactions.ResultsWe here obtain, by rigorous means and in closed-form, a reduced linear Langevin equation description of the stochastic dynamics of monostable biochemical networks in conditions characterized by small intrinsic noise and timescale separation. The slow-scale linear noise approximation (ssLNA), as the new method is called, is used to calculate the intrinsic noise statistics of enzyme and gene networks. The results agree very well with SSA simulations of the non-reduced network of elementary reactions. In contrast the conventional heuristic SSA is shown to overestimate the size of noise for Michaelis-Menten kinetics, considerably under-estimate the size of noise for Hill-type kinetics and in some cases even miss the prediction of noise-induced oscillations.ConclusionsA new general method, the ssLNA, is derived and shown to correctly describe the statistics of intrinsic noise about the macroscopic concentrations under timescale separation conditions. The ssLNA provides a simple and accurate means of performing stochastic model reduction and hence it is expected to be of widespread utility in studying the dynamics of large noisy reaction networks, as is common in computational and systems biology.

Global sensitivity analysis in stochastic simulators of uncertain reaction networks

“…Examples where these features would be of special importance include the analysis of system robustness, model reduction, and parameter calibration. 4,[34][35][36][37][38] The application of the proposed method to complex stochastic systems remains challenging because of the sampling effort. Even though the method is trivial to implement in parallel, future works should focus on the improvement of the methods to lower its computational complexity.…”

Reduced linear noise approximation for biochemical reaction networks with time-scale separation: The stochastic tQSSA+

“…19 However, validity of this method still remains under investigation. 20,22,[25][26][27] The theoretical analysis of the CME is challenging due to the large system size and the lack of appropriate analytical tools. Therefore, several approximations to the chemical master equation have been developed under the assumptions that the system's volume and the molecular counts are sufficiently large.…”

“…10,35 Recently, model order reduction methods for the LNA have been developed using projection operators 27,36 or singular perturbation analysis. 37 However, in these studies, the error between the full and reduced-order models is not analytically quantified.…”

General transient solution of the one-step master equation in one dimension

“…Full analytical solutions to master equations are rare. However, there are several approximate analytical methods, including the system size expansion [4], the Fokker-planck equation and corresponding Langevin equation [5][6][7], moment closure approximations [8][9][10][11], and time-scale separation [12,13].…”