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

Abstract

Biochemical reaction networks often involve reactions that take place on different time scales, giving rise to “slow” and “fast” system variables. This property is widely used in the analysis of systems to obtain dynamical models with reduced dimensions. In this paper, we consider stochastic dynamics of biochemical reaction networks modeled using the Linear Noise Approximation (LNA). Under time-scale separation conditions, we obtain a reduced-order LNA that approximates both the slow and fast variables in the system. We mathematically prove that the first and second moments of this reduced-order model converge to those of the full system as the time-scale separation becomes large. These mathematical results, in particular, provide a rigorous justification to the accuracy of LNA models derived using the stochastic total quasi-steady state approximation (tQSSA). Since, in contrast to the stochastic tQSSA, our reduced-order model also provides approximations for the fast variable stochastic properties, we term our method the “stochastic tQSSA+”. Finally, we demonstrate the application of our approach on two biochemical network motifs found in gene-regulatory and signal transduction networks.

Balanced truncation for model reduction of biological oscillators

“…Model reduction alleviates these issues by constructing reduced order models whose behavior captures that of the original system (Antoulas 2005). Motivated by the goal of designing biological devices of increasing size and complexity (Del Vecchio et al 2018), model reduction of biological systems has recently attracted renewed interest in the rapidly emerging field of systems and synthetic biology (Gómez-Uribe et al 2008;Anderson et al 2011;Thomas et al 2012;Kang and Kurtz 2013;Rao et al 2014;Sootla and Anderson 2014;Radulescu et al 2015;Del Vecchio and Murray 2015;Herath et al 2016;Sootla and Anderson 2017;Herath and Del Vecchio 2018). Given the pressing need for compositional modeling frameworks in mathematical biology (Del Vecchio et al 2018), the present paper seeks to develop a model reduction framework compatible with modeling and interconnection of open systems whose behavior is not restricted to the stability of a single equilibrium.…”

Postponing production exponentially enhances the molecular memory of a stochastic switch

“…The key insight of [29], on which we expand in this work, is that a stochastic autoregulatory circuit extended by a large delay exhibits a near-deterministic behaviour of the inactive protein (the production of which has been initiated but not yet completed). A classical means of eliminating noise in a stochastic model is to increase its system size/volume [30]; the near-deterministic behaviour in large-volume systems can be described using the linear noise approximation (LNA) [31][32][33][34][35] and the Wentzel-Kramers-Brillouin (WKB) approximation [36][37][38][39][40]. These two approaches are complementary: the LNA applies on finite temporal domains [41]; the WKB approximation covers slow metastable dynamics such as transitions between deterministically stable steady states or to a fixation/extinction point [42].…”

Stochastic time-dependent enzyme kinetics: closed-form solution and transient bimodality

“…Research efforts in the area of stochastic chemical kinetics can be, broadly speaking, categorized into three types: (i) The search for a solution of the CME for the MM reaction and its various extensions, i.e., obtaining a closed-form solution for the time-dependent or steady-state probability distribution of the molecule numbers of each species in the reaction system [9,10]. (ii) The reduction of the CME and the construction of the stochastic equivalent of deterministic approximations (such as the fast equilibrium, quasi steadystate and total quasi steady-state approximations) and understanding their regime of validity [11][12][13][14][15][16][17][18][19][20][21][22][23][24]. (iii) The derivation of exact or approximate expressions for the mean of the stochastic rate of product formation and an investigation of the differences or similarities from the predictions of the deterministic Michaelis-Menten equation [25][26][27][28][29][30][31].…”