Reduction of multiscale stochastic biochemical reaction networks using exact moment derivation

Abstract

Biochemical reaction networks (BRNs) in a cell frequently consist of reactions with disparate timescales. The stochastic simulations of such multiscale BRNs are prohibitively slow due to high computational cost for the simulations of fast reactions. One way to resolve this problem uses the fact that fast species regulated by fast reactions quickly equilibrate to their stationary distribution while slow species are unlikely to be changed. Thus, on a slow timescale, fast species can be replaced by their quasi-steady state (QSS): their stationary conditional expectation values for given slow species. As the QSS are determined solely by the state of slow species, such replacement leads to a reduced model, where fast species are eliminated. However, it is challenging to derive the QSS in the presence of nonlinear reactions. While various approximation schemes for the QSS have been developed, they often lead to considerable errors. Here, we propose two classes of multiscale BRNs which can be reduced by deriving an exact QSS rather than approximations. Specifically, if fast species constitute either a feedforward network or a complex balanced network, the reduced model based on the exact QSS can be derived. Such BRNs are frequently observed in a cell as the feedforward network is one of fundamental motifs of gene or protein regulatory networks. Furthermore, complex balanced networks also include various types of fast reversible bindings such as bindings between transcriptional factors and gene regulatory sites. The reduced models based on exact QSS, which can be calculated by the computational packages provided in this work, accurately approximate the slow scale dynamics of the original full model with much lower computational cost.

Basal leakage in oscillation: Coupled transcriptional and translational control using feed-forward loops

“…Several studies recently have shown examples where using approximation QSSA can lead to considerable errors in the stochastic simulation [ 98 – 101 ]. Since then, a lot of effort has been done to find rigorous condition to check the validity of stochastic QSSA [ 99 , 100 ] or deriving an exact QSS of fast species rather than using approximate QSS [ 101 ]. Hence, depending the purpose of the study, one need to choose more carefully which simplification technique they should use.…”

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

“…Thus, it follows that the region of attraction for the system (B34)-(B36) is given by R b1 ×R 3 where R b1 is the region of attraction of the equilibrium point z 0 γ 1 (x 0 , 0 . Thus, the assumptions of Tikhonov's theorem are satisfied and applying the theorem to the set of moment equations in (B1)-(B5) and (B9)-(B12) yields the result (19)- (25).…”

“…9,10 There have been several studies that obtain reduced-order representations of the CME under time-scale separation conditions. [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] Among these, a common approach used to reduce the complexity in the simulations of the CME is to approximate the fast variables by their deterministic quasi-steady state expressions. 19 However, validity of this method still remains under investigation.…”

“…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.…”

Stochasticity in multi-phosphorylation and quasi steady state approximation in stochastic simulation

“…In order to improve the efficiency of SSA various methods have been developed exploiting the firing rate or the timescale separation of chemical reactions. Methods such as tau leaping, slow-scale SSA, stochastic quasi steady state approximation on enzyme catalyzed reactions are useful in this context with their own advantages and disadvantages (33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43) .…”