Reduction of multiscale stochastic biochemical reaction networks using exact moment derivation

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.Author summary: Molecules inside a cell undergo various transformations via biochemical reactions with disparate rates. For instance, while transcriptional factors bind and unbind gene promoters in a time scale of seconds, mRNA transcription takes at least several minutes. For such systems regulated by both fast and slow reactions together, most of the computation time in stochastic simulations is spent on simulating the fast reactions, even if our interest is in the dynamics of slow reactions such as transcription. This problem can be resolved by deriving a reduced system, which can accurately approximate the slow dynamics of the original system without simulating fast reactions. However, when nonlinear reactions exist, the accurate reduction is often impossible. Here, we describe that two classes of nonlinear systems, both of which appear frequently as models of natural biochemical reaction networks, can be effectively reduced, and we provide a computational package to carry out the reduction. We find that the resulting reduced systems accurately capture the stochastic dynamics of the original system, while saving considerable simulation time.