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.
Highlights
As opposed to deterministic models, employing timescale separation for model order reduction remains an ongoing area of research for stochastic models of biological systems
We addressed the problem of model order reduction for biochemical reaction networks with time-scale separation, where the system dynamics are modeled with the Linear Noise Approximation (LNA)
After transforming the system into a standard singular perturbation form, we developed a reduced-order model that approximates the slow and fast dynamics of the full system when the time-scale separation is large
Summary
As opposed to deterministic models, employing timescale separation for model order reduction remains an ongoing area of research for stochastic models of biological systems. The Linear Noise Approximation (LNA) is another approximate model for the CME, where stochasticity is represented as random fluctuations around a deterministic trajectory using stochastic differential equations or partial differential equations.[10,35] Recently, model order reduction methods for the LNA have been developed using projection operators[27,36] or singular perturbation analysis.[37] In these studies, the error between the full and reduced-order models is not analytically quantified. The application of our approach is demonstrated on two biochemical network motifs found in gene-regulatory networks and signal transduction cascades Through these examples, we illustrate the practical applications of the reduced-order models and the necessity of both slow and fast variable approximations for analysis. Notation: E[·] denotes the expected value of a random variable. · denotes the Euclidean norm for vectors and · F denotes the Frobenius norm for matrices
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