[Paper Review] Separation of time-scales and model reduction for stochastic reaction networks

Hye-Won Kang, Thomas G. Kurtz|arXiv (Cornell University)|Nov 7, 2010

Gene Regulatory Network Analysis16 references4 citations

TL;DR

This paper develops a systematic framework for model reduction in stochastic reaction networks by exploiting multiple time-scales through a one-parameter scaling of species numbers and rate constants. It establishes rigorous limits as the scaling parameter tends to infinity, enabling approximation of complex networks via lower-dimensional stochastic, deterministic, or hybrid models based on averaging and separation of time-scales principles.

ABSTRACT

A stochastic model for a chemical reaction network is embedded in a one-parameter family of models with species numbers and rate constants scaled by powers of the parameter. A systematic approach is developed for determining appropriate choices of the exponents that can be applied to large complex networks. When the scaling implies subnetworks have different time-scales, the subnetworks can be approximated separately providing insight into the behavior of the full network through the analysis of these lower dimensional approximations.

Motivation & Objective

• To address the challenge of modeling large, complex biochemical reaction networks with widely varying species counts and reaction rates.
• To develop a rigorous mathematical framework for identifying and exploiting multiple time-scales in stochastic reaction networks.
• To enable model reduction by deriving limiting models—stochastic, deterministic, or hybrid—through scaling and averaging techniques.
• To provide a systematic method for selecting appropriate scaling exponents for species numbers and rate constants in large networks.
• To establish convergence results for the limiting processes using advanced stochastic analysis, including martingale properties and weak convergence in the Jakubowski topology.

Proposed method

• Formalize a one-parameter family of stochastic reaction network models by scaling species numbers and rate constants by powers of a large parameter $ N $.
• Use stochastic integral equations of the form $ X(t) = X(0) + \sum_k Y_k\left(\int_0^t \lambda_k(X(s))\,ds\right) \zeta_k $ to describe the Markov jump process dynamics.
• Apply time-scale separation by identifying distinct time-scales via exponents $ \gamma $ such that species and reaction dynamics exhibit nondegenerate limits as $ N \to \infty $.
• Implement averaging techniques inspired by Khasminskii’s theory to eliminate fast variables and derive effective dynamics for slow components.
• Utilize the Jakubowski topology for weak convergence of processes, particularly when Skorohod topology fails due to coalescing jumps.
• Establish convergence of rescaled processes under conditions (A.7) and (A.8), ensuring the limiting process satisfies $ R_0 = \sum_{k=1}^m R_k $ with intensities $ \lambda_k(t) = \frac{\mu_k(t)}{\sum_l \mu_l(t)} \lambda_0(t) $.

Experimental results

Research questions

• RQ1 How can one systematically identify and separate multiple time-scales in large stochastic reaction networks?
• RQ2 What scaling exponents for species numbers and rate constants yield nondegenerate, well-behaved limiting models?
• RQ3 Under what conditions does the limiting process of a fast-slow system converge to a hybrid or averaged model?
• RQ4 How can the convergence of rescaled counting processes be rigorously established in the absence of strict monotonicity or regularity?
• RQ5 What is the role of the Jakubowski topology in ensuring relative compactness and convergence of processes with potential jump coalescence?

Key findings

• The limiting process satisfies $ R_0 = \sum_{k=1}^m R_k $, with the $ R_k $ being pairwise orthogonal counting processes.
• The intensity of each $ R_k $ in the limit is $ \lambda_k(t) = \frac{\mu_k(t)}{\sum_l \mu_l(t)} \lambda_0(t) $, reflecting proportionate splitting of the total intensity.
• Relative compactness of the sequence $ \{(R_0^N, R_1^N, \ldots, R_m^N)\} $ is established in the Jakubowski topology, even when Skorohod compactness fails.
• The time-changed processes $ R_k^N \circ \gamma_N $ converge in distribution, enabling the derivation of limiting dynamics via time-change and martingale techniques.
• The limiting model can be stochastic, deterministic, or hybrid, depending on the scaling regime and the interplay between species and reaction time-scales.
• The framework rigorously justifies the quasi-steady state assumption and averaging methods in stochastic reaction networks, extending prior heuristic approaches.