An Analytical Framework for Studying Small-Number Effects in Catalytic Reaction Networks: A Probability Generating Function Approach to Chemical Master Equations.

Masaki Nakagawa,Yuichi Togashi

doi:10.3389/fphys.2016.00089

Abstract

Cell activities primarily depend on chemical reactions, especially those mediated by enzymes, and this has led to these activities being modeled as catalytic reaction networks. Although deterministic ordinary differential equations of concentrations (rate equations) have been widely used for modeling purposes in the field of systems biology, it has been pointed out that these catalytic reaction networks may behave in a way that is qualitatively different from such deterministic representation when the number of molecules for certain chemical species in the system is small. Apart from this, representing these phenomena by simple binary (on/off) systems that omit the quantities would also not be feasible. As recent experiments have revealed the existence of rare chemical species in cells, the importance of being able to model potential small-number phenomena is being recognized. However, most preceding studies were based on numerical simulations, and theoretical frameworks to analyze these phenomena have not been sufficiently developed. Motivated by the small-number issue, this work aimed to develop an analytical framework for the chemical master equation describing the distributional behavior of catalytic reaction networks. For simplicity, we considered networks consisting of two-body catalytic reactions. We used the probability generating function method to obtain the steady-state solutions of the chemical master equation without specifying the parameters. We obtained the time evolution equations of the first- and second-order moments of concentrations, and the steady-state analytical solution of the chemical master equation under certain conditions. These results led to the rank conservation law, the connecting state to the winner-takes-all state, and analysis of 2-molecules M-species systems. A possible interpretation of the theoretical conclusion for actual biochemical pathways is also discussed.

Highlights

Biochemical systems consist of a variety of chemicals, including proteins, nucleic acids, and small metabolites
In subsequent Sections (2.3.1–2.3.3), we show that the generating function equation (GFE) introduces the time evolution equation of the first-order and second-order moments of concentrations, and the second-order moment expression of time-averaged concentrations (SME)
Consider an abstract catalytic reaction network consisting of M chemical species and N molecules in a well-stirred reactor of volume V, as in Awazu and Kaneko (2007)

Summary

Introduction

Biochemical systems consist of a variety of chemicals, including proteins, nucleic acids, and small metabolites. Because each chemical consists of molecules, the concentration of each species should be a discrete variable The effects of such discreteness as well as finite-size fluctuations in stochastic reactions become non-negligible if the number of molecules in the system is small. In theory, situations such as these can result in phenomena that cannot be described by rate equations (as well as those equations with additional noise; Togashi and Kaneko, 2001, 2007; Awazu and Kaneko, 2007, 2009)

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