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

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.

Stationary distributions of systems with discreteness-induced transitions

“…To make this effect quantitatively apparent, in the general setting where (11) holds but not necessarily (12), we prove that for V → 0 the stationary distribution concentrates at the boundaries by showing that the conditional probability π(0|n) + π(n|n) tends to one, for any n. Indeed, inserting (11) into (10), we get α = α V and…”

“…Molecule counts and mass-action rates can be scaled with the volume V in such a way that the scaled stochastic system X(t)/V converges for large V to the solution 7) from simulation (grey histograms) and from ( 9) in Theorem 1 (red lines). Parameters are chosen according to (11) with…”

“…Several attempts have been made to underpin the phenomenon theoretically, at least in simplified toy models, through derivation of analytic expressions, without resorting to simulation or approximation. Examples in this direction are [12][13][14][15][16][17], though many questions remain unsolved. The stationary distributions of autocatalytic systems of a similar nature have been studied in [18], but their theory does not apply to the Togashi-Kaneko (TK) model and similar systems where mass is not conserved.…”

“…Several attempts have been made to underpin the phenomenon theoretically, at least in simplified toy models, through derivation of analytic expressions, without resorting to simulation or approximation. Examples in this direction are [10][11][12][13], though many questions are still unsolved.…”

Modeling of Nanomachine/Micromachine Crowds: Interplay between the Internal State and Surroundings

“…If the system includes positive feedback (such as the regulatory product in Model 2), effects from small numbers of molecules may appear, as observed in autocatalytic reaction systems. [53][54][55][56][57] In the current model, a single machine that runs the cycle ahead may facilitate the operation of others through mechanical interference, as seen in Fig. 7.…”