Motif Analysis for Small-Number Effects in Chemical Reaction Dynamics

TL;DR

This paper defines and analyzes small-number effects in chemical reaction networks at the mesoscopic level, identifying motifs that exhibit significant fluctuations like power laws and bimodal distributions, which are crucial for biological functions and system design.

Contribution

It introduces a quantitative criterion for small-number effects and systematically derives motifs capable of exhibiting these effects in chemical reaction networks.

Findings

Identified motifs that show power law distributions.

Found motifs with bimodal distribution behavior.

Provided a criterion to determine small-number effects.

Abstract

The number of molecules involved in a cell or subcellular structure is sometimes rather small. In this situation, ordinary macroscopic-level fluctuations can be overwhelmed by non-negligible large fluctuations, which results in drastic changes in chemical-reaction dynamics and statistics compared to those observed under a macroscopic system (i.e., with a large number of molecules). In order to understand how salient changes emerge from fluctuations in molecular number, we here quantitatively define small-number effect by focusing on a `mesoscopic' level, in which the concentration distribution is distinguishable both from micro- and macroscopic ones, and propose a criterion for determining whether or not such an effect can emerge in a given chemical reaction network. Using the proposed criterion, we systematically derive a list of motifs of chemical reaction networks that can show small-number effects, which includes motifs showing emergence of the power law and the bimodal distribution observable in a mesoscopic regime with respect to molecule number. The list of motifs provided herein is helpful in the search for candidates of biochemical reactions with a small-number effect for possible biological functions, as well as for designing a reaction system whose behavior can change drastically depending on molecule number, rather than concentration.