Motif analysis for small-number effects in chemical reaction dynamics

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.

Embedding dual function into molecular motors through collective motion

“…These findings are related to those reported in the growing literature on the small-number effect in chemical reaction systems3233.…”

Stationary Distributions and Condensation in Autocatalytic Reaction Networks

“…Our results are a step towards the rigorous analysis of the product-form stationary distribution of non-weakly-reversible ergodic stochastic CRNs of arbitrary deficiency. We exhibit product-form stationary distributions π N for a large class of autocatalytic mass preserving CRNs, including the models in [43,40,6,5,33,34] (which were studied via simulation and approximations) and generalizing results of [28,24]. The emerging infinite family of product-form functions in the stationary distributions (with Poisson form as a particular case) is also possibly interesting from the view of natural computation, where it extends the range of designable probability distributions of CRNs; see, e.g., [11,44].…”

“…The relation between the deterministic and the stochastic model as well as their differences are a focus of current research [6,5,28,43,40,10,3]. Kurtz [35] linked the short term behavior of the adequately scaled continuous-time Markov chain to the dynamics of the ODE model.…”

“…CRNs with stochastic behavior differing from the behavior of the deterministic CRN due to molecular discreteness and stochasticity were identified. The mathematical analysis is based on approximations [43,40,6,5,33,34] in the ergodic case, or on the analysis of absorbing states for absolute concentration robust CRNs [4,2,15]. In the ergodic case such behavior appeared in the literature as noise-induced bi-/multistability [6,5], small-number effect [43,40], or noise-induced transitions [27].…”

“…The mathematical analysis is based on approximations [43,40,6,5,33,34] in the ergodic case, or on the analysis of absorbing states for absolute concentration robust CRNs [4,2,15]. In the ergodic case such behavior appeared in the literature as noise-induced bi-/multistability [6,5], small-number effect [43,40], or noise-induced transitions [27]. Our setting includes examples from [6,5,33] and some examples of [43,40].…”