Discreteness-induced concentration inversion in mesoscopic chemical systems

Abstract

molecular discreteness is apparent in small-volume chemical systems, such as biological cells, leading to stochastic kinetics. Here we present a theoretical framework to understand the effects of discreteness on the steady state of a monostable chemical reaction network. We consider independent realizations of the same chemical system in compartments of different volumes. Rate equations ignore molecular discreteness and predict the same average steadystate concentrations in all compartments. However, our theory predicts that the average steady state of the system varies with volume: if a species is more abundant than another for large volumes, then the reverse occurs for volumes below a critical value, leading to a concentration inversion effect. The addition of extrinsic noise increases the size of the critical volume. We theoretically predict the critical volumes and verify, by exact stochastic simulations, that rate equations are qualitatively incorrect in sub-critical volumes.

Effects of small particle numbers on long-term behaviour in discrete biochemical systems

“…Because COT is based on the stoichiometric information of RNs alone, only constellations of species in the potential steady states are computed. We do not consider information on amounts or ratio of amounts as addressed in (Ramaswamy et al , 2012). Furthermore, we do not say what the concentration or amounts in the long run might be.…”

“…Cao and Liang (2008); Schultz et al (2007) already made some efforts to analyse small stochastic networks by analysing effects of small numbers of molecules on the stability of switches and by proposing an algorithm to enumerate the state space for spaces with small copy numbers with a limited number of newly produced molecules, respectively. Ramaswamy et al (2012) discovered that for small sizes of reactors, the ratio between the amounts of species in steady state can change, and hence, called this effect “discreteness-induced inversion effect”.…”

Motif analysis for small-number effects in chemical reaction dynamics

“…[17][18][19] Moreover, how such microscopic molecular discreteness can contribute to cellular functions at a larger scale has gathered much interest, and the effect induced by the discreteness is expected to provide a novel concept to understand cellular behaviors and function. 17,18,[20][21][22] In a macroscopic system, i.e., when the volume size of a system and the number of contained molecules are large, the overall behaviors of the system can be described by the deterministic rate equation of reaction dynamics for the average concentration of chemicals or a Langevin equation that takes into account small Gaussian fluctuation around it.…”

Mathematical Models Light Up Plant Signaling

“…The modeling of processes at the cellular or subcellular level presents distinct challenges; the low copy number of many molecular players inside cells (Grima and Schnell, 2008) means that stochasticity must be considered. The models described in this review are mostly deterministic and, hence, ill-suited to tackle such challenges; indeed, it has been shown that some phenomena cannot be captured even qualitatively by deterministic approaches (Ramaswamy et al, 2012). Stochastic modeling frameworks have been developed in the past few decades, notably the stochastic simulation algorithm (Gillespie, 1977) and its various approximations (Thomas et al, 2012), which have led to means to efficiently probe single-cell dynamics.…”