Discrete flux and velocity fields of probability and their global maps in reaction systems

Abstract

Stochasticity plays important roles in reaction systems. Vector fields of probability flux and velocity characterize time-varying and steady-state properties of these systems, including high probability paths, barriers, checkpoints among different stable regions, as well as mechanisms of dynamic switching among them. However, conventional fluxes on continuous space are ill-defined and are problematic when at boundaries of the state space or when copy numbers are small. By re-defining the derivative and divergence operators based on the discrete nature of reactions, we introduce new formulations of discrete fluxes. Our flux model fully accounts for the discreetness of both the state space and the jump processes of reactions. The reactional discrete flux satisfies the continuity equation and describes the behavior of the system evolving along directions of reactions. The species discrete flux directly describes the dynamic behavior in the state space of the reactants such as the transfer of probability mass. With the relationship between these two fluxes specified, we show how to construct time-evolving and steady-state global flow-maps of probability flux and velocity in the directions of every species at every microstate, and how they are related to the outflow and inflow of probability fluxes when tracing out reaction trajectories. We also describe how to impose proper conditions enabling exact quantification of flux and velocity in the boundary regions, without the difficulty of enforcing artificial reflecting conditions. We illustrate the computation of probability flux and velocity using three model systems, namely, the birth-death process, the bistable Schlögl model, and the oscillating Schnakenberg model. Keywords: Stochastic biochemical reaction networks, discrete flux and velocity fields of probability INTRODUCTIONBiochemical reactions in cells are intrinsically stochastic [1][2][3][4]. When the concentrations of participating molecules are small or the differences in reaction rates are large, stochastic effects become prominent [3,[5][6][7]. Many stochastic models have been developed to gain understanding of these reaction systems [8][9][10][11][12]. These models either generate time-evolving landscapes of probabilities over different microstates [9][10][11][12], or generate trajectories along which the systems travel [8,13]. Vector fields of probability flux and probability velocity are also of significant interest, as they can further characterize time-varying properties of the reaction systems, including that of the non-equilibrium steady states [14][15][16][17][18][19]. For example, determining the probability flux can help to infer the mechanism of dynamic switching among different attractors [20,21]. Quantifying the probability flux can also help to characterize the departure of non-equilibrium reaction systems from detailed balance [16,22,23], and can help to identify barriers and checkpoints between different stable cellular states [24]. Computing probability fluxes and velocity fields has found ...

Unifying deterministic and stochastic ecological dynamics via a landscape-flux approach

“…The statistical patterns are typically relatively regular and the associated probabilistic dynamics can be predicted since they typically follow the linear evolution law dictated by the associated Fokker-Planck or Master equation. Although the probabilistic evolution equation and the corresponding Langevin equation for the stochastic trajectories are usually mathematically equivalent in terms of the statistics, the individual trajectories as a result of the nonlinear interactions and fluctuations are often impossible to reliably predict (1,15,16). These trajectories can be compared with observations…”

Unraveling the stochastic transition mechanism between oscillation states by the landscape and the minimum action path theory

“…The probability flux has been proposed to play an important role and reflect the non-equilibrium characteristics in oscillatory and multistable systems. 34,38 So we decompose the driving force into two components: F flux and F gradient (see Method B). The flux drives the system to move cyclically (white arrows in Fig.…”