Discrete Flux and Velocity Fields of Probability and Their Global Maps in Reaction Systems

TL;DR

This paper introduces a novel discrete flux framework for reaction systems that accurately captures probability flow and dynamics, especially at boundaries and small copy numbers, improving understanding of system behavior.

Contribution

It develops new formulations of discrete probability fluxes based on redefined derivatives, fully accounting for the discrete nature of reactions and state space, and constructs global flow-maps for reaction systems.

Findings

Accurately models probability fluxes in reaction systems.

Constructs global flow-maps of probability flux and velocity.

Demonstrates methods on birth-death, bistable, and oscillating models.

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 Schl\"ogl model, and the oscillating Schnakenberg model.