Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1810.07263

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

Anna Terebus·Chun Liu·Jie Liang
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

q-bio.MN

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Anna Terebus, Chun Liu, Jie Liang. 2018-10-16. Discrete Flux and Velocity Fields of Probability and Their Global Maps in Reaction Systems. https://doi.org/10.1063/1.5050808

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Minimality in Reflexive and Stoichiometric Autocatalysis

Autocatalysis, the ability of a chemical subsystem to sustain its own constituents when supplied with sufficient food molecules, has been closely related to the origin of life on Earth. Emerging from Wilhelm Ostwald's considerations about an explicit autocatalytic reaction, different notions of autocatalysis have been developed over the years. The two most prominent are reflexively autocatalytic F-generated sets (RAFs) and stoichiometric autocatalysis. After having shown that each RAFs is, under reasonable conditions, in general stoichiometrically autocatalytic, we examine here the relationship between the two notions of minimality: irreducible RAFs and autocatalytic cores. To this end, we overcome the obstacle that RAFs and stoichiometric autocatalysis have been formalized in distinct systems of chemical reactions, i.e., catalytic reaction systems (CRS) and chemical reaction networks (CRNs), respectively. We show that reactions in a CRS constitute equivalence classes of reactions of the corresponding CRNs w.r.t. their specific catalyzations. Using the fact that CRN and CRS can be canonically identified whenever each CRS reaction is associated with a single catalyzation, we demonstrate that the Kőnig graph of a monocatalyzed, irreducible RAF is composed of strong blocks devoid of food and waste species that are separated by reaction vertices, each of which contains an autocatalytic core. In fact, a single irreducible RAF can, in general, contain exponentially many autocatalytic cores.

q-bio.MN↗
arXiv2026

Implementation of Linear Regression and Linear Interpolation using Reaction Networks

Statistical inference is a fundamental component of data science. In this work, we focus on two classical inference techniques: regression and interpolation. We propose a reaction-network-based framework for implementing linear regression, including both univariate and multivariate settings, as well as linear interpolation. Our approach encodes the outputs of these inference techniques in the steady-state concentrations of species within the reaction network. A key ingredient of the construction is a novel generalized division module capable of handling division involving negative numbers. We validate the proposed framework through in silico implementations on standard synthetic datasets and obtain the expected regression and interpolation outputs.

q-bio.MN↗
arXiv2026

Mapping disease-regulatory flux through eQTL-based causal gene networks: a complex-trait framework applied to coronary artery disease

A substantial proportion of inherited susceptibility to common diseases is mediated through tissue-specific regulatory variation. The omnigenic model hypothesizes that much of this risk arises from distant (trans) regulatory effects that are propagated via gene-regulatory networks (GRNs) from numerous regulatory genes onto a relatively small set of core genes. Two major challenges have hindered an empirical assessment of this model. First, transcriptome-wide association studies primarily assess gene expression that is regulated locally (in cis), and thus miss trans-acting effects. Second, trans regulation has so far been examined only in aggregate, without identifying which regulatory genes transmit disease-associated signals to which target genes. Here, we present a framework that decomposes each gene's disease association into a local cis component and a set of trans components attributable to its upstream regulators, propagated along a directed causal GRN. For each gene, we estimate sparse Bayesian expression models and we assess model performance using out-of-sample prediction. Propagating trans signals through the inferred network yields a disease-regulatory flux map: a directed, signed representation that quantifies the contribution of each regulator to the disease association of each target gene. Applying this framework to seven tissues relevant to coronary artery disease, we demonstrate how genetic variation flows through the GRN to affect disease risk. Outgoing regulatory influences from individual genes are directionally heterogeneous, whereas disease-associated genes tend to integrate convergent input from multiple regulators; these convergent targets are enriched for cardiovascular-related biological processes. Consequently, each gene's disease association can be reinterpreted as a detailed allocation of the disease signal among its contributing regulators.

q-bio.MN↗