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Entropy
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Probability Distributions and Maximum Entropy in Stochastic Chemical Reaction Networks
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Probability Distributions and Maximum Entropy in Stochastic Chemical Reaction Networks

A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (28 February 2019) | Viewed by 16617

Special Issue Editor

Dr. Yiannis N. Kaznessis

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Guest Editor
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN, USA
Interests: stochastic kinetics; statistical thermodynamics; molecular simulations; antimicrobials

Special Issue Information

Dear Colleagues,

According to the second law of thermodynamics, when isolated systems are at an equilibrium state, their entropy is maximum. Numerous attempts have been made to establish a similar criterion for non-equilibrium steady states (NESS). For decades, the question has been posed whether NESS are stable when the entropy or the rate of entropy production is maximum or minimum. A satisfactory answer has yet to be provided, and there is no established criterion for NESS.

Chemical reaction networks away from the thermodynamic limit have drawn considerable attention in the past two decades with the advent of numerous stochastic simulation algorithms. Instead of the canonical continuous-deterministic models of reaction kinetics, reactions with small numbers of reactants are modeled with discrete-stochastic (e.g., Gillespie algorithms) or continuous-stochastic (e.g., Langevin equations) models. The outcome of these models is a probability distribution of the number of molecules for each of the reactants or products in the system. Because of the probabilistic nature of these networks, which can be studied in non-equilibrium steady states, the entropy may be computed and parallels may be drawn with molecular thermodynamics of systems at equilibrium, as well as with information theory arguments.

In this Special Issue, we welcome papers reporting on the progress of stochastic kinetic models for reaction networks. We welcome review and original papers in subjects that include, but are not limited to, the following areas:

  • NESS of stochastic reaction networks
  • Stochastic simulation algorithms
  • Hybrid and multiscale modeling formalisms
  • Chemical master equations
  • Steady state probabilities of reaction networks
  • Entropy and entropy production in reaction networks
Dr. Yiannis N. Kaznessis
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Stochastic chemical reactions
  • Non-equilibrium steady states (NESS)
  • Probability distributions
  • Entropy and entropy production in NESS

Published Papers (5 papers)

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Research

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18 pages, 908 KiB  
Article
On the Properties of the Reaction Counts Chemical Master Equation
by Vikram Sunkara
Entropy 2019, 21(6), 607; https://doi.org/10.3390/e21060607 - 19 Jun 2019
Cited by 2 | Viewed by 3230
Abstract
The reaction counts chemical master equation (CME) is a high-dimensional variant of the classical population counts CME. In the reaction counts CME setting, we count the reactions which have fired over time rather than monitoring the population state over time. Since a reaction [...] Read more.
The reaction counts chemical master equation (CME) is a high-dimensional variant of the classical population counts CME. In the reaction counts CME setting, we count the reactions which have fired over time rather than monitoring the population state over time. Since a reaction either fires or not, the reaction counts CME transitions are only forward stepping. Typically there are more reactions in a system than species, this results in the reaction counts CME being higher in dimension, but simpler in dynamics. In this work, we revisit the reaction counts CME framework and its key theoretical results. Then we will extend the theory by exploiting the reactions counts’ forward stepping feature, by decomposing the state space into independent continuous-time Markov chains (CTMC). We extend the reaction counts CME theory to derive analytical forms and estimates for the CTMC decomposition of the CME. This new theory gives new insights into solving hitting times-, rare events-, and a priori domain construction problems. Full article
(This article belongs to the Special Issue Probability Distributions and Maximum Entropy in Stochastic Chemical Reaction Networks)
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20 pages, 4886 KiB  
Article
Non-Equilibrium Thermodynamics and Stochastic Dynamics of a Bistable Catalytic Surface Reaction
by Miguel Pineda and Michail Stamatakis
Entropy 2018, 20(11), 811; https://doi.org/10.3390/e20110811 - 23 Oct 2018
Cited by 3 | Viewed by 3238
Abstract
Catalytic surface reaction networks exhibit nonlinear dissipative phenomena, such as bistability. Macroscopic rate law descriptions predict that the reaction system resides on one of the two steady-state branches of the bistable region for an indefinite period of time. However, the smaller the catalytic [...] Read more.
Catalytic surface reaction networks exhibit nonlinear dissipative phenomena, such as bistability. Macroscopic rate law descriptions predict that the reaction system resides on one of the two steady-state branches of the bistable region for an indefinite period of time. However, the smaller the catalytic surface, the greater the influence of coverage fluctuations, given that their amplitude normally scales as the square root of the system size. Thus, one can observe fluctuation-induced transitions between the steady-states. In this work, a model for the bistable catalytic CO oxidation on small surfaces is studied. After a brief introduction of the average stochastic modelling framework and its corresponding deterministic limit, we discuss the non-equilibrium conditions necessary for bistability. The entropy production rate, an important thermodynamic quantity measuring dissipation in a system, is compared across the two approaches. We conclude that, in our catalytic model, the most favorable non-equilibrium steady state is not necessary the state with the maximum or minimum entropy production rate. Full article
(This article belongs to the Special Issue Probability Distributions and Maximum Entropy in Stochastic Chemical Reaction Networks)
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16 pages, 440 KiB  
Article
Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers
by Michail Vlysidis and Yiannis N. Kaznessis
Entropy 2018, 20(9), 700; https://doi.org/10.3390/e20090700 - 12 Sep 2018
Cited by 1 | Viewed by 3297
Abstract
The time evolution of stochastic reaction networks can be modeled with the chemical master equation of the probability distribution. Alternatively, the numerical problem can be reformulated in terms of probability moment equations. Herein we present a new alternative method for numerically solving the [...] Read more.
The time evolution of stochastic reaction networks can be modeled with the chemical master equation of the probability distribution. Alternatively, the numerical problem can be reformulated in terms of probability moment equations. Herein we present a new alternative method for numerically solving the time evolution of stochastic reaction networks. Based on the assumption that the entropy of the reaction network is maximum, Lagrange multipliers are introduced. The proposed method derives equations that model the time derivatives of these Lagrange multipliers. We present detailed steps to transform moment equations to Lagrange multiplier equations. In order to demonstrate the method, we present examples of non-linear stochastic reaction networks of varying degrees of complexity, including multistable and oscillatory systems. We find that the new approach is as accurate and significantly more efficient than Gillespie’s original exact algorithm for systems with small number of interacting species. This work is a step towards solving stochastic reaction networks accurately and efficiently. Full article
(This article belongs to the Special Issue Probability Distributions and Maximum Entropy in Stochastic Chemical Reaction Networks)
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26 pages, 442 KiB  
Article
Gradient and GENERIC Systems in the Space of Fluxes, Applied to Reacting Particle Systems
by D. R. Michiel Renger
Entropy 2018, 20(8), 596; https://doi.org/10.3390/e20080596 - 09 Aug 2018
Cited by 8 | Viewed by 3101
Abstract
In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager–Machlup relations. Of particular interest is the case where the microscopic system [...] Read more.
In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager–Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or GENERIC system in the space of fluxes. In a general setting we study how flux gradient or GENERIC systems are related to gradient systems of concentrations. This shows that many gradient or GENERIC systems arise from an underlying gradient or GENERIC system where fluxes rather than densities are being driven by (free) energies. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well. Full article
(This article belongs to the Special Issue Probability Distributions and Maximum Entropy in Stochastic Chemical Reaction Networks)
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Review

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37 pages, 483 KiB  
Review
Chemical Kinetics Roots and Methods to Obtain the Probability Distribution Function Evolution of Reactants and Products in Chemical Networks Governed by a Master Equation
by José-Luis Muñoz-Cobo and Cesar Berna
Entropy 2019, 21(2), 181; https://doi.org/10.3390/e21020181 - 14 Feb 2019
Cited by 4 | Viewed by 3081
Abstract
In this paper first, we review the physical root bases of chemical reaction networks as a Markov process in multidimensional vector space. Then we study the chemical reactions from a microscopic point of view, to obtain the expression for the propensities for the [...] Read more.
In this paper first, we review the physical root bases of chemical reaction networks as a Markov process in multidimensional vector space. Then we study the chemical reactions from a microscopic point of view, to obtain the expression for the propensities for the different reactions that can happen in the network. These chemical propensities, at a given time, depend on the system state at that time, and do not depend on the state at an earlier time indicating that we are dealing with Markov processes. Then the Chemical Master Equation (CME) is deduced for an arbitrary chemical network from a probability balance and it is expressed in terms of the reaction propensities. This CME governs the dynamics of the chemical system. Due to the difficulty to solve this equation two methods are studied, the first one is the probability generating function method or z-transform, which permits to obtain the evolution of the factorial moment of the system with time in an easiest way or after some manipulation the evolution of the polynomial moments. The second method studied is the expansion of the CME in terms of an order parameter (system volume). In this case we study first the expansion of the CME using the propensities obtained previously and splitting the molecular concentration into a deterministic part and a random part. An expression in terms of multinomial coefficients is obtained for the evolution of the probability of the random part. Then we study how to reconstruct the probability distribution from the moments using the maximum entropy principle. Finally, the previous methods are applied to simple chemical networks and the consistency of these methods is studied. Full article
(This article belongs to the Special Issue Probability Distributions and Maximum Entropy in Stochastic Chemical Reaction Networks)
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