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Kinetic and Thermodynamic Descriptions of Open Systems of Complex Chemical Reactions with Multiple Scales

Liu Hong, Hong Qian

Abstract

The general theory of a complex system of nonlinear chemical reactions is a primary language of chemistry that includes chemical engineering and cellular biochemistry. Its significance as an analytical framework, however, has not been fully appreciated outside the community of physical chemists. In this review, we discuss the latest advances in the kinetics and Gibbsian thermodynamics of chemical reactions in a spatially homogeneous aqueous solution with a multiscale perspective on complex systems. From the microscopic level of single reaction events which are purely stochastic in continuous time, one at a time among a set of molecules, to the macroscopic chemical reaction systems in bulk in terms of deterministic rate equations, the mathematical descriptions of kinetic models for chemical reactions at different levels are presented in detail, with rigorous mathematical justifications presented. In parallel with the kinetics of chemical reactions, the irreversible thermodynamics of open systems and the stochastic thermodynamics along reactions trajectories are reviewed thoroughly. As a novel feature, the mathematical theory of large deviations is shown to play a pivotal role in the thermodynamics of chemical reactions in equilibrium and in irreversible processes. This review is expected to stimulate interests in and help defining multiscale phenomena and nonequilibrium thermodynamics in many research fields on population dynamics of interacting species using chemical reactions as an analytic paradigm.

Kinetic and Thermodynamic Descriptions of Open Systems of Complex Chemical Reactions with Multiple Scales

Abstract

The general theory of a complex system of nonlinear chemical reactions is a primary language of chemistry that includes chemical engineering and cellular biochemistry. Its significance as an analytical framework, however, has not been fully appreciated outside the community of physical chemists. In this review, we discuss the latest advances in the kinetics and Gibbsian thermodynamics of chemical reactions in a spatially homogeneous aqueous solution with a multiscale perspective on complex systems. From the microscopic level of single reaction events which are purely stochastic in continuous time, one at a time among a set of molecules, to the macroscopic chemical reaction systems in bulk in terms of deterministic rate equations, the mathematical descriptions of kinetic models for chemical reactions at different levels are presented in detail, with rigorous mathematical justifications presented. In parallel with the kinetics of chemical reactions, the irreversible thermodynamics of open systems and the stochastic thermodynamics along reactions trajectories are reviewed thoroughly. As a novel feature, the mathematical theory of large deviations is shown to play a pivotal role in the thermodynamics of chemical reactions in equilibrium and in irreversible processes. This review is expected to stimulate interests in and help defining multiscale phenomena and nonequilibrium thermodynamics in many research fields on population dynamics of interacting species using chemical reactions as an analytic paradigm.
Paper Structure (30 sections, 9 theorems, 194 equations, 6 figures, 3 tables)

This paper contains 30 sections, 9 theorems, 194 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Chemical Reaction Systems at Different Scales
  3. Elementary Reactions as Poisson Random Events at Individual Molecule Level
  4. Chemical Master Equation for Molecular Number Distribution
  5. Chemical Rate Equation in Macroscopic Limit
  6. Diffusion Process and Fokker-Planck Equation for Concentration at Mesoscale
  7. Chemical Langevin Equation for Concentration Fluctuations
  8. Reaction-Diffusion Equations for Spatial Heterogeneity
  9. NESS Thermodynamics for Open Chemical Reaction Systems
  10. Equilibrium Chemical Thermodynamics in a Nutshell and Beyond
  11. Chemical Potential, Affinity, and Gibbs Free Energy for Nonequi-librium States
  12. Thermodynamics for General Discrete-State Markov Processes
  13. Nonequilibrium Thermodynamics for Continuous-Space Diffusion Processes
  14. Emergent Macroscopic Thermodynamics by Large Deviations Principle
  15. Large Deviations Principle for Chemical Master Equations
  16. ...and 15 more sections

Key Result

Theorem 1

A renewal process is a sequence of time points $T_1,\cdots,T_n,\cdots$, where successive waiting times $X\equiv T_{i+1}- T_i\ge 0$ are independent and follow a common probability density function $f_X(x)$. Let $\{T_i^{(k)}; i\ge 1,\, 1\le k\le m\}$ be $m$ i.i.d. renewal processes all having the dist is a Poisson process in the limit of $m\rightarrow\infty$, with rate parameter $\mathbb{E}^{-1}[X]$

Figures (6)

  • Figure 1: Macroscopic deterministic chemical reaction kinetics vs. Newtonian mechanics with contradistinctions: While most comparisons are pedantic, we call attention to that the force with a potential $F(\vec{x})=-\nabla U(\vec{x})$ gives rise to Hamiltonian conservative dynamics; and the existence of a Gibbs potential $G(\vec{x})$ guarantees detailed balance, thus chemical equilibrium, among reactions. The pair of chemical reactions, $I+S\longrightarrow 2I$, $I\longrightarrow R$, represent the synthesis of $I$ from its precursor $S$ and the degradation of $I$. As for Hamilton-Jacobi-Hu equation$^a$, see Eq. \ref{['HJE-Hu']} and Ref. gang1986lyapounov.
  • Figure 2: Kinetic and thermodynamic descriptions of chemical reactions at multiple scales. Left: microscopic counting of each and every reaction and individual molecules, one at a time xie. Middle: mesoscopic concentrations with stochastic fluctuations elson. Right: macroscopic deterministic kinetics and Gibbs' thermodynamics.
  • Figure 3: A cyclic reaction network. When $k_1^-=k_2^-=k_3^-=0$ and $k_1^+,k_2^+,k_3^+>0$, the system is complex balanced and contains a clockwise net flux even at the stationary state. If and only if $\frac{k_1^+k_2^+k_3^+}{k_1^-k_2^-k_3^-}=1$, the system satisfies the condition of detailed balance, and there is no net flux at the equilibrium state.
  • Figure 4: A mechanical state is ${\bf x}=(\vec{q},\vec{p})$. A Newtonian system is determined by the initial total energy $E$; thus the entire level set $\{{\bf x}: H({\bf x})=E\}$. A thermodynamic state is the totality of a such level set represented by $E$, together with $V$ and $N$ that are implicit in $H({\bf x})$ and $\Omega$. A macroscopic thermodynamic system then is characterized by a Helmholtz free energy function $F(T,V,N)$, where $T$ is temperature; or Boltzmann's thermodynamic probability $\Gamma(E,V,N)$. State changes are characterized by work, heat, and quasi-static processes. In Gibbs' theory that bridges mechanics and macroscopic $F$, the latter is recognized as a functional of the entire function $H(\cdot)$. The collection of all possible $H(\cdot)$ is a function space $\mathcal{H}(\Omega)$. Chemical reactions joint two $H(\cdot)$'s into one or decompose one $H(\cdot)$ into two.
  • Figure 5: Relation between chemical reaction rate and activation energy barrier in the picture of Kramers' theory.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem 1: Superposition of Renewal Processes cox1954superpositionkhinchin1960mathematical
  • Theorem 2: Uniqueness of NESS for Finite-State Markov Process schnakenberg1976
  • Theorem 3: LLN for Chemical Master Equationskurtz1972kurtz1981cltkurtz1978strong
  • Theorem 4: Zero Deficiency for Complex-Balanced Systemhorn1972necessaryfeinberg1972complexfeinberg
  • Theorem 5: Uniqueness of NESS for Complex-Balanced Systemhorn1972general
  • Theorem 6: Existence and Uniqueness of Classical Solution to FP Equation friedman1964
  • proof
  • Remark 1
  • Theorem 7: Condition of Detailed Balancejiang2004mathematical
  • Theorem 8: Emergent macroscopic chemical thermodynamics Ge2016Mesoscopicge2016
  • ...and 3 more