A Graphic Formulation of Nonisothermal Chemical Reaction Systems and the Analysis of Detailed Balanced Networks

In this paper, we provide a graphic formulation for modeling non-isothermal reaction systems based on the classical chemical reaction network theory (CRNT) and apply it to analyzing dynamic properties of detailed balanced network systems. To model thermal effects, we extend the classical chemical reaction network (CRN) formulation by adding two parameters to each direct (reaction) edge, depicting, respectively, the instantaneous internal energy change after the firing of the reaction and the variation of the reaction rate with respect to the temperature. For systems possess thermodynamic equilibria, a compact dynamic equation is provided exhibiting at the same time the graphic topology and thermodynamic information. With this formula, the Legendre transformation, and some mild conditions, we show non-isothermal detailed balanced network systems to admit some fundamental properties: dissipativeness, the detailed balancing of each equilibrium, the existence and uniqueness of the equilibrium, and the asymptotic stability. In general, the analysis and results of this work provide insights into the research of non-isothermal chemical reaction systems.

[1]  B. Ydstie,et al.  Thermodynamic Potentials from Stationary Probabilities , 2019, IFAC-PapersOnLine.

[2]  I. Prigogine,et al.  Book Review: Modern Thermodynamics: From Heat Engines to Dissipative Structures , 1998 .

[3]  David F. Anderson,et al.  A Proof of the Global Attractor Conjecture in the Single Linkage Class Case , 2011, SIAM J. Appl. Math..

[4]  Chuanhou Gao,et al.  Lyapunov Function Partial Differential Equations for Chemical Reaction Networks: Some Special Cases , 2019, SIAM J. Appl. Dyn. Syst..

[5]  B. C. Garrett,et al.  Current status of transition-state theory , 1983 .

[6]  K. Laidler,et al.  Development of transition-state theory , 1983 .

[7]  Samuel Glasstone,et al.  Elements of Physical Chemistry , 1993 .

[8]  M. Feinberg,et al.  Structural Sources of Robustness in Biochemical Reaction Networks , 2010, Science.

[9]  Arjan van der Schaft,et al.  A graph-theoretical approach for the analysis and model reduction of complex-balanced chemical reaction networks , 2012, Journal of Mathematical Chemistry.

[10]  Arjan van der Schaft,et al.  A network dynamics approach to chemical reaction networks , 2015, Int. J. Control.

[11]  Ezra Miller,et al.  A Geometric Approach to the Global Attractor Conjecture , 2013, SIAM J. Appl. Dyn. Syst..

[12]  James Clerk Maxwell.  Illustrations of the Dynamical Theory of Gases , 1860 .

[13]  M. Feinberg.  Chemical reaction network structure and the stability of complex isothermal reactors—II. Multiple steady states for networks of deficiency one , 1988 .

[14]  Alexander N Gorban,et al.  Extended detailed balance for systems with irreversible reactions , 2011, 1101.5280.

[15]  M. Feinberg.  Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems , 1987 .

[16]  H. Eyring.  The Activated Complex in Chemical Reactions , 1935 .

[17]  Chuanhou Gao,et al.  Realizations of quasi-polynomial systems and application for stability analysis , 2017, Journal of Mathematical Chemistry.

[18]  Ludwig Boltzmann,et al.  Lectures on Gas Theory , 1964 .

[19]  Chuanhou Gao,et al.  Complex Balancing Reconstructed to the Asymptotic Stability of Mass-Action Chemical Reaction Networks with Conservation Laws , 2019, SIAM J. Appl. Math..

[20]  Martin Feinberg,et al.  Multiple Equilibria in Complex Chemical Reaction Networks: Ii. the Species-reactions Graph , 2022 .

[21]  James Clerk Maxwell,et al.  V. Illustrations of the dynamical theory of gases.—Part I. On the motions and collisions of perfectly elastic spheres , 1860 .

[22]  J. Sethna.  Statistical Mechanics: Entropy, Order Parameters, and Complexity , 2021 .

[23]  A. Schaft,et al.  Irreversible port-Hamiltonian approach to modeling and analyzing of non-isothermal chemical reaction networks , 2016 .

[24]  Shodhan Rao,et al.  On the network thermodynamics of mass action chemical reaction networks , 2013 .

[25]  H. Qian,et al.  Thermodynamic constraints for biochemical networks. , 2004, Journal of theoretical biology.

[26]  Uri Alon,et al.  Sensitivity and Robustness in Chemical Reaction Networks , 2009, SIAM J. Appl. Math..

[27]  M. Feinberg.  The existence and uniqueness of steady states for a class of chemical reaction networks , 1995 .

[28]  B. Erik Ydstie,et al.  Passivity based control via the second law , 2002 .

[29]  Antonio A. Alonso,et al.  Stabilization of distributed systems using irreversible thermodynamics , 2001, Autom..

[30]  M. Feinberg.  Complex balancing in general kinetic systems , 1972 .

[31]  A. Schaft,et al.  Complex and detailed balancing of chemical reaction networks revisited , 2015, Journal of Mathematical Chemistry.

[32]  H. Callen.  Thermodynamics and an Introduction to Thermostatistics , 1988 .

[33]  Antonio A. Alonso,et al.  Process systems and passivity via the Clausius-Planck inequality , 1997 .

[34]  Arjan van der Schaft,et al.  On the Mathematical Structure of Balanced Chemical Reaction Networks Governed by Mass Action Kinetics , 2011, SIAM J. Appl. Math..

[35]  H. Qian,et al.  Thermodynamic-based computational profiling of cellular regulatory control in hepatocyte metabolism. , 2005, American journal of physiology. Endocrinology and metabolism.

[36]  David F. Anderson,et al.  Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks , 2008, Bulletin of mathematical biology.

[37]  R. Jackson,et al.  General mass action kinetics , 1972 .

[38]  J. Stoer,et al.  Convexity and Optimization in Finite Dimensions I , 1970 .

[39]  G. Szederkényi,et al.  Finding complex balanced and detailed balanced realizations of chemical reaction networks , 2010, 1010.4477.

[40]  David F. Anderson,et al.  Global Asymptotic Stability for a Class of Nonlinear Chemical Equations , 2007, SIAM J. Appl. Math..

[41]  David Angeli,et al.  New Approach to the Stability of Chemical Reaction Networks: Piecewise Linear in Rates Lyapunov Functions , 2014, IEEE Transactions on Automatic Control.

[42]  Gheorghe Craciun,et al.  Toric Differential Inclusions and a Proof of the Global Attractor Conjecture , 2015, 1501.02860.

[43]  Daniele Cappelletti,et al.  Tier structure of strongly endotactic reaction networks , 2018, Stochastic Processes and their Applications.

[44]  J. Gibbs.  On the equilibrium of heterogeneous substances , 1878, American Journal of Science and Arts.

[45]  Badal Joshi,et al.  Simplifying the Jacobian Criterion for Precluding Multistationarity in Chemical Reaction Networks , 2011, SIAM J. Appl. Math..