Liangrong Peng

We present a systematic derivation of asymptotic expansions of nonequilibrium thermodynamics for chemical reaction networks (CRNs) based on singular perturbation theory. For a general reversible CRN with fast–slow kinetics, we obtain the first and second laws of thermodynamics for the asymptotic expansion model. We derive composite expansions of the enthalpy, entropy, entropy production rate, and relative entropy. The slow-varying outer parts of these thermodynamic quantities capture the long-time trend, while the fast-varying corrected inner parts decay to zero as the fast time variable tends to infinity. The convergence order of these quantities is determined by the local Lipschitz properties of the respective functions. The enthalpy retains the same convergence order as the kinetic variables, whereas the entropy, relative entropy, and entropy production rate involve logarithmic terms that cause their gradients to diverge when some concentrations approach zero, reducing their theoretical convergence order by one. The general theory is validated on the reversible Michaelis–Menten reaction, for which both leading-order and first-order matched asymptotic expansions are obtained analytically. Numerical simulations confirm the uniform accuracy of the composite thermodynamic approximations and further reveal that the entropy production rate converges with a higher order than theoretically predicted. The results demonstrate that the composite expansion provides a rigorous and physically consistent tool for analyzing energy and entropy balances in multiscale CRNs.

Maximum entropy principle (MEP) offers an effective and unbiased approach to inferring unknown probability distributions when faced with incomplete information, while neural networks provide the flexibility to learn complex distributions from data. This paper proposes a novel neural network architecture, the MEP-Net, which combines the MEP with neural networks to generate probability distributions from moment constraints. We also provide a comprehensive overview of the fundamentals of the maximum entropy principle, its mathematical formulations, and a rigorous justification for its applicability for non-equilibrium systems based on the large deviations principle. Through fruitful numerical experiments, we demonstrate that the MEP-Net can be particularly useful in modeling the evolution of probability distributions in biochemical reaction networks and in generating complex distributions from data.

Previous studies have primarily focused on the nonequilibrium thermodynamics of chemical reaction networks (CRNs) occurring in closed systems. In contrast, CRNs in open systems exhibit much richer nonequilibrium phenomena due to sustained matter and energy exchange. Here, we bridge the quantitative relationships between essential thermodynamic quantities --including the steady state, enthalpy, intrinsic Gibbs free energy, and entropy production rate -- in original mass-action equations and their PEA- or QSSA-reduced counterparts for open CRNs. Our analysis demonstrates that the thermodynamic structure, especially the second law of thermodynamics, of the full CRNs may not be preserved in reduced models when algebraic relations are imposed. Specifically, PEA-reduced models lose monotonicity in the intrinsic Gibbs free energy, whereas QSSA retains this property. These theoretical findings are further validated through analytical and numerical studies of two archetypal open systems: the Michaelis-Menten reactions and the phosphorylation-dephosphorylation cycle (PdPC). Our results provide a systematic framework for evaluating the fidelity of reduced models.

Polymeric aggregates play a significant role in biology and chemical engineering. In order to make a clear description of their underlying formation procedure, simplified models are crucial because the original mass-action equations involve numerous variables, complicating analysis and understanding. While the dynamical aspects of simplified models have been widely studied, their thermodynamic properties are less understood. In this study, we explore the Maximum Entropy Principle (MEP)-reduced models, initially developed for dynamical analysis, from a brand-new thermodynamic perspective. Analytical expressions, along with numerical simulations, demonstrate that the discrete MEP-reduced model strictly retains laws of thermodynamics, which holds true even when the aggregate size transits from discrete values to continuous real numbers. Our findings not only clarify the thermodynamic consistency between the MEP-reduced models and the original models of polymeric aggregates for the first time but also suggest avenues for future research into the model-reduction thermodynamics.

Previous studies have primarily focused on the nonequilibrium thermodynamics of chemical reaction networks (CRNs) occurring in closed systems. In contrast, CRNs in open systems exhibit much richer nonequilibrium phenomena due to sustained matter and energy exchange. Here, we bridge the quantitative relationships between essential thermodynamic quantities -- including the steady state, enthalpy, intrinsic Gibbs free energy, and entropy production rate -- in original mass-action equations and their PEA- or QSSA-reduced counterparts for open CRNs. Our analysis demonstrates that the thermodynamic structure, especially the second law of thermodynamics, of the full CRNs may not be preserved in reduced models when algebraic relations are imposed. Specifically, PEA-reduced models lose monotonicity in the intrinsic Gibbs free energy, whereas QSSA retains this property. These theoretical findings are further validated through analytical and numerical studies of two archetypal open systems: the Michaelis-Menten reactions and the phosphorylation-dephosphorylation cycle (PdPC). Our results provide a systematic framework for evaluating the fidelity of reduced models.

To manipulate the protein concentration at a certain functional state through chemical stabilizers is crucial for protein-related studies. It not only plays a key role in protein structure analysis and protein folding kinetics, but also affects protein functionality to a large extent and thus has wide applications in medicine, food industry, etc. However, due to concerns about side effects or financial costs of stabilizers, identifying optimal strategies for enhancing protein stability with a minimal amount of stabilizers is of great importance. Here, we prove that either for the fixed terminal time (including both finite and infinite cases) or for the free one, the optimal control strategy for stabilizing the folding intermediates with a linear strategy for stabilizer addition belongs to the class of bang–bang controls. The corresponding optimal switching time is derived analytically, whose phase diagram with respect to several key parameters is explored in detail. The bang–bang control will be broken when nonlinear strategies for stabilizer addition are adopted. Moreover, the above theory is applied to the stabilization of erythropoietin by ten different kinds of chemicals, providing theoretical guidance for the selection and rational usage of stabilizers. Our current study on optimal strategies for protein stabilizers not only offers deep insights into the general picture of protein folding kinetics but also provides valuable theoretical guidance on treatments for protein-related diseases in medicine.

The partial equilibrium approximation (PEA) and the quasi-steady-state approximation (QSSA) are two classical methods for reducing complex macroscopic chemical reactions into simple computable ones. Previous studies have primarily focused on the accuracy of solutions before and after applying model reduction. In this paper, however, we adopt a thermodynamic view and aim to establish a quantitative connection on the essential thermodynamic quantities, such as entropy production rate, free-energy dissipation rate, and entropy flow rate, between the original chemical mass-action equations and the reduced models obtained by PEA or QSSA. Our findings reveal that the reduced models by PEA and QSSA may not preserve the thermodynamic structure of the original full model, particularly when algebraic relations are used instead of differential equations. This is evident in the loss of non-negativity of the free-energy dissipation rate. To validate our results, we apply them to the Michaelis-Menten reactions as a prototype, both analytically and numerically. We anticipate that our study will inspire a reevaluation of the effectiveness of various model reduction or approximation methods from the perspective of nonequilibrium thermodynamics. Published by the American Physical Society 2024

Partial equilibrium approximation (PEA) and quasi-steady-state approximation (QSSA) are two classical methods for reducing complex macroscopic chemical reactions into simple computable ones. Previous studies mainly focus on the accuracy of solutions before and after applying model reduction. While, in this paper we start from a thermodynamic view, and try to establish a quantitative connection on the essential thermodynamic quantities, like entropy production rate, free energy dissipation rate and entropy flow rate, between the original reversible chemical mass-action equations and the reduced models by either PEA or QSSA. Our results reveal that the PEA and QSSA do not necessarily preserve the nice thermodynamic structure of the original full model during the reduction procedure (e.g. the loss of non-negativity of free energy dissipation rate), especially when adopting the algebraic relations in replace of differential equations. These results are further validated though the application to Michaelis-Menten reactions analytically and numerically as a prototype. We expect our study would motivate a re-examination on the effectiveness of various model reduction or approximation methods from a new perspective of non-equilibrium thermodynamics.