We consider growing open chemical reaction systems (CRSs), in which autocatalytic chemical reactions are encapsulated in a finite volume and its size can change in conjunction with the reactions. The thermodynamics of growing CRSs is indispensable for understanding biological cells and designing protocells by clarifying the physical conditions and costs for their growing states. In this work, we establish a thermodynamic theory of growing CRSs by extending the Hessian geometric structure of non-growing CRSs. The theory provides the environmental conditions to determine the fate of the growing CRSs; growth, shrinking or equilibration. We also identify thermodynamic constraints; one to restrict the possible states of the growing CRSs and the other to further limit the region where a nonequilibrium steady growing state can exist. Moreover, we evaluate the entropy production rate in the steady growing state. The growing nonequilibrium state has its origin in the extensivity of thermodynamics, which is different from the conventional nonequilibrium states with constant volume. These results are derived from general thermodynamic considerations without assuming any specific thermodynamic potentials or reaction kinetics; i.e., they are obtained based solely on the second law of thermodynamics.
The stochastic thermodynamics of a dilute, well-stirred mixture of chemically-reacting species is built on the stochastic trajectories of reaction events obtained from the Chemical Master Equation. However, when the molecular populations are large, the discrete Chemical Master Equation can be approximated with a continuous diffusion process, like the Chemical Langevin Equation or Low Noise Approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the Chemical Master Equation. We find that a stochastic-thermodynamic descr…
All current formulations of nonequilibrium thermodynamics of open chemical reaction networks rely on the assumption of non-interacting species. We develop a general theory which accounts for interactions between chemical species within a mean-field approach using activity coefficients. Thermodynamic consistency requires that rate equations do not obey to standard mass-action kinetics, but account for the interactions with concentration dependent kinetic constants. Many features of the ideal formulations are recovered. Crucially, the thermodynamic potential and the forces driving non-ideal chem…
In this chapter, we illustrate recently obtained thermodynamic bounds for a number of enzymatic networks by focusing on simple examples of unicyclic or multi-cyclic networks. We also derive complementary relations which constrain the fluctuations of first-passage times to reach a threshold current.
We generalize thermodynamic uncertainty relation (TUR) and thermodynamic speed limit (TSL) for deterministic chemical reaction networks (CRNs). The scaled diffusion coefficient derived by considering the connection between macroscopic CRNs and mesoscopic CRNs plays an essential role in our results. The TUR shows that the product of the entropy production rate and the ratio of the scaled diffusion coefficient to the square of the rate of concentration change is bounded below by 2. The TSL states a trade-off relation between speed and thermodynamic quantities, the entropy production and the time…
We establish a Hessian geometric structure in chemical thermodynamics which describes chemical reaction networks (CRNs) with equilibrium states. In our setup, the ideal gas assumption and mass action kinetics are not required. The existence and uniqueness condition of the equilibrium state is derived by using the Legendre duality inherent to the Hessian structure. The entropy production during a relaxation to the equilibrium state can be evaluated by the Bregman divergence. Furthermore, the equilibrium state is characterized by four distinct minimization problems of the divergence, which are o…
Biomolecular processes are typically modeled using chemical reaction networks coupled to infinitely large chemical reservoirs. A difference in chemical potential between these reservoirs can drive the system into a non-equilibrium steady state (NESS). In reality, these processes take place in finite systems containing a finite number of molecules. In such systems, a NESS can be reached with the help of an externally driven pump for which we introduce a simple model. Crucial parameters are the pumping rate and the finite size of the chemical reservoir. We apply this model to a simple biochemica…
We investigate the thermodynamic implications of two control mechanisms of open chemical reaction networks. The first controls the concentrations of the species that are exchanged with the surroundings, while the other controls the exchange fluxes. We show that the two mechanisms can be mapped one into the other and that the thermodynamic theories usually developed in the framework of concentration control can be applied to flux control as well. This implies that the thermodynamic potential and the fundamental forces driving chemical reaction networks out of equilibrium can be identified in th…
We construct an information-geometric structure for chemical thermodynamics, applicable to a wide range of chemical reaction systems including non-ideal and open systems. For this purpose, we explicitly construct dual affine coordinate systems, which completely designate an information-geometric structure, using the extent of reactions and the affinities of reactions as coordinates on a linearly-constrained space of amounts of substances. The resulting structure induces a metric and a divergence (a function of two distributions of amounts), both expressed with chemical potentials. These quanti…
We develop a thermodynamic framework for closed and open chemical networks applicable to non-elementary reactions that do not need to obey mass action kinetics. It only requires the knowledge of the kinetics and of the standard chemical potentials, and makes use of the topological properties of the network (conservation laws and cycles). Our approach is proven to be exact if the network results from a bigger network of elementary reactions where the fast-evolving species have been coarse grained. Our work should be particularly relevant for energetic considerations in biosystems where the char…
It can be fruitful to view two-component physical systems of attractive monomers, A and B, ``chemically'' in terms of a reaction A + B <-> C, where C = AB is an associated pair or complex. We show how to construct free energies in the three-component or chemical picture which, under mass-action equilibration, exactly reproduce any given two-component or ``physical'' thermodynamics. Order-by-order matching conditions and closed-form chemical representations reveal the freedom available to modify the A-C, B-C, and C-C interactions and to adjust the association constant. The theory (in the simple…
We study a connection between chemical thermodynamics and information geometry. We clarify a relation between the Gibbs free energy of an ideal dilute solution and an information-geometric quantity called an $f$-divergence. From this relation, we derive information-geometric inequalities that give a speed limit for a changing rate of the Gibbs free energy and a general bound of chemical fluctuations. These information-geometric inequalities can be regarded as generalizations of the Cramér--Rao inequality for chemical reaction networks described by rate equations, where unnormalized concentrati…
Chemical reaction networks (CRNs) are prototypical complex systems because reactions are nonlinear and connected in intricate ways, and they are also essential to understand living systems. Here, I discuss how recent developments in nonequilibrium thermodynamics provide new insight on how CRNs process energy and perform sophisticated tasks, and describe open challenges in the field.
We build a rigorous nonequilibrium thermodynamic description for open chemical reaction networks of elementary reactions. Their dynamics is described by deterministic rate equations satisfying mass action law. Our most general framework considers open networks driven by time-dependent chemostats. The energy and entropy balances are established and a nonequilibrium Gibbs free energy is introduced. The difference between this latter and its equilibrium form represents the minimal work done by the chemostats to bring the network in its nonequilibrium state. It is minimized in nondriven detailed-b…
Chemical waves constitute a known class of dissipative structures emerging in reaction-diffusion systems. They play a crucial role in biology, spreading information rapidly to synchronize and coordinate biological events. We develop a rigorous thermodynamic theory of reaction-diffusion systems to characterize chemical waves. Our main result is the definition of the proper thermodynamic potential of the local dynamics as a nonequilibrium free energy density and establishing its balance equation. This enables us to identify the dynamics of the free energy, of the dissipation, and of the work spe…
We introduce an infinite particle system dynamics, which includes stochastic chemical kinetics models, the classical Kac model and free space movement. We study energy redistribution between two energy types (kinetic and chemical) in different time scales, similar to energy redistribution in the living cell. One example is considered in great detail, where the model provides main formulas of chemical thermodynamics.
This paper studies a mathematical formalism of nonequilibrium thermodynamics for chemical reaction models with $N$ species, $M$ reactions, and general rate law. We establish a mathematical basis for J. W. Gibbs' macroscopic chemical thermodynamics under G. N. Lewis' kinetic law of entire equilibrium (detailed balance in nonlinear chemistry kinetics). In doing so, the equilibrium thermodynamics is then naturally generalized to nonequilibrium settings without detailed balance. The kinetic models are represented by a Markovian jumping process. A generalized macroscopic chemical free energy functi…
Onsager's phenomenological equations successfully describe irreversible thermodynamic processes. They assume a symmetric coupling matrix between thermodynamic fluxes and forces. It is easily shown that the antisymmetric part of a coupling matrix does not contribute to dissipation. Therefore, entropy production is exclusively governed by the symmetric matrix even in the presence of antisymmetric terms. In this work we focus on the antisymmetric contributions which describe isentropic oscillations and well-defined equations of motion. The formalism contains variables that are equivalent to momen…
Nonequilibrium thermodynamics of a system situated in a sustained environment with influx and efflux is usually treated as a subsystem in a larger, closed "universe". It remains a question what the minimally required description for the surrounding of such an open driven system is, so that its nonequilibrium thermodynamics can be established solely based on the internal stochastic kinetics. We provide a solution to this problem using insights from studies of molecular motors in a chemical nonequilibrium steady state (NESS) with sustained external drive through a regenerating system, or in a qu…
Chemical processes in closed systems are poorly controllable since they always relax to equilibrium. Living systems avoid this fate and give rise to a much richer diversity of phenomena by operating under nonequilibrium conditions. Recent experiments in dissipative self-assembly also demonstrated that by opening reaction vessels and steering certain concentrations, an ocean of opportunities for artificial synthesis and energy storage emerges. To navigate it, thermodynamic notions of energy, work and dissipation must be established for these open chemical systems. Here, we do so by building upo…
Phase separation is the thermodynamic process that explains how droplets form in multicomponent fluids. These droplets can provide controlled compartments to localize chemical reactions, and reactions can also affect the droplets' dynamics. This review focuses on the tight interplay between phase separation and chemical reactions originating from thermodynamic constraints. In particular, simple mass action kinetics cannot describe chemical reactions since phase separation requires non-ideal fluids. Instead, thermodynamics implies that passive chemical reactions reduce the complexity of phase d…
Colloidal motors without moving parts can be propelled by self-diffusiophoresis, coupling molecular concentration gradients generated by surface chemical reactions to the velocity slip between solid Janus particles and the surrounding fluid solution. The interfacial properties involved in this propulsion mechanism can be described by nonequilibrium thermodynamics and statistical mechanics, disclosing the fundamental role of microreversibility in the coupling between motion and reaction. Among other phenomena, the approach predicts that propulsion by fuel consumption has the reciprocal effect o…
A Minimal Thermodynamically Consistent Chemical Oscillator
University of Luxembourg（卢森堡大学）；University of Strasbourg（斯特拉斯堡大学）；Heidelberg University（海德堡大学）；University of Trieste（的里雅斯特大学）；Northwestern University（西北大学）
The dynamics of microbial communities is incredibly complex, determined by competition for metabolic substrates and cross-feeding of byproducts. Species in the community grow by harvesting energy from chemical reactions that transform substrates to products. In many anoxic environments, these reactions are close to thermodynamic equilibrium and growth is slow. To understand the community structure in these energy-limited environments, we developed a microbial community consumer-resource model incorporating energetic and thermodynamic constraints on an interconnected metabolic network. The cent…
We briefly review the recently developed, Markov process based isothermal chemical thermodynamics for nonlinear, driven mesoscopic kinetic systems. Both the instantaneous Shannon entropy {\boldmath $S[p_α(t)]$} and relative entropy {\boldmath $F[p_α(t)]$}, defined based on probability distribution {\boldmath $\{p_α(t)\}$}, play prominent roles. The theory is general; and as a special case when a chemical reaction system is situated in an equilibrium environment, it agrees perfectly with Gibbsian chemical thermodynamics: {\boldmath $k_BS$} and {\boldmath $k_BTF$} become thermodynamic entropy an…
We consider the efficiency of chemical energy extraction from the environment by the growth of a copolymer made of two constituent units in the entropy-driven regime. We show that the thermodynamic nonlinearity associated with the information processing aspect is responsible for a branching of the system properties such as power, speed of growth, entropy production, and efficiency, with varying affinity. The standard linear thermodynamics argument which predicts an efficiency of 1/2 at maximum power is inappropriate because the regime of maximum power is located either outside of the linear re…
From a mathematical model that describes a complex chemical kinetic system of $N$ species and $M$ elementrary reactions in a rapidly stirred vessel of size $V$ as a Markov process, we show that a macroscopic chemical thermodynamics emerges as $V\rightarrow\infty$. The theory is applicable to linear and nonlinear reactions, closed systems reaching chemical equilibrium, or open, driven systems approaching to nonequilibrium steady states. A generalized mesoscopic free energy gives rise to a macroscopic chemical energy function $φ^{ss}(\vx)$ where $\vx=(x_1,\cdots,x_N)$ are the concentrations of t…
Information thermodynamics relates the rate of change of mutual information between two interacting subsystems to their thermodynamics when the joined system is described by a bipartite stochastic dynamics satisfying local detailed balance. Here, we expand the scope of information thermodynamics to deterministic bipartite chemical reaction networks, namely, composed of two coupled subnetworks sharing species, but not reactions. We do so by introducing a meaningful notion of mutual information between different molecular features, that we express in terms of deterministic concentrations. This a…
Chemical reaction systems operating in nonequilibrium open-system states arise in a great number of contexts, including the study of living organisms, in which chemical reactions, in general, are far from equilibrium. Here we introduce a theorem that relates forward and re-verse fluxes and free energy for any chemical process operating in a steady state. This rela-tionship, which is a generalization of equilibrium conditions to the case of a chemical process occurring in a nonequilibrium steady state, provides a novel equivalent definition for chemical reaction free energy. In addition, it is …
General expression for the thermodynamic potential of the model of semi-infinite jellium is obtained. By using this expression, the surface energy for infinite barrier model is calculated. The behavior of the surface energy and of chemical potential as functions of the Wigner-Seitz radius and the influence of the Coulomb interaction between electrons on the calculated values is studied. It is shown that taking into account the Coulomb interaction between electrons leads to growth of the surface energy. The surface energy is positive in the entire area of the Wigner-Seitz radius. It is shown th…
We investigate the nonequilibrium stationary states of systems consisting of chemical reactions among molecules of several chemical species. To this end we introduce and develop a stochastic formulation of nonequilibrium thermodynamics of chemical reaction systems based on a master equation defined on the space of microscopic chemical states, and on appropriate definitions of entropy and entropy production, The system is in contact with a heat reservoir, and is placed out of equilibrium by the contact with particle reservoirs. In our approach, the fluxes of various types, such as the heat and …
Nonequilibrium thermodynamics (NET) investigates processes in systems out of global equilibrium. On a mesoscopic level, it provides a statistical dynamic description of various complex phenomena such as chemical reactions, ion transport, diffusion, thermochemical, thermomechanical and mechanochemical fluxes. In the present review, we introduce a mesoscopic stochastic formulation of NET by analyzing entropy production in several simple examples. The fundamental role of nonequilibrium steady-state cycle kinetics is emphasized. The statistical mechanics of Onsager's reciprocal relations in this c…
The thermodynamic Euler equation for high-energy states of large-$N$ gauge theories is derived from the dependence of the extensive quantities on the number of colors $N$. This Euler equation relates the energy of the state to the temperature, entropy, number of degrees of freedom and its chemical potential, but not to the volume or pressure. In the context of the gauge/gravity duality we show that the Euler equation is dual to the generalized Smarr formula for black holes in the presence of a negative cosmological constant. We also match the fundamental variational equation of thermodynamics …
In this and a companion paper we outline a general framework for the thermodynamic description of open chemical reaction networks, with special regard to metabolic networks regulating cellular physiology and biochemical functions. We first introduce closed networks "in a box", whose thermodynamics is subjected to strict physical constraints: the mass-action law, elementarity of processes, and detailed balance. We further digress on the role of solvents and on the seemingly unacknowledged property of network independence of free energy landscapes. We then open the system by assuming that the co…
Current formulations of nonequilibrium thermodynamics of open chemical reaction networks only consider chemostats as free-energy sources sustaining nonequilibrium behaviours. Here, we extend the theory to include incoherent light as a source of free energy. We do so by relying on a local equilibrium assumption to derive the chemical potential of photons relative to the system they interact with. This allows us to identify the thermodynamic potential and the thermodynamic forces driving light-reacting chemical systems out of equilibrium. We use this framework to treat two paradigmatic photochem…
For chemical reaction networks described by a master equation, we define energy and entropy on a stochastic trajectory and develop a consistent nonequilibrium thermodynamic description along a single stochastic trajectory of reaction events. A first-law like energy balance relates internal energy, applied (chemical) work and dissipated heat for every single reaction. Entropy production along a single trajectory involves a sum over changes in the entropy of the network itself and the entropy of the medium. The latter is given by the exchanged heat identified through the first law. Total entropy…
In the framework of irreversible thermodynamics, we study autonomous systems of reaction-diffusion equations to show how the entropy and free energy of an open and irreversible reactor depend on concentrations. To do this, we find a Lyapunov function that depends directly on the eigenvalues and eigenvectors of the linearized problem valid for linear and asymptotically stable systems. By suggesting the role of this Lyapunov function as internal free energy, we were able to reflect the properties of the dynamical system directly into the second law of thermodynamics making a redefinition of the …
In order to understand thermodynamical properties of N D-branes with chemical potentials associated with R-symmetry charges, we study a one dimensional large N gauge theory (bosonic BFSS type model) as a first step. This model is obtained through a dimensional reduction of a 1+D dimensional SU(N) Yang-Mills theory and we use a 1/D expansion to investigate the phase structure. We find three phases in the μ-T plane. We also show that all the adjoint scalars condense at large D and obtain a mass dynamically. This dynamical mass protects our model from the usual perturbative instability of massles…
Using the chemical potential of a solid in a dissolved state or the corresponding component of the chemical potential tensor at equilibrium with the solution, a new concept of grand thermodynamic potential for solids has been suggested. This allows generalizing the definition of Gibbs' quantity $σ$ (surface work often called the solid-fluid interfacial free energy) at a planar surface as an excess grand thermodynamic potential per unit surface area that (1) does not depend on the dividing surface location and (2) is common for fluids and solids.
We investigate nonequilibrium chemical reaction systems from the view point of steady state thermodynamics proposed by Oono and Paniconi [Prog. Theor. Phys. Suppl. 130, 29 (1998)]. The concentrations of some compounds are operated by an external system, so that a transition from a steady state to other steady state takes place. We show that an analogue of Clausius inequality holds macroscopically for the operation processes. This implies that the second law of thermodynamics can be generalized, including nonequilibrium steady states.
The analysis of non-equilibrium steady states of biochemical reaction networks relies on finding the configurations of fluxes and chemical potentials satisfying stoichiometric (mass balance) and thermodynamic (energy balance) constraints. Efficient methods to explore such states are crucial to predict reaction directionality, calculate physiologic ranges of variability, estimate correlations, and reconstruct the overall energy balance of the network from the underlying molecular processes. While different techniques for sampling the space generated by mass balance constraints are currently ava…
We discuss an effective Polyakov loop model for QCD thermodynamics with a chemical potential. Using high temperature expansion techniques the partition sum is mapped exactly onto the partition sum of a flux model. In the flux representation the complex action problem is resolved and a simulation with worm-type algorithms becomes possible also at finite chemical potential.
Institute of Physics, School of Basic Sciences, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院基础科学学院物理研究所）；Center for Systems Biology Dresden（德累斯顿系统生物学中心）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Institute of Bioengineering, School of Life Sciences, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院生命科学学院生物医学工程研究所）；Department of Physics and Astronomy “G. Galilei”, University of Padova（帕多瓦大学“伽利略”物理学与天文学系）
We study the following paradox associated with networks growing according to superlinear preferential attachment: superlinear preference cannot produce scale-free networks in the thermodynamic limit, but there are superlinearly growing network models that perfectly match the structure of some real scale-free networks, such as the Internet. We obtain an analytic solution, supported by extensive simulations, for the degree distribution in superlinearly growing networks with arbitrary average degree, and confirm that in the true thermodynamic limit these networks are indeed degenerate, i.e., almo…
