Thermodynamics of stoichiometric biochemical networks in living systems far from equilibrium

The principles of thermodynamics apply to both equilibrium and nonequilibrium biochemical systems. The mathematical machinery of the classic thermodynamics, however, mainly applies to systems in equilibrium. We introduce a thermodynamic formalism for the study of metabolic biochemical reaction (open, nonlinear) networks in both time-dependent and time-independent nonequilibrium states. Classical concepts in equilibrium thermodynamics-enthalpy, entropy, and Gibbs free energy of biochemical reaction systems-are generalized to nonequilibrium settings. Chemical motive force, heat dissipation rate, and entropy production (creation) rate, key concepts in nonequilibrium systems, are introduced. Dynamic equations for the thermodynamic quantities are presented in terms of the key observables of a biochemical network: stoichiometric matrix Q, reaction fluxes J, and chemical potentials of species mu without evoking empirical rate laws. Energy conservation and the Second Law are established for steady-state and dynamic biochemical networks. The theory provides the physiochemical basis for analyzing large-scale metabolic networks in living organisms.     

Kinetorheological aspects of biorheology

The relationship between stress and strain is the rheological equation of state. In the case of sophisticated systems such as biological tissue, this is rarely a simple relationship. The relationship is seen to be even more complex when it is recalled that in most living tissues, the tissue is not in chemical equilibrium, but is at best in some controlled steady state. At worst, it is undergoing major fluctuations or transitions because the chemical reactions or fluxes are altering the system. It is shown, in particular, that in addition to the changes in composition, the effective rheological relaxation times of the system are shortened due to contributions deriving from the reaction rate constants. These and other points are illustrated by considering a process of irreversible monomolecular degradation of a large macromolecular species.     

Nonequilibrium chemical potentials and free energies for enzyme-catalyzed reactions

Using the statistical theory of nonequilibrium thermodynamics we explore the nature of nonequilibrium corrections to chemical potentials in simple enzyme-catalyzed reactions. The statistical definition of the chemical potential, which pertains to systems that are at stable steady states, is applied to the Michaelis-Menten reaction scheme in a cellular-sized compartment that communicates with out-side reservoirs. Calculations based on the kinetic parameters for hexokinase and triose phosphate isomerase show that substantial corrections to the chemical potential of product (the order of 25 mV) are possible if the reaction is sufficiently far from equilibrium. The dependence of the corrections to the chemical potentials on the size of the cellular compartment are explored, and the relevance of the corrections for understanding the thermodynamics of metabolites is discussed.     

Kudos to Reviewers for the JGP: You Make Our Science Better

Rate-equilibrium free energy relationship (REFER) analysis provides information on transition-state structures and has been applied to reveal the temporal sequence in which the different regions of an ion channel protein move during a closed–open conformational transition. To date, the theory used to interpret REFER relationships has been developed only for equilibrium mechanisms. Gating of most ion channels is an equilibrium process, but recently several ion channels have been identified to have retained nonequilibrium traits in their gating cycles, inherited from transporter-like ancestors. So far it has not been examined to what extent REFER analysis is applicable to such systems. By deriving the REFER relationships for a simple nonequilibrium mechanism, this paper addresses whether an equilibrium mechanism can be distinguished from a nonequilibrium one by the characteristics of their REFER plots, and whether information on the transition-state structures can be obtained from REFER plots for gating mechanisms that are known to be nonequilibrium cycles. The results show that REFER plots do not carry information on the equilibrium nature of the underlying gating mechanism. Both equilibrium and nonequilibrium mechanisms can result in linear or nonlinear REFER plots, and complementarity of REFER slopes for opening and closing transitions is a trivial feature true for any mechanism. Additionally, REFER analysis provides limited information about the transition-state structures for gating schemes that are known to be nonequilibrium cycles.     

Basic statistical recipes for the emergence of biochemical discernment

An essential step towards understanding life would be to identify the very basic mechanisms responsible for the discerning behaviour of living biochemical systems, absent from randomly reacting chemical soups. One intuitively feels that this question goes beyond the particular nature of the biological molecules and should relate to general physical principles. The pre-eminent physicist Ludwig Boltzmann early envisioned life as a struggle for entropy, in concordance with the subsequent principle of self-organization out of equilibrium. Re-examination of elementary steady state biochemical systems from a statistical perspective supports this view and shows that sigmoidal responses arising from microstates elimination, are sufficient to explain innermost characteristics of life, including its capacity to convert random molecular interactions into accurate biological reactions. A primary operating strategy to achieve this goal is the introduction of time-irreversible transitions in molecular state conversion cycles by injection of free energy, which confers decisional capacity to single macromolecules. Selected examples from various fields of molecular biology such as enzymology and gene expression, are provided to show that these non-equilibrium steady state mechanisms remain important in contemporary biochemical systems. But in addition, information archiving allowed the emergence of the time-reversible counterparts of these mechanisms, mediated by evolutionary pre-organized macromolecular complexes capable of generating discernment in a non-dissipative manner.     

The free energies of metabolic reactions (ΔG) are not positive

Biochemistry textbooks often erroneously state that some metabolic reactions have positive free energies (ΔG), which suggests that they are spontaneous in the reverse direction. This incorrectly implies that reverse fluxes can occur through some steps in a metabolic pathway while the overall flux through the pathway is in the forward direction. In fact, at steady state all the reactions of a metabolic pathway proceed in the forward direction with the same flux. During periods of transition between steady states a positive free energy for any one reaction in a pathway means that the pathway is not actually proceeding from beginning to end.     

Phenomenon of Life: Between Equilibrium and Non-Linearity

A model of ordering applicable to biological evolution is presented. It is shown that a steady state (more precisely approaching to a steady state) system of irreversible processes, under conditions of disproportionation of entropy, produces a lower-entropy product, that is, ordering. The ordering is defined as restricting of degrees of freedom: freedom of motion, interactions etc. The model differs from previous ones in that it relates the ordering to processes running not far from equilibrium, described in the linear field of non-equilibrium thermodynamics. It is shown that a system, which includes adenosine triphosphate (ATP) to adenosine diphosphate (ADP) conversion meets the demands of the physical model: it provides energy maintaining steady state conditions, and hydrolysis of ATP proceeding with consumption of water can be tightly conjugated with the most important reactions of synthesis of organic polymers (peptides, nucleotide chains etc.), which proceed with release of water. For these and other reasons ATP seems to be a key molecule of prebiotic evolution. It is argued that the elementary chemical reaction proceeding under control of an enzyme is not necessarily far from equilibrium. The experimental evidence supporting this idea, is presented. It is based on isotope data. Carbon isotope distribution in biochemical systems reveals regularity, which is inherent to steady state systems of chemical reactions, proceeding not far from equilibrium. In living organisms this feature appears at the statistical level, as many completely irreversible and non-linear processes occur in organisms. However not-far-from-equilibrium reactions are inherent to biochemical systems as a matter of principle. They are reconcilable with biochemical behavior. Extant organisms are highly evolved entities which, however, show in their basis the same features, as the simplest chemical systems must have had been involved in the origin of life. Some consequences following from the model, which may be significant for understanding the origin of life and the mechanism of biological evolution, are pointed out.     

Equilibrium and steady state thermodynamics of active transport systems studied on simple models simulating Ca2+ transport through sarcoplasmic reticulum membranes

Equilibrium and steady state conditions of primary active transport systems are analyzed in models simulating well known characteristics of calcium transport through sarcoplasmic reticulum membranes. The model for the equilibrium simulations is a closed system with two compartments and a vectorial chemical reaction coupling Ca transport and ATP breakdown. The chemical potential difference for Ca (delta mu Ca) is calculated as a function of the total amount of Ca (Cat) and nucleotides (Nt) in the system. Results are obtained by successive approximations along the thermodynamic pathway of the reaction, up to minimizing free energy of the system, since the solution of the explicit equations cannot be obtained with computers of current precision for data within physiological ranges. delta mu Ca and [Caout] are extremely dependent on Cat and Nt for certain combinations of the variables, i.e. [Caout] can be raised from 10(-8) to 10(-6) M when Cat varies from 0.998 to 1.002 mM, therefore, the running force of the spontaneous reaction is largely shifted by tiny changes in the parameters of the system. For steady state simulations, ATP supply to the system, ADP and Pi drainage, and Ca diffusion through the barrier, are assumed. Again, conditions within physiological ranges can be found where tiny changes in Cat, the rate of ATP supply, diffusion, the ratio between the volumes of the compartments, or a relative uncoupling between the transport and hydrolytic reactions, largely shifts delta mu Ca and [Caout], thus making the steady state highly unstable and therefore well designed to operate as an amplifier of physiological signals. The equilibrium model describes some physicochemical characteristics of the system; the steady state model is more useful to simulate several physiological situations.     

The hill coefficient for the Ca2+-activation of striated muscle contraction

The following arguments are presented for the observation that curves relating free Ca2+ and force development of thin filament regulated myofilaments of skinned muscle fibers have Hill coefficient (n) greater than 4, which is the number of Ca2+ binding sites on troponin: Activation of the myofilaments is a process relaxing to a nonequilibrium steady state or stationary state. Systems operating at nonequilibrium stationary states are known to display Hill coefficients greater than the number of interacting sites and similar results have been obtained for Ca2+ activation of myofilament isometric force. The size of the basic subunit of thin filament regulated muscle may be the entire thin filament rather than seven actins, one tropomyosin, and one troponin. In this case the number of interacting sites may be on the order of hundreds. Hysteresis in the Ca2+ activation of isometric force might result from multiple stationary states and also might give rise to Hill coefficients greater than 4.     

Distance of the reduced Brusselator from equilibrium

The new concept of a nonequilibrium parameter P is applied to a reduced Brusselator, considered as a kinetic model for cellular processes. The reduction corresponds to omitting a monomolecular reaction. The deviation from equilibrium is due to a fixed nonequilibrium value of an extracellular concentration B, responsible for energizing and determining the steady-state value of the overall chemical affinity ?. The value of ? is insensitive to different steady states possible for a given set of rate constants. In contrast, the parameter P is state dependent. In particular, it may jump together with state variables. Two limiting cases of high ? are investigated, B→ 0 and B→ 1. In the first case P grows monotonically with ?. In the second case there is always a steady state solution with P→ 0. The physical interpretation of this effect of “equilibrium far from equilibrium” reveals the real predictive power of the parameter P. Relaxation regimes are investigated for a doubly reduced Brusselator. Both P and ? are in general time dependent and have jumps in their time derivatives. The canonical form of P is compared with the noncanonical one in the context of robustness of the new concept with respect to incomplete information about the system studied. These forms of P are different in relaxation to a nonequilibrium state and coincide in relaxation to an equilibrium state.     

Mathematical model of compartmentalized energy transfer: Its use for analysis and interpretation of 31P-NMR studies of isolated heart of creatine kinase deficient mice

A mathematical model of the compartmentalized energy transfer in cardiac cells is described and used for interpretation of novel experimental data obtained by using phosphorus NMR for determination of the energy fluxes in the isolated hearts of transgenic mice with knocked out creatine kinase isoenzymes. These experiments were designed to study the meaning and importance of compartmentation of creatine kinase isoenzymes in the cells in vivo. The model was constructed to describe quantitatively the processes of energy production, transfer, utilization, and feedback between these processes. It describes the production of ATP in mitochondrial matrix space by ATP synthase, use of this ATP for phosphocreatine production in the mitochondrial creatine kinase reaction coupled to the adenine nucleotide translocation, diffusional exchange of metabolites in the cytoplasmic space, and use of phosphocreatine for resynthesis of ATP in the myoplasmic creatine kinase reaction. It accounts also for the recently discovered phenomenon of restricted diffusion of adenine nucleotides through mitochondrial outer membrane porin pores (VDAC). Practically all parameters of the model were determined experimentally. The analysis of energy fluxes between different cellular compartments shows that in all cellular compartments of working heart cells the creatine kinase reaction is far from equilibrium in the systolic phase of the contraction cycle and approaches equilibrium only in cytoplasm and only in the end-diastolic phase of the contraction cycle.Experimental determination of the relationship between energy fluxes by a ³¹P-NMR saturation transfer method and workload in isolated and perfused heart of transgenic mice deficient in MM isoenzyme of the creatine kinase, MM -/- showed that in the hearts from wild mice, containing all creatine kinase isoenzymes, the energy fluxes determined increased 3-4 times with elevation of the workload. By contrast, in the hearts in which only the mitochondrial creatine kinase was active, the energy fluxes became practically independent of the workload in spite of the preservation of 26% of normal creatine kinase activity. These results cannot be explained on the basis of the conventional near-equilibrium theory of creatine kinase in the cells, which excludes any difference between creatine kinase isoenzymes. However, these apparently paradoxical experimental results are quantitatively described by a mathematical model of the compartmentalized energy transfer based on the steady state kinetics of coupled creatine kinase reactions, compartmentation of creatine kinase isoenzymes in the cells, and the kinetics of ATP production and utilization reactions. The use of this model shows that: (1) in the wild type heart cells a major part of energy is transported out of mitochondria via phosphocreatine, which is used for complete regeneration of ATP locally in the myofibrils - this is the quantitative estimate for PCr pathway; (2) however, in the absence of MM-creatine kinase in the myofibrils in transgenic mice the contraction results in a very rapid rise of ADP in cytoplasmic space, that reverses the mitochondrial creatine kinase reaction in the direction of ATP production. In this way, because of increasing concentrations of cytoplasmic ADP, mitochondrial creatine kinase is switched off functionally due to the absence of its counterpart in PCr pathway, MM-creatine kinase. This may explain why the creatine kinase flux becomes practically independent from the workload in the hearts of transgenic mouse without MM-CK. Thus, the analysis of the results of studies of hearts of creatine kinase-deficient transgenic mice, based on the use of a mathematical model of compartmentalized energy transfer, show that in the PCr pathway of intracellular energy transport two isoenzymes of creatine kinase always function in a coordinated manner out of equilibrium, in the steady state, and disturbances in functioning of one of them inevitably result in the disturbances of the other component of the PCr pathway. In the latter case, energy is transferred from mitochondria to myofibrils by alternative metabolic pathways, probably involving adenylate kinase or other systems.     

Optimal Control of Transitions between Nonequilibrium Steady States

Biological systems fundamentally exist out of equilibrium in order to preserve organized structures and processes. Many changing cellular conditions can be represented as transitions between nonequilibrium steady states, and organisms have an interest in optimizing such transitions. Using the Hatano-Sasa Y-value, we extend a recently developed geometrical framework for determining optimal protocols so that it can be applied to systems driven from nonequilibrium steady states. We calculate and numerically verify optimal protocols for a colloidal particle dragged through solution by a translating optical trap with two controllable parameters. We offer experimental predictions, specifically that optimal protocols are significantly less costly than naive ones. Optimal protocols similar to these may ultimately point to design principles for biological energy transduction systems and guide the design of artificial molecular machines.     

Modeling Free Energy Availability from Hadean Hydrothermal Systems to the First Metabolism

Off-axis Hydrothermal Systems (HSs) are seen as the possible setting for the emergence of life. As the availability of free energy is a general requirement to drive any form of metabolism, we ask here under which conditions free energy generation by geologic processes is greatest and relate these to the conditions found at off-axis HSs. To do so, we present a conceptual model in which we explicitly capture the energetics of fluid motion and its interaction with exothermic reactions to maintain a state of chemical disequilibrium. Central to the interaction is the temperature at which the exothermic reactions take place. This temperature not only sets the equilibrium constant of the chemical reactions and thereby the distance of the actual state to chemical equilibrium, but these reactions also shape the temperature gradient that drives convection and thereby the advection of reactants to the reaction sites and the removal of the products that relate to geochemical free energy generation. What this conceptual model shows is that the positive feedback between convection and the chemical kinetics that is found at HSs favors a greater rate of free energy generation than in the absence of convection. Because of the lower temperatures and because the temperature of reactions is determined more strongly by these dynamics rather than an external heat flux, the conditions found at off-axis HSs should result in the greatest rates of geochemical free energy generation. Hence, we hypothesize from these thermodynamic considerations that off-axis HSs seem most conducive for the emergence of protometabolic pathways as these provide the greatest, abiotic generation rates of chemical free energy.     

Differentiation between equilibrium and nonequilibrium kinetic systems by noise analysis.

Methods for the differentiation between equilibrium and nonequilibrium steady-state kinetic mechanisms based on fluctuation and noise analysis are discussed. Specifically, the "sharpening" in the auto noise power spectrum is shown to be a useful indicator in identifying a nonequilibrium steady state.     

The steady state kinetics of some biological systems: III. Thermodynamic aspects

Some thermodynamic aspects of steady systems are considered. The time rates of changes, “flux”, of various thermodynamic quantities are formulated. In particular the free energy flux in the steady state, the difference between the free energy flux in the steady and time dependent states and the change in free energy flux upon transition between steady states are discussed. Equations are derived which exhibit the formal similarities and differences between the free energy flux and the conventional free energy change. The temperature dependence of the steady state rate is examined and conditions for “mastery” by a single step discussed. A brief discussion of the role ofrate in the coupling of exergonic and endergonic reactions is given.     

Fluorescence correlation spectroscopy: past, present, future

In recent years fluorescence correlation spectroscopy (FCS) has become a routine method for determining diffusion coefficients, chemical rate constants, molecular concentrations, fluorescence brightness, triplet state lifetimes, and other molecular parameters. FCS measures the spatial and temporal correlation of individual molecules with themselves and so provides a bridge between classical ensemble and contemporary single-molecule measurements. It also provides information on concentration and molecular number fluctuations for nonlinear reaction systems that complement single-molecule measurements. Typically implemented on a fluorescence microscope, FCS samples femtoliter volumes and so is especially useful for characterizing small dynamic systems such as biological cells. In addition to its practical utility, however, FCS provides a window on mesoscopic systems in which fluctuations from steady states not only provide the basis for the measurement but also can have important consequences for the behavior and evolution of the system. For example, a new and potentially interesting field for FCS studies could be the study of nonequilibrium steady states, especially in living cells.     

Oscillations and efficiency in glycolysis

We suggest that temporal oscillations of concentrations of intermediates in biochemical reaction systems may enhance the efficiency of free energy conversion (reduce dissipation) in those reactions. Experiments on glycolysis are used to estimate the Gibbs free energy changes along the glycolysis mechanism, and to postulate a construct for the glycolysis "machine" which involves: the PFK reaction as the primary oscillophor; the GAPDH reaction as a phase-shifting device; and the PK reaction with the property of intrinsic oscillatory response at resonance with the driving frequency. Analysis of a simple reaction mechanism with these postulated properties shows that the conversion of free energy from reactants to products is more efficient in an oscillatory than a steady state operation. The efficiency of free energy conversion in glycolysis from glucose + ADP to products + ATP is estimated to be increased by 5--10% due to oscillations. This may have been relevant for the evolutionary development of oscillations such as in glycolysis, especially in anaerobic cells.     

Effect of fluctuating surface structure and free energy on the growth of linear tubular aggregates.

Simple linear tubular aggregates with up to eight strands are studied theoretically at equilibrium and under conditions of steady growth or shortening. The surface structure and free energy at an end of the polymer fluctuate as a consequence of the gain or loss of individual subunits. The surface free energy governs the probability distribution of surface structures at equilibrium. At steady state, on and off rate constants are crucial for this purpose; these depend on the gain or loss of neighbor interactions at the polymer end when a subunit is gained or lost. The observed on and off rate constants are averages of microscopic rate constants. A consequence of this is that the subunit flux onto the polymer end is, in general, not a linear function of the free subunit concentration, as is usually assumed. Monte Carlo calculations are needed at steady state for three or more strands. The general approach can be applied to microtubules, which have 13 strands. Actin is a special case, included here, with two strands.     

Metabolic systems maintain stable non‐equilibrium via thermodynamic buffering

Here, we analyze how the set of nucleotides in the cell is equilibrated and how this generates simple rules that help the cell to organize itself via maintenance of a stable non‐equilibrium state. A major mechanism operating to achieve this state is thermodynamic buffering via high activities of equilibrating enzymes such as adenylate kinase. Under stable non‐equilibrium, the ratios of free and Mg‐bound adenylates, Mg²⁺ and membrane potentials are interdependent and can be computed. The adenylate status is balanced with the levels of reduced and oxidized pyridine nucleotides through regulated uncoupling of the pyridine nucleotide pool from ATP production in mitochondria, and through oxidation of substrates non‐coupled to NAD⁺ reduction in peroxisomes. The set of adenylates and pyridine nucleotides constitutes a generalized cell energy status and determines rates of major metabolic fluxes. As the result, fluxes of energy and information become organized spatially and temporally, providing conditions for self‐maintenance of metabolism.     

Characterization of the steady-state calcium fluxes in skeletal sarcoplasmic reticulum vesicles. Role of the Ca2+ pump

Unidirectional Ca2+ fluxes (influx and efflux), supported by ATP as a phosphate-donor substrate, were measured without alteration of the lumenal Ca2+ content in longitudinal sarcoplasmic reticulum vesicles. The referred fluxes are dependent on extravesicular Ca2+, ATP and ADP. They are unaffected by ruthenium red but inhibited by quercetin. The Ca2+ fluxes at steady state are drastically diminished when ATP is substituted by acetylphosphate although the addition of 10 microM ADP increases the apparent rate constants more than eight fold. The observed fluxes appear to be dependent on Ca2(+)-ATPase phosphoenzyme transitions. The results indicate that: (a) the slow Ca2+ release, due to the passive permeability of the membrane, is a minor component of the total Ca2+ efflux, and (b) the ATPase protein is basically operating as a Ca2+/Ca2+ exchanger at steady state. Kinetic resolution of the Ca2+ fluxes, measured by isotopic tracer and rapid filtration techniques can be recreated by computer simulation of the ATPase reaction cycle featuring some modifications to account for the fast Ca2+/Ca2+ exchange and the uncoupling effect observed at steady state.