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.     

Linear nonequilibrium thermodynamics describes the dynamics of an autocatalytic system.

A model simulating oscillations in glycolysis was formulated in terms of nonequilibrium thermodynamics. In the kinetic rate equations every metabolite concentration was replaced with an exponential function of its chemical potential. This led to nonlinear relations between rates and chemical potentials. Each chemical potential was then expanded around its steady-state value as a Taylor series. The linear (first order) term of the Taylor series sufficed to simulate the dynamic behavior of the system, including the damped and even sustained oscillations at low substrate input or high free-energy load. The glycolytic system is autocatalytic in the first half. Because oscillations were obtained only in the presence of that autocatalytic feed-back loop we conclude that this type of kinetic nonlinearity was sufficient to account for the oscillatory behavior. The matrix of phenomenological coefficients of the system is nonsymmetric. Our results indicate that this is the symmetry property and not the linearity of the flow-force relations in the near equilibrium domain that precludes oscillations. Given autocatalytic properties, a system exhibiting liner flow-force relations and being outside the near equilibrium domain may show bifurcations, leading to self-organized behavior.     

Stoichiometric network theory for nonequilibrium biochemical systems.

We introduce the basic concepts and develop a theory for nonequilibrium steady-state biochemical systems applicable to analyzing large-scale complex isothermal reaction networks. In terms of the stoichiometric matrix, we demonstrate both Kirchhoff's flux law sigma(l)J(l)=0 over a biochemical species, and potential law sigma(l) mu(l)=0 over a reaction loop. They reflect mass and energy conservation, respectively. For each reaction, its steady-state flux J can be decomposed into forward and backward one-way fluxes J = J+ - J-, with chemical potential difference deltamu = RT ln(J-/J+). The product -Jdeltamu gives the isothermal heat dissipation rate, which is necessarily non-negative according to the second law of thermodynamics. The stoichiometric network theory (SNT) embodies all of the relevant fundamental physics. Knowing J and deltamu of a biochemical reaction, a conductance can be computed which directly reflects the level of gene expression for the particular enzyme. For sufficiently small flux a linear relationship between J and deltamu can be established as the linear flux-force relation in irreversible thermodynamics, analogous to Ohm's law in electrical circuits.     

Competitive Calcium Binding: Implications for Dendritic Calcium Signaling

Action potentials evoke calcium transients in dendrites of neocortical pyramidal neurons with time constants of <100 ms at physiological temperature. This time period may not be sufficient for inflowing calcium ions to equilibrate with all present Ca²⁺-binding molecules. We therefore explored nonequilibrium dynamics of Ca²⁺ binding to numerous Ca²⁺ reaction partners within a dendritelike compartment using numerical simulations. After a brief Ca²⁺ influx, the reaction partner with the fastest Ca²⁺ binding kinetics initially binds more Ca²⁺ than predicted from chemical equilibrium, while companion reaction partners bind less. This difference is consolidated and may result in bypassing of slow reaction partners if a Ca²⁺ clearance mechanism is active. On the other hand, slower reaction partners effectively bind Ca²⁺ during repetitive calcium current pulses or during slower Ca²⁺ influx. Nonequilibrium Ca²⁺ distribution can further be enhanced through strategic placement of the reaction partners within the compartment. Using the Ca²⁺ buffer EGTA as a competitor of fluo-3, we demonstrate competitive Ca²⁺ binding within dendrites experimentally. Nonequilibrium calcium dynamics is proposed as a potential mechanism for differential and conditional activation of intradendritic targets.     

The “cratic correction” and related fallacies

The “cratic correction” for removing the contribution of translational motion from standard free energies and entropies of chemical reactions is examined. Although removal of translational effects is useful in interpreting free energy and entropy changes in molecular terms, this particular correction does not do so. No basis for it is found in thermodynamics or statistical mechanics. The appropriate correction for center-of-mass translational motion is derived from classical statistical mechanics. However, the importance of recognizing the proper molecular meaning of the remaining free energy is stressed. In any case, this latter correction, however legitimate, is shown to be futile, since it has long been recognized from classical statistical mechanics that all terms involving momenta (thermal wavelength terms) —be they classified as translational, rotational, vibrational, or whatever—necessarily cancel out informing the standard free energy change of any chemical reaction from the constituent standard chemical potentials. © 1995 John Wiley & Sons, Inc.     

Thermodynamically feasible kinetic models of reaction networks

The dynamics of biological reaction networks are strongly constrained by thermodynamics. An holistic understanding of their behavior and regulation requires mathematical models that observe these constraints. However, kinetic models may easily violate the constraints imposed by the principle of detailed balance, if no special care is taken. Detailed balance demands that in thermodynamic equilibrium all fluxes vanish. We introduce a thermodynamic-kinetic modeling (TKM) formalism that adapts the concepts of potentials and forces from irreversible thermodynamics to kinetic modeling. In the proposed formalism, the thermokinetic potential of a compound is proportional to its concentration. The proportionality factor is a compound-specific parameter called capacity. The thermokinetic force of a reaction is a function of the potentials. Every reaction has a resistance that is the ratio of thermokinetic force and reaction rate. For mass-action type kinetics, the resistances are constant. Since it relies on the thermodynamic concept of potentials and forces, the TKM formalism structurally observes detailed balance for all values of capacities and resistances. Thus, it provides an easy way to formulate physically feasible, kinetic models of biological reaction networks. The TKM formalism is useful for modeling large biological networks that are subject to many detailed balance relations.     

Thermodynamics in the present progressive mode and its role in the context of the origin of life.

The origin and evolution of biological organizations proceeding on Earth are put in a nonequilibrium thermodynamic framework within a cosmological context. The dynamic process responsible for chemical evolution leading to the origin of biological being depends upon consumer-dominating thermodynamics, in which the heat sink is taken to be active in extracting heat energy from a body at a higher temperature. Consumer-dominating thermodynamics follows from the fact that when a small hot body contacts a cold heat sink, it decreases the temperature at the possible fastest rate. The fastest temperature drop, when applied to chemical products being synthesized through the energy supplied from an external heat source, is selective in keeping only those products that can decrease the temperature at the fastest rate among the available alternatives. Synthesis of small organic molecules in the small ice grains in interstellar diffuse clouds irradiated by ultraviolet radiation is a representative case of consumer-dominating thermodynamics, in which diffuse clouds serve as cold heat sinks in the cosmological context. Another case of consumer-dominating thermodynamics predominant on Earth especially in the perspective of the origin and evolution of life is with submarine hydrothermal vents, in which the surrounding cold seawater constantly serves as the cold heat sink.     

The thermodynamics of thiol sulfenylation

Protein sulfenic acids are essential cysteine oxidations in cellular signaling pathways. The thermodynamics that drive protein sulfenylation are not entirely clear. Experimentally, sulfenic acid reduction potentials are hard to measure, because of their highly reactive nature. We designed a calculation method, the reduction potentials from electronic energies (REE) method, to give for the first time insight into the thermodynamic aspects of protein sulfenylation. The REE method is based on the correlation between reaction path-independent reaction energies and free energies of a series of analogous reactions. For human peroxiredoxin (Tpx-B), an antioxidant enzyme that forms a sulfenic acid on one of its active-site cysteines during reactive oxygen scavenging, we found that the reduction potential depends on the composition of the active site and on the protonation state of the cysteine. Interaction with polar residues directs the RSO(-)/RS(-) reduction to a lower potential than the RSOH/RSH reduction. A conserved arginine that thermodynamically favors the sulfenylation reaction might be a good candidate to favor the reaction kinetics. The REE method is not limited to thiol sulfenylation, but can be broadly applied to understand protein redox biology in general.     

Simulation of electron-proton coupling with a Monte Carlo method: application to cytochrome c3 using continuum electrostatics.

A new method is presented for simulating the simultaneous binding equilibrium of electrons and protons on protein molecules, which makes it possible to study the full equilibrium thermodynamics of redox and protonation processes, including electron-proton coupling. The simulations using this method reflect directly the pH and electrostatic potential of the environment, thus providing a much closer and realistic connection with experimental parameters than do usual methods. By ignoring the full binding equilibrium, calculations usually overlook the twofold effect that binding fluctuations have on the behavior of redox proteins: first, they affect the energy of the system by creating partially occupied sites; second, they affect its entropy by introducing an additional empty/occupied site disorder (here named occupational entropy). The proposed method is applied to cytochrome c3 of Desulfovibrio vulgaris Hildenborough to study its redox properties and electron-proton coupling (redox-Bohr effect), using a continuum electrostatic method based on the linear Poisson-Boltzmann equation. Unlike previous studies using other methods, the full reduction order of the four hemes at physiological pH is successfully predicted. The sites more strongly involved in the redox-Bohr effect are identified by analysis of their titration curves/surfaces and the shifts of their midpoint redox potentials and pKa values. Site-site couplings are analyzed using statistical correlations, a method much more realistic than the usual analysis based on direct interactions. The site found to be more strongly involved in the redox-Bohr effect is propionate D of heme I, in agreement with previous studies; other likely candidates are His67, the N-terminus, and propionate D of heme IV. Even though the present study is limited to equilibrium conditions, the possible role of binding fluctuations in the concerted transfer of protons and electrons under nonequilibrium conditions is also discussed. The occupational entropy contributions to midpoint redox potentials and pKa values are computed and shown to be significant.     

Using the Unfolded State as the Reference State Improves the Performance of Statistical Potentials

Distance-dependent statistical potentials are an important class of energy functions extensively used in modeling protein structures and energetics. These potentials are obtained by statistically analyzing the proximity of atoms in all combinatorial amino-acid pairs in proteins with known structures. In model evaluation, the statistical potential is usually subtracted by the value of a reference state for better selectivity. An ideal reference state should include the general chemical properties of polypeptide chains so that only the unique factors stabilizing the native structures are retained after calibrating on reference state. However, reference states available as of this writing rarely model specific chemical constraints of peptide bonds and therefore poorly reflect the behavior of polypeptide chains. In this work, we proposed a statistical potential based on unfolded state ensemble (SPOUSE), where the reference state is summarized from the unfolded state ensembles of proteins produced according to the statistical coil model. Due to its better representation of the features of polypeptides, SPOUSE outperforms three of the most widely used distance-dependent potentials not only in native conformation identification, but also in the selection of close-to-native models and correlation coefficients between energy and model error. Furthermore, SPOUSE shows promising possibility of further improvement by integration with the orientation-dependent side-chain potentials.     

Models of active transport of ions in biomembranes of various types of cells

Nonequilibrium statistical models of the active transport of ions in biomembranes have been constructed. Differences of chemical potentials of the ATP-ADP reaction and the electrochemical potential of ions were taken as the thermodynamic forces responsible for the flow of ions through the membrane. The active transport of ions was viewed as a cross phenomenon arising from the chemical reaction of the ATP hydrolysis. These models provide independent calculations of the resting potential at the biomembrane and concentrations of ions in a cell on the assumption the free energy of the ATP-ADP reaction is fully (without the dissipation loss) converted to the free energy of transported ions. They take into account the presence of nonpenetrating ions in a cell. It was shown that different concentrations of nonpenetrating ions have a considerable effect on the resting potential. The proposed models were compared with experimental data obtained for different types of cells including neurons, muscular cells, bacteria, plants, and mitochondria. Calculated values of the membrane potential and ion concentrations were in good qualitative agreement with experimental data.     

Relationship between thermodynamic driving force and one-way fluxes in reversible processes

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 reverse fluxes and free energy for any chemical process operating in a steady state. This relationship, which is a generalization of equilibrium conditions to the case of a chemical process occurring in a nonequilibrium steady state in dilute solution, provides a novel equivalent definition for chemical reaction free energy. In addition, it is shown that previously unrelated theories introduced by Ussing and Hodgkin and Huxley for transport of ions across membranes, Hill for catalytic cycle fluxes, and Crooks for entropy production in microscopically reversible systems, are united in a common framework based on this relationship.     

The influence of conductivity changes in boundary element compartments on the forward and inverse problem in electroencephalography and magnetoencephalography.

Source localization based on magnetoencephalographic and electroencephalographic data requires knowledge of the conductivity values of the head. The aim of this paper is to examine the influence of compartment conductivity changes on the neuromagnetic field and the electric scalp potential for the widely used three compartment boundary element models. Both the analysis of measurement data and the simulations with dipoles distributed in the brain produced two significant results. First, we found the electric potentials to be approximately one order of magnitude more sensitive to conductivity changes than the magnetic fields. This was valid for the field and potential topology (and hence dipole localization), and for the amplitude (and hence dipole strength). Second, changes in brain compartment conductivity yield the lowest change in the electric potentials topology (and hence dipole localization), but a very strong change in the amplitude (and hence in the dipole strength). We conclude that for the magnetic fields the influence of compartment conductivity changes is not important in terms of dipole localization and strength estimation. For the electric potentials however, both dipole localization and strength estimation are significantly influenced by the compartment conductivity.     

Logical elements in living cells

Recognition processes with enhanced accuracy (as performed by structures like enzymes or ribosomes) are investigated using elementary ideas of statistical mechanics and related concepts of thermodynamics. The analysis starts from a formal definition of recognition and provides a correspondence with appropriate physical properties of the macromolecular logical elements. Transitions of the recognizing system between different modifications are a necessary feature of a more exacting recognition process. Rearrangement steps provide the process with higher accuracy by performing two physical operations: (1) rearranging the phase space of the system so that the "correct" states be better separated from the "wrong" states and the probability of occupation of the "correct" states be enhanced, (2) directing the process toward the more favourable modifications thus formed. Both operations are related to changes in the physical properties of the recognizing system. These changes can be expressed as differences of macromolecular Gibbs energy levels; if ligand binding or release participate in a step, directivity of the step depends also on the actual chemical potentials of the ligands in solution. The two operations just mentioned resemble two basic operations known to be necessary in electronic digital networks: directivity of control and signal standardization. An analysis of the entire reaction catalysed by a macromolecular logical element takes into account the requirements imposed by the logical functions as well as the need that the chemical potential of the product be not restricted to very low values. To satisfy these conditions, the reaction must be supported by a so-called non-specific reaction, usually implemented by the cleavage reaction of a nucleoside triphosphate.     

Modeling microbial consortiums as distributed metabolic networks

Biogeochemistry is the study of how living systems in combination with abiotic reactions process and cycle mass and energy on local, regional, and global scales (Schlesinger, 1997). Understanding how these biogeochemical cycles function and respond to perturbations has become increasingly important, as anthropogenic impacts have significantly altered many of these cycles (Galloway and Cowling, 2002; Houghton et al., 2002). Biogeochemistry is strongly governed by microbial processes, and it appears to closely follow thermodynamic constraints in that electron acceptor (O(2), NO(3)(-), SO(4)(2-), etc.) utilization closely follows a priori expectations based on energetics (Vallino et al., 1996; Hoehler et al., 1998; Jakobsen and Postma, 1999; Amend and Shock, 2001). Consortiums of microorganisms seem to have evolved to exploit chemical potentials wherever they exist in the environment, as manifested by the recent discovery of anaerobic methane oxidation by sulfate (Boetius et al., 2000) or sulfide oxidation by nitrate (Schulz et al., 1999). Three and a half billion years of natural selection have produced living systems capable of degrading most chemical potentials. We may therefore ask: If all ecosystem niche space is filled, is the biogeochemistry we observe in the environment dependent on the organisms that occupy that environment, or is the biogeochemistry determined by fundamental forces, with the evolution of living systems being the implementation of those forces? Recent developments in nonequilibrium thermodynamics (NET) are beginning to support the latter alternative, and advances in genomics are allowing us to explore microbial consortiums in detail. Taking advantage of ideas being suggested by NET, we have developed a modeling framework that views microbial consortiums as an inter-species distributed metabolic network. When combined with experimental observations, the model should help us test hypotheses that govern how living systems function.     

Relating a calcium indicator signal to the unperturbed calcium concentration time-course

Carbohydrate restriction as a strategy for control of obesity is based on two effects: a behavioral effect, spontaneous reduction in caloric intake and a metabolic effect, an apparent reduction in energy efficiency, greater weight loss per calorie consumed. Variable energy efficiency is established in many contexts (hormonal imbalance, weight regain and knock-out experiments in animal models), but in the area of the effect of macronutrient composition on weight loss, controversy remains. Resistance to the idea comes from a perception that variable weight loss on isocaloric diets would somehow violate the laws of thermodynamics, that is, only caloric intake is important ("a calorie is a calorie"). Previous explanations of how the phenomenon occurs, based on equilibrium thermodynamics, emphasized the inefficiencies introduced by substrate cycling and requirements for increased gluconeogenesis. Living systems, however, are maintained far from equilibrium, and metabolism is controlled by the regulation of the rates of enzymatic reactions. The principles of nonequilibrium thermodynamics which emphasize kinetic fluxes as well as thermodynamic forces should therefore also be considered. Here we review the principles of nonequilibrium thermodynamics and provide an approach to the problem of maintenance and change in body mass by recasting the problem of TAG accumulation and breakdown in the adipocyte in the language of nonequilibrium thermodynamics. We describe adipocyte physiology in terms of cycling between an efficient storage mode and a dissipative mode. Experimentally, this is measured in the rate of fatty acid flux and fatty acid oxidation. Hormonal levels controlled by changes in dietary carbohydrate regulate the relative contributions of the efficient and dissipative parts of the cycle. While no experiment exists that measures all relevant variables, the model is supported by evidence in the literature that 1) dietary carbohydrate, via its effect on hormone levels controls fatty acid flux and oxidation, 2) the rate of lipolysis is a primary target of insulin, postprandial, and 3) chronic carbohydrate-restricted diets reduce the levels of plasma TAG in response to a single meal. In summary, we propose that, in isocaloric diets of different macronutrient composition, there is variable flux of stored TAG controlled by the kinetic effects of insulin and other hormones. Because the fatty acid-TAG cycle never comes to equilibrium, net gain or loss is possible. The greater weight loss on carbohydrate restricted diets, popularly referred to as metabolic advantage can thus be understood in terms of the principles of nonequilibrium thermodynamics and is a consequence of the dynamic nature of bioenergetics where it is important to consider kinetic as well as thermodynamic variables.     

Comparative energy measurements in single molecule interactions

Single molecule experiments have opened promising new avenues of investigations in biology, but the quantitative interpretation of results remains challenging. In particular, there is a need for a comparison of such experiments with theoretical methods. We experimentally determine the activation free energy for single molecule interactions between two synaptic proteins syntaxin 1A and synaptobrevin 2, using an atomic force microscope and the Jarzynski equality of nonequilibrium thermodynamics. The value obtained is shown to be reasonably consistent with that from single molecule reaction rate theory. The temperature dependence of the spontaneous dissociation lifetime along with different pulling speeds is used to confirm the approach to the adiabatic limit. This comparison of the Jarzynski equality for intermolecular interactions extends the procedure for calculation of activation energies in nonequilibrium processes.     

Impact of external influences on characteristics of domains in biological membranes

Living systems continually interact with the environment. External influences evoke complex processes in the entire system. In this work, the impact of external influences on the behavior of the domains in biological membranes is considered both in general and on special examples. A general approach of nonequilibrium statistical thermodynamics proposed by L.A. Stratonovich along with the same approach adapted to stochastic storage processes are applied. Both the Stratonovich theory and the storage models allow characterizing specific features of the stationary non-equilibrium states as well as the behavior of the highly non-equilibrium processes in open systems, such as live organisms.     

A minimalist's reactive potential for efficient molecular modelling of chemistry

We review a minimalist's reactive force field, reactive state summation (RSS) potential. The essence of RSS potential scheme is to model each reactive state by individual non-reactive force fields, then modulate each term by a reaction-coordinate-dependent weight function, finally sum together to obtain the reactive potential. Compared with existing reactive potentials, RSS potential is easier to formulate and parameterise and is computationally efficient, at the expense of lesser accuracy. Thus, RSS potential can be regarded as a ‘reactive Lennard-Jones’ potential. Three exemplary RSS potentials are described in the context of their respective chemical systems: RSS-nitrogen for modelling detonation, RSS-carbon for modelling pyrolysis of activated carbon and RSS-fuel-catalyst for modelling catalytic chemical reaction.