Fundamental equation of thermodynamics for protein-ligand binding

For the internal energy and every thermodynamic potential that can be defined by a Legendre transform, there is a fundamental equation that contains all the thermodynamic information about a system. For a system involving the binding of molecular oxygen and hydrogen ions by a protein, fundamental equations are given for the Gibbs energy G, the transformed Gibbs energy G' at specified pH, and the further transformed Gibbs energy G" at specified pH and specified concentration of molecular oxygen. The Maxwell equations for these various Gibbs energies are important because they provide the connection with experimentally determined properties and increase our understanding of these properties. Measurements of the average number of oxygen molecules bound as a function of T, pH and concentration of molecular oxygen make it possible to calculate Delta(f)G"(o) of the reactant. Maxwell equations make it possible to calculate the average number of hydrogen ions bound, Delta(f)S"(o), Delta(f)H"(o) and their partial derivatives. These relations are illustrated with numerical calculations on a simple reaction system.     

Systems of biochemical reactions from the point of view of a semigrand partition function

Semigrand partition functions contain all the thermodynamic information on reaction systems. When they are written for systems at specified pH, they yield the transformed Gibbs energy G' of the system and the thermodynamic properties that can be calculated from G'. When they are written for systems at specified pH and specified concentrations of coenzymes, they yield the further transformed Gibbs energy G" and properties that can be calculated from G". This is illustrated by considering: (1) a reactant that is a weak monoprotic acid at a specified pH; (2) a reaction between two pseudoisomer groups at a specified pH; and (3) the first five reactions of glycolysis. Equilibrium compositions in glycolysis are calculated at pH 7 and different steady-state concentrations of ATP and ADP.     

Levels of thermodynamic treatment of biochemical reaction systems.

Equilibrium calculations on biochemical reaction systems can be made at three levels. Level 1 is the usual chemical calculation with species at specified temperature and pressure using standard Gibbs energies of formation of species or equilibrium constants K. Level 2 utilizes reactants such as ATP (a sum of species) at specified T, P, pH, and pMg with standard transformed Gibbs energies of formation of reactants or apparent equilibrium constants K'. Calculations at this level can also be made on the enzymatic mechanism for a biochemical reaction. Level 3 utilizes reactants at specified T, P, pH, and pMg, but the equilibrium concentrations of certain reactants are also specified. The fundamental equation of thermodynamics is derived here for Level 3. Equilibrium calculations at this level use standard transformed Gibbs energies of formation of reactants at specified concentrations of certain reactants or apparent equilibrium constants K". Level 3 is useful in calculating equilibrium concentrations of reactants that can be reached in a living cell when some of the reactants are available at steady-state concentrations. Calculations at all three levels are facilitated by the use of conservation matrices and stoichiometric number matrices for systems. Three cases involving glucokinase, glucose-6-phosphatase, and ATPase are discussed.     

Thermodynamics of systems of biochemical reactions

When a reaction system described in terms of species is in a certain state, the Gibbs energy G provides the means for determining whether each reaction will go to the right or the left, and the equilibrium composition of the whole system can be calculated using G. When the pH is specified, a system of biochemical reactions is described in terms of reactants, like ATP (a sum of species), and the transformed Gibbs energy G' provides the means for determining whether each reaction will go to the right or the left. The equilibrium composition of the whole system can be calculated using G'. Since metabolism is complicated, the thermodynamics of systems of reactions like glycolysis and the citric acid cycle can also be considered at specified concentrations of coenzymes like ATP, ADP, NAD(ox), and NAD(red). This is of interest because coenzymes tend to be in steady states because they are involved in many reactions. When the concentrations of coenzymes are constant, the further transformed Gibbs energy G" provides the means for calculating whether each reaction will go to the right or the left, and the equilibrium composition of the whole system can be calculated using G". Under these conditions, a metabolic reaction system can be reconceptualized in terms of sums of reactants; for example, glycolysis can be represented by C(6)=2C(3), where C(6) is the sum of the reactants with six carbon atoms and C(3) is the sum of the reactants with three carbon atoms. These calculations can also be described by use of semigrand partition functions. Semigrand partition functions have the advantage of containing all the thermodynamic information on a series of reactions at specified pH or at specified pH and specified concentrations of coenzymes.     

The role of water in the thermodynamics of dilute aqueous solutions

Water plays a role in the thermodynamics of dilute aqueous solutions that is unusual in two ways. First, knowledge of hydration equilibrium constants of species is not required in calculations of thermodynamic properties of biochemical reactants and reactions at specified pH. Second, since solvent provides an essentially infinite source of oxygen atoms in a reaction system where water is a reactant, oxygen atoms are not conserved in the reaction system in dilute aqueous solutions. This is related to the fact that H₂O is omitted in equilibrium expressions for dilute aqueous solutions. Calculations of the standard transformed Gibbs energies of formation of total carbon dioxide and total ammonia at specified pH are discussed, and the average bindings of hydrogen ions by these reactants are calculated by differentiation. Since both of these reactants are involved in the urease reaction, the apparent equilibrium constants and changes in the numbers of hydrogen ions bound are calculated for this reaction as functions of pH.     

Thermodynamics of the purine nucleotide cycle

Since the standard Gibbs energies of formation are known for all the species in the purine nucleotide cycle at 298.15 K, the functions of pH and ionic strength that yield the standard transformed Gibbs energies of formation of the ten reactants can be calculated. This makes it possible to calculate the standard transformed Gibbs energies of reaction, apparent equilibrium constants, and changes in the binding of hydrogen ions for the three reactions at desired pHs and ionic strengths. These calculations are also made for the net reaction and a reaction that is related to it. The equilibrium concentrations for the cycle are calculated when all the reactants are initially present or only some are present initially. Since the concentrations of GTP, GDP, and P(i) may be in steady states, the equilibrium concentrations are also calculated for the system at specified steady-state concentrations.     

Standard apparent reduction potentials of biochemical half reactions and thermodynamic data on the species involved

Standard apparent reduction potentials E' degrees of half reactions of enzyme-catalyzed reactions are useful because they provide a global view of the apparent equilibrium constants of redox reactions. A table of E' degrees at a specified pH shows at a glance whether a given half reaction will drive another half reaction or be driven by it. This table can be used to calculate apparent equilibrium constants. Standard Gibbs energies of formation of species in a half reaction can be used to calculate E' degrees values at pHs in the range 5-9 and ionic strengths in the range of 0-0.35 M. My previously published values of E' degrees values for 42 half reactions has been extended by 22 new E' degrees values in this paper. When DeltafG degrees and DeltafH degrees are both known for all the species in an enzyme-catalyzed reaction at 298.15 K, it is possible to calculate all the standard transformed thermodynamic properties of the reaction over a range of pHs, ionic strengths, and temperatures.     

Equilibrium calculations on systems of biochemical reactions at specified pH and pMg.

The use of G' in discussing the thermodynamics of biochemical reactions at a specified pH and pMg is justified by use of a Legendre transform of the Gibbs energy G. When several enzymatic reactions occur simultaneously in a system, the standard transformed Gibbs energies of reaction delta rG'0 can be used in a computer program to calculate the equilibrium composition that minimizes the transformed Gibbs energy at the specified pH and pMg. The calculation of standard transformed Gibbs energies of formation of reactants at pH 7 and pMg 3 is described. In addition a method for calculating the equilibrium concentrations of reactants is illustrated for a system with steady state concentrations of some reactants like ATP and NAD.     

Recommendations for terminology and databases for biochemical thermodynamics

Chemical equations are normally written in terms of specific ionic and elemental species and balance atoms of elements and electric charge. However, in a biochemical context it is usually better to write them with ionic reactants expressed as totals of species in equilibrium with each other. This implies that atoms of elements assumed to be at fixed concentrations, such as hydrogen at a specified pH, should not be balanced in a biochemical equation used for thermodynamic analysis. However, both kinds of equations are needed in biochemistry. The apparent equilibrium constant K' for a biochemical reaction is written in terms of such sums of species and can be used to calculate standard transformed Gibbs energies of reaction Δ(r)G'°. This property for a biochemical reaction can be calculated from the standard transformed Gibbs energies of formation Δ(f)G(i)'° of reactants, which can be calculated from the standard Gibbs energies of formation of species Δ(f)G(j)° and measured apparent equilibrium constants of enzyme-catalyzed reactions. Tables of Δ(r)G'° of reactions and Δ(f)G(i)'° of reactants as functions of pH and temperature are available on the web, as are functions for calculating these properties. Biochemical thermodynamics is also important in enzyme kinetics because apparent equilibrium constant K' can be calculated from experimentally determined kinetic parameters when initial velocities have been determined for both forward and reverse reactions. Specific recommendations are made for reporting experimental results in the literature.     

Standard thermodynamic formation properties for the adenosine 5'-triphosphate series.

The criterion for chemical equilibrium at specified temperature, pressure, pH, concentration of free magnesium ion, and ionic strength is the transformed Gibbs energy, which can be calculated from the Gibbs energy. The apparent equilibrium constant (written in terms of the total concentrations of reactants like adenosine 5'-triphosphate, rather than in terms of species) yields the standard transformed Gibbs energy of reaction, and the effect of temperature on the apparent equilibrium constant at specified pressure, pH, concentration of free magnesium ion, and ionic strength yields the standard transformed enthalpy of reaction. From the apparent equilibrium constants and standard transformed enthalpies of reaction that have been measured in the adenosine 5'-triphosphate series and the dissociation constants of the weak acids and magnesium complexes involved, it is possible to calculate standard Gibbs energies of formation and standard enthalpies of formation of the species involved at zero ionic strength. This requires the convention that the standard Gibbs energy of formation and standard enthalpy of formation for adenosine in dilute aqueous solutions be set equal to zero. On the basis of this convention, standard transformed Gibbs energies of formation and standard transformed enthalpies of formation of adenosine 5'-trisphosphate, adenosine 5'-diphosphate, adenosine 5'-monophosphate, and adenosine at 298.15 K, 1 bar, pH = 7, a concentration of free magnesium ions of 10(-3) M, and an ionic strength of 0.25 M have been calculated.     

Equilibrium concentrations for pyruvate dehydrogenase and the citric acid cycle at specified concentrations of certain coenzymes

It is of interest to calculate equilibrium compositions of systems of biochemical reactions at specified concentrations of coenzymes because these reactants tend to be in steady states. Thermodynamic calculations under these conditions require the definition of a further transformed Gibbs energy G" by use of a Legendre transform. These calculations are applied to the pyruvate dehydrogenase reaction plus the citric acid cycle, but steady-state concentrations of CoA, acetyl-CoA and succinyl-CoA cannot be specified because they are involved in the conservation of carbon atoms. These calculations require the use of linear algebra to obtain further transformed Gibbs energies of formation of reactants and computer programs to calculate equilibrium compositions. At specified temperature, pH, ionic strength and specified concentrations of several coenzymes, the equilibrium composition depends on the specified concentrations of the coenzymes and the initial amounts of reactants.     

Thermodynamic properties of oxidoreductase, transferase, hydrolase, and ligase reactions

Transferases formally couple together two oxidoreductase reactions or two hydrolase reactions. Therefore the thermodynamic properties of transferase reactions can be calculated from differences between thermodynamic properties of two oxidoreductase or two hydrolase reactions. Ligases couple together two hydrolase reactions, and so their thermodynamic properties can be calculated from differences between two hydrolase reactions. These relationships are demonstrated by calculating standard transformed Gibbs energies of reaction and the changes in binding of hydrogen ions at pHs 5-9 of a number of oxidoreductase, transferase, hydrolase, and ligase reactions by use of the data base BasicBiochemData2 and its recent extensions. Coupling is not restricted to two reactions, and an example is given of the coupling of three reactions.     

Thermodynamics of the reactions of carbamoyl phosphate

Two measurements of equilibrium constants by Marshall and Cohen make it possible to calculate standard Gibbs energies of formation of the species of carbamate and carbamoyl phosphate. Carbamate formation from carbon dioxide and ammonia does not require an enzyme, and the equilibrium concentrations of carbamate in ammonium bicarbonate are calculated. Knowing the values of standard Gibbs energies of formation of species of carbamate and carbamoyl phosphate make it possible to calculate the dependencies of the standard transformed Gibbs energies of formation of these reactants on pH and ionic strength and to calculate apparent equilibrium constants for several enzyme-catalyzed reactions and several chemical reactions. These calculations are sufficiently complicated that computer programs in Mathematica are used to make tables and plots. The dependences of apparent equilibrium constants on pH are consequences of the production or consumption of hydrogen ions, which are shown in plots. As usual the increase in the number of enzyme-catalyzed reactions for which apparent equilibrium constants can be calculated is larger than the number of reactions required to obtain the thermodynamic properties of the species involved.     

Electron-spin-resonance evidence for enzymic reduction of oxygen to a free radical, the superoxide ion

1. An electron-spin-resonance signal with g( parallel)2.08 and g( perpendicular)2.00 is observed by the rapid-freezing technique during the oxidation of substrates by molecular oxygen catalysed by xanthine oxidase at pH10. 2. The intensity of this signal is shown to depend on oxygen rather than on enzyme concentration, indicating that it is due to an oxygen free radical and not to the enzyme. 3. The same species is shown to be produced in the reaction at pH10 between hydrogen peroxide and periodate ions. Studies with this system have facilitated comparison of the properties of the oxygen radical with data in the literature on the products of pulse radiolysis of oxygenated water over a wide pH range. 4. It is concluded that the species observed is the superoxide ion, O(2) (-), and that the stability of this ion is greatly increased in alkaline solution. A mechanism explaining the alkaline stability is proposed. 5. The importance of O(2) (-) in the enzymic reaction is discussed.     

APUAMA: a software tool for reaction rate calculations

APUAMA is a free software designed to determine the reaction rate and thermodynamic properties of chemical species of a reagent system. With data from electronic structure calculations, the APUAMA determine the rate constant with tunneling correction, such as Wigner, Eckart and small curvature, and also, include the rovibrational level of diatomic molecules. The results are presented in the form of Arrhenius-Kooij form, for the reaction rate, and the thermodynamic properties are written down in the polynomial form. The word APUAMA means “fast” in Tupi-Guarani Brazilian language, then the code calculates the reaction rate on a simple and intuitive graphic interface, the form fast and practical. As program output, there are several ASCII files with tabulated information for rate constant, rovibrational levels, energy barriers and enthalpy of reaction, Arrhenius-Kooij coefficient, and also, the option to the User save all graphics in BMP format.     

Catalytic and ligand binding properties of bovine trypsinogen and its complex with the effector dipeptide Ile-Val

Steady-state and pre-steady-state kinetic data for the trypsinogen catalyzed hydrolysis of a series of synthetic substrates (i.e. p-nitrophenyl esters of N-alpha-carbobenzoxy-L-amino acids) have been obtained as a function of pH (3.4-8). Moreover, the effect of ethylamine on the hydrolysis of a neutral substrate and benzamidine binding have been extensively studied. In order to obtain direct information on the transition of trypsinogen to a beta-trypsin-like structure, the role of the effector dipeptide Ile-Val on the catalytic and ligand binding properties of the zymogen has been investigated. Kinetic and thermodynamic data for beta-trypsin and alpha-chymotrypsin are also reported for the purpose of an homogeneous comparison of the various (pro)enzymes. Under all the experimental conditions, kinetic data for (pro)enzyme catalysis are consistent with the minimum three-step mechanism: (formula; see text) involving the acyl intermediate E X P. In the presence of Ile-Val dipeptide, trypsinogen assumes catalytic and ligand binding properties that are reminiscent of activated beta-trypsin. This is at variance with free trypsinogen, which shows a alpha-chymotrypsin-like behavior. The large differences in the results of kinetic and thermodynamic measurements for free trypsinogen, as compared to its binary adduct with Ile-Val, can be ascribed to the substantial differences in the two molecular species, which include the spatial orientation of Asp189.     

Thermodynamic molecular switch in macromolecular interactions

It is known that most living systems can live and operate optimally only at a sharply defined temperature, or over a limited temperature range, at best, which implies that many basic biochemical interactions exhibit a well-defined Gibbs free energy minimum as a function of temperature. The Gibbs free energy change, deltaG(o) (T), for biological systems shows a complicated behavior, in which deltaG(o)(T) changes from positive to negative, then reaches a negative value of maximum magnitude (favorable), and finally becomes positive as temperature increases. The critical factor in this complicated thermodynamic behavior is a temperature-dependent heat capacity change (deltaCp(o)(T) of reaction, which is positive at low temperature, but switches to a negative value at a temperature well below the ambient range. Thus, the thermodynamic molecular switch determines the behavior patterns of the Gibbs free energy change, and hence a change in the equilibrium constant, Keq, and/or spontaneity. The subsequent, mathematically predictable changes in deltaH(o)(T), deltaS(o)(T), deltaW(o)(T), and deltaG(o)(T) give rise to the classically observed behavior patterns in biological reactivity, as demonstrated in three interacting protein systems: the acid dimerization reaction of alpha-chymotrypsin at low pH, interaction of chromogranin A with the intraluminal loop peptide of the inositol 1,4,5-triphosphate receptor at pH 5.5, and the binding of L-arabinose and D-galactose to the L-arabinose binding protein of Escherichia coli. In cases of protein unfolding of four mutants of phage T4 lysozyme, no thermodynamic molecular switch is observed.     

Thermodynamics of the arginine kinase reaction.

The effect of temperature, pH, free [Mg(2+)], and ionic strength on the apparent equilibrium constant of arginine kinase (EC 2.7.3.3) was determined. At equilibrium, the apparent K' was defined as [see text] where each reactant represents the sum of all the ionic and metal complex species. The K' at pH 7.0, 1.0 mM free [Mg(2+)], and 0. 25 M ionic strength was 29.91 +/- 0.59, 33.44 +/- 0.46, 35.44 +/- 0. 71, 39.64 +/- 0.74, and 45.19 +/- 0.65 (n = 8) at 40, 33, 25, 15, and 5 degrees C, respectively. The standard apparent enthalpy (DeltaH degrees') is -8.19 kJ mol(-1), and the corresponding standard apparent entropy of the reaction (DeltaS degrees') is + 2. 2 J K(-1)mol(-1) in the direction of ATP formation at pH 7.0, free [Mg(2+)] =1.0 mM, ionic strength (I) =0.25 M at 25 degrees C. We further show that the magnitude of transformed Gibbs energy (DeltaG degrees ') of -8.89 kJ mol(-1) is mostly comprised of the enthalpy of the reaction, with 7.4% coming from the entropy TDeltaS degrees' term (+0.66 kJ mol(-1)). Our results are discussed in relation to the thermodynamic properties of its evolutionary successor, creatine kinase.     

Quantitative assignment of reaction directionality in a multicompartmental human metabolic reconstruction

Reaction directionality is a key constraint in the modeling of genome-scale metabolic networks. We thermodynamically constrained reaction directionality in a multicompartmental genome-scale model of human metabolism, Recon 1, by calculating, in vivo, standard transformed reaction Gibbs energy as a function of compartment-specific pH, electrical potential, and ionic strength. We show that compartmental pH is an important determinant of thermodynamically determined reaction directionality. The effects of pH on transport reaction thermodynamics are only seen to their full extent when metabolites are represented as pseudoisomer groups of multiple protonated species. We accurately predict the irreversibility of 387 reactions, with detailed propagation of uncertainty in input data, and manually curate the literature to resolve conflicting directionality assignments. In at least half of all cases, a prediction of a reversible reaction directionality is due to the paucity of compartment-specific quantitative metabolomic data, with remaining cases due to uncertainty in estimation of standard reaction Gibbs energy. This study points to the pressing need for 1), quantitative metabolomic data, and 2), experimental measurement of thermochemical properties for human metabolites.     

Protein phase diagrams II: nonideal behavior of biochemical reactions in the presence of osmolytes

In the age of biochemical systems biology, proteomics, and high throughput methods, the thermodynamic quantification of cytoplasmatic reaction networks comes into reach of the current generation of scientists. What is needed to efficiently extract the relevant information from the raw data is a robust tool for evaluating the number and stoichiometry of all observed reactions while providing a good estimate of the thermodynamic parameters that determine the molecular behavior. The recently developed phase-diagram method, strictly speaking a graphical representation of linkage or Maxwell Relations, offers such capabilities. Here, we extend the phase diagram method to nonideal conditions. For the sake of simplicity, we choose as an example a reaction system involving the protein RNase A, its inhibitor CMP, the osmolyte urea, and water. We investigate this system as a function of the concentrations of inhibitor and osmolyte at different temperatures ranging from 280 K to 340 K. The most interesting finding is that the protein-inhibitor binding equilibrium depends strongly on the urea concentration--by orders-of-magnitude more than expected from urea-protein interaction alone. Moreover, the m-value of ligand binding is strongly concentration-dependent, which is highly unusual. It is concluded that the interaction between small molecules like urea and CMP can significantly contribute to cytoplasmic nonideality. Such a finding is highly significant because of its impact on renal tissue where high concentrations of cosolutes occur regularly.