Calculating apparent equilibrium constants of enzyme-catalyzed reactions at pH 7

Apparent equilibrium constants K' of biochemical reactions at pH 7 and standard apparent reduction potentials of half reactions at pH 7 can be calculated using a table of standard transformed Gibbs energies of formation Delta(f)G'(0) at pH 7. A table is provided for 136 reactants at 25 degrees C, pH 7, and ionic strengths of 0, 0.10, and 0.25 M. Examples are given to illustrate the use of the table.     

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.     

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.     

Thermodynamic properties of nucleotide reductase reactions

Recent thermodynamic measurements have made it possible to calculate the apparent equilibrium constants of the ribonucleoside diphosphate reductase reaction and the ribonucleoside triphosphate reductase reaction with various reducing agents. Third law heat capacity measurements on crystals of d-ribose and other calorimetric measurements make it possible to calculate Delta(f)G degrees for D-ribose and two species of D-ribose 5-phosphate. The experimental value of the apparent equilibrium constant K' for the deoxyribose-phosphate aldolase reaction makes it possible to calculate the standard Gibbs energies of formation Delta(f)G degrees for two protonation states of 2'-deoxy-D-ribose 5-phosphate. This shows that Delta(f)G degrees (2'-deoxy-D-ribose 5-phosphate(2)(-)) - Delta(f)G degrees (D-ribose 5-phosphate(2)(-)) = 147.86 kJ mol(-1) at 298.15 K and zero ionic strength in dilute aqueous solutions. This difference between reduced and oxidized forms is expected to apply to D-ribose, D-ribose 1-phosphate, ribonucleosides, and ribonucleotides in general. This expectation is supported by two other enzyme-catalyzed reactions for which apparent equilibrium constants have been determined. The availability of Delta(f)G degrees values for the species of 2'-deoxy-D-ribose and its derivatives makes it possible to calculate standard transformed Gibbs energies of formation of these reactants, apparent equilibrium constants for their reactions, changes in the binding of hydrogen ions in these reactions, and standard apparent reduction potentials of the half reactions involved as a function of pH and ionic strength at 298.15 K. The apparent equilibrium constant for ADP + thioredoxin(red) = 2'-deoxyADP + H(2)O + thioredoxin(ox) is 1.4 x 10(11) at 298.15 K, pH 7, and 0.25 M ionic strength.     

Thermodynamic properties of enzyme-catalyzed reactions involving cytosine, uracil, thymine, and their nucleosides and nucleotides

The standard Gibbs energies of formation of species in the cytidine triphosphate series, uridine triphosphate series, and thymidine triphosphate series have been calculated on the basis of the convention that Delta(f)G=0 for the neutral form of cytidine in aqueous solution at 298.15 K at zero ionic strength. This makes it possible to calculate apparent equilibrium constants for a number of reactions for which apparent equilibrium constants have not been measured or cannot be measured because they are too large. This paper adds fifteen reactants to the database BasicBiochemData3 at MathSource that includes 199 reactants. The standard transformed Gibbs energies of formation of these fifteen reactants are used to calculate apparent equilibrium constants at 298.15 K, ionic strength 0.25 M, and pHs 5, 6, 7, 8, and 9 for thirty two reactions. The pKs, standard Gibbs energies of hydrolysis, and standard Gibbs energies of deamination are given for these fifteen reactants.     

Effect of temperature on the creatine kinase equilibrium.

The effect of temperature on the apparent equilibrium constant of creatine kinase (ATP:creatine N-phosphotransferase (EC 2.7.3.2)) was determined. At equilibrium the apparent K' for the biochemical reaction was defined as [formula: see text] The symbol sigma denotes the sum of all the ionic and metal complex species of the reactant components in M. The K' at pH 7.0, 1.0 mM free Mg2+, and ionic strength of 0.25 M at experimental conditions was 177 +/- 7.0, 217 +/- 11, 255 +/- 10, and 307 +/- 13 (n = 8) at 38, 25, 15, and 5 degrees C, respectively. The standard apparent enthalpy or heat of the reaction at the specified conditions (delta H' degree) was calculated from a van't Hoff plot of log10K' versus 1/T, and found to be -11.93 kJ mol-1 (-2852 cal mol-1) in the direction of ATP formation. The corresponding standard apparent entropy of the reaction (delta S' degree) was +4.70 J K-1 mol-1. The linear function (r2 = 0.99) between log10 K' and 1/K demonstrates that both delta H' degree and delta S' degree are independent of temperature for the creatine kinase reaction, and that delta Cp' degree, the standard apparent heat capacity of products minus reactants in their standard states, is negligible between 5 and 38 degrees C. We further show from our data that the sign and magnitude of the standard apparent Gibbs energy (delta G' degree) of the creatine kinase reaction was comprised mostly of the enthalpy of the reaction, with 11% coming from the entropy T delta S' degree term. The thermodynamic quantities for the following two reference reactions of creatine kinase were also determined. [formula: see text] The delta H degree for Reaction 2 was -16.73 kJ mol-1 (-3998 cal mol-1) and for Reaction 3 was -23.23 kJ mol-1 (-5552 cal mol-1) over the temperature range 5-38 degrees C. The corresponding delta S degree values for the reactions were +110.43 and +83.49 J K-1 mol-1, respectively. Using the delta H' degree of -11.93 kJ mol-1, and one K' value at one temperature, a second K' at a second temperature can be calculated, thus permitting bioenergetic investigations of organs and tissues using the creatine kinase equilibria over the entire physiological temperature range.     

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.     

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.     

Changes in binding of hydrogen ions in enzyme-catalyzed reactions

Most enzyme-catalyzed reactions produce or consume hydrogen ions, and this is expressed by the change in the binding of hydrogen ions in the biochemical reaction, as written in terms of reactants (sums of species). This property of a biochemical reaction is important because it determines the change in the apparent equilibrium constant K' with pH. This property is also important because it is the number of moles of hydrogen ions that can be produced by a biochemical reaction for passage through a membrane, or can be accepted from a transfer through a membrane. There are two ways to calculate the change in binding of hydrogen ions for an enzyme-catalyzed reaction. The first, which has been used for a long time, involves calculating the partial derivative of the standard transformed Gibbs energy of reaction with respect to pH. The second involves calculating the average numbers of hydrogen ions in each reactant and adding and subtracting these average numbers. The changes in binding of hydrogen ions calculated by the second method at pHs 5, 6, 7, 8, and 9 are given for 23 enzyme-catalyzed reactions. Values are given for 206 more reactions on the web. This database can be extended to include more reactions for which pKs of reactants are known or can be estimated.     

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.     

Standard transformed Gibbs energies of coenzyme A derivatives as functions of pH and ionic strength

The best way to store data on apparent equilibrium constants for enzyme-catalyzed reactions is to calculate the standard Gibbs energies of formation of the species involved at 298.15 K and zero ionic strength so that equilibrium constants can be calculated at the desired pH and ionic strength. These calculations are described for CoA, acetyl-CoA, oxalyl-CoA, succinyl-CoA, methylmalonyl-CoA, malyl-CoA and CoA-glutathione. The species properties are then used to calculate standard transformed Gibbs energies of formation for these reactants as functions of pH at ionic strength 0.25 M. The species data also make it possible to calculate apparent equilibrium constants of 23 enzyme-catalyzed reactions as a function of pH, including some that cannot be determined directly because they are so large.     

Standard apparent reduction potentials for biochemical half reactions as a function of pH and ionic strength

Standard apparent reduction potentials are important because they give a more global view of the driving forces for redox reactions than do the standard transformed Gibbs energies of formation of the reactants. This paper emphasizes the effects of pH on biochemical half reactions in the range pH 5 to 9, but it also shows the effect of ionic strength. These effects can be calculated if the pKs of acid groups in the reactants are known in the range pH 4 to 10. Raising the pH decreases the standard apparent reduction potentials of half reactions when it has an effect, and the slope is proportional to minus one times the ratio of the change in binding of hydrogen ions in the half reaction to the number of electrons transferred. These effects are discussed for 19 biochemical reactions. This effect is most striking for the nitrogenase reaction, where the apparent equilibrium constant is proportional to 10(-10 pH) and is unfavorable for nitrogen fixation above pH 8.     

Thermodynamic properties of enzyme-catalyzed reactions involving guanine, xanthine, and their nucleosides and nucleotides

The standard Gibbs energies of formation of species in the guanosine triphosphate and the xanthosine triphosphate series have been calculated on the basis of the convention that the standard Gibbs energy of formation for the neutral form of guanosine is equal to zero in aqueous solution at 298.15 K and zero ionic strength. This makes it possible to calculate apparent equilibrium constants for a number of enzyme-catalyzed reactions for which apparent equilibrium constants have not been measured or cannot be measured directly because they are too large. The eventual elimination of this convention is discussed. This adds ten reactants to the database BasicBiochemData3 that has 199 reactants. The standard transformed Gibbs energies of formation of these ten reactants are used to calculate apparent equilibrium constants at 298.15 K, 0.25 M ionic strength, and pHs 5, 6, 7, 8, and 9. The pKs, standard Gibbs energies of hydrolysis, and standard Gibbs energies of deamination are given for the reactants in the ATP, IMP, GTP, and XTP series.     

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.     

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.     

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.     

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.     

Thermodynamics and kinetics of the glyoxylate cycle

Because the standard Gibbs energies of formation of all the species of reactants in the glyoxylate cycle are known at 298.15 K, it is possible to calculate the apparent equilibrium constants of the five reactions in the cycle in the pH range 5-9 and ionic strengths from 0 to approximately 0.35 M. In making calculations on such a system, it is convenient to specify concentrations of coenzymes like NADox and NADred because they are involved in many reactions and may be in steady states. Calculations are given for [NADox] = 1000[NADred] and [NADox] = 10[NADred]. Equilibrium compositions are calculated using computer programs when all the reactants are present initially and when only glyoxylate and CoA are present initially. The kinetics of the reactions in the glyoxylate cycle at specified concentrations of NADox and NADred are calculated by numerical solution of the steady-state rate equations for the case where the reactant concentrations are below their Michaelis constants and only glyoxylate and CoA are present initially.     

A group contribution method for the estimation of equilibrium constants for biochemical reactions

Using Gibbs Energies of compounds, as well as Gibbs Energy changes and equilibrium constants of biochemical reactions, the contributions of functional groups to the Gibbs Energy (in aqueous solution, temperature 25°C, and pH=7) have been estimated. These contributions allow the estimation of the Gibbs Free Energy and the equilibrium constant of a biochemical reaction, given the structure of the reactants and products.     

Group contributions for estimating standard gibbs energies of formation of biochemical compounds in aqueous solution

A method is presented for the estimation of the standard Gibbs energies of formation of biochemical compounds (and hence the Gibbs energies and equilibrium constants of biochemical reactions) from the contributions of groups. The method employs a large set of groups and special corrections. The contributions were estimated via multiple linear regression, using screened and weighted literature data. For most of the data employed, the error is less than 2 kcal/mol. The method provides a useful first approximation to Gibbs energies and equilibrium constants in biochemical systems.