The term biochemical reactions refers to enzyme-catalyzed reactions at a specified pH. When the pH is specified, the Gibbs energy G does not provide the criterion for spontaneity and equilibrium, and so it is necessary to define a transformed Gibbs energy G' that does. The reason it is necessary to introduce a new thermodynamic property, which is minimized at equilibrium, is that many biochemical reactants are weak acids so that H+ is a reacting species. When the concentration of a reacting species is specified at equilibrium, the equilibrium composition will depend on the specified concentration, as well as on temperature and pressure. When biochemical reactants have a pK near the pH of the equilibrium measurement, they consist of more than one species, so that it is convenient to think of the sum of species as the reactant at that pH. The apparent equilibrium constant K' is written in terms of concentrations of reactants (that is sums of species), without a term for hydrogen ions. When the apparent equilibrium constant is a function of pH, the standard thermodynamic properties calculated from K' are functions of pH. This is very different from chemical thermodynamics, where equilibrium constants K are written in terms of species (including H+) and where the equilibrium constants and standard thermodynamic properties calculated from them are not functions of pH.

Studies of the equilibrium constants K of chemical reactions and their dependence on temperature (or calorimetric measurements of heats of reaction) yield the standard reaction Gibbs energy DrGo, standard reaction enthalpy DrHo, and standard reaction entropy DrSo for the reaction. These measurements lead to the standard formation properties (DfGo, DfHo, and DfSo) of the species that are involved. When the pH is specified, studies of the apparent equilibrium constants K' of biochemical reactions and their dependence on temperature (or calorimetric measurements of heats of reaction) yield the standard transformed reaction Gibbs energy DrG'o, standard transformed reaction enthalpy DrH'o, and standard transformed reaction entropy DrS'o. These measurements lead to the standard transformed formation properties (DfG'o, DfH'o, and DfS'o) of the reactants (sums of species) involved. Thus when the pH is specified, you are in a new world of thermodynamics that is similar to the thermodynamics of chemical reactions, but is different in very significant ways. These two worlds of thermodynamics are connected by mathematical equations. If your know DfGo and DfHo values for the species of a reactant, DfG'o and DfH'o of the reactant can be calculated at any desired pH. On the other hand, if you know DfG'o and DfH'o for a reactant at a specified pH and the acid dissociation constants and the heats of dissociation, the DfGo and DfHo values of the species can be calculated.

Specifying the pH affects more than the values of the thermodynamic properties because it affects the way biochemical reactions are written. There is a relationship between the way a chemical reaction is written and the expression for K. There is the same kind of relationship between the way a biochemical reaction is written and the expression for K'. Biochemical reactions are written in terms of sums of species, which contain different numbers of hydrogen atoms. Biochemical reactions should not show hydrogen ions as reactants, even though biochemical reactions may produce or consume H+. Because of these major differences in the expressions for K and K', it is important to be able to distinguish chemical equations from biochemical equations at a glance. Chemical equations deal with species, but biochemical equations deal with sums of species, and the abbreviations used in writing these two types of equations need to be different enough to indicate this fundamental distinction. This distinction is clear in

Chemical reaction: ATP4- + H2O = ADP3- + HPO42- + H+

Biochemical reaction at a specified pH: ATP + H2O = ADP + Pi

Chemical equations balance all atoms and charges. Biochemical reactions balance all atoms except hydrogen and do not balance charges. A common error in biochemical text books is to write ATP + H2O = ADP + Pi + H+, which is stoichiometrically incorrect. The amount of H+ produced or consumed per mole of ATP hydrolyzed depends on the pH, and at pH 7 this biochemical reaction produces about 0.6 mole of H+ per mole of ATP hydrolyzed. Reactions involving nicotinamide adenine dinucleotide or carbon dioxide raise further problems in writing chemical reactions and biochemical reactions so that they can be clearly distinguished from each other. Biochemists need both kinds of reaction equations because chemical reactions and their equilibrium constants are used to discuss the chemical mechanisms of enzyme catalysis in terms of K and biochemical reactions are used to discuss metabolism at a specified pH in terms of K'.

Biochemical reactions may also bind metal ions. When the binding of metal ions is significant, the apparent equilibrium constant K' depends on the concentratin of free metal ions in addition to the pH. In this case DrG'o, DrH'o, DfG'o, and DfH'o depend on the free concentration of the metal ion, and metal ions may be produced or consumed in the enzyme-catalyzed reaction. If the products bind metal ions about as strongly as the reactants, this effect may be negligible.

The purpose of this web site is not to give derivations or explain details, but to serve as a guide to the literature on transformed thermodynamic properties. A list of references is given in order of date, and these references are referred to in answers to questions that may be asked about this new development.

I am indebted to NIH for support of this research and to the Dreyfus Foundation for the support of curriculum development.

Where have these new developments been reviewed?

Early review: 93ALBa

IUBMB/IUPAC Recommendations: 94ALB/COR

Major review for biochemists: 94ALBc

Chapter in Physical Chemistry textbook: 97ALB/SIL

Chapter in encyclopedia: 98ALBf

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Where do the transformed Gibbs energy G', transformed enthalpy H', and transformed entropy S' come from and how are they related to G, H, and S?

The transformed Gibbs energy G' is defined

by

G' = G - nc(H)m(H+)

where nc(H) is the amount of the hydrogen component (95ALBc, 97ALB/SIL) in the system (the total amount of hydrogen atoms) and m(H+) is the chemical potential of hydrogen ions at the specified pH (92ALBa, 92ALBb). The chemical potential m(H+) is used in derivations, but it is replaced by DfG(H+) in actual calculations. The chemical potential of H+ depends on the pH. This kind of definition is called a Legendre transform. Another example of a Legendre transform is G = H - TS. The explanation of why a Legendre transform is used is given in 94ALBd and 97ALBb. The derivations of equations used in biochemical thermodynamics are reviewed in 94ALBc.

The transformed entropy S of a system is defined by

S' = S - nc(H)

where is the molar entropy of hydrogen ions at the specified pH, which in actual calculations is replaced by DfS.

The transformed enthalpy of a system is defined by

H' = H - nc(H)

where is the molar enthalpy of hydrogen ions at the specified pH, which in actual calculations is replaced by DfH. Note that G' = H' - TS' for a system because G = H - TS and = -T.

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What are Legendre transforms and why is it necessary to use them?

A Legendre transform is the definition of a new thermodynamic potential that has convenient natural variables and can be used as a criterion of spontaneity and equilibrium at specified values of these natural variables (94ALBb, 97ALB/SIL). Examples of thermodynamic potentials are U(internal energy), H, S, and G, which are related by H = U + PV and G = H - TS. The natural variables of these thermodynamic potentials are indicated by U(S,V), S(U,V), H(S,P), and G(T,P). If a thermodynamic potential can be determined as a function of its natural variables, all of the thermodynamic properties of the system can be calculated. The criterion of spontaneous change and equilibrium in terms of the enthalpy is (dH)S,P 0, but this is not very useful because in general the entropy S cannot be held constant. The criterion of spontaneous change and equilibrium in terms of the Gibbs energy is (dG)T,P 0, which is useful because temperature and pressure can be held constant. Note that the Legendre transform involves the subtraction of a product of conjugate variables (for example, T and S, or P and V). When the pH is held constant the conjugate variables are the amount of the hydrogen component nc(H), the total amount of hydrogen atoms in the system, and the specified chemical potential of hydrogen ions m(H+). The transformed Gibbs energy G' and the transformed enthalpy H' at a specified pH, or pH and pMg, was introduced in 92ALBa, 92ALBb, and reviewed in 94ALBc. A Legendre transform involving [ATP] and [ADP] is discussed in 93ALBb, and a Legendre transform involving the electric potential is discussed in 95ALBa and 97ALBd.

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How are changes in G', H', and S' in a biochemical reaction related to experimental values of apparent equilibrium constants?

The standard transformed thermodynamic properties of a reaction are the changes in these properties when the separated reactants at 1 molar concnetration are converted to separated products at 1 molar concentration. These standard properties are defined in 92ALBa and 92ALBb.

DrG'o = - RT ln K'

DrH'o = RT2 ()P,pH

DrG'o = DrH'o - TDrS'o

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How are standard reaction properties DrGo and DrHo related to the standard transformed formation properties DfGo and DfHo of species? How are standard transformed reaction properties DrG'o and DrH'o at a specified pH related to the standard transformed formation properties DfG'o and DfH'o for reactants?

For a chemical reaction,

DrGo = S niDfGio

DrHo = S niDfHio

where the ni are the stoichiometric numbers (positive for products and negative for reactants) for the species. DfGio is the standard Gibbs energy for the formation of species i from its elements in their reference states.

For a biochemical reaction (92ALBa, 92ALBb, 94ALBc),

DrG'o = S ni'DfGi'o

DrH'o = S ni'DfHi'o

where the ni' are the stoichiometric numbers (positive for products and negative for reactants) for the reactants. The primes are used to distinguish these stoichiometric numbers from thos of the underlying chemical reactions. DfGi'o is the standard transformed Gibbs energy for the formation of reactant i from its elements in their reference states, where the reference state for hydrogen is hydrogen ions at the specified pH.

________________________________________________________________________ How are standard transformed formation properties of species calculated from standard formation properties?

When the DfGo and DfHo are known (57BUR, 69WIL, 77THA, 90MIL/SMI, 82WAG/EVA, 89COX/WAG, 97BUR, 69WIL), these values are adjusted by an amount proportional to the total number of hydrogen atoms in the species using (92ALBa, 92ALBb)

DfGi'o = DfGio - NH(i){DfGo(H+) + RT ln [H+]}

DfHi'o = DfHio - NH(i)DfHo(H+)

DfGo(H+) = DfHo(H+) = 0 at zero ionic strength, but not at finite ionic strengths.

When pMg is specified, there are additional terms in these equations. These equations were first used with inorganic phosphate, H2O, glucose, and glucose 6-phosphate (92ALBa, 92ALBb), and then they were used to calculate the standard thermodynamic transformed properties in the ATP series (92ALB/GOL).

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How are standard transformed formation properties of reactants calculated from standard transformed properties of species at a specified pH?

If a reactant is made up of a single species at the specified pH, DfGi'oand DfHi'o are the properties of the reactant. If the reactant is made up of more than one species at the specified pH, the group of species is said to form a pseudoisomer group. The standard transformed Gibbs energy of formation of this pseudoisomer group is calculated using

DfG'o(iso) = - RT ln S exp(-DfGi'o/RT)

The terms in the summation form a kind of partition function. Note that DfG'o(iso) is always more negative than the most negative DfGi'o of a species because the pseudoisomer group is more stable than the most stable species in the group. The value of DfG'o(iso) can also be calculated from the more fraction-weighted average of the DfGi'o values by adding an adjustment for the entropy of mixing. The standard transformed enthalpy DfH'o(iso) is calculated using

DfH'o(iso) = S riDfHi'o

where the mole fractions ri of the species in the pseudoisomer group are given by

ri = exp{[DfG'o(iso) - DfGi'o]/RT}

Isomer group thermodynamics is discussed by 82SMI/MIS, 83ALB, and 97ALB/SIL. The equations given above were first used for biochemical reactants in 92ALBa and 92ALBb.

When DfGo and DfHo values are not known for all the species of a reactant, their standard transformed formation properties can be calculated from experimental values of apparent equilibrium constants and apparent heats of reaction using (98ALBa, 98ALBb)

DrG'o = S ni'DfGi'o

DrH'o = S ni'DfHi'o

The DfGo and DfHo values of species can be calculated from DfG'o and DfH'o of a reactant if the acid dissociation constants and heats of dissociation are known (98ALBe).

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Where are DfGo and DfHo values for species of biochemical interest to be found?

Basic tables of thermical thermodynamic properties include the NBS Tables (82WAG/EVA) and the CODATA Tables (89COX/WAG). The values of special interest to biochemists were first tabulated by Burton (57BUR). This table was considerably extended by Wilhoit (69WIL). Additional values are to be found in 77THA. Goldberg and Tewari (89GOL/TEW) published tables on carbohydrates and their monophosphates. Miller and Smith-Magowan (90MIL/SMI) published values for the Krebs cycle and related compounds.

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Where can you find apparent equilibrium constants K' in the literature?

Goldberg and Tewari (93GOL/TEW, 94GOL/TEWa. 94GOL/TEWb, 95GOL/TEWa, 95GOL/TEWb) searched the literature for apparent equilibrium constants and heats of reaction DrH'o and have critically evaluated these data for about 500 biochemical reactions, involving about 1000 reactants. In principle, DfG'o values of these 1000 reactants can be calculated at any desired pH and ionic strength. The calculations of DfG'o values require acid dissociation constants of the reactants in the neighborhood of the specified pH and dissociation constants of complex ions formed by species with metal ions. For some purposes the effect of metal ions may be negligible when the effects cancel in the reactants and products (98ALBb, 98ALBc). It is important to note that when the apparent equilibrium constant of an enzyme-catalyzed reaction is very large, it is not possible to obtain an accurate value by experimental measurement; such values can be obtained by adding DrG'o values for other biochemical reactions. The 1992 recommendation of the IUBMB Nomenclature Committee (92WEB) lists about 3500 biochemical reactions, involving about 10,000 reactants. It is hoped that the group additivity will make it possible to calculate apparent equilibrium constants for many of these reactions (98ALBd).

Where are values of DfG'o and DfH'o at 298.15 K, pH 7, and ionic strengths of 0, 0.10, and 0.25 M to be found?

The values for 136 biochemical reactants are to be found in 98ALBb and 98ALBc.

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When carbon dioxide is involved, a biochemical reaction can be written in terms of the equilibrium partial pressure of CO2(g) or in terms of the sum of the equilibrium concentrations of CO2(aq), H2CO3, HCO3-, and CO32-. What is the relationship between these two apparent equilibrium constants and how are standard thermodynamic properties for this sum of species calculated?

For example, consider the apparent equilibrium constants of

pyruvate + H2O + CO2(g) + ATP = oxaloacetate + Pi + ADP

and

pyruvate + TotCO2 + ATP = oxaloacetate + Pi + ADP

where TotCO2 is the sum of the concentrations of the four dissolved species, which equilibrate with each other in about a second at pH 7. The ratio of these two apparent equilibrium constants is equal to the pH-dependent Henry's law constant: KH' = P(CO2,g)/[TotCO2]. The values of this apparent Henry's law constant at a series of pH values and ionic strengths and the standard transformed Gibbs energies and enthalpies of TotCO2 are discussed in 95ALBb, 97ALBe, and 98ALBb.

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How are the effects of ionic strength taken into account?

Equilibrium constants K for chemical reactions in aqueous solutions can be written in terms of activities of species, where the activity is the product of the activity coefficient gi and the concentration of the species. Equilibrium constants written in this way are independent of the ionic strength I. However, in working in the laboratory, it is more convenient to use the equilibrium constant of a reaction written in terms of concentrations at the experimental ionic strength. The value of this equilibrium constant depends on the ionic strength when ions are involved. Therefore, the standard Gibbs energy of reaction in aqueous solution, calculated using DrGo = - RT ln K, depends on the ionic strength. This approach is used in the research and tables described here.

The activity coefficients gi at 298.15 K at ionic strengths I of 0-0.25 M can be estimated using the extended Debye-Hckel theory, which yields

log10gi = - 0.51065 zi2I1/2/(1 + BI1/2)

where zi is the charge number of ion i and B is 1.6 L1/2 mol-1/2 (97ALB/SIL). The use of activity coefficients is avoided by making the standard thermodynamic properties of species functions of the ionic strength (80CLA/GLE, 89GOL/TEW, 91GOL/TEW, 93LAR/TEW)

DfHio(I) = DfHio(I = 0) + 1.4775zi2I1/2/(1 + BI1/2)

DfGio(I) = DfGio(I = 0) - 2.91482zi2I1/2/(1 + BI1/2)

These equations can also be applied to chemical reactions

DrHo(I) = DrHo(I = 0) + (1.4775 I1/2) S nizi2/(1 + BI1/2)

DrGo(I) = DrGo(I = 0) - (2.91482 I1/2) S nizi2/(1 + BI1/2)

logK(I) = logK(I = 0) + (0.51065 I1/2 S vizi2) /(1 + BI1/2)

where S nizi2 is the change in zi2 in the reaction and reaction properties are expressed in kJ mol-1.

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How can the equilibrium composition be calculated for a system of biochemical reactions at a specified pH?

When there is a single biochemical reaction, the equilibrium composition corresponding with a certain initial composition can be calculated by solving a polynomial equation that involves the apparent equilibrium constant. When there are two or more biochemical reactions, the problem becomes much more difficult. The equilibrium composition has to satisfy apparent equilibrium constant expressions for a set of independent biochemical reactions that are involved and conservation equations for all of the elements except for hydrogen. In the Newton-Raphson method this is done by guessing at a possible equilibrium composition and using an iterative method to find the actual equilibrium composition (82SMI/MIS). This solution can be checked by substituting it into the equilibrium expressions and conservation equations. Krambeck (91KRA) has written a computer program equcalc in Mathematica (96WOL) that does this for gas reactions and has shown how it can be modified to equcalcc (uses a conservation matrix) and eqrxc (uses a matrix of stoichiometric numbers) for solution reactions (97ALBa). These programs calculate a first approximation to the equilibrium composition, so that it is not necessary to guess the equilibrium composition.

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In what sense do the stoichiometric numbers of a biochemical reaction form a vector and the stoichiometric numbers of a series of biochemical equations form a matrix? Why is this important?

Biochemical equations, like chemical equations, are mathematical equations and are related to the corresponding equations for the conservation of atoms (82SMI/MIS, 91ALBb, 92ALBc, 92ALBd, 94ALBb). The difference between chemical equations and biochemical equations is that chemical equations balance all atoms and electric charges, while biochemical equations conserve all atoms except hydrogen atoms (and possibly metal atoms). The matrix n of stoichiometric numbers for a system of biochemical reactions has a column for each reaction and a row for each reactant. The pathway for a certain net reaction in a system of reactions is expressed by a vector s which gives the number of times the individual reactions have to go to produce the net reaction. The net reaction for pathway s can be calculated by nultiplying the stoichiometric number matrix by the pathway vector.

n s = nnet

The pathway can be calculated for a given net reaction by using a mathematical program in a personnel computer (96ALBa, 97ALBc). In Mathematica (96WOL)

s = LinearSolve[n,nnet]

This approach becomes increasingly useful as larger systems of biochemical reactions are considered.

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How can standard apparent reduction potentials E'o for half-cell reactions at a specified pH be calculated from DfG'o values?

Standard apparent reduction potentials are just another way of expressing the thermodynamic properties of reactants at specified T, P, and pH. Standard apparent reduction potentials are useful in determining whether a given biochemical redox reaction goes to the right or the left. The standard apparent reduction potential for a galvanic cell is given by

Ecell'o = ER'o - EL'o

where the standard apparent reduction potentials of the right and left electrodes are given by (98ALBc)

E'o = - S ni'DfGi'o

where n is the number of electrons involved, F is the Faraday constant (96,485 coulombs mol-1), and the ni' are the stoichiometric numbers in the reduction half reaction. The convention is that DfG'o(e-) = 0, where e- is the formal electron used in the reduction reaction for the half cell.

It is not possible to directly measure standard apparent reduction potentials for most biochemical half reactions because the reactants do not transfer electrons to and from metallic electrodes reversibly. There is a good deal of data that can be used to calculate E'o values because K' values are known for about 200 reactions involving nicotinamide adenine dinucleotide, for which the standard apparent reduction potential is known (93ALBc).

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How can the change in binding of hydrogen ions and magnesium ions (a) be determined from measurements of K' and (b) be calculated without making measurements of K'? (c) How are the bindings of hydrogen ions and magnesium ions linked?

(a) The change in the binding of hydrogen ions DrN(H+) is given by (94ALBc)

DrN(H+) = - ()P,pMg

DrN(Mg2+) = - ()P,pH

Note that the amount of H+ produced per mole of reaction is - DrN(H+).

(b) If the acid dissociation constants and dissociation constants of complex ions containing magnesium ions are known for all of the reactants, the average bindings of H+ and Mg2+ by each of the reactants can be calculated. Taking the differences yields DrN(H+) and DrN(Mg2+).

(c) The linkage relation is

()P,pH = ()P,pMg

These relations are discussed in 98ALBa. The nitrogenase reaction is extremely pH-dependent because 10H+ are consumed per mole of N2 reduced (94ALBa). Ror a discussion of the importance of the pH dependence of K' see 93ALB/COR.

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How are standard transformed enthalpies of reaction DrH'o determined experimentally?

Since

DrH'o = RT2()P,pH

measurements of K' at a series of temperatures can be used, but more frequently calorimetric measurements are used. When there is no change in the binding of H+ or metal ions, DrH'o is obtained directly from the experiment. But when H+ is produced or consumed there is a heat effect from the reaction of the buffer with H+. The contribution to this buffer heat effect is proportional to the change in binding of hydrogen ions DrN(H+) in the enzyme-catalyzed reaction and the heat of dissociation of the buffer. The experimental heat of reaction for the enzyme-catalyzed reaction has to be corrected in this way to obtain DrH'o for the biochemical reaction. When metal ions are bound by reactants there is a further complication which is taken care of in the same way. See 93ALB/GOL.

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Can standard transformed Gibbs energies of formation DfG'o be obtained for proteins, as well as reactants with low molar masses?

DfG'o values of oxidixed and reduced forms of several proteins have been calculated because K' or standard aparent reduction potantials E'o have been measured: ferredoxinox and ferredoxinred (94ALBa), cytochromecox and cytochromecred (98ALBc), thioredoxinox and thioredoxinred (98ALBc). Since the thermodynamic properties of these reactants cannot be connected with the elements DfGo was set at zero for ferredoxin+, cytochromec(Fe3+), and thioredoxin(oxidized). The DfG'o values at 21.6 oC, pH 7.4,
[Cl-] = 0.2M, and ionic strength 0.2 M have been calculated (96ALBb) for (a) the dimer subunit of hemoglobin and the two oxygenated forms and for (b) the tetramer and its four oxygenated forms. These values are based on the convention that DfG'o = 0 for the tetramer. The values of DfG'o have also been calculated for TotT and TotD at several concentrations of molecular oxygen, where Tot T is the sum of five species and TotD is the sum of three species.