Calculation of Equilibrium Compositions of Biochemical Reaction

When water is a reactant in a system of reactions, there is a sense in which oxygen atoms are not conserved because they can be brought into reactants...
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J. Phys. Chem. B 2001, 105, 1109-1114

1109

Calculation of Equilibrium Compositions of Biochemical Reaction Systems Involving Water as a Reactant Robert A. Alberty Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ReceiVed: September 27, 2000; In Final Form: NoVember 9, 2000

In the thermodynamic treatment of reactions involving water as a reactant in dilute aqueous solutions, the activity of water is taken as unity in expressions for equilibrium constants, but the calculation of equilibrium constants using tables involves the standard Gibbs energy of formation of water. This convention, which is quite satisfactory for simple systems, causes problems in the thermodynamics of biochemical reaction systems in dilute aqueous solutions. One reason is that the identification of components in these systems is very important, and this involves the use of conservation matrixes and stoichiometric number matrixes. These matrixes can be interconverted, and this is especially useful in computer programs for calculating equilibrium compositions. When water is a reactant in a system of reactions, there is a sense in which oxygen atoms are not conserved because they can be brought into reactants or expelled from reactants without altering the activity of water in the solution. These problems can be avoided by using a Legendre transform to define a further transformed Gibbs energy that provides the criterion for spontaneous change and equilibrium when a(H2O) ) 1. The equations that are derived here are illustrated by equilibrium calculations on a system of three enzyme-catalyzed reactions in the citric acid cycle. This system of reactions also illustrates the fact that the mechanisms of enzyme-catalyzed reactions may introduce constraints in addition to element balances.

Introduction

Niso

The criterion for spontaneous change and equilibrium in aqueous solutions at specified pH is provided by the transformed Gibbs energy G′ that is defined by the Legendre transform1,2

G′ ) G - nc(H)µ(H+)

(1)

where G is the Gibbs energy of the reaction system and µ(H+) is the specified chemical potential of hydrogen ions. The amount of the hydrogen component in the system is given by nc(H) ) Σ NH(i)ni, where NH(i) is the number of hydrogen atoms in species i and ni is the amount of species i. In eq 1 and later equations, thermodynamic properties in dilute aqueous solutions are taken to be functions of ionic strength so that, for example, µ(H+) ) µ°(H+) + RT ln 10-pH, where pH ) - log[H+]. The Gibbs energy G and µ°(H+) are functions of ionic strength.3,4 If the equilibrium composition of a reaction system is affected by the concentration of free metal ions that are bound reversibly, the definition of the transformed Gibbs energy can be extended by subtracting products of conjugate variables; for example, if magnesium ions are involved, nc(Mg)µ(Mg2+) can be subtracted from the right side of eq 1. Equation 1 leads to the following expression for the standard transformed Gibbs energy of formation of species i:

∆fGi′° ) ∆fGi° - NH(i)(∆fG°(H+) + RT ln 10-pH) (2) When this adjustment of the standard Gibbs energy of formation of a species is made, the species becomes a pseudoisomer of other species that differ from it only with respect to the number of hydrogen atoms they contain, and so, the standard transformed Gibbs energy of formation of the pseudoisomer group is calculated using the usual partition function5

∆fG′°(iso) ) - RT ln{

exp[-∆fGi′°/RT]} ∑ i)1

(3)

where Niso is the number of pseudoisomers in the pseudoisomer group. Whenever there is a pseudoisomer group, it is aggregated in this way to avoid redundant columns in the conservation matrixes discussed later. At equilibrium, the mole fraction ri of a pseudoisomer in a group is given by the distribution function

ri ) exp[(∆fG′°(iso) - ∆fGi′°)/RT]

(4)

Equation 3 can also be written in terms of the binding polynomial (partition function) P, which has been used by Wyman6 and others.7 This is done by taking the first term out of the summation in eq 3

∆fG′°(iso) ) ∆fG1′° - RT ln P

(5)

P ) 1 + [H+]/K1 + [H+]2/K1K2 + ...

(6)

where

This form has been used in this investigation. The apparent equilibrium constant K′ for a biochemical reaction at specified pH is given by

∆rG′° ) - RT ln K′ )

∑νi′∆fGi′°

(7)

Tables of standard transformed Gibbs energies of formation ∆fG′° have been calculated at 298.15 K, pH 7, and ionic strengths of 0, 0.10, and 0.25 M for about 150 biochemical reactants,8,9 and they can be used to calculate the apparent equilibrium constant K′ for any biochemical reaction between reactants in the table. The criterion for spontaneous change and

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1110 J. Phys. Chem. B, Vol. 105, No. 5, 2001

Alberty

equilibrium is given by dG′ e 0 at specified, T, P, pH, and amounts of apparent components. The fundamental equation for the transformed Gibbs energy G′ for a system of biochemical reactions at specified pH is10 N′

dG′ ) -S′ dT + V dP +

µi′ dni′ + RT ln(10) nc(H) dpH ∑ i)1

(8)

where the transformed entropy of the system is given by

S′ ) S - nc(H)Sh(H+)

(9)

The molar entropy of hydrogen ions at the specified pH and ionic strength is represented by Sh(H+). N′ is the number of reactants (pseudoisomer groups) in the system, and ni′ is the amount of reactant i. Since eq 8 involves the concept of a component, its use is closely related to the apparent conservation matrix A′ and the apparent stoichiometric number matrix ν′ for the system. The adjective apparent is used to distinguish these matrixes, which omit the hydrogen ion, with those for the underlying chemical reactions. It is important to understand that enzyme-catalyzed reactions may involve conservation equations in addition to element balances.11 Matrix operations5 are useful for testing the independence of linear equations in a set, in identifying and selecting components,11 in expressing noncomponents in terms of components, and in calculating equilibrium compositions.12,13 The number C′ of apparent components at a specified pH can be determined by the row reduction of the apparent conservation matrix since C′ ) rank A′. The number R′ of independent reactions at specified pH can be determined by the row reduction of the apparent stoichiometric number matrix because R′ ) rank ν′. Note that N′ ) C′ + R′. These two types of matrixes can be interconverted by use of

A′ν′ ) 0

and

(ν′) (A′) ) 0 T

T

(10)

where the superscript T indicates the transpose. Thus, a basis for the apparent stoichiometric number matrix ν′ can be obtained by calculating the null space of A′. One way to do this is to arrange the columns so that an identity matrix is obtained on the left. The remaining matrix is then represented by Z′. A basis for the apparent stoichiometric matrix is obtained from5

ν′ )

[ ] -Z′ IR′

transformed Gibbs energy G′′ that takes advantage of the fact that oxygen atoms are available to a reaction system from an essentially infinite reservoir when dilute aqueous solutions are considered at specified pH. Further Legendre transforms of this type have been used before.13,14 The solution of the problem encountered with enzyme-catalyzed reactions where H2O is a reactant is to define the thermodynamic potential G′′ that provides the criterion for spontaneous change and equilibrium when oxygen atoms are provided from H2O at a(H2O) ) 1. The Legendre transform that does this is

G′′ ) G′ - nc(O)µ′°(H2O)

The amount of the oxygen component in the system is given by nc(O) ) Σ NO(i)ni, where NO(i) is the number of oxygen atoms in reactant i. µ′°(H2O) is the standard transformed chemical potential for H2O at the specified pH and ionic strength. Following the same steps as in the derivation1,2 of eq 2, the standard further transformed Gibbs energy of formation of reactant i can be shown to be given by

∆fGi′′° ) ∆fGi′° - NO(i)∆fG′°(H2O)

where IR′ is the identity matrix of rank R′. This relation is especially useful with small conservation matrixes, but for larger matrixes, it is easier to use a computer program. MathematicaR has been used in the calculations presented here.14 The use of eq 7 involves the usual convention that when H2O is a reactant in dilute aqueous solutions, ∆fGi′° (H2O) is involved in the summation, but the activity of water is set at unity in the expression for K′. This is a practical convention in treating systems with several reactions, but in treating larger systems, it is almost a necessity to use matrixes, linear algebra, and a computer. Linear algebra can be used to convert sets of stoichiometric equations to sets of conservation equations and vice versa, but these operations are incompatible when H2O is a reactant and the convention that a(H2O) ) 1 in dilute aqueous solutions is used. The problem encountered in using A′ and ν′ matrixes can be avoided by using a Legendre transform to define a further

(13)

where ∆fG′°(H2O) is given by eq 2. Note that ∆fG′′°(H2O) ) 0. When this adjustment of the standard transformed Gibbs energy of formation of a reactant is made, this reactant becomes a pseudoisomer of other species that differ from it only with respect to the number of oxygen atoms they contain, and so, the standard further transformed Gibbs energy of formation of the pseudoisomer group has to be calculated using the analogue of eq 3. The apparent equilibrium constant K′′ for a biochemical reaction at specified pH and a(H2O) ) 1 is given by

∆rG′′° ) - RT ln K′′ )

∑νi′′∆fGi′′°

(14)

There is no term for H2O in the summation. When the pH is specified and a(H2O) ) 1, the criterion for spontaneous change and equilibrium is dG′′ e 0 at specified T, P, pH, a(H2O) ) 1, and amounts of apparent components. Note that oxygen is no longer a component. The fundamental equation for the further transformed Gibbs energy G′′ for a system of biochemical reactions when a(H2O) ) 1 and the pH is specified is given by N′′

dG′′ ) -S′′ dT + V dP + (11)

(12)

µi′′ dni′′ + ∑ i)1 RT ln(10) nc(H) dpH (15)

where the further transformed entropy of the system is given by

S′′ ) S′ - nc(O)Sh′°(H2O)

(16)

The standard transformed entropy of H2O at the specified pH and ionic strength is given by Sh′° (H2O). In eq 15, N′′ is the number of reactants (pseudoisomer groups) in the system, and ni′′ is the amount of reactant i at a(H2O) ) 1. Note that there is no term for H2O in the summation in eq 15. To identify components when a(H2O) ) 1 and the pH is specified, apparent conservation matrixes A′′ and apparent stoichiometric number matrixes ν′′ are needed. A′′ is obtained from A′ by deleting the row and column for H2O because its activity is constant. Therefore, ν′′ does not involve H2O as a reactant. Thus, the inconsistency between A′ and ν′ is eliminated in A′′ and ν′′. The number C′′ of apparent components can be

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J. Phys. Chem. B, Vol. 105, No. 5, 2001 1111

Figure 1. Apparent conservation matrix A′ for reactions 18, 19, and 20 at specified pH when H2O is treated the same way as the other reactants.

determined by the row reduction of A′′ since C′′ ) rank A′′. The number R′′ of independent reactions can be determined by the row reduction of ν′′ because R′′ ) rank ν′′. Note that N′′ ) C′′ + R′′. These two types of matrixes can be interconverted by the use of

A′′ν′′ ) 0

and

(ν′′)T(A′′)T ) 0

(17)

The apparent stoichiometric number matrix ν′′ can be obtained from the row-reduced form of A′′ by the use of the analogue of eq 10 or by calculating a basis for the null space using a computer program. Further transformed Gibbs energies of formation are especially useful in calculating equilibrium compositions using computer programs that accept conservation matrixes and vectors of initial amounts, like the equcalcc and equcalcrx used here.12,13 In these programs, which are given in the Appendix, Lagrange multipliers are improved by an iterative process using the Newton-Raphson method to obtain the equilibrium composition. The use of these equations will be clarified by the consideration of a specific example in the next section. Application to a System of Three Reactions in the Citric Acid Cycle that Involve Water as a Reactant The following three successive reactions in the citric acid cycle involve H2O as a reactant:

citrate ) cis-aconitate + H2O

(18)

cis-aconitate + H2O ) isocitrate

(19)

Figure 2. Row-reduced form of the apparent conservation matrix A′ for the reaction system at a specified pH when H2O is treated the same way as the other reactants.

Figure 3. Row-reduced form of the transpose of the apparent stoichiometric number matrix ν′ obtained from the apparent conservation matrix A′ in Figure 1 or Figure 2.

and con2 are [NADox] + [oxoglutarate] ) const and [NADox] + [CO2tot] ) const. These equations are not unique, but they are independent. The matrix in Figure 1 is not unique, but its row-reduced form is unique for a given order of columns. The row-reduced form of the apparent conservation matrix in Figure 1 is given in Figure 2. Figure 2 contains the same information as Figure 1, but the conservation equations are expressed in terms of components that are the groups of atoms in citrategp, H2O, NADox, CO2tot, and NADred. The fact that an identity matrix is obtained on the left side indicates that the five conservation equations are independent. This matrix is very important because it shows how the noncomponents (aconitate and oxoglutarate) are made up of the five components. The calculation of standard further transformed Gibbs energies of formation uses the fact that aconitate involves -1 H2O and oxoglutarate involves 1 H2O. A basis for the apparent stoichiometric matrix ν′ for the reaction system can be obtained by calculating the null space of the apparent conservation matrix (see eq 9), and the result is given in Figure 3. The two independent biochemical reactions for this system can be written in many different ways, but the most familiar way to write these equations is

citrate group ) aconitate + H2O

(21)

isocitrate + NADox + H2O ) CO2tot + NADred + oxoglutarate (20)

citrate group + NADox + H2O ) CO2tot + NADred + oxoglutarate (22)

These biochemical equations are written in terms of reactants, which in some cases are sums of species that are in equilibrium with each other at the specified pH. Note that the abbreviations used do not show ionic charges. Since the pH is held constant, these reactions do not balance hydrogen atoms or electric charges. The abbreviation CO2tot represents8,17,18 the sum of the species H2CO3, CO2(aq), HCO3-, and CO32-. The most convenient17 way to calculate ∆fG′°(CO2tot), is to follow the convention of the NBS Tables.19,20 The apparent conservation matrix A′ for the system involving reactions 18-20 is given in Figure 1. Since citrate and isocitrate are isomers, they are represented by a single column in the conservation matrix, which is labeled citrategp. If the only conservation equations for this system were for carbon, oxygen, and nitrogen atoms, four biochemical reactions would be expected from R′ ) N′ - C′. However, the mechanism of the third reaction introduces two more conservation equations labeled con1 and con2. This is not unusual for systems of biochemical reactions.11 Conservation equations con1

The row-reduced form of the stoichiometric number matrix for these two reactions is the same as that in Figure 3, and so the conservation matrixes in Figure 1 and Figure 2 are equivalent to reactions 21 and 22. The apparent equilibrium constants K′ of biochemical eqs 21 and 22 can be used to calculate the equilibrium composition for specified amounts of the five components, as shown in the next section. There is a problem since conservation matrix A′ indicates that the expressions for K′ for the two reactions should involve a term in [H2O], but that term is unity. In treating larger system using computers, this is a fatal error. As indicated in the Introduction, this problem can be solved by making a Legendre transform that defines a further transformed Gibbs energy G′′ that provides the criterion for spontaneous change and equilibrium when a(H2O) ) 1. Oxygen atoms are conserved in the reaction system, but the activity of H2O remains at unity when reactions involving H2O are studied in dilute aqueous solution so that the solvent is effectively an infinite reservoir for oxygen atoms.

1112 J. Phys. Chem. B, Vol. 105, No. 5, 2001

Alberty TABLE 1: Standard Transformed Gibbs Energies of Formation of Reactants and Equilibrium Concentrations at 298.15 K, pH 7, and Ionic Strength 0.25 M

Figure 4. Row-reduced form of the apparent conservation matrix A′′ for the reaction system at a specified pH that takes into account the fact that a(H2O) ) 1.

Figure 5. Row-reduced form of the apparent conservation matrix A′′′ for the reaction system at a specified pH, a(H2O) ) 1, and specified [NADox] and [NADred].

Deleting the row and column for H2O in Figure 2 produces a conservation matrix with identical columns for the citrategp and aconitate. One of these columns is deleted in A′′. These reactants are represented by the first column, which is labeled citacon. The standard transformed Gibbs energy of formation of citacon is calculated using eq 3. The new apparent conservation matrix A′′ is given in Figure 4. There are now four components because there is no conservation equation for H2O. Figure 4 indicates that reaction in the system can be represented by a single equation:

citacon + NADox ) CO2tot + NADred + oxoglutarate (23) In the next section, this reaction is used to calculate the equilibrium composition when the initial concentrations of the components citacon and NADox are 0.01 M. In considering large systems of biochemical reactions,21 it is of interest to consider equilibria that can be reached at steadystate concentrations of coenzymes such as NADox and NADred. When this is done, the rows and columns for these reactants in Figure 4 are deleted to obtain the conservation matrix given in Figure 5. There are now two components since NADox and NADred are not conserved. This conservation matrix corresponds with the reaction (see eq 11)

citacon ) CO2tot + oxoglutarate

(24)

When NADox and NADred are held constant, Figure 4 indicates that no adjustments need to be made on the standard further transformed Gibbs energies of formation of citacon and CO2tot, but oxoglutrate contains the components 1 NADox and -1 NADred so that its standard further transformed Gibbs energy needs to be adjusted. The reaction system involves eight reactants and three biochemical reactions (five components), but after two Legendre transforms, only three reactants and one biochemical reaction (two components) remain. When this is done with a larger reaction system, there is a considerable simplification in thinking about the reaction system and calculating equilibrium compositions and effects of changing the pH and concentrations of coenzymes. Calculation of the Equilibrium Composition at a Specified pH When the pH is specified, the equilibrium composition of the system under discussion can be calculated using the equilibrium expressions for reactions 18-20. However, since citrate and isocitrate lead to redundant columns in A′, the

citrate isocitrate citrate group cis-aconitate H2O NADox NADred CO2tot oxoglutarate

∆fG′°/kJ mol-1

equil. conc./M

-966.23 -959.58 -966.39 -802.12 -155.66 1059.11 1120.09 -547.10 -633.59

(0.00120) (0.000082) 0.00128 4.0 × 10-5 0.00132 0.00868 0.00868 0.00868

reactions are written in terms of the citrate group, as shown in reactions 21 and 22 and the value of ∆fG′°(citgp) is calculated using eq 3. The standard transformed Gibbs energies of formation of all of these reactants at 298.15 K, pH 7, and ionic strength 0.25 M have been published,8,9 and so, the following calculations are for these conditions. The apparent equilibrium constants for reactions 21 and 22 are 0.310 and 0.389, respectively. The equilibrium composition can be calculated using Solve in MathematicaR by also including the four conservation equations in Figure 2. If initially the concentrations of citrate and NADox are 0.01 M, the four conservation equations for a volume of 1 L will be

n(citgp) + n(aconitate) + n(oxoglutarate) ) 0.01 mol (25) n(NADox) + n(oxoglutarate) ) 0.01 mol

(26)

n(CO2tot) - n(oxoglutarate) ) 0 mol

(27)

n(NADred) - n(oxoglutarate) ) 0 mol

(28)

as can be seen from Figure 2. The program Solve yields the equilibrium composition given in Table 1. Since this equilibrium calculation yields the concentration of the citrate group, eq 4 is used to calculate the equilibrium concentrations of citrate and isocitrate, which are given in parentheses in the table. Calculation of the Equilibrium Composition at Specified pH and a(H2O) ) 1 When a Legendre transform is used to take H2O out of the conservation matrix, the thermodynamic properties of aconitate and oxoglutarate have to be adjusted using eq 13, as shown in Figure 2

∆fGi′′°(aconitate) ) ∆fGi′°(aconitate) + ∆fG′°(H2O) (29) ∆fGi′′°(oxoglutarate) ) ∆fGi′°(oxoglutarate) - ∆fG′°(H2O) (30) Now the citrate group and aconitate are pseudoisomers, and so, eq 3 is used to calculate ∆fG′°(citacon) for the pseudoisomer group. Under these conditions, chemical change in the system is represented by eq 23, for which K′′ ) 0.377. The equilibrium concentrations calculated using Solve are given in Table 2. This table shows that the use of standard further transformed Gibbs energies of formation yields the same equilibrium composition as the preceding section. These equilibrium concentrations can also be calculated using MathematicaR with equcalcc[as_,lnk_,no_] or equcalcrx[nt_,lnkr_,no_], which are given in the Appendix. In equcalcc, as_ is the conservation matrix in Figure 4, lnk_ is calculated from ∆fG′′° values in Table 2, and no_ is the vector of initial amounts of reactants. In equcalcrx, nt is the transposed sto-

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J. Phys. Chem. B, Vol. 105, No. 5, 2001 1113

TABLE 2: Standard Further Transformed Gibbs Energies of Formation of Reactants and Equilibrium Concentrations at 298.15 K, pH 7, a(H2O) ) 1, and Ionic Strength 0.25 M citrate group aconitate citacon NADox NADred CO2tot oxoglutarate

∆fG′′°/kJ mol-1

equil. conc./M

-966.39 -957.78 -966.47 1059.11 1120.09 -547.10 -477.93

(0.00128) (0.00004) 0.00132 0.00132 0.00868 0.00868 0.0868

TABLE 3: Standard Further Transformed Gibbs Energies of Formation of Reactants and Equilibrium Concentrations at 298.15 K, pH 7, Ionic Strength 0.25 M, and [NADox] ) [NADred] ) 0.01 M citacon CO2tot oxoglutarate

∆fG′′′°/kJ mol-1

equil. conc./M

-966.47 -547.10 -411.40

0.00166 0.00834 0.00834

ichiometric number matrix {-1, -1, 1, 1, 1}, lnkr is the vector of logarithms of apparent equilibrium constants of reaction 23 (that is, ln 0.377), and no_ is the vector of initial amounts of reactants. The advantage of these programs over the use of Solve is that the inputs are in matrix form, so they can be expanded indefinitely for large systems of biochemical reactions. The point of this article is that when H2O is a reactant, programs of this type cannot be used with matrixes A′ and ν′ because they are inconsistent. The use of equcalcc and equcalcrx is much more convenient than the use of Solve because it is not necessary to put in the conservation equations. In studying systems of enzyme-catalyzed reactions, we find equcalcrx to be the more useful because it avoids the problem of identifying constraints introduced by the enzyme mechanisms. This process can be taken one step further when the concentrations of the coenzymes NADox and NADred are in steady states, as indicated in the discussion after eq 24. The further transformed Gibbs energy of formation of oxoglutarate is calculated using

∆fGi′′′°(oxoglutarate) ) ∆fGi′′°(oxoglutarate) (∆fG′°(NADox) + RT ln[NADox]) + (∆fG′°(NADred) + RT ln[NADred]) (31) When the steady-state concentrations of the coenzymes are [NADox] ) 0.001 M and [NADred] ) 0.009 M, the equilibrium composition calculated using eq 24 and the following two conservation equations from Figure 5 are given in Table 3

n(citacon) + n(oxoglutarate) ) 0.01 mol

(32)

n(citacon) - n(oxoglutarate) ) 0 mol

(33)

These equilibrium concentrations are different from those in Tables 1 and 2 because the steady-state concentrations of NADox and NADred were chosen arbitrarily. However, the equilibrium concentration of the citrate-isocitrate-aconitate group can be used to calculate the corresponding equilibrium concentrations of the individual reactants. Thus, no information is lost in going to this more global view of the thermodynamics of this reaction system. Discussion When the pH is held constant in studying an aqueous reaction system where hydrogen ions are involved, it is necessary to use a Legendre transform to define a transformed Gibbs energy G′ that provides the criterion for spontaneous change and

equilibrium. In studying biochemical reaction systems in dilute aqueous solutions where H2O is a reactant, the convention is that the activity of water is taken as unity, but the standard transformed Gibbs energy of formation of H2O is involved in the calculation of the apparent equilibrium constant K′. This convention can be used in considering small systems of reactions, but in considering large systems, it is advantageous to use apparent conservation matrixes and apparent stoichiometric matrixes that are interconvertible using the operations of linear algebra. The use of conservation matrixes and stoichiometric number matrixes involving H2O is inconsistent with omitting [H2O] from the expression for the apparent equilibrium constants. In dilute aqueous solutions, the solvent provides essentially an infinite reservoir for H2O, and so, a Legendre transform can be used to define a further transformed Gibbs energy G′′ that provides the criterion for spontaneous change and equilibrium at specified pH and a(H2O) ) 1. When this is done, the apparent conservation matrix A′′, which does not include the conservation of H2O, becomes consistent with the apparent stoichiometric number matrix ν′′, which does not include the stoichiometric number for H2O. The use of this method has been illustrated by quantitative calculations on the equilibria in a system of three consecutive reactions in the citric acid cycle, where each involves H2O as a reactant. This example also illustrates the fact that the mechanisms of enzyme-catalyzed reactions may introduce constraints in addition to the conservation of atoms of elements. The use of matrix operations shows how these constraits can be expressed in terms of amounts of reactants that are chosen as components. This process is taken one step further by calculating the equilibrium composition for the system of three reactions when the concentrations of NADox and NADred are in steady states. This type of equilibrium calculation is useful in studying metabolism because the effects of changing the concentrations of coenzymes can be studied. However, when concentrated aqueous solutions are considered, the approximation that a(H2O) ) 1 cannot be used. A byproduct of the use of these Legendre transforms is that a more global view of a complicated reaction system can be obtained. In the system considered, there are eight reactants and three reactions, but after making two Legendre transforms there are three reactants and one reaction. Yet the equilibrium calculations on this one reaction can be used to obtain the equilibrium concentrations of all eight reactants. These discussions have been focused on Gibbs energies, but they can be readily extended to the corresponding enthalpies, entropies, and heat capacities by use of the fundamental equation. This approach can be used to investigate the effects of changing the pH, ionic strength, and concentrations of coenzymes on the equilibrium composition that can be reached. Appendix equcalcc[as_,lnk_,no_]:)Module[{l,x,b,ac,m,n,e,k}, (*as)conservation matrix lnk)-(1/RT)(Gibbs energy of formation vector at T) no)initial composition vector *) (*Setup*) {m,n})Dimensions[as]; b)as.no; ac)as; (*Initialize*) l)LinearSolve[ as.Transpose[as],-as.(lnk+Log[n]) ]; (*Solve*) Do[e)b-ac.(x)E∧(lnk+l.as)); If[(10∧-10)>Max[ Abs[e] ], Break[] ];

1114 J. Phys. Chem. B, Vol. 105, No. 5, 2001 l)l+LinearSolve[ac.Transpose[as*Table[x,{m}]],e], {k,100}]; If[ k)100,Return[“Algorithm Failed”] ]; Return[x] ] equcalcrx[nt_,lnkr_,no_]:)Module[{as,lnk}, (*nt)transposed stoichiometric number matrix lnkr)ln of equilibrium constants of rxs (vector) no)initial composition vector*) (*Setup*) lnk)LinearSolve[nt,lnkr]; as)NullSpace[nt]; equcalcc[as,lnk,no] ] Acknowledgment. I am indebted to Dr. Frederich Krambeck (Mobil Research) and Dr. Robert N. Goldberg (NIST) for many helpful discussions. I am indebted to NIH for support of this research by Award 5-R01-GM48358-05. References and Notes (1) Alberty, R. A. Biophys. Chem. 1992 42, 117. (2) Alberty, R. A. Biophys. Chem. 1992 43, 239. (3) Clark, E. C.; Glew, D. N. Trans. Faraday Soc. 1966 62, 539. (4) Goldberg, R. N.; Tewari, Y. Biophys. Chem. 1991 40, 241. (5) Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis: Theory and Algorithm; Wiley-Interscience: New York, 1982. (6) Wyman, J.; Gill, S. J. Binding and Linkage: Functional Chemistry of Biological Macromolecules; University Science Books, Mill Valley, CA, 1990.

Alberty (7) Di Cera, E. Thermodynamic Theory of Site-Specific Binding Processes in Biological Macromolecules; Cambridge University Press: Cambridge, 1995. (8) Alberty, R. A. Arch. Biochem. Biophys. 1998 353, 116. (9) Alberty, R. A. Arch. Biochem. Biophys. 1998 358, 25. (10) Alberty, R. A. J. Phys. Chem. 1992, 96, 9614. (11) Alberty, R. A. Biophys. Chem. 1994 49, 251. (12) Krambeck, F. J. Presented at the 71st Annual Meeting of the AIChE, Miami Beach, FL, Nov. 16, 1978. (13) Krambeck, F. J., In Chemical Reactions in Complex Systems; Sapre, A. M., Krambeck, F. J., Eds.; Van Nostrand Reinhold: New York, 1991. (14) Wolfram Research, Inc., 100 World Trade Center, Champaign, IL 61820-7237. (15) Alberty, R. A. Biophys. J. 1993 65, 1243. (16) Alberty, R. A. J. Phys. Chem. B 2000 104, 650. (17) Alberty, R. A. J. Phys. Chem. 1995, 99, 11028. (18) Alberty, R. A. Arch. Biochem. Biophys. 1997 348, 116. (19) Wagman, D. D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, I.; Bailey, S. M.; Churney, K. L.; Nuttall, R. L. The NBS Tables of Chemical Thermodynamic Properties. J. Phys. Chem. Ref. Data 1982 11, Suppl. 2. (20) In the NBS Tables of Chemical Thermodynamic Properties,17 some species in aqueous solution are listed with two or more formulas that differ only by the use of molecules of water contained in them. An example is CO2(ao) and H2CO3(ao), where H2CO3(ao) is equivalent to H2CO3(ao) + H2O(l). This is equivalent to setting the equilibrium constant for A(aq) + nH2O(l) ) A‚nH2O equal to unity. In dilute aqueous solutions, the thermodynamic treatment is independent of the value of this equilibrium constant. Thus, the equilibria H2CO3(ao) ) H+ + HCO3- and HCO3- ) H+ + CO32- can be used even though they appear to ignore CO2(aq) and H2CO3(aq), where these two symbols represent actual species. (21) Alberty, R. A. J. Phys. Chem. B 2000 104, 4807.