Apparent Equilibrium Constants and Standard Transformed Gibbs Energies of Biochemical Reactions Involving Carbon Dioxide

ARCHIVES OF BIOCHEMISTRY AND BIOPHYSICS

Vol. 348, No. 1, December 1, pp. 116–124, 1997 Article No. BB970403

Apparent Equilibrium Constants and Standard Transformed Gibbs Energies of Biochemical Reactions Involving Carbon Dioxide1 Robert A. Alberty2 Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received June 19, 1997, and in revised form September 17, 1997

When carbon dioxide is produced in a biochemical reaction, the expression for the apparent equilibrium constant K* can be written in terms of the partial pressure of carbon dioxide in the gas phase or the total concentration of species containing CO2 in the aqueous phase, referred to here as [TotCO2]. The values of these two apparent equilibrium constants are different because they correspond to different ways of writing the biochemical equations. Their dependencies on pH and ionic strength are also different. The ratio of these two apparent equilibrium constants is equal to the apparent Henry’s law constant K*H . This article provides derivations of equations for the calculation of the standard transformed Gibbs energies of formation of TotCO2 and values of the apparent Henry’s law constant at various pH levels and ionic strengths. These equations involve the four equilibrium constants interconnecting the five species [CO2(g), CO2(aq), H2CO3 , 20 HCO0 3 , and CO3 ] of carbon dioxide. In the literature there are many errors in the treatment of equilibrium data on biochemical reactions involving carbon dioxide, and so several examples are discussed here, including calculation of standard transformed Gibbs energies of formation of reactants. This approach also applies to net reactions, and the net reaction for the oxidation of glucose to carbon dioxide and water is discussed. q 1997 Academic Press Key Words: apparent equilibrium constants; standard transformed Gibbs energies of reaction; standard transformed Gibbs energies of formation; Henry’s law constants; carbon dioxide; isocitrate dehydrogenase.

When carbon dioxide is produced in a biochemical reaction, the reaction can be written to show the forma1 Acknowledgment is made to the National Institutes of Health for the support of this research (NIH-1-R01-GM48358-01A1). 2 Address correspondence to author. Fax: 617-253-7030.

tion of gaseous carbon dioxide or the formation of TotCO2 , which represents of sum of CO2 , H2CO3 , HCO30 , and CO320 in the aqueous phase, as discussed by Alberty (1). For example, a generalized biochemical reaction involving CO2 at a specified pH in dilute solutions can be written as

A Å B / CO2(g)

[B]PCO2 [A]

[1]

or

A / H2O Å B / TotCO2

K*[2] Å

[B][TotCO2] , [2] [A]

where A and B represent sums of species. Reaction [2] has the advantage that the apparent equilibrium constant K* can be used to calculate the equilibrium concentrations of all of the reactants in the aqueous phase. The H2O in Reaction [2] is required to balance oxygen atoms. When the pH is specified, a biochemical equation does not balance hydrogen atoms or electric charge, but it must balance atoms of other elements (2). In the literature, there are many errors in the treatment of the equilibrium data on biochemical reactions of this type. These include the assumption that HCO30 is the only species containing CO2 that is present in the aqueous phase, that the biochemical reaction can be written as A Å B / CO2(aq), or that the biochemical reaction can be written A Å B / (CO2(aq) / H2CO3). Except for CO2(g), all the species are in the aqueous phase, but CO2(aq) is used to distinguish it from CO2(g). The H2O in Reaction [2] is often omitted, and this error is fatal when standard transformed Gibbs energies of formation of reactants are calculated because Df G*0(H2O) is involved. The standard trans-

116

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K*[1] Å

0003-9861/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.

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BIOCHEMICAL REACTIONS INVOLVING CARBON DIOXIDE

formed Gibbs energies of both Reactions [1] and [2] are given by

DrG*0 Å 0RT ln K* Å ∑ niDf G*i 0 ,

[3]

where the ni are the stoichiometric numbers for the reactants (sums of species) in a balanced biochemical equation and the Df Gi=0 are the standard transformed Gibbs energies of formation of the reactants. The previous treatment of TotCO2 (1) utilized a Legendre transform to introduce pH as a natural variable, but it is shown here that the same conclusions can be reached using the four equilibrium constants for the reactions between the five species containing CO2 . Equation [3] is applied here to specific biochemical reactions involving carbon dioxide, and standard transformed Gibbs energies of reaction and standard transformed Gibbs energies of formation of reactants are calculated. THERMODYNAMICS OF REACTIONS INVOLVING THE PRODUCTION OF CARBON DIOXIDE

The equilibria involved in Reactions [1] and [2] can be discussed when A and B each represent sums of two species in equilibrium with each other with HA Å H/ / A0

KHA Å [H/][A0]/[HA]

to remember that KHA and KHB depend on the ionic strength (3). However, there are advantages in writing a biochemical reaction in terms of concentrations of all of the reactants in the aqueous phase so that the equilibrium concentrations of all of the reactants in the aqueous phase can be calculated. Thus, the biochemical reaction and corresponding equilibrium expression can be represented as shown in Eq. [2]. The apparent equilibrium constant for Reaction [2] at specified pH is given by ([B0] / [HB])([CO2(aq)] / [H2CO3] / [HCO30] / [CO320]) K*[2] Å . [8] [A0] / [HA] All of these species are in the aqueous phase, but the symbol CO2(aq) is used because CO2 also exists in the gas phase, where it is represented by CO2(g). This equation can be written in terms of equilibrium constants by introducing the following three chemical reactions (1): CO2(aq) / H2O Å H2CO3 Kh Å

[4]

and

[H2CO3] Å 2.584 1 1003 [9] [CO2(aq)]

(note that Kh is independent of the ionic strength), HB Å H/ / B0

KHB Å [H/][B0]/[HB].

[5]

Strictly speaking, the right-hand sides of Reactions [2], [4], and [5] should have a c0 in the denominator, where c0 is the standard state concentration (1 M), and Eq. [1] should have a P0 in the denominator, where P0 is the standard state pressure (1 bar), so that the equilibrium constants are dimensionless and their logarithms can be taken. As a simplification, c0 and P0 will be omitted, but the equilibrium constants will be considered to be dimensionless. The apparent equilibrium constant for Reaction [1] at specified pH is given by K*[1] Å

([B0] / [HB])PCO2 [A0] / [HA]

Å

K[7](1 / [H/]/KHB) , [6] 1 / [H/]/KHA

where K[7] is the equilibrium constant for the chemical reaction A0 Å B0 / CO2(g)

K[7] Å

[B0]PCO2 0

[A ]

.

H2CO3 Å H/ / HCO30 KH2CO3 Å

HCO30 Å H/ / CO320 K2 Å

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[7]

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[H/][CO320] Å 4.685 1 10011. [11] [HCO30]

The values for the equilibrium constants are for 298.15 K, 1 bar, and zero ionic strength. Concentrations are in moles per liter. The usual first acid dissociation constant is not used in this paper, but it is given by K1 Å KH2CO3 /(1 / 1/Kh). Equation [8] can be rearranged to the form

K[7] is not a function of pH or ionic strength. Equation [6] shows how K*[1] depends on pH, and it is important

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[H/][HCO30] Å 1.6681 1 1004, [10] [H2CO3]

and

K*[2] Å

AID

117

K[13](1 / [H/]/KHB) 1 (1 / Kh(1 / KH2CO3 /[H/] / KH2CO3 K2/[H/]2)) 1 / [H/]/KHA

,

[12]

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ROBERT A. ALBERTY

where K[13] is the equilibrium constant for the chemical reaction A0 Å B0 / CO2(aq)

K[13] Å

[B0][CO2(aq)] . [13] [A0]

Note that in contrast with Reaction [2], this chemical equation does not require H2O to balance oxygen. The equilibrium constant of Reaction [13] is not a function of pH or ionic strength. It can be expressed in terms of the standard Gibbs energies of formation of the three species involved, as shown by

F

K[13] Å exp

01 (Df G0(B0) RT

G

e

e

.

0Df G0(A0)/RT

0

When the dissociation constants KHA and KHB are expressed in terms of the standard Gibbs energies of formation of the species involved, it is readily shown that these two equations are equivalent to

[14]

(1 / [H/]/KHB)}{ fe0Df G (CO2(aq))/RT} , 0 0 {e0Df G (A )/RT(1 / [H/]/KHA)} [15]

where factor f is given by

0

0

0)0RT ln(1/[H/]/K HB))/RT

e

0

}{eln fe0Df G (CO2(aq))/RT

0(Df G0(A0)0RT ln(1/[H/]/KHA))/RT

.

The apparent equilibrium constant of biochemical Reaction [2] can also be expressed in terms of the standard transformed Gibbs energies of formation of the four reactants:

F

01 (Df G*0(B) / Df G*0(TotCO2) RT

0 Df G*0(A) 0 Df G*0(H2O)) 0

where NH(i) is the number of hydrogen atoms in species i. At zero ionic strength, Df G0(H/) Å 0, but this is not true at higher ionic strengths. Substituting Eqs. [19] and [20] into Eq. [17] yields 0

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0

e0Df G= (B)/RT{elnfe0Df G (CO2(aq))/RT} . 0 e0Df G= (A)/RT

[24]

To make this equation look like Eq. [18], it is necessary to multiply and divide by 0

0

0

e0Df G= (H2O)/RT Å e0[Df G (H2O)02(Df G (H

/)/RT ln100pH)]/RT]

. [25]

This yields 0

0

e0Df G= (B)/RT{elnfe0Df G (CO2(aq))/RT} 0 0 / 0pH 1 e0[Df G (H2O)02(Df G (H )/RT ln10 )]/RT] K*[2] Å . [26] 0 0 e0Df G= (A)/RTe0Df G= (H2O)/RT Comparison of Eq. [26] with Eq. [18] shows that

G

Df G*0(TotCO2) Å Df G0(CO2(aq)) 0 RT ln f / Df G0(H2O) 0 2(Df G0(H/)

0

e0Df G= (B)/RTe0Df G= (TotCO2)/RT Å 0D G=0(A)/RT 0D G=0(H O)/RT . 2 e f e f

AID

0

/ e0Df G= (HB)/RT), [22]

Df G=i 0 Å Df G0i 0 NH(i)(Df G0(H/) / RT ln 100pH), [23]

K*[2] Å

[17]

K*[2] Å exp

/ e0Df G= (HA)/RT) [21]

where Df G=0(A0) Å Df G0(A0) and Df G=0(HA) Å Df G0(HA) 0 (Df G0(H/) / RT ln 100pH), which are examples of the general equation for calculating the standard transformed Gibbs energy of formation of a species (2):

Equation [15] can be rearranged to {e0(Df G (B

0)/RT

Df G=0(B) Å 0RT ln(e0Df G= (B

f Å 1 / Kh(1 / KH2CO3 /[H/] / KH2CO3K2 /[H/]2). [16]

K*[2] Å

0

0)/RT

0

0)/RT

{e0Df G (B

Df G=0(B) Å Df G0(B0) 0 RT ln(1 / [H/]/KHB). [20]

and

When Eq. [14] is substituted in Eq. [12], the expression for the apparent equilibrium constant for Reaction [2] can be written as K*[2] Å

and

0

0Df G0(B0)/RT 0Df G0(CO2(aq))/RT

e

Df G=0(A) Å Df G0(A0) 0 RT ln(1 / [H/]/KHA) [19]

Df G=0(A) Å 0RT ln(e0Df G= (A

/ Df G0(CO2(aq)) 0 Df G0(A0)) Å

Equations [17] and [18] provide two equivalent ways to express the apparent equilibrium constant of Reaction [2]. The terms for reactants A and B in Eqs. [17] and [18] can be set equal to obtain

[18]

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/ RT ln 100pH)

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BIOCHEMICAL REACTIONS INVOLVING CARBON DIOXIDE

use of Eq. [27] are given in Table III. Programs written in Mathematica (4) were used to make these calculations.

TABLE I

Standard Gibbs Energies of Formation of Species at 298.15 K and I Å 0 Reactant

D f G0/kJ mol01

Reference

C6H12O6(aq)(glucose) H2O(l) O2(g) O2(aq) CO2(g) CO2(aq) C6H5O73-(isocitrate) C5H4O52-(2-oxoglutarate) NAD0 NADH20

0915.90 0237.19 0 16.40 0394.36 0385.97 01156.04 0793.41 0 22.65

(15) (16) (5) (5) (5) (1) (12) (12) (10) (10)

HENRY’S LAW CONSTANTS FOR CARBON DIOXIDE

It is necessary to distinguish between three different Henry’s law constants for carbon dioxide and aqueous solutions. The first is the equilibrium constant KHsp , written in terms of species for the following reaction (1): CO2(aq) Å CO2(g)

KHsp Å

PCO2 [CO2(aq)]

Å 29.466. [29]

The second is the equilibrium constant KH for the reaction Å Df G0(CO2(aq)) 0 RT ln f / Df G=0(H2O),

[27]

where Df G=0(H2O) Å Df G0(H2O) 0 2(Df G0(H/) / RT ln 100pH). Equation [27] can be used to calculate Df G=0(TotCO2) as a function of pH and ionic strength. The previous article (1) followed a general procedure of thermodynamics, which is to (a) use a Legendre transform to introduce the pH as a natural variable, (b) calculate standard transformed Gibbs energies of formation by subtracting adjustments for the pH, and (c) use isomer group thermodynamics to calculate the standard transformed Gibbs energy of formation of TotCO2 . This yielded

Df G=0(TotCO2) Å 0RT ln[exp(0Df G=0(CO2(aq))/RT) / exp(0Df G=0(H2CO3)/RT) / exp(0Df G=0(HCO30)/RT) / exp(0Df G=0(CO320)/RT)].

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Å

KH Å

PCO2 [CO2(ao)]

PCO2 [CO2(aq)] / [H2CO3]

Å

KHsp Å 29.39, 1 / Kh

K*H Å

[28]

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[30]

where ao indicates the sum of CO2(aq) and H2CO3 (5) and Kh is defined by Eq. [9]. These are the values of the equilibrium constants at 298.15 K. Note that KHsp and KH are not functions of ionic strength. KH is the Henry’s law constant that is calculated from the entries in the NBS tables, which are based on the convention that the equilibrium constant for Reaction [9] is unity. The third type of Henry’s law constant is the apparent Henry’s law constant K*H at a specified pH. It is the apparent equilibrium constant for the biochemical reaction TotCO2 Å CO2(g) / H2O

Equations [27] and [28] look different, but they are actually identical, as can be shown by replacing the equilibrium constants in Eq. [27] with expressions utilizing the standard Gibbs energies of formation of the species and writing Eq. [28] in terms of standard Gibbs energies of formation of species. The value of Df G=0(TotCO2) can be calculated either way, but the method used in this article is based directly on the equilibrium constants defined in Eqs. [9]–[11]. The standard Gibbs energies of formation at 298.15 K and I Å 0 used in this article are given in Table I. The standard transformed Gibbs energies of formation that are calculated from Table I at 298.15 K, pH 7, and I Å 0.25 M are given in Table II. The standard Gibbs energies of formation of TotCO2 calculated at 298.15 K, 1 bar, and ionic strengths of 0, 0.05, and 0.25 M by

AID

CO2(ao) Å CO2(g)

PCO2 [TotCO2]

. [31]

The value of K*H depends on T, pH, and ionic strength,

TABLE II

Standard Transformed Gibbs Energies of Reactants at 298.15 K, pH 7, and I Å 0.25 M Reactant

D f G*o/kJ mol01

Reference

Glucose(aq) H2O(l) O2(aq) TotCO2 Isocitrate 2-Oxyglutarate NADox NADred

0426.70 0155.66 16.40 0547.16 0959.47 0633.57 1059.10 1120.09

(15) (15) (5) (1) This paper This paper (10) (10)

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ROBERT A. ALBERTY TABLE III

Standard Transformed Gibbs Energies of Formation of TotCO2 and H2O at 298.15 K as a Function of pH and Ionic Strength D f G*o(TotCO2)/kJ mol01

D f G*o(H2O)/kJ mol01

pH

IÅ0

I Å 0.05 M

I Å 0.25 M

IÅ0

I Å 0.05 M

I Å 0.25 M

5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

0566.19 0560.70 0555.56 0551.09 0547.39 0544.20 0541.23 0538.37 0535.58

0565.28 0559.87 0554.93 0550.73 0547.24 0544.15 0541.23 0538.40 0535.70

0564.67 0559.34 0554.55 0550.53 0547.16 0544.12 0541.24 0538.45 0535.85

0180.11 0174.40 0168.69 0162.99 0157.28 0151.57 0145.86 0140.15 0134.45

0179.15 0173.44 0167.73 0162.03 0156.32 0150.61 0144.90 0139.19 0133.49

0178.49 0172.78 0167.07 0161.37 0155.66 0149.95 0144.24 0138.53 0132.83

and it is 3.168 at 298.15 K, pH 7, and 0.25 M ionic strength. The standard transformed Gibbs energy of reaction for Reaction [31] is given by DrG=0[31] Å 02.86 kJ mol01. This apparent Henry’s law constant, which is a function of the pH and ionic strength, is important in biochemistry because it is the factor for calculating the apparent equilibrium constant of Reaction [2] from the apparent equilibrium constant for Reaction [1]. If PCO2 in Eq. [1] is replaced using Eq. [31], [B][TotCO2]K*H K*[1] Å Å K*[2]K*H . [A]

K*[1] K*H

.

[32]

[33]

Note that Reaction [2] is equal to Reaction [1] minus Reaction [31]. The value of the apparent Henry’s law constant can be calculated by treating Reaction [31] as a biochemical reaction:

DrG*0 Å 0RT ln K*H Å Df G*0(CO2(g)) / Df G*0(H2O) 0 Df G*0(TotCO2),

Å Å

Å

Thus,

K*[2] Å

K*H Å

[34]

PCO2 [TotCO2] PCO2 [CO2(aq)] / [H2CO3] / [HCO30] / [CO320] PCO2 [CO2(aq)](1 / Kh(1 / KH2CO3 /[H/] / KH2CO3 K2 /[H/]2)) KHsp . [35] (1 / Kh(1 / KH2CO3 /[H/] / KH2CO3 K2 /[H/]2))

K*H is plotted versus pH in Fig. 1 for three values of the ionic strength. This plot is not useful for interpolating values above pH 7 because K*H becomes so small. However, the plot of DrG*0[31] shown in Fig. 2 is useful over the whole pH range. The standard transformed Gibbs energy of Reaction [2] can be obtained by substituting Eq. [33] in DrG*0[2] Å 0RT ln K*[2]:

DrG*0[2] Å 0RT ln K*[2] Å 0RT ln K*[1] / RT ln K*H .

At T Å 298.25 K, pH 7, and I Å 0.25 M, K*H Å 3.168, and so

DrG*0[2] Å 0RT ln K*[1] / 2.86 kJ mol01. 0

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[37]

0

where Df G* (CO2(g)) Å Df G (CO2(g)) because this term is concerned with the gas phase. Values of K*H and DrG*0[31] are given in Table IV as a function of pH and I. The values of Df G*0(TotCO2) and DfG*0(H2O) used in the calculation are given in Table III. The equation for calculating the dependence of K*H on pH and ionic strength can also be obtained from

AID

[36]

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Thus, the standard transformed Gibbs energy of Reaction [3] is a little more positive than for Reaction [1], which is in agreement with K*3 õ K*1 . There is another way to look at this difference of 2.86 kJ mol01 and that is in terms of the changes in the niDf G*0i terms in Eq. [3]. In going from Reaction [1] to Reaction [3], we sub-

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BIOCHEMICAL REACTIONS INVOLVING CARBON DIOXIDE TABLE IV

Apparent Henry’s Law Constants and Standard Transformed Gibbs Energies of the Biochemical Reaction TotCO2 Å CO2(g) / H2O at 298.15 K as a Function of pH and Ionic Strength DrG*0/kJ mol01

K*H pH

IÅ0

I Å 0.05 M

I Å 0.25 M

IÅ0

I Å 0.05 M

I Å 0.25 M

5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

28.18 25.87 20.55 12.46 5.544 2.011 0.6650 0.2115 0.0652

27.64 24.49 17.99 9.786 4.005 1.394 0.4524 0.1415 0.0421

27.15 23.30 16.09 8.132 3.168 1.078 0.3456 0.1063 0.0303

08.28 08.07 07.49 06.25 04.25 01.73 1.01 3.85 6.77

08.23 07.93 07.16 05.65 03.44 00.82 1.97 4.85 7.85

08.18 07.81 06.89 05.20 02.86 00.19 2.63 5.56 8.67

tract 0394.36 kJ mol01 for CO2(g), add 0547.16 for TotCO2 , and subtract 0155.66 for H2O; this is a change of /2.86 kJ mol01, in agreement with the /2.86 kJ mol01 in Eq. [37].

The apparent equilibrium constant at a specified pH is expressed by K*[38] Å

[2-oxoglutarate][NADred]PCO2 [isocitrate][NADox]

.

[39]

APPLICATIONS TO BIOCHEMICAL REACTIONS

Goldberg et al. (7, 8) have evaluated literature data on thermodynamic measurements on a number of biochemical reactions involving CO2 and the following examples have been chosen from their publications. Londesborough and Dalziel (9) measured the equilibrium partial pressure of carbon dioxide and obtained an equilibrium constant for the isocitrate dehydrogenase reaction (EC 1.1.1.4) (6): isocitrate / NADox Å 2-oxoglutarate / NADred / CO2(g).

[38]

FIG. 1. Apparent Henry’s law constant as a function of pH at 298.15 K, 1 bar, and I Å 0, 0.05, and 0.25 M. The upper curve is for I Å 0.

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Londesborough and Dalziel found that the apparent equilibrium constant for this reaction at 298.15 K, pH 7, and I Å 0.10 M is 24.2 atm. However, then they calculated K*c by dividing 24.2 atm by Henry’s law constant of 29.1 atm/M for CO2 in pure water to obtain K*c Å 0.83 M. This equilibrium constant does not correspond with a balanced biochemical equation because the carbon dioxide in the solution, that is CO2(aq), is about 10% of TotCO2 at pH 7. It is incorrect to use [CO2(aq)] in an equilibrium expression because the CO2

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FIG. 2. Standard transformed Gibbs energy of reaction in kJ mol01 for TotCO2 Å CO2(g) / H2O as a function of pH at 298.15 K and 1 bar for I Å 0, 0.05, and 0.25 M. The lower curve is for I Å 0.

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ROBERT A. ALBERTY

formed comes to equilibrium with the other species containing CO2 in a couple of seconds. When isocitrate is oxidized, a stoichiometric amount of TotCO2 is formed, but not a stoichiometric amount of CO2(aq). The correct stoichiometry for the reaction in the aqueous phase is represented by

relations because HCO30 is 4% of TotCO2 at pH 5, 30% at pH 6, and 80% at pH 7. This means that if [HCO30] is used in calculating K*, there is an error by a factor of 25 at pH 5, 3.3 at pH 6, and 1.2 at pH 7. Fortunately, they determined K* Å 123 at 377C, pH 8.15, and I Å 0.067 M, and so the error was less than 3%.

isocitrate / NADox / H2O Å 2-oxoglutarate / NADred / TotCO2 .

[40]

Note that H2O has been added to the left-hand side to balance oxygen. The apparent equilibrium constant expression for Reaction [40] is K*[40] Å

[2-oxoglutarate][NADred][TotCO2] . [41] [isocitrate][NADox]

To obtain the value of K*[40], K*[38] should be divided by K*H Å 3.5 at pH 7 and I Å 0.1 M. Therefore, the value of the apparent equilibrium constant for Reaction [41] is (24.2/3.5)(760/750) Å 7.0, rather than 0.83. Thus, Df G*0[40] Å 04.82 kJ mol01. Since the standard thermodynamic properties of isocitrate30 and 2-oxoglutarate20 are known at 298.15 K (12), the standard transformed Gibbs energies of formation of these species can be calculated at pH 7 and I Å 0.25 M, and they are given in Table II. The standard transformed Gibbs energies of formation of NADox and NADred are known (10), and so the apparent equilibrium constants of Reactions [38] and [40] can be calculated: for Reaction [38], K* Å 20.36 and DrG*0 Å 07.47 kJ mol01, and for reaction [40], K* Å 6.267 and DrG*0 Å 04.55 kJ mol01. These calculations are based on the assumption that the effects of the highest pK terms of isocitrate and 2-oxoglutarate cancel. There is a further complication since the Londesborough and Dalziel experiments involved 0.133 mM Mg2/, and we do not know the dissociation constants of the complexes with isocitrate and 2-oxoglutarate that are probably formed. Thus, the experimental value is probably better for the actual conditions than the value calculated using data in Table II. Villet and Dalziel (11) used the same methods in determining the two types of apparent equilibrium constant for the 6-phosphogluconate dehydrogenase reaction, and so the above remarks apply to it also. Halenz et al. (13) determined the apparent equilibrium constants for the propional-CoA carboxylase reaction (EC 6.4.1.3) and wrote it as ADP / Pi / (S)-methylmalonyl-CoA / H2O Å ATP / propanoyl-CoA / HCO30 .

[42]

THE NET REACTION FOR THE OXIDATION OF GLUCOSE TO CARBON DIOXIDE AND WATER

The same considerations apply to net reactions that produce CO2 , even though their equilibrium constants may not be obtainable by direct experiment. An important example is the oxidation of glucose to carbon dioxide and water. As a simplification, the production of ATP from ADP and Pi , which occurs in enzymatic systems, is not included. This reaction can be written as a chemical reaction: C6H12O6(aq) / 6O2(g) Å 6CO2(g) / 6H2O(l). [43] The standard Gibbs energy of the reaction at 298.15 K and 1 bar is given by

DrG0[43] Å 6(0394.36) / 6(0237.19) / 915.90 Å 02 873.4 kJ mol01.

For dilute solutions the expression for the equilibrium constant for Reaction [43] is K[43] Å

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P6CO2 [C6H12O6(aq)]P6O2

.

[45]

Reaction [43] can be written slightly differently to indicate that the pH is considered to be an independent variable. When the pH is specified, all possible ionized species of the reactants are included, and that is indicated in the following biochemical reaction: glucose(aq) / 6O2(g) Å 6CO2(g) / 6H2O(l). [46] The standard transformed Gibbs energy DrG*0 for Reaction [46] is equal to DrG0 for Reaction [43] because in dilute aqueous solutions the equilibrium constant for this reaction is not a function of pH in the region pH 2–10. The standard transformed Gibbs energy of this reaction at pH 7 can be calculated from standard transformed Gibbs energies of formation of glucose and water given in Table II.

DrG*0[46] Å 6(0394.36) / 6(0155.66) / 426.70 Å 02 873.4 kJ mol01.

This hybrid equation does not represent stoichiometric

AID

[44]

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[47]

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BIOCHEMICAL REACTIONS INVOLVING CARBON DIOXIDE

For dilute solutions the expression for the apparent equilibrium constant K* of Reaction [46] is K*[46] Å

P6CO2 6 O2

[glucose]P

.

[48]

K*[46] Å

3.1686[TotCO2]6 . [glucose]746.696[O2(aq)]6

Thus, K*[49] (defined by Eq. [50]) is given by K*[49] Å K*[46]

The value of this apparent equilibrium constant is independent of the ionic strength. However, for equilibrium calculations on the intermediate steps in this biological oxidation, it is more useful to write the biochemical reaction as glucose(aq) / 6O2(aq) Å 6TotCO2 .

S D

746.9 6 . 3.168

[55]

The corresponding DrG*0[49] is given by

S D

DrG*0[49] Å 0RT ln K*[46] 0 6RT ln

[49]

The expression for the apparent equilibrium constant K* of Reaction [49] is

[54]

746.9 3.168

Å 02873.4 0 81.2 Å 02954.6 kJ mol01. [56] This agrees with Eq. [51] within rounding errors.

6

K*[49] Å

[TotCO2] . [glucose][O2(aq)]6

[50]

This apparent equilibrium constant is a function of pH and ionic strength in the neutral pH range because DfG*0(TotCO2) is a function of pH and ionic strength. The standard transformed Gibbs energy of Reaction [49] at pH 7 and I Å 0.025 M can be calculated using the standard transformed Gibbs energies of formation of glucose, TotCO2 , and O2(aq):

DrG*0[49] Å 6(0547.10) / 426.70 0 6(16.4) Å 02954.3 kJ mol01.

[51]

There is another way to calculate the value of DrG*0[49] that yields the same result as the preceding paragraph. The Henry’s law expressions for carbon dioxide and molecular oxygen can be substituted into Eq. [48] to obtain the value of the apparent equilibrium constant for Reaction [49]. The Henry’s law constant for molecular oxygen in dilute solutions is independent of the pH and ionic strength: O2(aq) Å O2(g)

[52]

and K*H(O2) Å

PO 2 [O2(aq)]

Å 746.69

DrG*0 Å 016.4 kJ mol01. [53] These values are obtained from the NBS tables (5). Substituting Eqs. [31] and [53] into Eq. [48] yields

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DISCUSSION

We are so used to writing chemical equations that balance atoms of each element and also charge that it is hard to get used to the fact that when the pH is specified, atoms of hydrogen are not conserved (14). Biochemical equations must balance atoms other than hydrogen, but it is more important to emphasize that they conserve groups of atoms. The reactants in a biochemical equation are often sums of species and in the case of carbon dioxide in aqueous solution the species are CO2 ,(aq), H2CO3 , HCO30 , and CO320 . The reactions between these species are equilibrated within several seconds, and the distribution between species depends on the T, pH, and ionic strength. When a biochemical reaction produces carbon dioxide, it is the sum of these species that is stoichiometrically related to the changes in amounts of other reactants. When carbon dioxide is produced in a biochemical reaction, the expression for the apparent equilibrium constant can be written in terms of PCO2 or [TotCO2], but the use of [TotCO2] has the advantage that the equilibrium concentrations of all of the reactants in the aqueous phase can be calculated. The value of the apparent equilibrium constant in terms of [TotCO2] can be obtained by dividing the apparent equilibrium constant in terms of PCO2 by the pH-dependent Henry’s law constant K*H . The biochemical equation corresponding with the apparent equilibrium constant written in terms of [TotCO2] is obtained by adding CO2(g) / H2O Å TotCO2 to the biochemical reaction in terms of PCO2 . The derivations given here show how Df G*0(TotCO2) and K*H depend on the four equilibrium constants between the five species of carbon dioxide involved. These equations have been used to calculate tables of values of thermodynamic properties that can be used to con-

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124

ROBERT A. ALBERTY

vert apparent equilibrium constants in terms of PCO2 to apparent equilibrium constant in terms of TotCO2 . Specific examples of reactions producing CO2 are discussed, and Df G*i 0 values for reactants are calculated. The calculation of Df G*0 for reactants provides a much better way to store information on apparent equilibrium constants of biochemical reactions than storing apparent equilibrium constants because each reactant contributes this same amount to DrG*0 for any reaction in which it is involved. The usefulness of a table of Df G*i 0 values under a specified set of conditions increases exponentially with the number of entries because a reactant may be involved in a very large number of reactions. Here the equilibrium composition is discussed in terms of CO2(aq), H2CO3 , HCO30 , and CO320 , but equilibrium compositions can also be discussed in terms of the species (CO2(aq) / H2CO3), HCO30 , and CO320 . This is the way this system is treated in the NBS tables. The same values of Df G*0(TotCO2) and K*H can be calculated using the entries in the NBS tables, which are based on the assumption that the equilibrium constant is unity for CO2(aq) / H2O Å H2CO3 . The NBS tables list 13 more pairs of species that are connected by the formal chemical relationship A(aq) / nH2O Å ArnH2O(aq).

[57]

The type of treatment given here to CO2/H2CO3 can also be applied to NH3/NH4OH, SO2/H2SO3 , and these other pairs of species. The rate of equilibration of the four species containing CO2 has been studied in some detail (17), and the half life is about 4 s at pH 6 and 7 at 257C. The halflife is shorter at higher and lower pH values. Although these reactions seem pretty fast, the equilibration of carbon dioxide in our lungs requires the presence of

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carbonic anhydrase to reduce the half-life. The calculations in this paper are concerned with equilibrium experiments that take longer than about 10 s. ACKNOWLEDGMENT I acknowledge Robert N. Goldberg for many helpful discussions.

REFERENCES 1. Alberty, R. A. (1995) J. Phys. Chem. 99, 11028–11034. 2. Alberty, R. A. (1994) Biochem. Biophys. Acta 1207, 1–11. 3. Clarke, E. C. W., and Glew, D. N. (1980) J. Chem. Soc. 176, 1911–1916. 4. Wolfram, S. (1996) The Mathematica Book, Cambridge Univ. Press, Cambridge. 5. Wagman, D. D., et al. (1982) J. Phys. Chem. Ref. Data 11,(Suppl. 2). 6. Webb, E. C. (1992) Enzyme Nomenclature, Academic Press, San Diego. 7. Goldberg, R. N., Tewari, Y. B., Bell, D., Fazio, K., and Anderson, E. (1993) J. Phys. Chem. Ref. Data. 22, 515–582. 8. Goldberg, R. N., and Tewari, Y. B. (1994) J. Phys. Chem. Ref. Data. 24, 1765–1801. 9. Londesborough, J. C., and Dalziel, K. (1968) Biochem. J. 110, 217–222. 10. Alberty, R. A. (1993) Arch. Biochem. Biophys. 307, 8–14. 11. Villet, R. H., and Dalziel, K. (1969) Biochem. J. 115, 633–638. 12. Wilhoit, R. C. (1969) in Biochemical Microcalorimetry (Brown, H. D., Ed.), pp. 305–317, Academic Press, New York. 13. Halenz, D. R., Feng, J-Y., Hegre, C. S., and Lane, M. D. (1962) J. Biol Chem. 237, 2140–2147. 14. Alberty, R. A., Cornish-Bowden,A., Gibson, Q. H., Goldberg, R. N., Hammes, G. G., Jencks, W., Tipton, K. F., Veech, R., Westerhoff, H. V., and Webb, E. C. (1994) Pure Appl. Chem. 66, 1641– 1666. [Republished in Eur. J. Biochem. 240, 1–14 (1996)]. 15. Alberty, R. A., and Goldberg, R. N. (1992) Biochemistry 31, 10610–10615. 16. Cox, J. D., Wagman, D. D., and Medvedev (1989) Codata Key Values for Thermodynamics, Hemisphere Publishing, New York. 17. Alberty, R. A., and Silbey, R. J. (1992) Physical Chemistry, Wiley, New York.

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