THE JOURNAL OF BIOLOGICAL CHEMISTRY Vol. 244, No. 12, Issue of June 25, PP. 329fF3302, 1969

Printed in U.S.A.

Standard Gibbs Free Energy, Enthalpy, and Entropy Changes

as a F-unction of pH and pMg for Several Reactions

Involving Adenosine Phosphates

(Received for publication, September 5, 1968)

ROBERT A. ALBERTY

From the Department of Chemistry, Massachusetts In,stitute of Technology, Cambridge, Massachuseits 0.2139

SUMMARY

The standard Gibbs free energy change AGZ,,, number (nn) of moles of H+ produced, number (nMg) of moles of Mg2+ produced, standard enthalpy change AH:,,, and standard entropy change AC&, have been calculated as functions of pH and pMg for the following phosphohydrolyase and phosphotransferase reactions at 25” and 0.2 ionic strength:

ATP + Hz0 = ADP + Pi (1)

ATP + AMP = 2ADP (2)

ATP + Hz0 = AMP + PPi (3)

ADP + Hz0 = AMP + Pi (4)

ATP + 2HzO = AMP + 2Pi (5)

PPi + Hz0 = 2Pi (6)

The values of these thermodynamic quantities are presented by means of contour diagrams for the range pH 4 to 10 and pMg 1 to 7. These diagrams make the general features of the pH and pMg dependences readily discernible and sum- marize the results of some 2500 calculations per diagram. There are significant changes in the heat evolved by these reactions over this range of the independent variables. Equations are derived which make it possible to calculate the standard entropy of reaction AS& from the entropy change of the reaction written in terms of particular ionic species, entropy changes of the various acid and metal ion dissociation reactions, the entropies of mixing of the various forms of each reactant and product, and the entropies of dilution of H+ and Mg2+. The relative contributions of enthalpy and entropy to the equilibrium constants of these reactions may be accessed from the diagrams as a function of pH and pMg.

If the equilibrium constant for a biochemical reaction is known at one temperature, ionic strength, pH, and known free concentrations of metal ions that form complexes with the reactants, the equilibrium constant may be calculated for

another pH and different metal ion concentrations at this temperature and ionic strength provided that the acid dissocia- tion constants and complex dissociation constants are known under these conditions. If the heat of reaction is also known at this temperature and ionic strength and for known pH and free metal ion concentrations, the heat of reaction may be calcu- lated at another pH and metal ion concentration, provided that the heats of dissociation of the acids and complexes are known. When the changes in Gibbs free energy and enthalpy are known under a given set of conditions the entropy change may also be calculated. Even if the required constants are known the labor of making the calculations encourages the introduction of approximations and short cuts. Fortunately, the modern digital computer eliminates this problem, and so it is possible to take six reactions and see what can be done to relate the observed thermodynamic quantities to the thermodynamic quantities for the individual ionic reactions which occur together. These calculations emphasize the need for certain types of experimental data. The basic pattern for these calculations was established in the treatment of the ATP phosphohydrolase reaction (1). Phillips, George, and Rutman (2) have made similar calculations.

For use by the experimentalist it is most convenient to define equilibrium constants for reactions such as we are interested in here in terms of the total concentrations of the reactants and products, excluding water. The six so-called “observed” equilibrium constants are defined by

K (ADPI (Pi)

lobs = (ATP) (7)

K (ADP)*

Poba = (ATP) (AMP) (8)

K (AMP) @‘Pi)

sobs = (ATP)

e-0

K (AMP) (Pi)

‘Oba = (ADP) (10)

K (AMP) (Pi) * ‘Ohs = (ATP) (11)

K (Pi)* --

“” - (PPJ (12)

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where the parentheses represent total concentrations of the various reactants and products in moles per liter. The concen- tration of Hz0 is omitted because the other reactants and products are assumed to be present at such low concentrations that the activity of Hz0 is a constant independent of the rela- tive amounts of these other reactants and products. Such observed equilibrium constants are a function of the tempera- ture, the ionic strength, the pH, and the concentrations of any metal ions which form complexes with species of the reactants or products. In this paper calculations are presented only for 25’, 0.2 ionic strength, and Mgz+ as the only metal ion present. Since various phosphate ions tend to bind Na+, K+, and even (CH3)4N+ (3), the calculations are carried out for an idealized supporting electrolyte with a non-complex-forming cation. This is perhaps most closely approximated by tetra-n-propyl am- monium chloride, and so these calculations are for that electro- lyte with the reactants and products making negligible contri- butions to the ionic strength. Thus complexes of the type Mg(ATP)#- formed only at relatively high ATP concentrations do not have to be taken into account at the low ATP concentra- tions involved. Writing the equilibrium expressions in terms of concentrations means that the chemical potentials p of the ions are given by the ideal solution equation p = /.LO + RT ln c, where c is concentration, in media of constant ionic strength, pH, and pMg.l

The standard Gibbs free energy change, AG!&, for a reaction is made up of two contributions, the standard enthalpy change, AHzbs, and the standard entropy change, AS&.z

AGO,,. = AH%,. - TASk (13)

A&,, is calculated from the ‘Lobserved” equilibrium constant -- K obs.

AG& = -RT In Kobs (14)

This is the Gibbs free energy change which occurs when the reactants appearing in the equilibrium constant expression, each at a hypothetical concentration of 1 mole per liter (except for water), are converted into the products, each at a hypothetical concentration of 1 mole per liter, all in a medium of the stated pH and pMg and a constant ionic strength. We say at a “hypothetical” concentration of 1 mole per liter because we do not use AG$,, to calculate the AG,b, value for such concen- trated solutions, only for dilute solutions. The standard enthalpy change AH& and the standard entropy change A$& are for the same process. AH$,* is equal to the heat evolved by the reaction at constant pressure and temperature when no work is done and, in contrast with AG& and AS&+, may be

1 pMg is defined for Mg2+ in a similar way to pH for H+ and is assumed to be obtained by use of a reversible divalent cation electrode.

2 The prime on AG8b, used in the preceding paper (1) to indicate that the standard Gibbs free energy change is for a finite ionic strength has been deleted in this paper to simplify the notation.

taken to be equal to AHohs (that is, the enthalpy change for any reactant and product concentrations) so long as the solu- tions are dilute.

THEORY

The expression of Kobs as a product of the equilibrium constant for the reaction written in terms of particular ion species and a function of (H+) and (Mg”+), involving the various acid and complex dissociation constants, is the key to the calculation of the thermodynamic functions.

nHz&[+]DM,=[y-$gMg (15)

(16)

The number, nn, of moles of H+ produced by the reaction and the number, nMs, of moles of Mg2+ produced are related by

(i-z&J, = (%).,, 09)

Thus at a particular pH and pMg the slope of the nH surface measured downward (increasing pMg) is equal to the slope of the nM, surface measured to the right (increasing pH). If you were a climber on surface nH and looked to the south you would see the same slope as if you were on surface nMg at the same coordinates looking to the east. This is an expression of the linkage of the Hf binding and Mg2f binding by the equi- librium system. The general theory has been described by Wyman (4) and Edsall and Wyman (5) and has been applied to Reaction 1 (1). Strictly speaking, Wyman considered only the binding by a single substance, but this concept is readily ex- tended to calculate the change in binding when a reactant is converted to a product with different binding properties.

By the use of Equation 17 AHzbs may be obtained as a func- tion of (H+) and (Mg*) involving the temperature coefficients of the dissociation constants of the various ionic reactions. These temperature coefficients may be expressed in terms of the standard enthalpy changes, AHiO, of these ionic reactions by use of

(20)

The detailed equations are given only for Reaction 1. It is convenient to introduce a symbol for the fractions, f, of the various reactants and products in one arbitrarily chosen form.

jp = (HPO:-1 -= (Pi)

+ (Mg”+) + (H+) 1 - - KM.GHATP KZATP II

+ (Mg”+) + (H+) -1 - -

KMBHADP &ADP

(21)

(22)

(2.3)

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TABLE I

Thermodynamic parameters for dissociationa reactions used in calculations at .W and 0.2 ionic strength ---------

Acids HPzO,a- = H+ + PzO,J-. ................... HATPB- = H+ + ATP+ .................... H2P207* = H+ + HPzOv+. ................. HzATP* = H+ + HATP3-. ................. HADP2- = H+ + ADP”. ................... H*POr’ = H+ + HPOP . . r . HAMPI- = H+ + AMP*-. .................. HzADP’- = H+ + HADP*. ................. HzAMPO = H+ + HAMP’. ..................

Complexes MgP207” = Mg2+ + PzO#-. ................. MgATP* = Mg2+ + ATPa-. ................. MgHPzO,l- = Mg2+ + HPzOr+. .............. MgHATPr- = Mg2+ + HATPZ-. ............. MgADP’- = Mg2+ + ADP3-. ................. Mg,PaO,O = Mg2+ + MgPsO," ............... MgHPOdO = Mg2+ + HPOf-. ................ MgAMPO = Mg2+ + AMP*. ................. MgHADPO = Mgz+ + HADP+. ..............

Constant

K IPP KIATP

KZPP K 2ATP

KIADP

&P

K IAMP

&ADP

KZAMP

KM~PP

KM,ATP

KM~HPP

KM~HATP K MgADP

K MgaPP K M.P KM~AMP

K MgHADP

PK AGO AHO

km1 mole-’ km1 mle-1

8.95 (6)b 12.21 0.40 (7)c 6.95 (8)d 9.48 -1.68 (9)” 6.12 (6)b 8.35 0.11 (7)6 4.06 (lo)/ 5.55 0 W)f

6.88 (8)a 9.39 -1.37 (9)” 6.78 (8)d 9.25 0.80 (11)~ 6.45 (8)d 8.80 -0.85 (9)’ 3.93 (10)’ 5.36 1.0 (1O)f

3.74 (lO)f 5.10 1.0 (10)f

5.41 (12)h 7.39 -3.5 (13)’ 4.00 (14)i 5.46 -3.3 (15>k 3.06 (12)h 4.17 -3.5 (13)’ 1.49 (8)d 2.04 -1.9 (15)k 3.01 (8)d 4.11 -3.6 (15)k 2.34 (12)h 3.19 -2.26 1.88 (S)d 2.56 -2.9 (16)” 1.69 (8)d 2.31 -2.9 (16)” 1.45 (8)d 1.98 -2.0 (15)k

-

--

-

AS0

cd aeg-' mole-'

-39.6 -37.4 -27.6 -18.6 -36.1 -28.4 -32.4 -14.6 -13.8

-36.5 -29.4 -25.7 -13.2 -25.9 -18.3 -18.3 -17.5 -13.3

0 pK = - log K, where K is a dissociation constant calculated with concentrations in moles per liter. b 0.1 M (CH&NCl. c 0.22 (CH&NCl. d 0.2 M (n-propyl),NCl. 6 Zero ionic strength. f 0.15 M NaCI. g 0.01 ionic strength. b 1 M (CH,),NCl. i 0.05 ionic strength. j 0.1 M KC1 (20’). k 0.1 ionic strength.

The arbitrarily chosen form is the most basic except for ortho- phosphate, where the last proton dissociation has a pK far beyond the range of interest here. The various dissociation constants are defined in Table I. Since

K &fm loba = (H+)fmfp

(24)

where K, is defined in Equation 29, the change in standard Gibbs free energy is given by

A~$,I,~ = -RT In &fATP

(H+) fmpfp (25)

The equations for nn and nM, have been given earlier (1). The heat evolved at constant temperature and pressure is given by

This equation looks complicated at first sight, but it is of such a form that AHfobs is equal to AH0 for the predominating ionic reaction, if there is a predominating ionic reaction, and it properly weights the AH% of all of the reactions that occur together. At high pH and in the absence of Mg*, AHibs = AHlO, the standard enthalpy change for Reaction 1 in Table II. As pH or pMg is reduced various terms in fATp, fADp and fp tend to dominate as the particular species that they represent tend to dominate. When particular species other than the reference forms (the forms in Table II) dominate, the AHi0 (or AH&) required to form those species from the reference forms are taken into account in Equation 26. Podolsky and Morales (17) gave the equation for the dependence of AHioba on pH with two species of each reactant. Phillips et al. (2) have

AH:abs = aI&0 - fADP (Mg”c) (H+) @I+) @f@+) KM~ADP

AH&r f &ADP &ADP + KIADpKMgHADp (AHkDp+AH&ADP)

(H+Y + &ADP&ADP

(AH:,,, -I- AHO,& F AH& -I- '2 AHip MOP

ELAH&ATP (26)

+ @+I &o

&ATT IATP

+ OX+) CM@+)

KIAT~KM~EAT~ (NATP + &,HATP) +

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TABLE II

Standard Gibbs free energy, enthalpy, and entropy changes used for calculations at 66” and 0.2 ionic strength

Concentrations are expressed in moles per liter. The number of significant figures is not meant to indicate the accuracy of these quantities, but shows the unrounded figures that were used in the calculations.

1. ATP” + Hz0 = ADP3- + HPO,*- + H+. . .

2. ATP+- + AMP+ = 2ADP3- 3. ATP+- + Hz0 = AMP+ +

PgO+- + 2H+. . . . 4. ADPa- + Hz0 = AMP+ +

HPOb* + Hf............ 5. ATP+ + 2HzO = AMP” +

2HPOd2- + 2H+. 6. PzO,4- + Hz0 = 2HPO?. .

AGO

km1 mole-’

0.276 -0.602

11.800

0.878

1.154 -10.646

-

_-

-

AHO

-4.70 -16.7 -1.46 -2.88

-4.54 -54.9

-3.24 -13.82

-7.94 -30.4 -3.40 25.3

described the calculation of AH” lobs without the species HsATP+ and HzADPl-.

An expression for AS!&, may be obtained by substituting Equations 25 and 26 in Equation 13 or by the use of Equation 18. The resulting equation can be arranged in many ways, but it is probably most instructive to arrange it to bring out the entropy of mixing terms. This was suggested by the demon- stration by King (18) that, when thermodynamic quantities are measured for a composite reaction, the entropy of mixing of the related species of reactants and products comes into the equa- tion for the standard entropy change of the composite reaction. The reactions considered here are good illustrations of this point.

The expression for the standard entropy change may be written as follows:

A.Sks = Afkwg + A&, ADP + A&, p - AsLi, ATP (27)

- nHR In (H+) - nMgR In (Mg*+)

The quantity A&,,, has exactly the same form as the expression for AHiobs, except that AH’s are replaced by AL%. The form of the mixing terms may be illustrated by giving just one of them, that for the mixing of the five ionic species of ATP.

negative (H+ is consumed) then -nHR In (H+) is the entropy change for the concentration of Hf from the experimental pH to 1 M. Phillips et al. (2) have given plots for Reaction 1 of n,RT In (H+) and nMgRT In (Mg*), which they refer to as the hydrogen ion driving force and the magnesium ion driving force.

In summary, the standard entropy of reaction AS& may be considered to be made up of three contributions:

1. Weighted Average of AS0 for Various Ionic Reactions-This term is independent of the choice of reaction in the first term of AS:,,, in these calculations the reactions in Table II.

2. Difference in Entropy of Mixing of Products and Reactants- The maximum value that one of these terms can have varies from 1.37 cal deg-l mole-l for two species to 3.56 cal deg-’ mole-l for six species. However, the maximum value occurs only in the rare circumstance that the species are all present at equal concentrations.

3. Entropy of Dilution of H+ and Mg* Formed by Reaction- The hydrogen ion term will be very large at high pH if nH is ~1 or f2. Since nM, is always equal to zero at high pMg values this term is always negligible at sufficiently high pMg values, but may be significant at some intermediate pMg.

The more positive that these various terms become, the more favorable AGtbs becomes for the forward reaction; that is, the forward reaction is favored by a predominating ionic reaction with a large positive ASo. It is also favored by high pH if nR is positive and by mixed ionic species for products (but not reactants).

CALCULATIONS

As shown in a previous paper (1) it is convenient to present thermodynamic functions for biochemical reactions as functions of pH and pMg by the use of contour diagrams which depict the height of a surface above a plane with pH plotted in one direction and pMg in the other. These calculations have been made practical by use of a digital computer. A typewriter terminal of the MIT CTSS system was used to allow the IBM 7094 computer to print out arrays of 2511 values of the desired thermo- dynamic function in response to a MAD program giving the necessary equations. The desired contour lines were then drawn in by hand.

Contour diagrams of log &bs, nH, and nM, are given in Fig. 1 and of AG&, AH&,,, and TAS& in Fig. 2.

The number (nH) of moles of H+ produced is simply equal to the slope of the log Kobs surface measured in the direction of

AS~~~ATP = -R @I+)

In ~ATP + ~ATP - K IATP

1nfATP gp + fATP el hfATP KFL

+ fATP (H+) (Mg*+) ln fATp (H+) Ok*+)

KIATP&IHATP KATPKM~HATP + fATP

&~~:IATP In fATp KJZZIATP]

(28)

This is the entropy change per mole of ATP of preparing a mixture with the following mole fractions: fATp of ATPa-, fATP(H+)/&ATP ofHATP3--,fATP (Mgz+)/KMgATPOfMgATP2-, f&H+) (Mg2f)/K~~~~f&g~~~P of MgHATP1-, and fATP (H+)2/K1~~p&~~p of HzATl?.

The last two terms of Equation 27 are entropy of dilution terms for Hf and Mg2+. For example, if the number, nH, of moles of H+ is positive (H+ is produced) then -n=R In (H+) is the entropy of dilution of nR moles of H+ from 1 M to the actual concentration under the experimental conditions. If nH is

increasing pH, and the number (n& of moles of Mg2+ produced is simply equal to the slope of the log Kobs surface measured in the direction of increasing pMg.

In order to calculate nH and nMg it is necessary to know only the dissociation constants for the various acid species and com- plex ions which have to be taken into account in the range of pH and pMg under consideration. The various pK values used in the present calculations are summarized in Table I along with the conditions under which they were ineasured. The required constants have all been measured at 0.1 to 0.2 ionic strength,

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I. ATP + Hz0 = ADP + Pi

nu oH

‘Og Kobt OH

9 4 s 6 I I

2 e -2

-3

-5

-6

2. ATP + AMP = 2ADP

nH

PH “Mg

PH 4 5 6 7 8 9 IO

i i i b lb PH

3. ATP + Hz0 = AMP + PPi

nH “m

FIG. 1. Contour diagrams for thermodynamic functions at 25” and 0.2 ionic strength tetra-n-propyl ammonium chloride 83 a func- tion of pH and pMg. Left, log &be; middle, number, nH, of moles of H+ produced; tight, number, nxs, of moles of Mgz+ produced. In the shaded regions the value is within 0.01 of the value zero.

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4. ADP + H,O = AMP + Pi

nti

‘Og Kobs

PH 4 5 6 7 8 9

5. ATP + 2H,O q AMP + 2P,

6. PPi + Hz0 = 2Pi

“H

PH 4 6 , 8 9 10

2

3

PM.3 4

5

5

7

“Mg PH

4 5 6 7 8 9

*-

3-

PW 4-

-2

-3

m -4

“Mg

PH 4 5 6 7 s 9 10

-2

-3

PW -4

-5

FIG. l-Continued

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3296 Thermodynamics of Reactions of Aclenosine Phosphates Vol. 244, No. 12

I. ATP + Hz0 = ADP + Pi

J

2. ATP + AMP = 2ADP

A&s

PH 4 5 6 7 8 9 10

PH 4 5 6 7 8 9 IC

I I I-

P- -2

‘. I-

-------o --

3- 3

Pm 4-

PM9

5-

4 5 i i i b io PH

3. ATP + Hz0 = AMP + PPi AHobs 0

1

2

3

PM9 4

5

6

7

Fm. 2. Contour diagrams for thermodynamic functions at 25’ in 0.2 ionic strength tetra-n-propyl ammonium chloride as a function of pH and pMg. Left, AG$,. in kilocalories per mole; middle, AHibs in kilocalories per mole; right, TA&% in kilocalories per mole.

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4. ADP + Hz0 = AMP + pi

0 AGobs

PH 7 8 ? 10

0 AHobs 0

TASobs

DH PH 4 5 6 7 8 9 10

6% Pm

5-

-45 -45 -40 -3.5

6- -6

7 I I I I 7

4 5 6 7 8 9 10 PH

I

2

3

VW 4

5

6

1

m 4

5

6

-7 IO

10 I

:-2

-3

I PM

t-4

I

-5

-6

---7

10

5. ATP t ZH,O = AMP + 2Pi

A&s

PH 4 5 6 7 8 9 10

I (

-2

-3

PM9 4-

Pm -4

5- -5

6- -s-p6

7 I I I

7

4 6 6 7 8 9 10

PH

0 AGobs

1 u I -2

d 3

PM 4

-25

5

7

5

6. PPi + Hz0 = 2Pi

AHobs 0 0 TASobs PH PH

4 5 6 7 8 9 10

I

6 6- 5.5 6 -6

7---

7 7 4 5 6 7 8 9 10 I I I I I 7

4 5 6 7 e 9 10 PH PH

Pm. 2-Continued

A&s

2

5

5

except for PLPP, P&,Q~PP, and pKMaHpp, which were meas- titrations of HPzO+- and HzPzO,* in 1 M and 0.1 M (CHJI- ured in 1 M (CHJfiCl (12). It would be desirable to correct NC1 suggests that these corrections may not be large. Lambert these values to 0.2 ionic strength, but we have no theoretical (19) found that pKlpp increases 0.18 and pKwp increases guidance at unit ionic strength. However, experience with 0.23 in going from 1 M to 0.1 M. Irani (6) found 0.21 and 0.14,

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respectively. Schuegraf, Ratner, and Warner (20) corrected pKMgpp, ~K~~nrn, and pKMMgePP determined by Lambert and Watters (12) in 1 M (CHS)~NC~ up 0.79, 0.56, and 0.31, respec- tively, to obtain values at 0.2 ionic strength. This includes a small temperature correction in going from 25’ to 37”. Since there does not seem to be adequate experimental data to justify such large corrections, uncorrected values have been used in the present research.

There seems to be no evidence of complexes MgHzPz070 or Mg,ATP”. If they exist their pK values for dissociation of Mg2+ would probably be less than 1.5 so that they would not play much of a role in the range of magnesium ion concentrations considered here. There is an uncertainty anyway along the edge where pMg = 1 because the ionic strength of 0.1 M MgClz is 0.3, not 0.2.

Experimental values of AH0 are available for all of the acid dissociations and for all of the magnesium ion dissociations but two (Mg2P207° and MgAMPO), and reasonable estimates may be made for these. Since AH0 may depend significantly on ionic strength and ionic composition (7, 21), there is need for calori- metric measurements at ionic strengths in the physiological range. The media to which the AH0 values in Table I apply are indicated. The AH0 values for acid dissociation are small and are almost equally divided between positive and negative values. Thus these acid dissociation constants are determined primarily by the entropy changes, which are unfavorable for acid dissociation because of the organization of the water struc- ture resulting from the dissociation of a proton and the increase in charge of the anion.

The heats of Mg2+ dissociation range only from -1.9 to -3.6 kcal mole-r. Although there is a decrease in enthalpy in each of these reactions, they do not have equilibrium constants greater than unity because of the unfavorable entropy change, which again results from the organization of the water structure by the ions produced by dissociation.

The reactions are arranged in order of decreasing charge of the parent ion, and there is a tendency for the magnitude of the entropy change to decrease in this order. However, this is not the whole story because the charges are distributed over the molecules in different ways. For example, in the adenine ionization (pK = 4) the proton dissociation takes place from a part of the molecule which is quite distant from the negative charges of the phosphate groups.

Since AH0 and AX0 have not been measured for the dissocia- tion of MgAMPO and MgLP2070, it is assumed that the heat of dissociation of MgAMPO is the same as for the magnesium com- plex of glucose l-phosphate (16). It is assumed that the entropy of dissociation of MgtP2Or’J is the same as for the uncharged species MgHPOdO, and the heat of dissociation is calculated with the use of this value for ASO.

In order to calculate AG&, as a function of pH and pMg it is necessary to have an experimental value of Koss at some particu- lar pH and pMg, ideally in a medium of 0.2 ionic strength con- taining known concentrations of free cations. The experimental value of Koss is used to calculate the value of an equilibrium constant K expressed in terms of particular ionized species. In order to clearly distinguish the K values from the Kobs values they are defined as follows for the reactions in Table II.

K 1

= (ADPa-) 0=02-) @+I (ATP4-)

cm

(ADPs-)2

K2 = (ATPI-) (AMP) (30)

K 3

= @P-l (Pe0~~3 @+I" (ATP4-)

(31)

K 4

= (AMP-) (HPOa2-) (H+) (ADPP) (32)

K 6

= (AMP2-) (HP04*)2 (H+)* (ATP”-)

(33)

(34)

where the parentheses represent concentrations in moles per liter.

The standard Gibbs free energy change, calculated from

AGi” = -RT In Ki (35)

where i = 1 to 6, is the free energy change which occurs when the reactants appearing in the equilibrium constant expression, each at a hypothetical concentration of 1 mole per liter, are converted into products, each at a hypothetical concentration of 1 mole per liter, all in a medium with an ionic strength of 0.2. The calculation of AGIO at 25” is described in Reference 1.

The value of AG# in Table II was calculated from the research of Eggleston and Hems (22) and pK values in Table I.

There are several ways in which the free energy change of Reaction 3 may be obtained. Atkinson and Morton (23) have presented calculations showing that AG& is about 1 kcal mole-l more negative than AGi,b, under physiological conditions. Jencks et al. (24-26) conclude that AG&b, is about 0.4 kcal mole-1 more negative than for Reaction 1 under physiological conditions. There is at present inadequate information about pMg in experiments on reactions which yield AG’&,. For the present calculations I have taken AG&ss to be 1 kcal mole+ more negative than AG~,I,, at pH 7.0 pMg 3.0 at 25”; that is, AGiobs = -9.75 kcal mole-r. This leads to a value of AG$ = 11.800 kcal mole-r for Reaction 3 in Table II.

With these values of AGrO, AGzO, and AGsO, the values of AG40, AGsO and AG$ have been calculated from

AG4O = AG1° - AG,O (36)

AGs” = 2 AGI” - AGzo (37)

AG,O = 2AG1° - AG,O - AGsO (33)

These are not the only free energy changes which may be calcu- lated from the equilibrium data for Reactions 1 to 3. For

example,

ATP’ + HP042- = ADP+ + P2014- + H+ (39)

AGO = AGlo - AGs” = 10.922 kcal mole-l

ADPE- + HP042- = AMP2- + P2014- + H+

AGO = AG40 - AC60 = 11.524 kcal mole-l (40)

SATP” + H20 = 2ADP3- + PzO?’ + 2Hf (41)

AGO - 2AG10 - AGaO = 11.198 kcal mole-l

Also, a large number of free energy changes may be calculated by adding or subtracting reactions in Table I from Reactions 1 to 6 in Table II.

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Issue of June 25, 1969 R. A. Alberty 3299

Since the shapes of the AG$,, surfaces in Fig. 1 are independent of the values of AGk” to AGsO, these graphs may readily be used to calculate AG& for Reactions 4, 5, and 6 for different values of AG1’J, AGzO, and AGaO. For example, if AG’&, is taken equal to AG’&,, at pH 7 pMg 3, AGzO = -8.75 kcal mole-1 and all of the AG& values become 1 kcal mole-1 more negative.

In order to calculate the heats of reaction AH& for these six reactions as functions of pH and pMg it is necessary to have the values of the six AHO’s in Table II. The values of AHlO, AHzO, and AH,+ were calculated from experimental data and the re- maining values were calculated with

AHa = 2AHl” - AHz” - AHe (42)

AHdo = AHlo - AHz” (43)

AHso = 2AH1° - AHzO (44)

The heat evolved in the ATP phosphohydrolase reaction has been measured by Podolsky and Sturtevant (27), Kitzinger and Benzinger (28), and Podolsky and Morales (17). Since these experiments were carried out with 0.6 M KCl, corrections need to be made for K+ binding and for the change in ionic strength in order to do calculations for 0.2 ionic strength tetraalkylam- monium halide solutions. However, adequate data do not exist for either correction, and so AH” = -4.7 kcal mole-l, ob- tained by Podolsky and Morales (17) at 20” and pH 8.0 in 0.6 M KCl, has simply been used for AHlO in Table II. Since they found that 0.15 M KC1 did not give significantly different AH&b, values from 0.6 M KCl, this may not introduce a very large error.

For the adenosine triphosphate-adenosine 5’-monophospho- transferase reaction, AH!,b, may be calculated from the data of Bowen and Kerwin (29) at 1” and 40”. This yields AH&Q,* = -1.25 kcal mole-1 at pH 6 in 0.06 M disodium succinate buffer. The value of AHzO is calculated as follows

HATPa- + HAMP’- = 2HADP2- AH0 = -1.25 kcal molewl 2HADP2- = 2H+ + 2ADP3- AH0 = -2.74 kcal mole-l H+ + AMP2- = HAMPI- AH0 = +0.85 kcal mole-l H+ + ATP4- = HATPS- AH0 = +1.68 kcal mole-l

ATPh- + AMP2- = 2ADP3- AHz0 = -1.46 kcal mole-l

The enthalpy change for the pyrophosphatase reaction has been measured by Ging and Sturtevant (30) and by Wu et al. (31). The value of AH$ was calculated from the measurement of Wu et al. (31) of AH0 = -3.8 kcal mole-l for the reaction

HP207@ + Hz0 = HP0d2- + HzP041-

at 25” and 0.07 ionic strength. Adding AH0 for the dissociation of HzP041- and subtracting AH0 for the dissociation of HPzO? yields AH,+ = -3.4 kcal mole-l. The problem resulting from the presence of Na+ in these experiments is neglected in the present calculation.

DISCUSSION

Under experimental conditions in which nn and ?%M, both have integer values, or the value zero, it is usually possible to write a simple ionic reaction which represents the reaction quite well. This has been done for Reaction 1 in an earlier paper (1) and George and Rutman (32) have discussed Reaction 6 in the absence of Mg* from this point of view. However, when one looks at the various contour diagrams of nu and nM, in Fig. 1, it is striking how few values of pH and pMg there are in which a single ionic reaction predominates. In general, therefore, it is

necessary to take several ionic reactions into account in order to understand the stoichiometry with respect to H+ and Mg2f and the effects of these ions on the equilibrium.

For the adenosine triphosphate-adenosine 5’-monophosphate phosphotransferase reaction, Reaction 2, the effects of changing pH and pMg are not large because they result from the relatively small differences between the acid and complex ion dissociation constants of the three adenosine phosphates.

For Reaction 3, the ATP pyrophosphohydrolase reaction, the equilibrium at pH 7 is not much affected by raising (Mg”+) until about low3 M is reached, but raising (Mg#) from 10V M to 10-l M increases AG’&,bs by 2 kcal mole-l. The high sensitivity of this equilibrium at pH 7 pMg 1 to both H+ and Mg2+ is indi- cated by nH = 2 and nMg = -1.7.

The effect of Mg* on AG&bs is even more striking. The calculations of nM, in Fig. 1 show that AG&bS changes at least 1 kcal mole-1 per pMg unit all the way from pMg 3.8 to 1.7 (that is from 0.16 mM to 20 mM). The effect of changing pH is rather small. Thus in studying this reaction experimentally in the neutral pH range at pMg below 4, it is more important to have a good measurement of pMg than pH. It is important to remem- ber that in determining AG&,bs from 2AGi,bs - AG.& - AGiobs or other series of reactions, the result is equally sensitive to PMg.

The number of equivalents of acid produced by Reactions 3 and 5 in the absence of Mg* and at pMg 2.27 has been calcu- lated by Schuegraf, Ratner, and Warner (20) with different values for pK, as mentioned above. They also pointed out that AG&bs is appreciably less negative in the presence of MgZ+.

In the present calculations AG&bs has been obtained from AG&, values for Reactions 1, 2, and 3. Thus it is of interest to compare the results with the experimental value of Stiller et al. (33). They determined AG&,* at 25’, pH 7.5,0.25 ionic strength at zero MgZt in experiments in which the reaction was catalyzed by yeast pyrophosphatase and the final concentrations of pyro- phosphate were determined by paper chromatography with x2P. Their values under these conditions (-5.8 to -7.4 kcal mole-l) are somewhat less negative than the value calculated here (-8.5 kcal mole-l).

Wood, Davis, and Lochmiiller (34) give AGO = 1.0 f 0.2 kcal mole@ for Reaction 1 in Table II. Since they used a value for the standard free energy of the ATP phosphohydrolase reaction which had not been corrected for the formation of the magnesium glutamate complex (l), their value for AF’ionio should be corrected by subtracting 1.365 log (1 + 10-1.45/10-1.go) = 0.80 kcal mole-l. With this correction for magnesium glutamate their value for AGIO becomes 0.2 f 0.2 kcal mole-1 which may be compared with the value of 0.276 kcal mole+ in Table II. Wood et al. (34) have calculated the free energy of hydrolysis of in- organic pyrophosphate by adding free energy changes for the four reactions catalyzed by: EC 2.7.1.40, ATP:pyruvate phosphotransferase; EC 3.6.1.4, ATP: phosphohydrolase; EC 4.1.1.38, pyrophosphorylase: oxaloacetate carboxylase; EC 4.1.1.3, oxaloacetate carboxylyase. For

HPsO,a- + Hz0 = 2HPO,+ + H’ (45)

they get AF’ionic = 2.4 f 0.3 kcal mole+ for 25” and 0.1 ionic strength. The value that I used for this reaction is obtained by adding the free energy of dissociation of HP20+- (12.21 kcal mole-1 in Table I) to the value for AG# (-10.646 kcal mole-1 in Table II) to obtain 1.56 kcal mole+. If their value for ATP

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3300 Thermodynamics of Reactions of A&no&e Phosphates Vol. 244, No. 12

TABLE III

AGf&, in kilocalories per mole at %5” in 0.1 M KC1 for PPi + Hz0 = BP<

(ME?+) Reference

rnM

0

0.5

5.0

345@ This research

345a This research

3450 This research

6.0

-8.6 -8.5 -8.3 -8.4 -6.7 -7.7

I

i&s at pH

7.0 8.0 ~-

-8.8 -9.8 -8.7 -9.6 -7.2 -6.7 -8.1 -8.1 -5.5 -5.0 -6.9 -6.8

a The values of Wood et al. (34) have been corrected for the Mg glutamate complex by subtracting 0.8 kcal mole-l.

phosphohydrolase is corrected by subtracting 0.8 kcal mole-l, our values for AGO for Reaction 45 agree. The differences in AG&,# between their calculations and mine are to be attributed entirely to the difference in the acid and complex dissociation constants. The direction and magnitude of this difference may be seen in Table III where their values, corrected by subtracting 0.8 kcal mole0, are compared with values calculated providing for 0.1 M KCl, with the use of their constants for potassium complexes but constants in Table I for the other species. At high pMg and low pH our calculations are in good agreement, but at low pMg and high pH the differences between our values range up to 1.8 kcal mole-l. This is a result of the fact that they used free energies of Mg* dissociation for MgPtO,*, MgHP20+-, and MgzP20$J which were 1.44,0.88, and 0.67 kcal mole-1 larger tha.n the values used in the present research.

Fig. 2 is designed to show the relative roles that enthalpy changes and entropy changes play in determining Kobs under different conditions of pH and pMg. The equilibrium is shifted to the right as AH& becomes more negative and AS&, becomes more positive. Excluding Reaction 2, which is a closely balanced system, AH& and AS!&, are individually favorable for a spon- taneous forward reaction, except that AH&bs is unfavorable at pH 7 to 10 at the highest magnesium ion concentrations and AX&ss is unfavorable in a narrow strip which lies above pMg 3 at pH 6 to 10. However, the relative contributions of AH,“,, and AS%,* shift tremendously. At pH 7 large shifts in the relative importance of AH& and AS& result from changing pMg. For reactions producing H+ the standard entropy change plays a larger role in determining Kobs at high pH than AH&. This results from the large contribution from the entropy of dilution of H+.

It is important to realize that even if there is a predominating ionic reaction, with a AS’ given by adding reactions in Tables I and II, AS& will have a different value, except under the condition that nn = nM, = 0. That is because AS& contains an entropy of dilution term 2.3 R(nnpH + n,,pMg). For example, at high pH values in the absence of Mg2+ the hydrolysis of ATP to ADP and Pi occurs via Reaction 1 in Table II and AS10 = -16.7 cal deg-r male-1. However, ASqohs at pH IO pMg 7 is 29.0 cal deg-* mole-l.

The heat of hydrolysis of ATP is quite sensitive to the con- centration of magnesium ions in the physiological range: d(AH!&,)/dpMg is approximately 2 kcal mole-1 pMg-1 in the range pMg 1 to 3.

The AHioba values may be compared with those calculated by Phillips et al. (2) at pH 6,7.5, and 9 at 0.1 ionic strength. They find the most negative heat of reaction at pH 6 to be at pMg 3, at pH 7.5 to be at pMg 3.8, and at pH 9 to be at pMg 4.2. There is reasonably good agreement between their values and those in Fig. 2.

Podolsky and Morales (17) did not find a significant change in AHqobs upon addition of lOma M Ca*, in the presence of 0.6 M

KC1 and 0.1 M Tris at pH 8.0 and 20”. Part of the Ca* was undoubtedly bound by the myosin which was present at a concentration of about 0.1 g/100 ml. Since Ca2+ is more weakly bound by the reactants than Mg2+, the effect of Ca2+ on AH!,b, would probably be expected at higher free concentrations than used by Podolsky and Morales.

The standard entropy change for the ATP phosphohydrolase reaction is positive and increases with increasing pH. Phillips et al. (2) have calculated a different type of standard entropy change for Reaction 1. In their entropy calculations H+ and Mg2+ have 1 M standard states as ATP, ADP, and Pi do.

Since AH& and AS&, may be expressed in terms of the values of AH0 and AS0 in Tables I and II and entropy of mixing and entropy of dilution terms, it is possible to see exactly what contributes to a favorable equilibrium constant. The composite reaction may be said to be understood to the extent that the individual ionic reactions in Tables I and II are understood.

The adenylate kinase reaction has a value of Kobs greater than unity under all conditions considered here except at pMg 3 between pH 8 and 10. At the higher pMg values Kobs > 1 because AH$,, is negative (although ASib, is unfavorable). At lower pMg values &bs > 1 because As!&, is positive (although AH& is unfavorable).

At pH 7 and pMg 4 enthalpy and entropy effects make ap- proximately equal contributions to the standard free energy of the ATP pyrophosphohydrolase reaction, but at higher magne- sium concentrations and at higher pH entropy effects tend to dominate.

In contrast with the other phosphohydrolase reactions ti& for the pyrophosphate phosphohydrolase reaction is not large at high pH and low pMg, and as a result the reaction is not very favorable under these conditions. This difference from the other phosphohydrolase reactions is a result of the fact that this reaction does not produce acid under these conditions.

The calculations of AHiobs are based on calorimetric measure- ments by Wu et al. (13), and so it is of interest to compare them with the measurement by Ging and Sturtevant (30) at pH 7.5, pMg 3.9, 25’, and 0.1 ionic strength. Their experimental value of -6.3 kcal mole-1 may be compared with the -5.0 kcal mole-1 in Fig. 2. There is evidently need for further calorimetric experiments to explore the effect of changing pH and metal ion concentrations.

The contour diagrams of TAs& give the heat absorbed (n) under conditions in which the reaction is harnessed to give t.he maximum non-PI’ work (WJ = -AG$,,) at constant temperature and pressure. For a reaction with a negligible volume change, the first law may be written

AH& = q - w (46)

Substitution of w = -AGO obB and comparison with Equation 13 yields q = T&f&,,. The contour diagrams show that in general these reactions absorb heat when they are harnessed to do maximum work.

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lssue of June 25, 1969 R. A. Alberty 3301

Figs. 3 and 4 are presented to give some idea of the way that experimental errors are propagated in these calculations. sup- pose redeterminations showed that the pK values for ATP in Table I are each 0.1 too high and the pK values for ADP and Pi are each 0.1 too low. The changes in AGiobs that would be obtained in changing to the new pK values are given in Fig. 3. The value of AGrO has been changed to give the experimental value of AGFobs at pH 7 pMg 1.45. This calculation gives too large a calculated error in the sense that the signs of the errors in the pK values have been taken so that they all add. If the errors were randomly distributed with respect to sign, they would tend to compensate each other. I f the experimental value of AGi,b, at pH 7 pMg 1.45 is incorrect, the whole surface would simply be raised or lowered without a change in shape.

In order to estimate the changes in the calculated values of AHqobs resulting from errors in the values of the various enthal- pies of dissociation, the changes in AHiobs were calculated for 0.2 kcal mole-1 changes in the values of the various enthalpies of dissociation and the results are presented in Fig. 4. The signs of the changes were taken in such a way that the errors all add; the enthalpies of dissociation for ADP and orthophosphate were all increased 0.2 kcal mole-r, and the enthalpies of dissociation of ATP were all decreased 0.2 kcal mole-l. I f the errors were randomly distributed with respect to sign there would be some compensation of errors, and so Fig. 4 gives too large a calcu- lated error for the assumed errors in the enthalpies of dissocia- tion. On the other hand, some of the enthalpies of dissociation are undoubtedly in error by more than 0.2 kcal mole-l. No errors in pK values have been included in the calculation of Fig. 4. Errors in pK values would simply shift the regions where AH& has the indicated errors.

One of the more serious problems in the determination of

PH 5 6 7 8 9 IO

I I I I / /

I

4

t l-

2-

3-

PMg 4-

5-

6-

7-

I 0 8 0.7 0.6 -0.5 --

:

/

--f 5

PMg

f

- I 7

PH

4

FIG. 4. Change in AI&,. (kilocalories per mole) resulting from changes in enthalpies of dissociation of 0.2 kcal mole-l, each change being taken with the sign to give the maximum error in AZ&X.

dissociation constants of magnesium complexes and in deter- mining equilibrium constants for the over-all reactions is lack of knowledge of the concentration of free Mg”+. The new electrodes for divalent cations3 should assist greatly in deter- mining pMg so that it does not have to be calculated.

Acknowledgments-I am indebted to Professor Roy Kaplow, Mrs. Barbara Boudreau, and Miss Sally Duren of Project MAC for programming assistance and to Professor Phillips W. Robbins of the Department of Biology at the Massachusetts Institute of Technology for advice about biochemical equilibria.

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FIG. 3. Change in AG!,ba (kilocalories per mole) resulting from changes in pK values for dissociation of 0.1, each change being

a Divalent cation activity electrode and calcium ion activity

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3302 Thermodynamics of Reactions of Adenosine Phosphates Vol. 244, No. 12

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Robert A. AlbertypH and pMg for Several Reactions Involving Adenosine Phosphates

Standard Gibbs Free Energy, Enthalpy, and Entropy Changes as a Function of

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