Robert A. Alberty Massachusetts Institute of Technology Cambridge, 02139
Thermodynamics of the Hydrolysis of Adenosine Triphosphate
The common currency of energy exchange in living things is the hydrolysis of adenosine triphosphate ATP to adenosine diphosphate ADP and orthophosphate Pi or to adenosine monophosphate AMP and pyrophosphate PPi (1, 9). The purpose of this article is to discuss certain aspects of the thermodynamics of the first reaction. ATP
+ HQO= ADP + Pi
(1)
The observed equilibrium constant Kobe,which is usually used by biochemists, is defined by
where [ATP] represents the total molar concentration of ATP including all the ionized and complexed forms. This type of equilibrium constant is convenient for use in the laboratory because the experimenter is interested in the relation between the equilibrium concentrations of ATP, ADP, and Pi in dilute solutions when the pH and electrolyte concentrations in the aqueous solution are held constant. The value of Kobeis also of interest because it can be used to calculate the amount of work (excluding PAT' work) which can be obtained in principle when a mole of ATP is hydrolyzed a t constant temperature and pressure. The hydrolysis of one mole of ATP under physiological conditions can yield as much as 5-10 kcal of work (that is, this is the value of -AGob., the decrease in Gibbs free energy), but the amount of work actually obtained depends on how this reaction is coupled with others. Since the maximum amount of work depends upon the pH and concentrations of metal ions, as well as the temperature, it is of interest to see how big these effects are and what causes them. Specifically we will be interested in the following types of questions: If Kobais known a t one p H and metal ion concentration, what is its value a t another p H and in the presence of another metal ion or a different metal ion concentration? If the heat evolved at constant temperature and pressure (AH, the enthalpy change) is determined at one pH and metal ion concentration, what will be the heat evolution under another set of conditions? Since the reaction may produce or consume H+ and metal ions, what is the stoichiometry under various conditions? What are the relative contributions of enthalpy and entropy change to the change in standard Gibbs free energy for the reaction? It will probably come as a surprise to you that Kob. is a function of pH and metal ion concentrations, even a t constant temperature, pressure, and total electrolyte concentration. This is a result of the way Kobeis defined in eqn. (2). Since Kobadeals with sums of ionized and complexed species, we will find that the thermo-
dynamic quantities consist of sums of terms and that these various contributions vary with pH and metal ion concentration. The basic ideas have been discussed in a number of recent articles (5-5). Theory The hydrolysis of ATP may be expressed in terms of particular ionized species ATP"
+ HzO = ADPa- + H P O P + H +
(3)
The corresponding equilibrium constant expression is
The hydrolysis of ATP could equally well be expressed in terms of other ionized species, but we will stick with eqn. (4) (which is also K, in the table) throughout this article. Since the choice of components is arbitrary in thermodynamics, we may calculate the Gibbs free energy of hydrolysis and other thermodynamic quantities using either Koba[eqn. (2) I or Ki [eqn. (4) 1. Comparison of these two points of view provides some excellent illustrations of elementary chemical thermodynamics. If all the known species of ATP, ADP, and P. are taken into account, the equations get a little long (5-5), and so we will consider an incomplete set of equilibria here so that the important basic principles will not be buried. To calculate the effecton reaction 1 of changing pH in the range 4-10 it is necessary to take into account two acid dissociations of ATP, two of ADP, and one of Pi, but we will ignore these ionizations and talk about effects of changing the concentration of magnesium ion at a constant pH of 9 and 25°C. Two magnesium complexes of ATP and two of ADP are known, but we will consider only the ATP complex forped at the lowest magnesium ion concentration and the ADP complex formed at the lowest magnesium ion concentration. Since the magnesium ion concentration may vary over many powers of ten, i t is convenient to use pMg = -log[Mg2+], where [MgZ+]is the magnesium ion concentration in moles per liter. If pMg is determined by use of a divalent cation electrode, we can think of [Mgz+] as the magnesium ion activity on the moles per liter scale. The equilibria we will consider may be summarized as follows: ATPa-
+ HIO k~ADPa- + HP0,'- + H+
n MgATPa Kl
KI
T~KI
Tl Ka
(5)
MgADPL- MgHPOP
The values of the equilibrium constants at 25'C and 0.2 ionic strength are summarized in the table (5). In the calculations it is assumed that there are no other cations in the electrolyte which form complexes with ATP4-, Volume 46, Number 11, November 1969
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Thermodynamic Parameters for lonic Reactions a t 25'C and 0.2 lonic Strength
+ +
MgATP2- = Mg2+ ATP4MgADP1- = Mgs+ + ADPaMgHPOP = Mg2+ HPOIZATP4- + HHIO= ADP3- + HP0.a- + H+ MgATP2H20 = ADP8- + HPO,%- H+ Mgat MgATP2H1O = MgADP1HPO2 H+ Mg2+ MgATPZ- HnO = MgADPL- MgHPOP H +
1. 2. 3. 4. 5. 6. 7.
+
+
a
pK
=
+
+
+
+ + + +
Constant
pKa
KI If% Ka K*
4.00 3.01 1.88
+
AH" kcd mole-'
AS' cal deg-1 mole-'
5.74
-3.3 -3.6 -2.9 -4.7 -8.0
1.63 -0.93
-1.5
-29.4 -25.9 -18.3 -16.7 -46.1 -20.2 - 19
AGO kcal mole-' 5.46 4.11 2.56 0.28
-4.4
-log K, where K is expressed in terms of concentrations in moles/l.
ADPa-, and HPOa2-. This means that Na+ and I) (14)
The values of -TASo,b, are given in Figure 2. It is convenient to give the entropy changes in this form so that the sum of the ordinates of the two lower plots gives the ordinate of the upper plot. The sign of the standard entropy change ASoOb,is favorable for hydrolysis of ATP at all values of pMg; that is, ASo.b, is positive. I n fact the entropy change contributes more to the negative value of AGOob.than does AHooba,except in the vicinity of pMg 3.5. It is of interest to note how the relative contributions of AHoOh,and ASooba shift with changing pMg.
If we only had a term
we would have ASomidop; SO let's add and subtract this term from eqn. (18). Thus eqn. (18) may he written R ln[l [MgW]/Kzl = ASomir*m
+
Similarly R lnll IMaP+l/Ktl= AS",~,ATF
+
Interpretation of ASoab.
To be sure eqn. (14) yields ASaoba,but it is disappointing that we cannot see in i t anything familiar or interesting. Perhaps the equation can he rearranged into a more meaningful form. King (11) has pointed out that when thermodynamic quantities are measured for a composite reaction, the entropy of mixing of the related species of reactants and products come into the equation for the standard entropy change of the composite reaction. Since the products and reactants in reaction 1 exist in multiple forms, we should expect to find mixing terms in eqn. (14). The entropy of mixing the two forms of ATP fATP4- and MgATP2-) is
+
Substituting these three equations into eqn. (14) and doing a little rearranging yields
[Mg2+]/K,) This is the entropy of mixing 1/(1 moles of ATP4- with ([Mg2+1/KJ/(1 [Mg2+l/K~) = AS',,. + AS0,i.mp A S o m i r ~- A S D m i x ~ ~ ~ moles Since the ~-~~ of MeATP2- to form an ideal mixture. R In [Hf1 - Rnu. In [Mg2+l (22) logarithms of the mole fractions are negative, ASo,i,mp is positive. Since AHo for mixing of ideal solutions to where nM,,the number of moles of magnesium ion proform an ideal solution is zero, AGO is negative and the duced, is given by eqn. (8) and mixing is a thermodynamically spontaneous process at constant temperature and pressure. The standard entropies of mixing the two species of ADP and the two species of Pi are correspondingly ~
+
-
Now the question is whether or not we can rearrange eqn. (14) so that it shows the combinations of terms given in the preceding three equations. I n eqn. (14) we have a term Rln [I [Mg2+]/K2]. What has to he done to this term to get the combination of terms defined as AS",i,ADp given in eqn. (16)? First we invert the
+
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+
which is of the same form as eqn. (12). If one ionic reaction predominates these four terms simply yield ASo of the predominating ionic reaction. Thus AS0.& contains the weighted average of the first four standard entropy changes from the table, and the three expected entropies of mixing, hut what are the lmt two terms? They must he related to the fact that the hydrolysis of ATP at pH 9 produces one mole of H+ and n ~ moles , of Mg2+per mole of ATP hydrolyzed. The last two terms in eqn. (22) are entropy of dilution terms. Before going on to discuss them it is worth noting how helpful algebra is in rearranging an equation which doesn't seem to have any meaning (eqn. (14)) into one that is full of meaning (eqn. (22)).
Entropy of Dilution
The entropy of expansion of one mole of an ideal gas from volume Vl (and concentraion CI = ~ / V I to ) volume V2 (and concentration cz = l/Vz) is given by (19)
Dilution is a spontaneons process in ideal solutions because AH = 0 and so AG5il.n = -2.3 RT(pH)
(28)
AGodilnM. = -2.3 nM. RT(pMg)
(29)
If n ~is. positive (Mg" is produced), AG,il.~Egis negaThis equation may also be applied to the dilution of a component of an ideal solution. If the intial concentration is 1M AS
=
-R in cs
(25)
tive as required for a spontaneous process a t constant temperature and pressure. Now we may rewrite eqn. (22) in the form where
Thus for the dilution of a mole of H + from 1M to [H+] = 10-vH M ASodil.x = -R In [HC] = -2.3Rlog 10-pH = 2.3 R(pH)
(26)
Similarly for the dilution of n ~moles . of Mg2+from 1M to [MgZ+]= 10W'"gM. ASod,l.arg = -nx.
R in [Mga+] = 2.3 n u . R(pMg)
(27)
In order to understand how these entropy of dilution terms arise, we can imagine reaction 1as taking place in steps with H+ and R!gZ+ being produced a t 1 M. Then the H + and Mg2+ions produced must be diluted from 1 M to 10-pH and 10-pMg M. These dilution processes may give rise to a considerable increase in entropy and may contribute significantly to making the over-all reaction spontaneous. Thus the Gihbs free energy of hydrolysis of ATP is more due to dilution of the H + produced than to a change in enthalpy.
The weighted average standard entropy change ASo.,. is plotted in Figure 3c. The values of ASomi, and its three component parts are plotted in Figure 4a. The Mg2+dilution term is plotted in Figure 4b, and the H + dilution term is simply equal to 2.3 (1.987) (9) = 41.2 cal deg-' mole-' since the pH is held constant a t 9. The entropy of dilution of Mg2+ is positive and hence makes a contribution to causing ATP to hydrolyze provided pMg > 2.3. At higher concentrations of Mg2+, n ~is. negative and A S W ~ Mopposes . the hydrolysis of A TP
. - - A .
It is interesting to note that eqn. (12) for AHoOb, does not have mixing and dilution terms because, as already noted, enthalpy changes are zero for these processes in ideal solutions. Inlerprelation of AGo.b.
Now that AS"',b.has been expressed in terms of average, mixing, and dilution terms, we need to go back to eqn. (6) and see whether it may be arranged in a form which shows these same types of terms. The Gibbs free energy of mixing of ATP4- and MgATP2- is
m9 Figure 3. a, Weighted overage stondord Gibbs free energy chong* AGO., in k c d mole-1 calruloted with eqn. 137). b. Weighted average rtondard entholpy change AH'.,. in kcal moleCLcalculated with eqn. (1 21. This is identical with AHoOb, because in ideal solutions there ore no enthdpio%of mixing and dilution. c, -TAS0., in kcal mole-',where ASosr. is the weighted overage stmdord entropy chmnge calculated using eqn. (231.
+
TASomiXp- T A S o r n i d ~ p Figur- 4. m, Plot of TASo,ir = T A S ' , : ~ D P in ksol mole-L. The three component curver calculated with eqnr. (151(17) are given as doshed liner. Since for ideal ~olvtionr there is no enthdpy of mixing, this graph may olro be interpreted or -AGo,i, - A G a m i x ~ ~ ~AGDmirp A G o r n i r ~ ~ pb,. Plot of TAS0dilohf. = -AGodilnwc in kcal mole-' s d c u l ~ t e dwith eqn. (271. TAS'di1.x = -AGodilna = 12.3 kcal mole-' at pH 9 (from eqn. 12611.
-
-
+
Volume 46, Number 1 1 , November 1969
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This is the Gibhs free energy change for the mixing of 1/[1 [Mg2+]/K~]moles of ATP4- with [[Mg2+]/ K1]/[l [Mg2+]/Kl] moles of MgATP2- to form an ideal mixture. Since the logarithm of a number less than unity is negative, AGomidm is always negative, and the mixing of ideal solutions is always spontaneous. Since ideal mixing does not involve a change in enthalpy, AGOmi, = -TASo,i. as may be illustrated by comparing eqns. (15) and (33). Similarly for ADP and Pi
+
+
AGomidm= RT [l
1
1
+ [Mg'+]/Kz In 1 + [Mg2+]/Kt [Mg'+I/Kn
[Mg2+1/Ks
=
RT [l
]
+ [ M ~ ' + ] / K +z ~[Mg2+l/Kn ~~ 1 1 + [Mga+]/KsIn 1 + [MgPt]/Ki
+1 AGomirp
LMgs+l/Ka In
+
hydrolysis reaction. The acid dissociations of ATP, ADP, and Pi involved in the range of pH of interest also need to he taken into account (5-5). Also magnesium complexes MgHATP1- and MgHADPo need to be introduced for a complete treatment of the effect of changing pMg. Under particular experimental conditions other cations which are bound may he present (for example, Na+ and I
