Standard Gibbs free energy, enthalpy, and entropy changes as a function of pH and pMg for several reactions involving adenosine phosphates.
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Page 1
THE
244, No.
JOURNAL
12, Issue
OF BIOLOGICAL
of June
Printed in
CHEMISTRY
Vol. 25, PP. 329fF3302,
U.S.A.
1969
Standard
as a Function
Involving
Gibbs Free
of pH
Energy,
and
Phosphates
Enthalpy,
for Several
and Entropy
Reactions
Changes
pMg
Adenosine
(Received for publication, September 5, 1968)
ROBERT
From the Department of Chemistry, Massachusetts In,stitute of Technology, Cambridge, Massachuseits 0.2139
A. ALBERTY
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:,,,
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:
and
ATP + Hz0 = ADP + Pi
ATP + AMP = 2ADP
ATP + Hz0 = AMP + PPi
ADP + Hz0 = AMP + Pi
ATP + 2HzO = AMP + 2Pi
(1)
(2)
(3)
(4)
(5)
(6)
PPi + Hz0 = 2Pi
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
known at one temperature,
concentrations
reactants, the
constant for a biochemical
ionic strength,
ions that form
constant
reaction is
pH, and known
complexes
may be calculated
free
the
for
of metal
equilibrium
with
another
temperature
tion
under
at this temperature
free metal ion concentrations,
lated at another
the heats of dissociation
When the changes in Gibbs free energy and enthalpy
under a given set of conditions the entropy
calculated.
Even if the required
of making the calculations
approximations and short cuts.
computer eliminates this problem,
six reactions and see what
thermodynamic quantities
the individual ionic reactions
calculations emphasize the need for certain types of experimental
data. The basic pattern for these calculations
in the treatment of the ATP
Phillips, George, and Rutman (2) have made similar calculations.
For use by the experimentalist
equilibrium constants for reactions
here in terms of the total concentrations
products, excluding water.
equilibrium constants are defined by
pH and
and ionic strength
constants and complex
these conditions.
different metal ion concentrations
provided that the acid dissocia
dissociation constants
If the heat of reaction
and ionic strength and for known
the heat of reaction
pH and metal ion concentration,
of the acids and complexes are known.
at this
are known
is also known
pH and
may be calcu
provided that
are known
change may also be
constants are known
encourages
Fortunately,
and so it is possible to take
can be done to relate the observed
to the thermodynamic
which occur
the labor
the introduction
the modern digital
of
quantities
together.
for
These
was established
reaction phosphohydrolase (1).
it is most convenient
such as we are interested
of the reactants
The six socalled
to define
in
and
“observed”
K
(ADPI (Pi)
(ATP) lobs =
(7)
K
(ADP)*
Poba = (ATP) (AMP)
(8)
K
(AMP) @‘Pi)
sobs =
(ATP)
e0
K
(AMP)
(ADP)
(Pi)
‘Oba =
(10)
K
(AMP) (Pi) *
‘Ohs = (ATP)
(11)
K
(Pi)*

 (PPJ “”
(12)
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Issue of June 25, 1969
R. A. Alberty
3291
where the parentheses represent total concentrations
various reactants and products in moles per liter.
tration of Hz0 is omitted because the other reactants
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
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 noncomplexforming
is perhaps most closely approximated
monium chloride, and so these calculations are for that electro
lyte with the reactants and products making negligible contri
butions to the ionic strength.
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,
AHzbs, and the standard entropy change, AS&.z
of the
The concen
and
and products. Such
cation. This
am by tetranpropyl
Thus complexes of the type
the standard enthalpy change,
AGO,,. = AH%,.  TASk (13)
A&,,

K obs.
is calculated from the ‘Lobserved” equilibrium constant
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
(iz&J, = (%).,,
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
09)
(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.
KZATP II
+ (Mg”+) + (H+)
1
 
KMBHADP &ADP
jp = (HPO:1
=
(Pi)
+ (Mg”+) + (H+) 1

KM.GHATP

(21)
(22)
(2.3)
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Page 3
3292
Thermodynamics of Reactions of Adenosine Phosphates
Vol. 244, No. 12
TABLE
I
Thermodynamic parameters for dissociationa
reactions used in calculations
at .W and 0.2 ionic strength

Acids
HPzO,a
HATPB
H2P207*
HzATP*
HADP2
H*POr’
HAMPI
HzADP’
HzAMPO
Complexes
MgP207”
MgATP*
MgHPzO,l
MgHATPr
MgADP’
Mg,PaO,O
MgHPOdO
MgAMPO
MgHADPO
=
=
H+
H+
+
+ ATP+ ....................
+ HPzOv+.
+ HATP3.
+ ADP”.
+ HPOP . .
+ AMP*.
+ HADP*.
+ HAMP’.
PzO,J. ...................
= H+
= H+
= H+
= H+
= H+
= H+
=
.................
.................
...................
r .
..................
.................
.................. H+
=
= Mg2+
= Mg2+
= Mg2+
= Mg2+
= Mg2+
= Mg2+
= Mg2+
= Mgz+
Mg2+ +
+
PzO#.
ATPa.
+ HPzOr+.
+ HATPZ.
ADP3.
+ MgPsO," ...............
+ HPOf.
+ AMP*.
+ 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 mle1
8.95
6.95
6.12
4.06
6.88
6.78
6.45
3.93
3.74
(6)b
(8)d
(6)b
(lo)/
(8)a
(8)d
(8)d
(10)’
(lO)f
12.21
9.48
8.35
5.55
9.39
9.25
8.80
5.36
5.10
0.40 (7)c
(9)”
(7)6
1.68
0.11
0
1.37
0.80
0.85
1.0
1.0
W)f
(9)”
(11)~
(9)’
(1O)f
(10)f
5.41
4.00
3.06
1.49
3.01
2.34
1.88
1.69
1.45
(12)h
(14)i
(12)h
(8)d
(8)d
(12)h
(S)d
(8)d
(8)d
7.39
5.46
4.17
2.04
4.11
3.19
2.56
2.31
1.98
3.5
3.3
3.5
1.9
3.6
2.26
2.9
2.9
2.0
(13)’
(15>k
(13)’
(15)k
(15)k
(16)”
(16)”
(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
b 0.1 M (CH&NCl.
c 0.22 (CH&NCl.
d 0.2 M (npropyl),NCl.
6 Zero ionic
M NaCI.
g 0.01 ionic
b 1 M (CH,),NCl.
i 0.05 ionic
j 0.1
k 0.1 ionic
=  log K, where
K is a dissociation constant calculated with concentrations in moles per liter.
strength.
f 0.15
strength.
strength.
(20’).
strength.
M KC1
The
phosphate,
beyond the range of interest here. The various dissociation
constants are defined in Table
I. Since
arbitrarily
chosen form is the most basic except for ortho
where the last proton dissociation has a pK far
K
&fm
(H+)fmfp loba =
(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
heat evolved at constant temperature and pressure is given by
(Mg”c)
KM~ADP
equations for nn and nM, have been given earlier (1). The
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
@I+) @f@+)
KIADpKMgHADp (AHkDp+AH&ADP)
AH:abs = aI&0  fADP
(H+)
&ADP AH&r f
&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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Issue of June 25, 1969
R. A. Alberty 3293
TABLE II
Standard
Gibbs free energy, enthalpy, and
for calculations
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.
entropy
ionic
changes used
strength at 66” and 0.2
1. ATP” + Hz0
+
AMP+
Hz0
+
Hz0
+
2HzO
+
Hz0
= ADP3
H+. . .
+
HPO,*
2. ATP+
3. ATP+
+
+
= 2ADP3
= AMP+
2H+.
= AMP+
Hf............
= AMP”
2H+.
=
+
. .
+
PgO+ .
4. ADPa +
HPOb*
5. ATP+ + +
2HPOd2
6. PzO,4 + 2HPO?. .
AGO
km1 mole’
0.276
0.602
11.800
0.878
1.154
10.646

_

AHO
4.70
1.46
16.7
2.88
4.54 54.9
3.24 13.82
7.94
3.40
30.4
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
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 ReactionsThis
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 degl molel for two species to 3.56 cal deg’ molel
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*
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).
In (H+) is the entropy
Formed by Reaction
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+)
 K IATP
In ~ATP + ~ATP 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
f&H+)
(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
the entropy of dilution of nR moles of H+ from 1 M to the actual
concentration under the experimental conditions. If nH is
ofHATP3,fATP (Mgz+)/KMgATPOfMgATP2,
of MgHATP1, (Mg2f)/K~~~~f&g~~~P and fATP
In (H+) 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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Page 5
3294 Thermodynamics of Reactions of Adenosine Phosphates
Vol. 244, No. 12
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
7 4 5 6 8 9
IO
i
i
PH
i b lb
3. ATP + Hz0 = AMP + PPi
nH
“m
FIG. 1. Contour
tion of pH and pMg.
the shaded
diagrams for thermodynamic
Left, log &be; middle, number, nH, of moles of H+ produced;
regions the value is within 0.01 of the value zero.
functions at 25” and 0.2 ionic strength tetranpropyl
tight, number, nxs, of moles of Mgz+ produced.
ammonium chloride 83 a func
In
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