J. Phys. Chem. 1995, 99, 1052-1059
1052
van’t Hoff Revisited: Enthalpy of Association of Protein Subunits Gregorio Weber School of Chemical Sciences, University of Illinois, 600 S. Mathews Ave., Urbana, Illinois 61801 Received: July 8, 1994; In Final Form: October 11, 1994@
The amounts of heat absorbed from and released to the environment in a chemical reaction are not experimentally separable. In consequence, entropy and enthalpy of reaction can only be computed with the help of specific hypotheses that relate them. We adopt here a previously described model in which enthalpy and entropy changes on reaction are determined by the thermally dependent probability of bond breakage, and we derive a simple function of that probability that gives the number of Boltzmann complexions associated to a set of bonds of arbitrary strength. The general relation proposed by Born for the dependence of bond energy upon distance is used to demonstrate that increase in pressure from atmospheric to 2.5 kbar produces much larger dissociating effects than increase in temperature from 0 to 40 “C. In protein oligomers the energies of protein-protein bonds are of the same order as the thermal energy, and both enthalpy and entropy of reaction must undergo significant changes with temperature. The “van’t Hoff enthalpies”, which are calculated on the assumption that enthalpy and entropy are constants independent of temperature, correspond to not one but two very different values of the enthalpies predicted by the model, but the effects of pressure upon the reaction can be used to choose between them. Application of these procedures to seven oligomeric proteins previously studied shows that the enthalpies so determined are remarkably alike with a mean value of +25 f 5 kcal per 1000 A2 of intersubunit surface, as compared with +7 f 6 kcal derived for the corresponding van’t Hoff enthalpies.
In a previous publication’ I have dealt with the thermodynamics of the association of protein subunits, examining the consequences of the theory there developed in relation to the effects of pressure upon the equilibria of aggregates and subunits, at constant temperature. In this publication I consider that theory in relation to the effects of temperature on the free energy of association of oligomers. In this respect, there is an all-important distinction to be made between those reactions that are wholly or largely determined by the exchange of covalent bonds and the macromolecular reactions of flexible polymers and proteins. In the former reactions the relevant bond energies are much larger than the thermal energy, but in the latter case the energies of the bonds involved are of the same order of magnitude of the thermal energy. In the first case enthalpy and entropy of reaction can be considered constant, at least over the reduced range of temperature necessary to characterize the thermal behavior, but in the second case enthalpy and entropy are indeed temperature-dependent quantities. It is then indispensable to examine in detail the assumptions that are made in the attempts to separate the enthalpy and entropy contributions to the free energy of reactions and to assess their variation with the temperature. Gibbs Free Energy and Balance of Heat and Work
The change in the Gibbs free energy function dG is defined as the difference between the change in total heat content d(TS) of the system of reagents and surroundings and the external work performed by the system at constant pressure, p dV.
In any process whatever the change in heat content comprises that intrinsic to the system d(TS)i and that of the passive environment (T ds),,the latter owing to effects starting originally @
Abstract published in Advance ACS Abstracts, January 1, 1995.
0022-365419512099-1052$09.00/0
in the system of reagents. Thus, following Planck2 we can write (1) as dG = -d(TS), - d(TSi) + p dV
(2)
Equation 2 expresses a simple conservation relation of heat and work that is valid in all circumstances, not only at equilibrium. In the application of eq 1 to chemical reactions, Gibbs3assumed that d(TS), resulted from a change of opposite sign in the “internal energy” of the system, dE,and as dH,the change in enthalpy at constant pressure, equals dE p dV the last equation becomes
+
dG = dH - d(TS), As d(TS)i = T dSi
(3)
+ Si dT, and at constant temperature dT = 0, d G = d H - TdS,
(4)
If eq 4 refers to the conversion of 1 mol of reactants into 1 mol of products, under conditions in which the composition of the system is maintained constant, at the stable values characteristic of the chemical equilibrium, eq 4 takes the familiar form AG=AH-
TAS
(5)
where the A’s are standard molar changes in free energy, enthalpy, and entropy respectively. I have gone in detail into the conceptual derivation of the general relation (4) and that applicable to equilibrium (5) to indicate that AG, apart from external work pAV, equals the diference between two quantities ofheat, those respectively released and absorbed by the reagents. The difficulty in the separation of these two quantities is evident from the start: Calorimetry can only measure the difference between them and reproduces dG with its proper sign, except for the contribution of p dV, while AG, which is derived from the proportions of the components at equilibrium, determines the magnitude of the difference between the heats absorbed and 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 3, I995 1053
Enthalpy of Association of Protein Subunits released by the system under the specific equilibrium conditions but does not decide what predominates in the reaction, the release of heat by the system into the surroundings (enthalpy driven reactions) or the absorption of heat by the system from the surroundings (entropy driven reactions). There is evidently no way of experimentally separating the changes in enthalpy and entropy in any chemical reaction carried out isothermally without some hypothesis as to the relations that must exist between them. This subject is more extensively examined below in relation to the van’t Hoff plot, which has been widely believed to accomplish the separation of the entropy and enthalpy contributions to the free energy change.
Relation of the Energies of the Exchanged Bonds to the Enthalpy and Entropy Changes In any isolated chemical system there are only two sources of energy: One is the exchangeable thermal energy or caloric, the product of the temperature T and a capacitive factor S, the entropy of the system; the other stems from the energies in the bonds that link the various particles of matter in the system. Of these, only the bond energies that are exchanged in a particular chemical reaction interest us in relation to (5). The relations between energy and entropy are determined by the interconversion of the caloric and the energy of the bonds of each chemical species involved in a reaction. This interconversion can be understood as the simplest expression of a “golden rule”: The replacement of stronger bonds by weaker ones, by increasing the probability of the bonds being absent, increases the total heat capacity of the reagents by the amount T dSi which is absorbed by the reagents from the environment. The replacement of weaker bonds by stronger ones has the opposite effect, diminishing the heat content of the system of reagents by the amount T d& that is released into the environment. These exchanges of heat owing to changes in the intrinsic entropy of the system add to those resulting from the exchange of the bonds themselves and determine together the direction of spontaneous change in the system according to (3). One would expect that this relation of entropy and bond strength ought to have been previously proposed by others, and more than once, but the only clear reference to it that I have seen is in a paper by W i d ~ m . ~ This rule is responsible for the often-noted compensation of enthalpy and entropy changes. I interpret the golden rule as implying that the entropy associated to the existence of bonds derives exclusively ffom the multiple complexions generated by their temporary presence or absence. The fundamental assumption that I make is that the entropy is determined by the average probability p of a bond being absent at a given temperature. Once this probability is given, for each of the reagents involved in the chemistry, we can unambiguously calculate the changes in entropy and energy that take place upon reaction. We can qualitatively appreciate that if the bonds are of sufficient strength with respect to the thermal energy, p becomes close enough to zero to make the entropy change negligible, and the direction of the chemical reaction is then determined by the heat released when weaker bonds are replaced by stronger ones (Berthelot’s rule). The simplest, and obvious, probability rule of bond breakage is that if the average energy of the bond in thermal energy units is EIRT, its average probability of transient disappearance is p = exp(-E/RT) or EIRT = -In@). It follows that the total energy ET of the M identical bonds of a reactant or product that are exchanged in the course of the reaction is ET=ME(l - p )
(6)
where E is the energy of the individual bond. As derived in a
previous paper,’ application of the Boltzmann formula to the case of a homogeneous population of M bonds gives for the exchangeable heat or caloric associated to the presence of the bonds
However, a simple and exact evaluation of that part of the Boltzmann entropy that relates exclusively to the bond exchanges in the chemical reaction is obtained as follows: If the probability of absence of a particular bond is one or zero, then its contribution to the number of complexions is a multiplier of 1, while a maximum multiplier of 2 applies if p = l / 2 . Thus, the number of complexions associated to a set of bonds of arbitrary strength is
with 0
IFi I 1.
Therefore
In W = ln(2)(F1
+ F2 + ... -tFM)= M ln(2)(F)
(9)
an equation that shows the entropy to depend only on the average value and not in any particular disposition of the individual probabilities of bond absence. We only require to express < F > in terms of the average probability p to obtain the entropy change on reaction as an explicit function of the strengths of the exchanged bonds. Evidently, (F)= 1 if p = l / 2 , (F)= 0 if p = 0 or 1, and (F)must be the same for probabilities p and 1 - p . These conditions are met if (F)is the normalized geometric mean of the conjugate probabilities p and 1 - p , so that (F)= [4p(l - P ) ] ’ / ~giving , finally TS, = MRT(1n 2)[4p(l - p)l1l2
(10)
Figure 1A exhibits the entropy associated to a set of 100 identical bonds according to eqs 7 and 10. It indicates that for M sufficiently large both equations give a similar dependence of entropy on bond energy. In both parts A and B of Figures 1 the entropy has been normalized by division by M so that they have In 2 as the ordinate maximum. The merits of eq 10, which make it preferable to (7), are several: (1) It shows that for all its complexity the Boltzmann entropy is proportional to the number of the bonds exchanged in the reaction, just like the total energy shown in (6), and that the ratio of entropy to enthalpy is independent of it5 and equals ln(2)(RT/E)[4p/(l - P ) ] ” ~ . (2) The manner of derivation of eq 9 implies proportionality to M , so that it is applicable equally well when M is only a few or even 1, and p may be taken then to refer to the average of the same bond in the direrent molecules of a very large population. By contrast, the probabilities of the complexions in eq 7 refer to a population of M members with equal p and cannot be applied without further qualification to a population of bonds of different strength. ( 3 ) For p = l / 2 there should be 2M complexions for any M , because then each member of the population generates two equally probable outcomes that multiply the already existing number of complexions. However, the number of complexions given by (7) for p = l / 2 is evidently dependent on M As shown in Figure lB, for M = 5 the number of complexions that it predicts at p = l / 2 falls short by about 40%, and at M = 50 there is still a 7% deficit. In the entropy calculations presented here I have uniformly used eq 10.
Weber
1054 J. Phys. Chem., Vol. 99, No. 3, 1995
temperature and pressure many pairs of P-P and P-W bond energies can combine to give a single experimentally observed free energy of reaction as shown in Figure 5 of ref 1. Selection from these necessitates further information. Calorimetry can inform us as to the sign of the heat exchanged on reaction and therefore indicate whether the reaction is entropy or enthalpy driven, but cannot go any further. The customary way of separating the enthalpy and entropy contributions to the free energy change on reaction applies the van’t Hoff plot to the data of the standard free energy change at a series of temperatures. In view of the inherent impossibility of separating the sources of the heat absorbed and evolved on reaction without additional hypotheses, it becomes indispensable to examine in detail the general validity of van’t Hoff plots for this purpose.
SET OF 100 BONDS
-
NORM. DIST. In (2)f I 4 p (1-p) 1 .
’
1
+
g0.17 3 I-
a:
z
H
t
a
g 0.52 I-
Limited Reliability of the van’t Hoff Plots
z W
If eq 5 is divided by T, one obtains
0.35
AGIT = AHIT - AS
(14)
0.17
and if; and only iJ; then
AH and AS are independent of temperature,
0.00
2.50
7.50
5.00
d(AGlT)ld(llT) = AH
BOND ENERGY IN RT UNITS Figure 1. (A, top) Entropy (TS) in RT units of a set of 100 identical bonds plotted against the bond energy in RT units according to eqs 7 and 9. It shows the general agreement of these equations for large M values. (B, bottom) A plot similar to (A) of eq 7 for increasing M values shows that the number of complexions given by the normal distribution reaches 2M at p = l/z, only for infinitely large M ,and a deficit from this value is apparent even at M = 200.
At atmospheric pressure the product pAV may be neglected E and the Gibbs free energy associated to an so that H ensemble of M bonds is G = ME(1 - p ) - MRT ln(2)[4p(l - p)]”’
(11)
The Gibbs free energy change AG that appears in eq 5 , as well as the constitutive enthalpy and entropy changes, can be calculated from the average bond energies attributed to the reactants and products by application of (11). For fitting purposes it will be often convenient to express G as a function of p alone, obtaining GIRT=M{[-ln(p)](l
- p ) -ln(2)[4p(l -p)]“’}
(11’)
While one can calculate the free energy difference AG at fixed temperature and pressure as that between reactants and products by (11) or (1 l’), after assigning values to the p ’ s or corresponding E’s, the problem that we face is the inverse one: the assignment to products and reactants of values of p appropriate to generate the experimental free energy observed at various temperatures or pressures. In the case of association of two identical subunits to form a protein dimer, we have the stoichiometric relation’ 2PW
-
P-P
+ w-w
(12)
and the free energy change is then given by AG = M[G(P-P)
+ G(W-W)
- 2G(P-W)]
(13)
in which we assume for simplicity that M is the same for all three reacting species. As the energy of the water-water bonds is specified, fitting consists in varying the bond energies P-P and P-W so as to satisfy the experimental free energy at constant temperature. Computation shows that at any fixed
(15)
the expression used in the determination of the standard enthalpy reaction by means of the van’t Hoff plot. In the more general case AH and AS are to be considered as temperature dependent and then d(AG/T)/d( 11T) = AH
+ (117) dAHld( 11T) dASld(l1T) = AH
+ T(T) (16)
where
T(T)
(117‘) d(Aff)/d( UT) - dAS/d( 11T)
(16’)
Evidently, (15) cannot apply except in the limiting case in which
r(q 0, and one can deny outright the general validity of van’t Hoff plots according to (15) to analyze the macromolecular reactions of proteins, that is, the folding of peptide chains and their subsequent association into oligomers. These cases stand in sharp contrast to the classical use of van’t Hoff plots in reactions involving the exchange of covalent bonds, of much larger energies, and therefore negligible changes of p over the entire temperature range used in the determination of AH, as in the examples given by Lewis and Randall6 in their textbook of 1923. It is not surprising that in the late 1920s the variation of AS and AH with temperature could be ignored because it was only after that time that the origin of bonds that are unstable
+
Enthalpy of Association of Protein Subunits
J. Phys. Chem., Vol. 99, No. 3, 1995 LO55
to room temperatures was made clear by the work of Debye on polar molecules7 and of London on apolar molecules.8 Insufficient examination of the problem has meant that (15) has continued to be employed without comment up to the present. It follows from (16) that the largest errors in the enthalpy obtained by means of van't Hoff plots ought to occur when AH approaches zero or, in general, when it is small in comparison with the separate H values of reactants and products. This situation obtains in the entropy driven association of protein subunits: The bond energies of reactants and products baiance each other to the largest extent and thus permit the comparatively small entropy change to impose the direction of chemical change. A quantitative decision as to the validity of the approximation (17) necessitates consideration of the effects of temperature on E, or p , besides the general relations of these with G shown in eqs 11 and 11'.
DEGREES CELSIUS
L t
0.343
a
m o 0.50E a
Variation of Bond Energies with Temperature and Pressure Temperature and pressure modify the energy E of the elementary bonds by altering the distance r between the interacting atoms or groups of atoms. Born proposed a formal description of the dependence of E upon r of the formgJO
The energy minimum Eo obtains at distance r, and is defined by the condition dEIdr = 0. Applying this condition to (18) permits to replace A and B by functions of E,, and r,,
- (SIT)]; B = E,,r:(S/T)/[l
..e--
--.--*
.____.--------
---
1.000
3.000
PRESSURE I N KILOBAR
E = Ar-S - Br-T
A = E,,r;/[l
__---
0.25-
- (SIT)]
(19)
Figure 2. Probabilities of bond absence as a function of temperature in the range 0-40 "C (upper panel) and pressure in the range of atmospheric to 2.5 kbar (lower panel), for bonds of E = 0.5, 1, 2, and 4 kcal mol-' at TO and po. The smallest energies correspond to the uppermost curves. For E = 4 kcal the probability of bond absence with temperature remains near zero even at 40 "C, and the change with pressure is barely noticeable at 2.5 kbar. TABLE 1: Parameters Adopted for the Lowest and Highest Energy Bonds Present in the Noncovalent Interactions of Proteins
so that
E = E,[(ro/r)s - (S/T)(r,,/r)T]/[l - (SIT)]
(20) ~~
Associating the volumes v = 9 and vo = rO3to the corresponding interaction distances gives
E = E,,[(V,,/~)~ - (s/t)(r,/r)']/[ 1 - (s/t)]
(21)
where evidently s = SI3 and t = T/3. The microscopic phenomenological relation of energy and volume of the Born relation (18) and its equivalent (21) are particularly well suited for use as an adjunct to the macroscopic equations of thermodynamics that relate the same variables. To describe the dependence of the bond volumes upon pressure and temperature, we write the preceding equation in the form
In this equation T and p are the actual temperature and pressure, To and po are the temperature and pressure at which E = E,,, and V,, is the volume V under these latter conditions. At any other temperature and pressure the volume is determined by the expansivity, a,and compressibility, x,at the bonds involved in the reaction. The four parameters a,x, s, and t determine the changes in volume with temperature or pressure. These parameters cannot be considered independent but primarily dependent upon E(T,,,po)itself. Therefore, if the values of a, x,s, and t are known for what we consider the extreme possible values of E(T,,po), and if E(To,po)is also specified for the reagents in the system, we can derive by linear interpolation the parameters a,x, s, and t for these reagents. In order to introduce such scaling of parameters, I shall assume that the low-energy end of the scale corresponds to weak dispersion
permanent dipole interactions dispersion forces
~~~
4 14
~~
4.3 9.5
8 1 10 2
forces of 1 RT unit at 0 "C and the high-energy end of the scale to interaction of permanent dipoles of 12 RT units at 0 "C. The expansivity and compressibility for the lowest bond strength will be taken to be those for hexane, and for the highest bond strength they will be assumed to be those of water. The t exponent in the Born equation is evidently 2 for the low-energy dispersion forces and 1 for the higher-energy interaction of permanent dipoles. From the expansivities and compressibilities for the two liquids," the s exponent can be shown to be 10 for hexane and 8 for water,' and these not very dissimilar values will be adopted for the low and high-energy limits of the scale respectively. The limiting parameters adopted are shown in Table 1. To characterize the effects of temperature or pressure on the final thermodynamic equilibrium, it is necessary to start by a computation of the change in the probability p of bond absence as a function of the energy of the bond with temperature or pressure in the separate reagents, as described by (22). By application of eq 11 to a set of A4 bonds of equal energy, we can then determine the values of AG, AH, and TAS for each reagent. Figure 2A plots the probability of bond absence over a pressure interval of 4 kbar, and Figure 2B plots it over a temperature interval of 0-40 "C for bond energies of 0.5-4 kcal/mol at 0 "C. A remarkable difference between the effects of pressure and temperature is immediately apparent: The changes with pressure are much steeper, and their magnitudes are much larger than those with temperature. Evidently, the main cause of these differences is to be found in the large slt
Weber
1056 J. Phys. Chem., Vol. 99, No. 3, I995
x,
t
0.0312
0,01'"4t /
ISOTHERMAL EXPANSION -
0.0000
=
0.00
large number of pairs of P-P and P-W energies that are compatible with the stated parameters. Application of eq 22 with the values of a, s, and t , assigned by scaling the corresponding bond energies, determines then the change in free energy with temperature for each of the P-P and P-W pairs thus selected. The knowledge of the bond energies P-P, P-W, and W-W over the range of temperatures makes it possible to compute the average enthalpy and entropy changes at each temperature and thus make a direct comparison of the enthalpy values predicted by eqs 15 and 16. Although neither AH nor AS is a constant with temperature, they do not vary much more than the free energy, and in order to compare the enthalpies derived by eqs 15 and 16, I have taken as representative of the latter their average value in the temperature range 0-20 "C. Figure 4 shows a plot of the "van't Hoff enthalpy" (eq 15) against the average enthalpy according to (16) for the values of AH compatible with a free energy of -12.9 kcal mol-' at 0 "C and M = 166, which are respectively the free energy of association of the monomers at 0 "C and the assumed number of intersubunit bonds of Rubisco.'* The figure shows that AH according to (16) takes two very different values for most values of AH according to (14). This situation can be understood by reference to (16'): As AH increases from negative to zero, TAS increases by a compensating amount and comes to determine the value of r and, therefore, the van't Hoff enthalpy. As the enthalpy increases now from zero to larger positive values, the enthalpy contribution to (16) becomes progressively more important, and the errors in the van't Hoff enthalpy decrease below the maximum. These changes are responsible for the parabolic character of the relation between the van't Hoff enthalpy and the true enthalpy of reaction, and this relation can be shown by computation to remain similar in shape down to M values of 10 and free energies of -1 kcal mol-' at 0 "C.
I
'
1.75
a I-
-
T
3.50
CL
= I
5.25
7.00
BONO ENERGY AT 0 CELSIUS, I N KCAL
Figure 3. Relative change in probability of bond absence with temperature depends linearly on the bond energy itself. The expansion at the bonds (lower curve) accounts for less than 5% of the total temperature effect (upper curve).
ratio of 5-8. The small linear changes in probability of bond absence with temperature for all bond energies are quite sufficient to explain the absence of reports in the literature of oligomer dissociation owing to increase in temperature. Figure 3 plots the average relative change in the probability of bond absence with temperature, (l/p)(dp/dT) computed over the temperature interval 0-40 "C, against the bond energy E,, for bond energies of 1-14 RT (or 0-7 kcal mol-') at 0 "C. From p = exp(-E/RT) it follows that (l/p)(dp/dr) = (E/R?)(dE/dT)
(23)
The slope in the plot is (l/RP)(dE/d7l and the two factors identify respectively the contributions from thermal agitation and bond expansion to the change in the probability of bond absence. When averaged over 0-40 "C, l / R F = 0.005 89 while the regression line of the points in Figure 3 has slope 0.006 21. Thus, for the parameters employed, the thermal disruption accounts for 94.8% and the expansion at the bonds for only 5.2% of the total effect. From these figures we can appreciate that only small changes in these proportions are to be expected by alteration of the parameters used in eq 22, and computation shows that making all expansivities equal to the maximum or minimum value (14 x and 4 x K-', respectively) changes the contribution of the bond expansion to the slope of the relative changes in p with temperature against bond energies to respectively 13.3% and 3.5%. Changes in s or t coefficients result in even smaller changes in that contribution. I conclude that the expansivity of the bonds provides only a minor correction to what is by a large margin the determinant of the change in probability of bond absence with temperature, the relationp = exp(-E/RT). This demonstration of the virtual independence of the changes in bond energy with temperature from anything but the characteristic energy EOitself gives us a degree of confidence in the resulting final computation of the changes in free energy with temperature, when both AH and AS are temperature dependent.
Computation of the Enthalpy Change When Neither AH nor AS Is Temperature Independent If we do not introduce other specificationsbesides the number M of exchanged bonds, the energy of the W-W bonds at temperature To,and the free energy of the association reaction, we obtain, by a systematic variation of the energies of P-P and P-W bonds and application of the relations 11 and 12, a
Enthalpies of Association of Pairs of Protein Subunits The enthalpies of association of seven oligomers: four dimers, one trimer, and two tetramers have been previously derived by Weber' employing van't Hoff plots for this purpose, and these results require reexamination in light of the concepts that we have just discussed. We recall again that the fitting of the experimental data to obtain the enthalpy values by means of eq 16 involves the successive steps: (1) Determination of the set of values of P-P and P-W bond energies that, together with W-W = 7 kcal mol-', can yield a free energy that differ from the experimental free energy at To by f 0 . 2 kcal mol-' or less. A set of corresponding enthalpies and entropies is thereby established. (2) Determination of the change of free energy with temperature expected from these bond energies and the values of a, s, and t associated to them (eq 22 and Table 1). (3) Selection of the cases that accord to the experimental van't Hoff plot (eq 15) and computation of the enthalpy values corrected according to (16'). As shown in Table 2, two widely different values of the enthalpy, one positive and the other negative, are possible for the experimental d(AG/T)/d(l/T) in each of the seven protein oligomers that we studied. Figure 4 places these values in a plot for a dimer with the free energy of association of Rubisco and Figure 5 for a pair of subunit interactions in the tetramer of glyceraldehyde phosphate dehydrogenase.
x,
Distinguishing between Entropy Driven and Enthalpy Driven Reactions by the Effects of Pressure As each value of the van't Hoff enthalpy corresponds to two very different values of the computed enthalpy, an unequivocal
Enthalpy of Association of Protein Subunits
J. Phys. Chem., Vol. 99, No. 3, 1995 1057
TABLE 2: Enthalpies of Association of Subunit Pairs in Oligomew enthalpies protein bonds van'tHoff AH(+) AH(-) yeast hexokinase 141 17 21.0 -12.6 E. coli BZtryptophan synthase 141 17.7 32.0 -5.4 Rhodobacter rubisco 166 6.0 39.6 -9.33 239 4.0 61.5 -7.0 glycogen phosphorylase A dimer allophycocyanine trimer (sp) 118 14 28.5 -3.3 glyceraldehyde phosphate dehydrogenase (sp) 120 -3.7 29.7 -9.5 239 8.25 67.8 -5.1 glycogen phosphorylase A tetramer (sp)
48,4i0.47
d
l
I Iz
COMPUTED ENTROPY I N KCAL. 9.05 18.57 28.09
l
11
65,8j.87
COMPUTED ENTROPY I N KCAL. 13.87 26.87 39.86
52.86 I
c
c r
l l
11
\I
RUBISCO DIMER AG=-12.9 AH(ll=39.55
/I
-27.28 -12.03
I
I
I
0.97 13.97 26.96 COMPUTED ENTHALPY I N KCAL.
I II 39.96
Figure 4. A plot of the "van't Hoff slope", Le., d(AG/T)/d(l/T),against the average enthalpy, or entropy, computed by the model described in text, in the range 0-20 "C, for a free energy of -12.9 kcal mol-' and M = 166 at 0 "C. choice between them is required. For this purpose we need an independent measure of the stability of the aggregate at the different temperatures that must be carried out under isothermal conditions. This is readily available by the shift in the pressure of middissociation with temperature, and the results obtained in a variety of aggregates are quite definitive in this respect:13 In the interval 0-40 "C an increase in temperature systematically increases the pressure necessary to reach a given degree of dissociation. Therefore, such reactions must all be entropy driven, and from the data of Table 2 they must all have AH > 0. As discussed elsewhere,' the physical reasons for the correlation between increase in volume and entropy-driven character on association and its reciprocal, decrease in volume and enthalpy-driven character on association, are to be found in the physical origin of the principle of Le Chatelier. An increase in volume on association results from the replacement of shorter, and therefore stronger, bonds in the reactants by longer, and therefore weaker, bonds in the products, and the reaction favoring the weaker bonds can only occur if driven by entropy. The dissociative effect of pressure originates in the differential compressibility of the bonds appearing in (1l), and
\I
SUBUNIT PAIR OF GAPOH TETRAMER AG= -8.0 AH(ll=29.33
x-6.66
=AH(+) and AH(-) are the two values of the enthalpy computed by (16) that correspond to the van't Hoff enthalpy, determined by (15). sp designates values for interactions between subunit pairs of the oligomers, of more than two subunits, assuming that all pairs are equivalent. The number of bonds is derived from the surfaces of contact of the subunits, calculated from their size and the assumption that each bond corresponds to the set of interactions over an area equal to the cross section of a water molecule (10 Az).
37.61
/I
-25.02 -8.47
I
I
I
1.05 10.57 20.09 COMPUTED ENTHALPY I N KCAL.
/
29.61
Figure 5. A plot similar to that in Figure 4 for the free energy of association of -8 kcal mol-' at 0 "C and M = 120 of a pair of subunits in glyceraldehyde phosphate dehydrogenase (GAPDH). according to eq 22 and Figure 2A, those of lower energy, the P-P bonds, are destabilized at pressures at which the P-W and W-W bonds are almost unaffected. I note that while the subunit association reaction involves the overcoming of the unfavorable positive enthalpy by the larger increase in entropy, the dissociating effect of pressure is brought about by the increasingly positive AH of the reaction, while the corresponding entropy of reaction is not much affected as shown in Figure 11 of ref 1. Similarly, the reversibility on release of pressure follows the same path in the opposite direction, the enthalpy of the P-P bonds regaining the initial value and thus permitting the excess entropy to drive the reaction toward association. With reference to the protein oligomers for which the results are gathered in Table 2, it appears that the enthalpy values calculated following the procedure outlined above, that is, taking into account that neither AH nor AS is a constant independent of temperature and employing the effects of hydrostatic pressure on the equilibrium to discard the inapplicable value, are far more uniform in character than those previously computed with help of the van't Hoff plot.' That analysis indicated that the enthalpy of association of glyceraldehyde phosphate dehydrogenase, unlike the other proteins studied, was negative. Figure 5 shows that, in the present analysis, this protein as well as the rest of them has a positive enthalpy of association. If the enthalpy values are normalized according to the assumed number of bonds, which we derive from the nominal area of contact of subunit pairs, we can obtain the enthalpy change for a standard area. The data of Table 2 yield an average of 25.2 f 4.7 kcal per 1000 A* of intersubunit surface. In contrast, the van't Hoff enthalpies previously reported had an average of 6.8 k 5.8 kcal mol- l. The correlation between volume change and entropy or enthalpy driven character of the reaction is not limited to proteins: a variety of complexes of small aromatic molecules are known in which stability is decreased by an increase in temperature, and they uniformly associate with a decrease in v01ume.~~J~ The contact surface between the aromatics involved in these complexes is only 2-3% of that of the protein subunits shown in Table 2 and is therefore less suited to a statistical analysis that employs average bond energies. Individual consideration of molecular details would be then required, but eq 9 which directly relates the probability of bond breakage to the entropy associated to each bond ought to permit calculations
Weber
1058 J. Phys. Chem., Vol. 99, No. 3, 1995
composite area is
COMPUTED ENTROPY IN KCAL.
2,1j0.81
-0.01
0.79
1.59
2.39
I
r
and
which defines the energy E of the system of elementary bonds. Similarly, we can use the property described by eqs 9 and 9‘ to define the entropy associated to the plurality of interactions present in the arbitrarily chosen area
E-0 2 U 8
c
MOLECULAR COMPLEX
z
W
AG= -1.0 A H [ l ) = 1.38 AH[2)=-1.71 d (AG/T) /d (l/T1=-2.00
LL U
2-2 c >
-3.43 -1.61
I
I
I
-1.01
-0.21
0.59
+ + + + FB, + ...) (27) + FA2+ ... + FBI + FB2 + ...)
W = WAWB...; In W = In 2(FA1 FAz ... FBI
In W = In 2(F); (F)= (FA1 1.39
(28)
COMPUTED ENTHALPY I N KCAL.
Figure 6. A plot like those of Figures 4 and 5 for a molecular complex held together by 10 bonds, with AG = -1 kcal mol-’. Note that, for the case of a negative volume change on association, the van’t Hoff enthalpy and the enthalpy calculated by the model are in reasonable quantitative agreement.
similar to the one that I have carried out for the proteins, but adapted to the particular cases. Figure 6 shows a plot similar to those of Figures 4 and 5 for a weak molecular complex held by 10 identical bonds with AG = -1 kcal mol-’ at TO. If pressure favors the association, as found for the aromatic complexes, the model predicts an enthalpy of - 1.7 kcal mol-’ for a van’t Hoff enthalpy of -2 kcal mol-’. For cases like these, of which there are many in the literature, the van’t Hoff enthalpy will be close to that predicted by the model. Then, disagreement between the enthalpies derived by the application of eqs 15 and 16 will not be nearly as evident as in the case of protein oligomer association, and aromatic association cannot provide as stringent a test of the general validity of the van? Hoff plot as is the case with entropy driven associations like those of protein subunits.
Bonds as Interactive Areas In the preceding treatment the word “bond” designates an arbitrary area of interaction between monomers, characterized by an energy of interaction E and a probability p of absence of interaction related by the formp = exp(-E/RT) or E = -In@). Because the reactions that we have discussed occur in water solution, it seemed natural to adopt for this area the cross section of a water molecule and to derive the parameter M from the nominal area of subunit contact that is replaced by water upon the oligomer association. It is evident that such arbitrary area of interaction between molecules is not necessarily simple and will often involve more than one atomic contact. If a plurality of atomic contacts exists, it becomes possible for some to be broken independently of others, and it is therefore important to define the relations of the probability of bond absence and the energy E of the interactive area with similar quantities that obtain at the ultimate microscopic level. If the “bond energy” E derives from a number of existing elementary interactions and to these we assign probabilities PA,p ~etc. , of bond absence or energies of interaction EA, EB, etc., then
Therefore, the average probability of bond absence for such
Equations 26 and 28 show respectively that the additive character of the microscopic interaction energies and the multiplicative character of the number of complexions apply to the elementary atomic bonds as well as to the fixed arbitrary areas of contact chosen to facilitate the computation of the total energy and entropy changes in the reactions. When we deal with probabilities of bond absence in the preceding text, it is to be understood that they have the composite character indicated by eqs 26 and 28. If we give values to the energy of the elementary interactions, a comparison can be made of the probabilities of bond absence for the whole bonding area defined by eqs 25 and 27. By assignment of random values between 0 and 0.3 to the elementary probabilities of bond absence, it is found that the probabilities of bond absence attributed to the bonding area by use of the two last mentioned equations differ from each other by less than 10%. Therefore, we can give p the significance of a composite parameter derived from the free energy changes at various temperatures, which allows the isolation of the contributions of the enthalpy and entropy to the experimentally derived free energy, under the hypotheses made on the general relations of enthalpy and entropy.
Extension to Protein Folding The folding of a peptide chain into a compact structure involves the formation of a large number of internal low-energy bonds of strength not very different from those between subunits, and like subunit association it appears to be also an entropy driven reaction.16 Thus, the ideas presented in this and the previous publication on subunit association are of possible application to protein folding, but one must recognize, nonetheless, the considerable differences in complexity between the two cases: The association of free subunits to form oligomers of defined stoichiometry is a well-defined reaction in which reactants and products can be unequivocally distinguished as to particle size by accepted physical methods. In that way the chemical equilibrium can be characterized at many intermediate extents of reaction, and free energy and enthalpy changes can be determined with some accuracy. The only limitation arises from the recognition that the free subunits cannot be given invariant chemical potentials independent of the extent of reaction on account of the “conformational drift” that they undergo when their mutual interactions are replaced by interaction of each of them with s o l ~ e n t However, . ~ ~ ~ ~such ~ ~drift ~ ~ is limited to 10% or less of the free energy of association at the highest degrees of dissociation and is probably negligible for degrees of dissociation below 0.5. Thus, the dissociation
Enthalpy of Association of Protein Subunits equilibrium of simple oligomers do not differ greatly from an ideal chemical reaction between reagents of fixed structural character, and the classical principles of chemical equilibrium can be used to determine the standard free energy change over a reasonable range of temperatures and pressures. Protein folding does not easily lend itself to similar examination: The structural characteristics of an unfolded protein must be greatly dependent upon the method used to achieve that condition: acid, urea, guanidin, temperature, pressure. The first three kinds depend upon addition or removal of ligands and may be expected to be quite unrelated to the unfolding by pressure or temperature. Folding consists of a number of independent, or quasi-independent,parallel processes which we cannot at present enumerate, so that extraction of the relevant free energies is not feasible. The denaturation of peptide chains by pressure and low temperaturelg is potentially capable of furnishing the most tractable data on protein unfolding, and the collection of these data in the near future may provide an impetus for employing in their interpretation the set of ideas that I have applied to subunit association. For the present it would be premature to attempt it.
Acknowledgment. The support of USPH through Grant GM11223 is acknowledged. References and Notes (1) Weber, G. J. Phys. Chem. 1993,97, 7108-7115. (2) Planck, M. Theory of Heat; Macmillan: London, 1932; pp 7483; English translation by L. Brose. (3) Gibbs, J. W. On the Equilibrium of Heterogeneous Substances. In Collected Works of J. Willard Gibbs,1876; p 85.
J. Phys. Chem., Vol. 99, No. 3, 1995 1059 (4) Widom, B. Two Ideas from Gibbs: The Entropy Inequality and the Dividing Surface. In Proceedings of the Gibbs Symposium, Caldi, D. G., Mostow, D. G., Eds.; AMs, AIP: 1989; pp 73-87. (5) In the previous paper of the author (ref 1) it is stated that if the number of bonds is indefinitely increased, one must derive correspondingly increasing entropies associated to the same total energy. Evidently this is incorrect, and one of the more important sources of uncertainty in the calculations there reported is thereby removed. (6) Lewis, G. N.; Randall, M. Thermodynamics; McGraw-Hill: New York, 1923; pp 298-301. (7) Debye, P. Polar Molecules; Dover: New York, 1929. (8) London, F. The General Theory of Molecular Forces. Trans. Faraday SOC.1936,33,8-16. (9) Fowler, R. H. Interatomic Forces. In Statistical Mechanics; Cambridge University Press: Oxford, 1936; Chapter 10. (10) Lennard-Jones,J. E. Cohesion. Proc. Phys. SOC. London 1931,43, 461-482. (1 1) Bridgman, P. W. The Physics of High Pressure; Dover: New York, 1931. (12) Erijman, L.; Lorimer, G. H.; Weber, G. Biochemistry 1993,32, 5 187-5 195. (13) Silva, L. J.; Weber, G. Pressure Stability of Proteins. Annu. Rev. Phys. Chem. 1993,44,89-113. (14) Heremans, K. High Pressure Effects on Proteins and Other Biomolecules. Annu. Rev. Biophys. Bioeng. 1982,1 I , 1-21. (15) Weber, G.; Drickamer, H. G. The Effects of Pressure on Proteins and Other Biomolecules. Q. Rev. Biophys. 1983,16, 89-112. (16) Freire, E.; van Osdol, W. W.; Mayorga, 0. L.; Sanchez-Ruiz, J. M. CalorimetricallyDetermined Dynamics of Complex Unfolding Transitions in Proteins. Annu. Rev. Biophys. Biophys. Chem. 1990,19,159-188. (17) Xu,G.-J.; Weber, G. Proc. Natl. Acad. Sci. U.S.A. 1982,79,52685271. (18) King, L.; Weber, G. Biochemistry 1986,25, 3632-3640. (19) Griko, Y. V.; Rivalov, P. L.; Sturtevant, J. M.; Venyaminov, S. Y. Proc. Natl. Acad. Sci. U.S.A. 1988,85, 3343-3347. JP941729V
