J. Phys. Chem. 1993,97, 7108-71 15
7108
Thermodynamics of the Association and the Pressure Dissociation of Oligomeric Proteins Gregorio Weber School of Chemical Sciences, University of Illinois, 1209 West California Street, Urbana, Illinois 61801 Received: December 2, 1992; In Final Form: March 4, 1993
A relation giving the distribution of the energies and the entropy of a set of identical bonds as a function of a suitably defined bond strength E is derived. The resulting normal distributions have energies that increase monotonically with E and entropies with a maximum a t E/RT = In 2 and almost negligible values a t E/RT > 5-7. Association of two protein subunits involves conversion of two sets of protein-water bonds (P-W) into separate water-water (W-W) and protein-protein (P-P) sets, and the constant W-W bond strength permits the determination of the P-P and P-W bond strengths from the experimental enthalpies and entropies of subunit association. Data on the dissociation of four dimers, one trimer, and two tetramers by hydrostatic pressure give Ep-p = 1.8 f 0.8 kcal and EP-w= 4.4 f 0.5 kcal. In contrast to prevalent views, the large entropy increase that drives these associations results from the conversion of the stronger P-W bonds into the weaker P-P bonds, while the conversion of P-W into W-W bonds does not make any appreciable contribution to it. The effects of temperature and pressure on the bond energies may be computed from the intermolecular potentials typical of dipolar and apolar interactions and by specification of the proportions of each type in the P-P and P-W bonds. The dissociating effect of pressure and the temperature stabilization of protein aggregates follow from the differential expansivities and compressibilities associated to the different bond strengths.
Entropy ( T S ) is that part of the energy of a body that is manifest in the heat exchanges with others. It is the energy equipartitioned among the possible degrees of freedom of the particles (atoms or molecules). On account of such equipartition two bodies at the same temperature brought in intimate contact cannot undergo changes in energy, or entropy, unless changes among the bonds linking the particles (chemical reaction) take place at the same time.' From such a definition it follows that "bonds" are, in the larger sense, the interactions of whatever nature among the particles that restricts the number and/or amplitude of their degrees of freedom. The disappearance of a bond on chemical reaction and the motions that become possible as a result will both produce an isothermal absorption of heat. If these two processes are viewed as independent of each other, the absorption of heat owing to the disappearance of the bond energy will be assigned to the enthalpy change, while the absorption of heat by the ensuing motions will be considered part of the entropy change. This distinction seems artificial because these two origins of isothermal absorption of heat are not experimentally separable, and particularly in the case of weak bonds, the energy that is attributed to the bond "in itself" must be modified, on account of energy conservation, by the restriction of those motions that will become active with the disappearance of the bond. Accordingly, I shall consider here that the total heat changeassociated with the bond is to beassigned to the bond energy. In this view the bond energy is a quantity that cannot be predefined but can only be determined by an analysis of the changes in the enthalpy and entropy of the particular chemical reaction in which it is modified. The success, as well as the limitations, of this definition of bond energy has to be measured by the value and consistency of the consequences that we derive by its means, but exact agreement with values of the bond energy defined by other criteria is not relevant to the present analysis. The practical outcome of this concept is that the change in entropy of chemical reaction, calculated from the bond energies thus defined, derives solely from the number of complexions of products, Nprd,and reactants, N,,,, according to the Boltzmann relation TAS = RTln (Nprd/Nreac), while the number and nature of the complexions depend only on the stationary probabilities of existence of the bonds involved. 0022-365419312097-7 108$04.00/0
Because of the interdependence of the bond energy and the motions that it restricts, as noted above, a fixed bond energy is an approximation valid only under specified pressure and temperature. To exemplify the situation, consider the hydrogen bonds in liquid water, to which a fixed strength may well be assigned from the data on the sublimation of ice. Yet the molecular dynamics simulations* show that the hydrogen bonds in the liquid lack the constancy in length and direction of those of ice and that the average energy that they would be assigned would differ somewhat from those in ice. In a similar manner the bond energies derived from the computation that we propose are those necessitated by the particular case and validity in other instances should be cautiously weighted.
Simplification of the Computations of Bond Energies Thermodynamics associates with isothermal, isopiestic chemical reactions two kinds of interdependent energy changes, those of enthalpy (AH) and of entropy (TAS),linked to the reaction equilibrium by the Gibbs free energy (AG):
AG=AH-TAs (1) The quantities appearing in eq 1 refer to molar (standard) equivalents which we write for simplicity as shown there. The reversible association equilibrium of two particles in solution involves separate processes of partial desolvation of the reactants and formation of the molecular adduct, on one hand, and of bonding between the released solvent molecules, on the other. These processes are illustrated in Figure 1, as they apply to a protein dimer generated by the association of two monomers. There will be three origins of the separate changes in enthalpy and entropy, and they will add to give the quantities that appear in eq 1: AH = AHpp + AHww - 2AHpw
As = As,,
+ A s w w - 2As,,
(3)
In eqs 2 and 3 the subscripts PP, WW, and PW designate interactions at the protein-protein, solvent-solvent, and proteinprotein solvent interfaces, respectively. The enthalpy of the solvent bonds ( M w w )is separately known, and if there existed a general 0 1993 American Chemical Society
Thermodynamics of Oligomeric Proteins
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7109
AH = H (P-P) +H (W-W) -2H (P-W)
AS
= S (P-P) +S (W-W) -2s (P-W)
b
Figure 1. Schematic of the bond changes that take place when monomers, hydrated at the subunit interfaces, associate to form a dimer.
relation between the changes in bond enthalpy and the ensuing entropy changes one could readily deduce the entropy changes pertaining to the solvent and thus fix also the value of ASWW.By applying the same rules of the relation of bond enthalpy and associated entropy to the P-P and P-W bonds we might reduce the problem to the determination of two quantities, A H p p and AHpw, from the two, AH and T U , that appear in eq 1.
Entropy and Bond Strength Consider M identical interactions holding an interface, like those shown in Figure 1. At any given time there is a single probabilityp of finding any one of these interactions broken and therefore a complementary probability 1 - p of live interaction. An individual interface characterized by the number j is held together by that number of bonds, and a collection of those interfaces make up a heterogeneous population described by a distributionfj of bonds. To construct this distribution we require a rule that specifies the dynamics of the population, that is, a rule governing the probabilities of the transitions j j 1 and j j - 1 that respectively increase and decrease by one the number of bonds present. The rule that we adopt is that there is a fixed probability p of the existence of the bond a t any given time and that changes in the fraction of active bonds follow the law of mass action. Then, the transition probability of a decrease in the number of bonds by one, j -* j - 1, is the product of the number of live bonds, j , and the a priori probability, p , that any one of them will be broken
-+ -
W(j+j-l) = p j (4) Similarly, the probability of an increase in the number of bonds depends only upon the number ( M - j ) of unmade bonds and the a priori probability of making one more of such bonds, 1 - p : W(j+j+l) = ( M - j ) ( l - p ) (5) Starting with any arbitrary distribution of values offj we can compute the final equilibrium distribution by means of a set of differential relations that gives the small change in fj in an appropriately small interval dt, in accordance with eqs 4 and 5: d& = {-f&( 1 - p )
+f i p ) dt
if j = 0
;3
;5
le io ;3 NUMBER OF BONDS
is
la
43
Figure 2. Progressive iterative generation of the equilibrium distribution of bonds by application of eqs 6, starting from a set of random fractional values, (A) with M = 50 and E / R T = 1. (B) After 100 iterations, (C) after 200 iterations, (D) after 1330 iterations. The envelope in D is the distribution calculated by eq 7.
p ) . The half-width of the distribution becomes a smaller fraction of M as M increases: The molecules become energetically more similar as the number of bonds increases, in the same manner as any collection of molecules defines an energetic average that increases in accuracy with their number. However, as Mincreases, the number of ways in which j can be formed by different sets of bonds in the molecule also increases. Thus, both the probability of energetic similarity and that of individual molecular conformations increase with M . Panels A-D of Figure 2 show the generation of the normal distribution, starting from a random distribution offj, by repeated application of eqs 6. The figure shows that the distribution is regenerative, in the sense that any arbitrarily chosen set isolated from it converges to the equilibrium distribution by a dynamic process that operates according to the transition probabilities stated in eqs 4 and 5. The equilibrium distribution has the analytical formulation (7) where (y)is the binomial coefficient or number of combinations of M objects taken j at a time, also including j = 0. In agreement with the definition of a bond as an interaction that limits the possible motions, its energy E is not directly related to any thermodynamically defined quantity but to intrinsic properties of the system that need not be analyzed further for the purpose of application. What is asserted here is that E is determined by a probability of existence that, a t constant volume, depends upon the temperature in a simple fashion, p = exp(EIRT). It is then relevant to analyze the effect of changes in p through those of EIRT, the anergy of the individual bond in thermal energy units. Figure 3 shows the total energy Et and the entropy S T of the distribution, both in RT units and each normalized to its maximum value in the range of the plot, as a function of EIRT. The energy is given by the relation M
d& = {-jj[(M-j)(l - P ) +jpl +jj-l(M-j + 1)(1 -P) + fJ+l(j+1 ) p J d t if j = 1 to M - I
E, =
EE& j=O
(6)
In calculating the entropy we use the classical Boltzmann relation
As may be expected, the assumption of complementary probabilities of a bond being present or absent generates, by application of eqs 6, a normal distribution with a maximum at j N M( 1 -
S=RInN (9) where N is the number of complexions. Taking into account that we often deal with a number of conformations of unequal
d& = (-fMMp +fM-l(l - p ) ) d t if j = M
Weber
7110 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993
OC.
bonds, in practice those bonds of energy less than 5 or 7 RT. The energy-nthalpy compensation that follows a moderate increase in temperature was first demonstrated in enzyme-catalyzed reactions by biochemists, and from the above discussion it follows that it is necessarily associated with molecules possessing a large number of weak bonds. Thus, it points unmistakably to proteins as having the necessary characteristics for the observation of such compensatory effects. An important premise in the derivation of this statistical relation of bond energy and entropy is that bonds are considered either as made or broken, while in reality bonds depend on interaction energies that vary continuously with distance. There is, however, no reason to believe that the relations of energy and entropy would be appreciably different in either circumstance, because both are subject to the constraints of energy conservation which in any case determines the fraction of lost bond energy that contributes to the increased entropy. A further limitation is apparent: By reason of the hypothesis made, the only source of entropy considered here is that due to the difference in the number of complexions attributed to products and reactants, and according to Figure 3, the contribution from the set of bonds with energies larger than 5-7 R T is negligible. Further unknown sources of entropy may be expected,on grounds already discussed, to monotonically decrease with bond energy, and as the experimental entropy is determined by the difference between the entropy contributions of the sets of bonds such a contribution would not significantly alter the resultant entropy change on reaction, so long as M is sufficiently large. For small M , as is the case in the enthalpy-driven complexes of small molecules, such additional sources of entropy might no longer be considered negligible. We assumed that E is a bond energy that satisfies fixed conditions of volume and temperature, and substitution of H by Et in eq 1 will then lead to the free energy at constant volume hA (Helmholz free energy). Correctionsfor the effects of changes in volume upon E are required to compare the effects at different temperatures and pressures. In what follows I shall identify the total energy Et with the enthalpy H of the system of bonds, as this identification is not expected to lead to appreciable errors or uncertainties in the case of a given isothermal equilibrium, and will further in this article consider the changes in E that follow the volume changes of the system.
probability3 the molar entropy of the distribution is
Entropy Change in the Association of Protein Monomers
I *'.
,/'. ,
I
6
I
I
2.5
0.0
10.0
BOND ENERG?.'IN RT UNI?; Figure 3. Dependence of entropy and enthalpy for bond distributions that follow eqs 8-10 or eq 6. Enthalpy is plotted as decreasing,entropy as increasing, upward. The ordinate scales are normalized to the
corresponding maxima.
-1.651
I 0.00
\ I
/ I
2.50
I
I
5.00
I
I
7.50
I
I1
10.00
BOND ENERGY/RT AT 0 CELSIUS Figure 4. Plot of the energy and entropy changes in the interval 0-20 OC of a set of 100 identical bonds against the energy of the bonds at 0
The energy approaches the expected value Et = ME(1 - p), increasing therefore monotonically with the number of bonds and their individual value. On the other hand, the entropy has an evident maximum for p = ' 1 2 , Le., E I R T = In 2. For values of EIRTsmaller than In 2 it drops rapidly and for values greater than In 2 it does so much more slowly (Figure 3). For sufficiently large bond energies the distribution approaches one restricted to almost a single value corresponding to no bonds broken. Simple inspection of Figure 3 also shows that the sign and magnitude of the entropy change in a chemical reaction is largely determined by the energy of the weaker bonds. The figure also shows that chemical reactions involving only bonds of considerable strength are accompanied by small, often negligible changes in entropy. Figure 3 clearly shows the origin of the often-observed enthalpyentropy compensation: A decrease in bond energy is necessarily followed by an increase in the number of complexions in the population, and in opposition, an increase in bond energy diminishes the number of possible complexions. Figure 4 displays the changes in energy and entropy of a population of 100 equal bonds, over an interval of 20 OC, as a function of the bond energy expressedinthermalunitsat OOC. Itisevidentthat themagnitude of the compensation is important only in the case of low-energy
A decrease in the number of particles by association, as is the case in Figure 1, results in the disappearance of three degrees of translational freedom and two or three degrees of rotational freedom per missing particle. Such a contribution amounts to 3 R T for a dimer and 9 R T for a tetramer compared to values of 50-150 R T of total entropy change. As these contributions oppose association,their considerationwould lead to larger values for the entropy change owing to multiple conformations, though not by a significant amount, and in the following I shall not take them into consideration. The stoichiometry of the association reaction is
-
+
2P-w P-P w-w and the energetics of the associationof the monomers is described by relations 2 and 3. The computation of the energy and entropy of the bond distribution that we have formulated for the protein is applicable to the solvent that surrounds it, with even more reason since the solvent-solvent bonds are more homogeneous as regards energy than those of the p r ~ t e i n .The ~ number of water molecules involved (cross section 10 A2)is determined by the surface exposed to solvent upon dissociation of the subunits and can be reasonably approximated from the known dimensions of the monomers. The solvent-solvent bonds broken in the process of creation of an interface of protein and water amount to two hydrogen bonds or -7 kcal/mol per pair of water molecules.
-
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7111
Thermodynamics of Oligomeric Proteins
I
I
3 000
I
3.625
I
I 4 250
I
I
4.875
I
I 5.500
PROTEIN-SOLVENT BOND ENERGY Figure 5. Lines joining points for which the free energy of association of a dimer at 0 “Cis -1 1 kcal mol-’. They correspond from right to left
to 200,100,75,50,25, and 12 bonds, the same for the P-W, W-W, and P-P species. The continuous portions of the lines join points of entropythe discontinuoussegments correspond driven reaction (ITAS1> la); to enthalpy-drivenreaction ( I T 4 < la). The points to the right of the dark circles have AH < 0, others AH > 0. The figure demonstrates that (1) a minimum number of bonds is required for an entropy driven association. (2) there is a restricted range of P-W and P-Pbond energies consistent with entropy-drivenreactions of positive enthalpy. Thus, if the size of the monomers is known the contribution of enthalpy and entropy from the solvent at temperature Tare fixed quantities, AHw-w and ASw-w, respectively, and relations 2 and 3 simplify to A H - AH,-, = AHp-p - 2AHp-w (12)
T h s - Thsw-w = TASp-p - 2 T u p - w
(1 3) If the numbers of protein-protein and protein-solvent bonds are now specified, the relations between AH and AS of eqs 8 and 10 permit the determination of AH and TAS in eqs 12 and 13 by a systematic variation of the energies of protein-protein and protein-solvent bonds over a range large enough to demonstrate the uniqueness of the solutions. Clearly, an important question is the assignment of the number of potential bonds on the protein-protein interfaces. If this number is indefinitely increased one must derive in the calculations proportionally smaller values of EP-P and EP-Wand thus correspondingly increasing entropies associated to the same total energies. Inevitably the choice of M i s arbitrary, but only within reasonable limits imposed by the known structure of the intersubunit surfaces. We expect that the major source of the protein interactions will be the external portion of the amino acid residues, with sizes of methyl or ethyl groups or the edge of an aromatic amino acid. We can assign these structures cross sections of interaction similar to that of a water molecule and greatly simplify the calculations by choosing for all contacting surfaces a single value of M equal to the number of water pairs formed on release of water from the protein surfaces. The energies of the P-P and P-W bonds in reaction 11, consistent with a free energy change of dimer association of -1 1 kcal mol-’, and of solvent-solvent bonds of 2 X 3.5 = 7 kcal mol-’, and decreasing values of M , the same for all interfaces, were calculated employing eqs 5-13, and the results are displayed in Figure 5. The bond values for which the entropy contribution predominates over the enthalpy, and for which additionally AH > 0, are those on the full lines to the left of the black points. The corresponding bond ranges are from 0.5 to 2.5 kcal for P-P and from 3.6 to 5.5 kcal for P-W. Thus, such reactions necessitate protein-solvent bonds of energy similar to those of the solvent and much weaker protein-protein bonds. Besides, the figure demonstrates that an entropy-driven molecular association with AH > 0 is possible only if the number of bonds is sufficiently large.
TABLE I: Enthalpy (AH) and Entropy (TAS) Contributions, in kcal mol-’, to the Free Energy of Association of Oligomers at 1 O C subunit mass protein ref no. in kDa bonds AH TAS yeast hexokinase 10 2 43 141 17 38 E.colia tryptophansynthase 8 2 43 141 17.7 28.4 Rhodobacter Rubiscob 14 2 55 166 6.0 18.9 phosphorylase A dimer 12 2 95 239 4 17 allophycocyanine 13 3 33 118 42 65 GAPDH 11 4 34 120 -14.1 17.8 phosphorylaseA tetramer 12 4 95 239 33 66 At 4 OC. At 15 OC. The free energies of association are obtained by extrapolation to atmospheric pressure thus minimizing the effects of “conformational drift”.7-11Bond numbers are assumed the same for solvent-solvent, protein-protein, and protein-solvent contacts. Interpretation of the Experimental Results of Oligomeric Associations Studies involving the perturbation of the dissociation equilibria of oligomeric aggregates by hydrostatic pressure over the last 10 year~5-I~ have permitted us to gather good quality data regarding the enthalpy and entropy changes in the corresponding association reactions. The data presented in Table I are readily summarized: The contribution of the enthalpy changes to the total free energy change on association is always smaller than the changes in entropy (TAS), and to judge from the existing examples they are much more often positive than negative. Table I shows six instances of the former and only one of the latter type. Therefore, the enthalpy changes almost always oppose association. In contrast, the changes in entropy (TAS) upon association are uniformly large and positive and are, in all the cases in Table I, responsible for the large preponderance of the aggregate over the isolated subunits, at the physiological concentrations of the oligomeric proteins. Consider the dissociation of a dimer protein like the first two entries in Table I. Each intersubunit surface contacts, on dissociation, 140 molecules of water, and the formation of the corresponding water-protein interface involves the disappearance of some 280 hydrogen bonds between the water molecules, amounting to a positive enthalpy change of ca. 980 kcal. As the total enthalpy change is 17 kcal mol-’ the positive enthalpy change owing to the disappearance of the water-water contacts is balanced within less than 2% of the total by the difference between the enthalpy of intersubunit bonds and that from the bonds of the exposed protein surfaces with the water. The conclusion is then inescapable that the interaction of the putative “hydrophobic” intersubunit surfaces and the water molecules involves bonds that are of a strength only fractionally different than those between the water molecules themselves. From the relations between the bond strength and the change in the number of complexions (eqs 8 and 10 and Figure 3), it follows that the entropy changes associated with the replacement of the waterprotein bonds by water-water bonds of similar strength cannot contribute an appreciable entropy increase and that entropy changes large enough to explain the existence of entropy-driven aggregation of protein subunits are only to be found in the difference in the entropy change associated to the conversion of protein-solvent bonds into protein-protein bonds. Table I1lists the strength of protein-protein and protein-solvent bonds that reproduce the entropy contributions to the free energy of association to better than 1%, when the bond enthalpy balance is exactly equal to the experimental enthalpy change (Figure 7). These computations have been made employing eqs 12 and 13, together with the general relation between changes in bond enthalpy and the associated entropy according to eqs 6-10. The average energy of the protein-protein bonds in Table I1 is 1.82 f 0.78 kcal mol-’, that of the protein-water bonds is 4.39 f 0.45
Weber
7112 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993
TABLE II: Calculated Energies, in kcal, of the Protein-Protein (P-P) and Protein-Solvent (P-W) Bonds That Satisfy the Energetic Data of Table I with an Entropy Error of 6( TAS)kcal mol-’ and Zero Enthalpy Error. P-P energy P-W energy 6( TAS) yeast hexokinase 0.76 3.84 0.21 (0.78)
(3.89)
trp synthase
1.25 (1.25)
4.12 (4.12)
0.04
phosphorylase A dimer
2.39 (2.60)
4.68 (4.80)
0.18
Rubisco
1.91 (1.92)
4.44 (4.46)
0.15
allophycocyanine
10.36 (1.35)
4.18 (4.17)
0.08
GAPDH
2.94 (2.92)
4.94 (4.96)
0.03
phosphorylase A tetramer
2.43 (2.42)
4.71 (4.71)
0.09
a The computations for trimers and tetramers have been made assuming equal contributions to the experimental enthalpies and entropies from three and four interfaces,respectively, and carryingout the computations as for a dimer. The quality of the computations used to derive the quoted bond energies is displayed in Figure 7. Numbers in parentheses are computed by disregardingthe enthalpy error and minimizing the entropy error. employed the one-variableprocedure that results from the use of relation i4.
I
0 75
,‘-
I
1.875
3.750
5 625
W-W
7
I
BOND ENERGY IN KILOCALORIES Figure6. Plot showing the negligiblecontributionof water bond changes to the increase in entropy that follows subunit association. The vertical lines, placed at the corresponding bond energies, are proportional to the entropies associated to the bond populations.
kcal mol-’. Figure 6 places these average values together with that of the bonds holding a pair of water molecules (7 kcal) in a plot like that of Figure 3, and from the placements it follows that the whole of the entropy change in the association arises from the conversion of P-W bonds into P-P bonds. We note that thequalitativeconclusions that may bederived from the calculated bond strengths are in agreement with the known physical propertiesof proteins: (1) Theconsiderable strengths of the bonds between the subunit surfaces and water, which approach that of the bonds between the water molecules themselves, are not surprising as both experiment15 and computation16 demonstrate the considerable polarity of the protein interior. (2) Kinetic observations of all kinds,l7-19 confirmed by calculations of molecular dynamics,20 demonstrate the existence of many weak bonds within proteins. Their conversion into the stronger proteinsolvent bonds is sufficient to account for the positive enthalpy change that opposes the association of the subunits. The data of Table I1 show that the values of the P-P and P-W
bonds obtained by a double variational procedure over the two quantities follow the relation
Ep-w= 3.5
+ Ep-,/2
This is a simple consequence of the constancy attributed to the water-water bonds and the number of bonds per surface, so that any energy changes that result in the decrease or increase of one of the bond energies, Ep-p or Ep-w, must, on account of energy conservation, produce a compensatory change in the other. It is therefore possible to replace the double variational procedure used to obtain the bond energies shown in Table I1 by a procedure involving a single variable and the fixed relation (14). Moreover, because of the fixed relations of entropy and enthalpy, clearly displayed in Figures 3 and 4, we may choose to minimize the entropy error alone. If we do so we obtain the set of P-W and P-P values shown in parentheses in Table 11. These latter values do not differ appreciably from those obtained by the double variational procedure, although in some of the cases the energies differ by a small fraction of a kilocalorie. The similarity of results obtained with a two-variable or one-variable minimization arises from the very large degree of compensation of the energies, which is a characteristic of entropy-driven reactions. Errors of energy compensation of 5 kcal distributed over 120 bonds would result in an uncertainty in the computed bond energy of 0.04 kcal assuming that all of it falls upon either the P-W or the P-P bond, and the uncertainty would be less than that if both P-P and P-W bonds contributed to it.
The Present Analysis and the “Hydrophobic Bond” Hypothesis The stability of the associations of the subunits of the capsid of tobacco mosaic virus, of hemoglobin S,actin, or tubulin have positive thermal coefficients. Because of their indefinite stoichiometries such protein associations do not lend themselves to the analysis that we have performed in the smaller, better-defined aggregates, and the observations made on them were rationalized by the supposition that contact of the protein’s apolar surfaces with water resulted in a large decrease in the entropy of the solvent and that the reversal of this process was promoted by the increase in water entropy. This interpretation was based on the large entropy changes computed from partition coefficients of small apolar molecules between apolar solvents and water. The increase in entropy of the water was supposed to provide most of the driving energy for the association of nonpolar solutes in water by what was designated a “hydrophobic bond”.21-25 The partition coefficient data do indicate a differencein entropy between the apolar molecule in the apolar solvent and the same molecule in water. This could be interpreted equally well as the result of a loss of entropy of the apolar molecule in water with respect to the entropy that it had in the apolar solvent, the result of a decrease in the entropy of the water on introduction of the apolar molecule with respect to the entropy previous to it, or by a mixture of both effects. From the point of view that we have adopted, the observed decrease in entropy results entirely from the substitution of the weaker bonds among the apolar molecules by stronger bonds with the water dipoles. Methane or ethane26 in water occupy volumes that are smaller, by, respectively, 50% and 36%, than those in an apolar solvent, a decrease in volume due to theexchangeof the weaker apolar bonds (dispersion forces) by the shorter and stronger dipole-induced dipole bonds. On the other hand, the replacement of the dipole-dipole bonds of the water by dipole-induced dipole bonds of similar strength ought to produce a much smaller entropy change, as shown in Figure 2. According to the hydrophobic bond hypothesis, the increase in entropy upon subunit association results from the conversion of P-W bonds into W-W bonds, a conclusion precisely opposite to the one derived from our computations, that the whole of the entropy change follows from the conversion of P-W bonds into P-P bonds, while the conversion of P-W bonds into W-W bonds produces no appreciable entropy change. Up to the present, no
Thermodynamics of Oligomeric Proteins specificcalculation of the thermodynamicparameters (enthalpy, entropy, or volume change) of any individual protein reaction, starting from the premises of the hydrophobic bond hypothesis is known to the author, although the hypothesis has produced an extensive literature of qualitative speculation, and imaginative hydrogen bond dispositions have been proposed to account for the assumed entropy changes of the water.27 If we dismiss this latter aspect of the hypothesis and reinterpret the original transfer experiments as explained above, one more example of the rule that a negative entropy change occurs when weaker bonds are substituted by stronger bonds, the protein association appears as a simple reversion of that process, a positive entropy change that evidently requires protein-protein bonds to be the weaker ones. It is also evident that the strong interactions of the subunit interfaces with water cannot be considered identical to those of truly apolar compounds: For virtually every amino acid residue in the protein there is a peptide arrangement with a dipole moment of 3.6 D, which can interact with the water dipoles with much higher energies than thosecharacteristic of a truly apolar surface. Even in the interior of the protein, where internal dipole and other interactions cancel the larger part of the dipole strength, both experiment's and computation16 have shown that this strength is far from negligible.
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7113 PROTEIN-PROTEIN BOND ENERGY 0.60
I'
1
0.96 I
1.69
1.32 1
1
I
1
2.05 I
I
0.c
3.75
4.15
3.95
4.35
4.55
PROTEIN-SOLVENT BOND ENERGY
Figure 7. Points are the entropy variance (=computed entropy experimental entropy)2 for the set of values at which computed enthalpy - experimental enthalpy = 0, for E. coli tryptophan synthase. (Tables I and 11). 1 277
2 553
3 830
I
1
1
WATER: 0.50.95 C
5 106
Pressure and Temperature Effects in Oligomeric Associations We have computed the strengths of the protein-protein and protein-solvent bonds by means of a theory that assigns to E a constant value, and the free energy calculated from it is therefore the Helmholz free energy, while the experimental free energy, obtained from data at constant pressure and variable temperature, is the Gibbs free energy. An estimation of the relative changes in E brought about by changes in volume following those of pressure and temperature should permit us to derive the Gibbs free energy as a function of pressure and thus to point out the manner in which applicationof pressure leads to thedestabilization of the association. Temperature and pressure produce changes in E through an alteration of the distances of interaction of the particles through thermal expansion or pressure compression of the interacting elements, the atoms or molecules. To determine the changes in the bond energy E, that appears in eqs 7-10, we require a relation that gives the dependence of the particle interactions upon their distanceor volume. With this purpose we consider the interaction energy as the result of an attraction that decays with a power t* of the distance and a repulsion that decays with a higher power of the distance s*, in the manner originally proposed by Born.28 We replace the two customary arbitrary constants by two determinable parameters: the energy EO and intermolecular distance ro that correspond to a minimum of potential energy, found at temperature TOand pressure p0.29 Moreover, we can replace the distances r and ro by volumes V and VO,respectively equal to r3 and r03, and obtain
where evidently s = s*/3 and t = t*/3. Equation 15 describes the relative volume changes of a homogeneous liquid at temperature T and pressure p with respect to VO the volume at temperature TOand pressure PO. The attraction exponent t is known to be 2 (t* = 6) for interaction of mutually induced dipoles and t = 1 (t* = 3) for permanent dipoles. Induction effects by permanent dipoles may be expected to have t fall between 1 and 2. The repulsion exponent s may be determined by the best fit of the compression curve of liquids to eq 15 over the range of s values. Figure 8 shows the agreement of the experimental volumes of hexane and water determined by Bridgman3O in the range of 1-5000 atm at the temperatures of 0,50, and 95 OC with those computed by eq 15 with TO= 0 OC and po = 1 bar. Values of s = 10 for hexane and s = 8 for water provide the best agreement
W
> + U
U
J1
( W I
W
51 J
0 >
0
Figure 8. Relative volumes of water (upper panel) and hexane (lower panel) in the range 1 bar to 5 kbar. The circles are the experimental values of Bridgman.3O The continuous lines are calculated by means of eq 15 with the exponents s = 10, t = 2 for hexane and s = 8 t = 1 for water.
between observed and computed pressures for a given volume. As indicated by Figure 9 the coefficient of variation of the computed volume-pressure relation was generally closer than 2% to the experimental values, for the best fitting s values. Introducing the coefficients s and t in eq 15 we can compute the changes in E with temperature and pressure, for bonds that have the apolar character of hexaneor thedipolar character of water. We consider that both P-P and P-W bonds have complementary fractions of apolar and dipolar character and that the dependence of E on the expansivity and compressibility of these bonds is best represented by the equation (16) E = YE(hexane) -k ( l - Y)E(water) where y and 1 - y are the fractions of truly apolar and dipolar character, respectively. Evidently, yw-w = 0, which leaves us with two arbitrary parameters yp-p and yp-w to fix the variation of Ep-p and EP-w,respectively. These parameters are chosen by the requirement to produce, for a given protein concentration, profiles of dissociation against applied pressure that agree with experiment on three counts: (1) The pressure of middissociation (a= 0.5); (2) A slope in the plot of AG against p that reproduces dG/dp = AV, the change in volume upon dissociation; and (3) A uniform increase in stability with temperature in the range -20-20 OC. Computations were performed employing the fixed
Weber
7114 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 I
WATER: 0, '50, 95 C
I
y I
/
H c
2
U .
o
.
B
*
~
-1.00 0 0
4
1.0
PRESSURE 2fON KILOBAR3" Figure 10. Relative destabilization of the P-P, P-W, and W-W bonds by pressure, at 0 OC, employing the bond energies quoted in Table I1 for tryptophan synthase. yp-p = 0.9, 7p-w = 0.223. Energies are set to -1
REPULSION
at 1 bar.
EXPONENT
Figure 9. Coefficients of variation of the curves of Figure 8, with respect to curves obtained by Newton interpolation of the experimental points of Bridgman (Figure 8), for variable s exponent. TABLE III: Correlated Character of the Apolar Fraction, 7 , of the P-P and P-W Interfaces. The Figures Were Com uted for To = 0 OC, po = 1 bar, and P-P = 1.25 kcal mol-rand Are Those That Accord with the Three Criteria Discussed in Text YP-P
YP-w
YP-P
YP-w
YP-P
1 0.9
0.268 0.223 0.184
0.7
0.145 0.111
0.4
0.8
0.6 0.5
0.080
YP-w
0.053
0.3
0.028
0.2
0.006
bond energy relations of eq 14, valid at 0 OC and atmospheric pressure, and varying Ep-p between 0.75 and 2.5 kcal mol-'. It became evident in these calculations that the three criteria just mentioned can be fulfilled when the fractional apolar character of the bonds varies within limits, but that there must necessarily be a difference in apolar character, always favoring the P-P bonds. Additionally the apolar character of the P-W bonds never exceeds 0.27,even when the apolar character of the P-P bonds is set to 1. A list of the correlated apolar characters of the bonds for the case in which Eo(p-p) = 1.25 kcal is shown in Table 111. Even more important than the numerical agreement with the experimental values is the light that the computations throw on the reasons for the increased stability with temperature and the decreased stability with pressure, features that have been observed in all the protein aggregates so far studied. These depend on characteristic differences of the molecular interactions of dipolar and apolar character as to thermal expansion and compression. We can clearly discern the microscopic circumstances that result in the dissociation of the aggregates when pressure is applied: As the volume decreases on compression, the Born repulsion increasingly predominates over the attraction. The weaker the bonds the stronger the compression and the larger the loss of bond energy. As shown in Figure 10 the bond destabilization is largest for the P-P bonds, smaller for the P-W bonds, and smallest for the W-W bonds. Figure 11 gives the changes in entropy and enthalpy that follow the compression of the three separate interfaces. While both energy and entropy increase, the enthalpy does so much more steeply, owing to the preferential destabilization of the weaker P-P bonds. We are able in this way to offer a simple explanation of the mechanism by which application of pressure leads to dissociation: Compression of the weaker bonds results in preferential destabilization of the subunit interactions and shifts the equilibrium toward the formation of the shorter subunit-water bonds. We no longer need to justify the dissociation of the aggregate by pressure after an appeal to the formal principle
& 37.5 ZE iI
W
a
ffl
30.0
n 0
1 4
u
0
1 c1 22.5 lL
I
0.0
I 1.0
I
I 2.0
I
I
3.0
I
1 4.
PRESSURE I N KILOBAR Figure 11. Changes in entropy and enthalpy with pressure at 0 OC, for
tryptophan synthase dimer, showing the origin of the dissociation upon compression. Theleveling beyond 3-4 kbar is the result of similar repulsive interactions for all bonds at the higher pressures. of Le Chatelier; the dissociation is the result of the differential destabilization of the bonds upon compression. Because of the progressive destabilization of all bonds as the pressure increases, the free energy of association becomes nearly constant and independent of pressure at about 3-4 kbar. The character of intersubunit bonds and protein-solvent bonds may not differ much from protein to protein, and in consequence we expect the dissociation of most protein aggregates to occur at pressures under 4 kbar, if it is to occur at all. The effects of the pressure at three different temperatures, -20, 0, and 20 OC are shown in Figure 12 by plots of the free energy of association against the applied pressure. Application of pressure must reverse the expansion a t the bonds, before reaching the range of distances at which repulsion leads to dissociation, and as the expansivity is greatest for the P-P bonds, the curves of free energy against pressure of Figure 12 show a progressive displacement toward higher pressures as the temperature is raised. In this way the increase in stability toward high pressure with an increase in temperature is explained. The increase in stability with temperature at atmospheric pressure results both from the decrease in the energy of the bonds owing to expansion and the decrease in EIRT. All bonds are weakened by the thermal expansion, but the enthalpy-entropy compensation (Figure 3) is the most pronounced for the P-P bonds, and the altered energetic balance results in increased stability of the aggregates with temperature. It is worthy of note that we could not have reached these conclusions as regards the effects of
~
Thermodynamics of Oligomeric Proteins
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7115 proteins were low-entropy entities and that the large entropy increases observed on association were to be found elsewhere, but this view is no longer tenable, as observation and calculation17-~0 have dismissed the ‘low-entropy protein”. Because the bonds linking the amino acid residues within the protein are of the same character as those between subunits, one is lead to the conclusion that the globularity of folded polypeptide chains must be similarly entropy driven. The view, that attributes a predominant effect in the stability of protein structures to the protein’s own entropy and limits the energetic participation of the solvent to that of providing a suitably matching enthalpy contribution, is so diametrically opposed to the one previously held as to beg reexaminationof protein functionslike folding, secretion,motility, and even catalysis in its light.
PRESSURE I N KILOBAR
Figure 12. Changes in Gibbs free energy with pressure for tryptophan synthase a t -20, 0, and 20 OC,showing the stabilization of the protein association with temperature. The inversion of the effects at the highest pressures results from the increase in repulsive interactions at all the bonds. temperature and pressure by considering enthalpy and entropy as two separate components of the internal energy but reached them only by examining the microscopic circumstances of their mutual influence. The relations of the energy of the bonds with their expansivity and compressibility show that the thermal stabilization and pressure destabilization of protein associations are not due to a peculiar characteristic of certain “hydrophobic” bonds but are the simple result of differences in the strengths of the bonds involved in the equilibrium.
Conclusions A theory is presented here of the relations of bond energy and entropy that can be applied to those cases where bonds exchanged in a chemical equilibrium are large in number and approximately homogeneous in energy, as it happens in the equilibrium of an oligomer and the constitutive monomers. The general procedure that I have adopted is the substitution of the actual system of associating protein subunits with its various kinds of bonds and their individual microscopic motions, by an equivalent constrained system, which is expected to yield average values for the bond energies involved, starting with the experimentally observed enthalpy and entropy. There is precedent for this: In the hydrodynamics of rigid particles an equivalent hydrodynamic ellipsoid is envisioned, the rotations and translations of which are the same as those of rigid particles of arbitrary shape. Although nobody well informed on the matter would expect the particles to have the actual shape of the equivalent ellipsoid, we are sure that in calculating the particle motions we will obtain the appropriate result by introducing the much simpler equivalent ellipsoid. In the same vein we do not expect the protein model used here to give more than an adequate representation of the characteristic changes in entropy and enthalpy as they relate to the average bond energies. Its best justification may be found in the experimentalobservation that, although thevarious proteins examined differ in their structure and amino acid composition as much as globular proteins generally do, the derived bond energies show considerable similarities, and the differences between the energies of protein-protein and protein-solvent bonds explain the experimentally observed changes in the stability of proteins aggregates toward changes in temperature and pressure. Conceptually, the most important conclusion of this examination is that the entropy-drivencharacter of protein associations results from theconsiderable number of weak protein bonds rather than from solvent contributions, as often postulated in the past. This latter view could be held while it was widely believed that
Acknowledgment. The author acknowledge the good advice of his colleagues Professors H. G. Drickamer, P. G. Wolynes, S. G. Sligar, and G. D. Reinhart, the many discussions with his collaborators Drs. Jerson Silva, Kancheng Ruan, Leonard0 Erijman, and Debora Foguel, whose experimental data have provided the main basis and stimulusfor this work, and the support of the National Institutes of Health through Grant GM11223. References and Notes (1) Adefinitionofentropyhereisnotsuperfluousasshown bythecomment “...no informed reader will presume that any one author that uses the word entropy means just what any other author does.” of C. Truesdell: Truesdell, C. In The Elements of Continuum Mechanics; Springer-Verlag: New York, 1966; p 234. (2) Stillinger, F. H. Science 1980, 209, 451457. (3) Equations 9 and 10 give the same result when Wis a large number, but for small numbers of complexions eq 10 is to be used, as discussed in the following: Yaglom, A. M.; Yaglom, I. M. Probabilite et Information; Dunod: Paris, 1959; pp 40-45. (4) Pauling, L. The Nature of the Chemical Bond; Cornel1 University Press: New York, 1944; p 303. Pauling calculated the residual entropy of ice by employing, in a more arbitrary fashion than that used here, a relation between bond enthalpy and entropy. (5) Paladini, A. A.; Weber, G. Biochemistry 1981, 20, 2587-2593. (6) King, L.; Weber, G. Biochemistry 1986, 25, 3632-3637. (7) Silva, J. L.; Miles, E. W.; Weber, G. Biochemistry 1986.25, 57815786.
(8) Weber, G. Pressure and Low-temperature Stability of Oligomeric Enzymes. In New Trends in Biological Chemistry; Ozawa, T., Ed.; J.S.S. Press: Tokyo, Springer: Berlin, 1991; pp 225-238. (9) Weber, G. Pressure Dissociation of the smaller oligomers. In High Pressurein Chemistry, Biochemistry andMaterialScience;Winter, R., Jonas, J., Eds.; NATO Advanced Study Institute Series C, Vol. 401; Kluwer, in press. (10) Ruan, K.; Weber, G. Biochemistry 1988.27, 3295-3301. (11) Ruan, K.; Weber, G. Biochemistry 1989, 28, 2144-2153. (12) Ruan, K.; Weber, G. Biochemistry, in press. (13) Foguel, D.; Weber, G. Manuscript in preparation. (14) Erijman,L.;Lorimer,G. H.; Weber,G.Biochemistry 1993,32,51875195. (15) Macgregor, R. B.; Weber, G. Nature 1986, 319, 70-73. (16) Warshel, A.; Russell, S. T.; Churg, A. K. Proc. Narl. Acad. Sci. U.S.A. 1984,81,4785-4789. (17) Lakowicz, R. J.; Weber, G. Biochemistry 1973, 12, 4171-4179. (18) Wagner, G.; Demarco, A.; Wuthrich, K. Biophys. Srruct. Mech. 1968, 2, 139-158. (19) Gratton, E.; Alcala, R. J.; Marriott, G. Biochem. Soc. Trans. 1986, 14,6785-6788. (20) Karplus, M.; McCammon, J. A. Crit. Rev. Biochem. 1981, 9,293349. (21) Nemethy, G.; Scheraga, H. A. J . Chem. Phys. 1962,36,2401-2417. (22) Ben Naim, A. Hydrophobic Interactions; Academic: New York, 1980. (23) Nemethy, G.; Peer, W. J.; Scheraga, H. A. Annu. Rev. Biophys. Bioeng. 1981, 10,459-497. (24) Alonso, D. 0. V.; Dill, K. A. Biochemistry 1991, 30, 5975-5985. (25) Hvidt, A.; Westh, P. Stabilization and Destabilization of Protein
Conformation. In Protein Interactions; Visser, H., Ed.; VCH Verlagsgesellshaft: Weinheim, 1992; pp 327-343. (26) Masterton, W. L. J . Chem. Phys. 1954, 22, 1830-1833. (27) Dill, K. A. Biochemistry 1990, 29, 7 133-7 15 1, particularly Figure 6. (28) Fowler, R. Statistical Mechanics; Cambridge University Press: London, 1955; Chapter X. (29) Weber, G. Protein Interactions; Chapman & Hall: New York and London, 1992; Chapter 10. (30) Bridgman, P. W. The Physics of High Pressure; G. Bell & Sons, 1931; pp 129-130.
