Dec., 1958
STABILITY OF HYDROGEN-BONDED POLYPEPTIDE STRUCTURES IN SOLUTION & = - - ET’( A P ) 6~r*qNL
(7)
The two transport rates may now be compared by taking the ratio of Q’ to Q
For close-packed hard spheres the fraction of the total volume that is occupied by the spheres is about 0.74. For a real liquid we can only estimate this ratio and we will take the convenient value of 4/9. Then, substitution in equation 8 for the volume in terms of the molecular radius shows that the two rates of transport would be equal if r =2r*
(9)
Therefore, within the limitations of the above assumptions, viscous flow across a membrane would predominate if * t h e capillary radius were much larger than the molecular radius. If, on the other hand, the capillary radius were only slightly larger than the molecular radius, diffusive flow would be important. Madras, et al., calculated the effective pore radius to be about 1.5 X lo-’ cm. for water permeation of their cellophane membranes. Their method of calculation (assumption of Poiseuille flow) gives values of 6 X 10-8 cm. for permeation by ethanol cm. of a water swollen membrane and 3.6 X for the permeation by butanol of a membrane previously swollen in 60% ethan0140% water. Thus, equation 8 suggests that for water viscous flow would predominate and for the alcohols both types of flow would occur, which agrees with the results of Tables 1 and 11. It would be difficult to separate the permeation into these types of flow, although it has been attempted for gaseous permeation of membranes. l1 (11) H. L. Frisch, THIEJOURNAL, 60, 1177 (1950).
1485
If viscous flow were the only transport mechanism, the permeation of a membrane by various liquids (assuming they did not change the pore radius) should yield the same permeability constant. Madras, et al., however, found permeability constants for water swollen membranes varying from 3.6 X 10-l6 cm.-2 for permeation by methanol to for hep1.8 x 1O-la for butanol and 0.8 X tane. On the other hand if the mechanism were partly viscous flow and partly diffusive flow, it would be expected that an increase in the molecular size of the permeant would cause a transfer from a convective to a diffusive mechanism. Thus, the observed results can be explained by a combined mechanism. The increase in the calculated diffusion coefficients with increased degree of swelling suggests an increase in the size of the capillaries and a transfer to more convective and less diffusive flow. Other investigators have found evidence for a diffusion mechanism in membrane permeation. Barrer and others’ have treated the gaseous permeation of organic membranes as a diffusive process. It has been shown by McBain and Stuewer12 that an ordinary cellophane membrane when swollen in water and used as an ultrafilter will partially hold back small molecules and ions such as sucrose, dextrose and potassium chloride. C. E. Reid and E. J. Breton13have reported on the use of cellulose acetate membranes for the ultrafiltration of a 0.5% aqueous NaCl solution; salt concentration was reduced by about 95%. These results suggest that, in general, permeation of membranes by liquids may be interpreted in part by a diffusion mechanism. (12) J. W. McBain and R.F. Stuewer, ibid., 40, 1157 (1936). (13) Charles E. Reid and Ernest J. Breton, “Water and Ion Flow through Cellulose Acetate Membranes,” a paper presented a t the 130th National Colloid Symposium, Madison, Wisc., June 18-22, 1956.
T H E FACTORS AFFECTING THE STABILITY OF HYDROGEN-BONDED POLYPEPTIDE STRUCTURES I N SOLUTION BY JOHN A. SCHELLMAN* Division of Physical Chemistry, Chenzistry Department, University of Minnesota, 1cf inneapolis, Minn. Received A p r i l 21, I968
A previous discussion of the thermal transition of helices is extended to include the effect of fluctuations, mixed organic solvent systems and pressure. It is found that though fluctuation effects are very large for helices of low stability, the average thermodynamic properties are altered in a relatively minor way. The fluctuation treatment reveals an asymmetry in transitions which has been verified experimentally. An essential difference between the transitions of helices in pure and mixed solvents is demonstrated. The form of the results gives an explanation for the inverted transitions which have been observed in proteins and polypeptides. The calculated properties of helices should have semiquantitative application to proteins.
It is now known that the various hydrogenbonded polypeptide structures which have been proposed1-a have a sufficiently limited stability in aqueous solution that a large number of the classical denaturation reactions can be qualitatively * Department of Chemistry, University of Oregon, Eugene, Oregon. (1) M . L. Huggins. C h e n . Revs., 3 2 , 195 (1943).
(2) L. Pauling, R. B. Corey and H. R. Eranson, PTOC.Nall. Acad. S c i . ( U . S.), 37,205 (1951). (4) B. Low and H. J. Grenville-Wells. ibid., 99, 785 (1953).
explained as the disruption of these weak structures by heating, denaturants, pH changes, e t ~ . ~ J This interpretation of denaturation is not always valid since cooperative transformations are known in which the configuration of the polypeptide back(4) W. Kauzman, “Rleohanism of Enzyme Action,” W. McElroy and B. Glass, Johns Hopkins Press, Baltimore, Md., 1954. (5) J. Schellman, Compl. rend. trav. Lab. Cadsberg. Ser. chin., 29, No. 15 (1955).
JOHN A. SCHELLMAN
1486
bone does not seem to be altered significantly,s but most such transitions are accompanied by the ehaiiges in viscosity, optical rotation and other properties which have both an empirical and a theoretical justification as being related to an unfolding of compact hydrogen-bonded configurations. Investigations of transformations of this type are now proceeding along two separate lines: (1) the unfolding of synthetic polypeptides, which has been extensively investigated by Bamford, Elliott and Hanby with respect to solvent composition, and Doty, Blout and Yang with respect to both solvent and composition and temperature; and (2) the reversible or irreversible denaturation of native proteins followed by one or more physical method of observation. The former investigations have the advantage that one knows very definitely the nature of the transition taking place: an or-helix is transformed to a disordered chain. The latter investigations clearly are connected most closely with the problem of the structure of native proteins; yet here the scheme of stabilization is so complex that the nature of the transition is not very clearly defined. It is the purpose of this investigation to consider the helix-coil transition of synthetic polypeptides in a sufficiently general manner that the related denaturation of proteins in aqueous solution may be included, at least by analogy.
Fluctuations in Helical Structure As a model for calculations we take the or-helix of Pauling and Corey though, as will be seen, the method is easily applicable t o other structures. I n a previous treatment the helix coil transition was assumed to represent the passage from a completely intact helix to a completely nonhydrogen-bonded structure.6 Thus, for example, a system a t the mid-point of such a transition was considered to contain equal numbers of helices and random coils with no intermediate configurations. We shall first explore the refinements which arise when this simplifying assumption is removed. The main body of this work will deal with simple polypeptides so that the folded configuration may be assumed to be the or-helix. We consider a single-chain polypeptide containing n identical amino acid residues. The two extreme configurations for this molecule are an or-helix containing n 4 hydrogen bonds and a random chain containing no hydrogen bonds except with solvent. Strictly speaking, the “random” chain includes configurations containing hydrogen bonds of the type O.......... ......... H
-
8-(”.C(HR).CO)i--N
I
where i has a value different from 3 which is characteristic of the a-helix. We are essentially ignoring such hydrogen bonds. This assumption will be evaluated after certain formulas are derived. The random chain containing no hydrogen bonds will be taken as the standard state of the molecule and its free energy set equal to zero. We ignore the effect of concentration on the free energy of the various states and in so doing deprive (6) M. Eisenberg and G . Schwert, J . Gen. Phgsiol., 84, 583 (1951).
Vol. 62
the results of any quantitative application to systems in which intermolecular interactions are of importance. With this reference state, the free energy of any of the hydrogen-bonded configurations may be set equal to the free energy of unfolding of this configuration, A F u n f , with opposite sign. The free energy of unfolding will be taken as a linear function of the number of residues, as done by Huggins, who demonstrated that this is a valid procedure for many long-chain series.’ Thus AFu,r = n’AFPepf C
(1)
where n’ represents the number of residues in the helix, which, since we are trying explicitly to take fluctuations into account, will in general be a smaller number than n. A F p e p is the change in the local free energy of a residue in the interior of the helix when it is transformed from the helical to the random configuration. The constant C arises because four N H and four CO groups of a helix are atypical, since they are on the leading edges of the helix. This is demonstrated in Fig. 1 of ref. 5. I t s value has been shown to be equal roughly to5 C = -4AHpep
where AHpepand usual manner
+ TASpep+ RT In 2
ASpep
AFpep
=
are related to
AHpep
-
AFpp
(2)
in the
TASpep
The factor -4AHpep, accounts for the deficit of four hydrogen bonds in the helix, the TASpepterm takes into account in a rough way the fact that there is freedom of rotation about the bonds beyond the first and last hydrogen bonds of the helix, and the RT In 2 is associated with the possibility of forming both right- and left-handed helices. Substituting (2) into (1) AF,,f = (n’ - 4)AHpep- T(n‘ - l)ASpep f RT In 2 = mAFPep- 3TaSPep R T l n 2 (3)
+
where we have replaced (n’ -4) by m,which will be used in this discussion to designate the number of hydrogen bonds in the helix. Since most recent evidence indicates that one type of helix is preferred,8-1’Jthe last term will be dropped, though naturally it must be retained for polyglycine systems where there can be no preference in helical sense. It is worth mentioning that two of our approximations, eq. 1 and the preference for helices of one sense, will be rather poor for very short helices; but that this will not affect the results very much since it will be shown that the contribution of short helices with only one or two hydrogen bonds to the thermodynamic properties of a polypeptide is slight. The probability of occurrence of a polypeptide with just one hydrogen bond will be given by PI =
(n
- 4)eAFpdRT e-3AS~sdR Q
J
(4)
where the exponentials are obtained from eq. 3 with m = 1, the factor n - 4 is the number of (7) M. L. Huggins, THISJOURNAL, 48, 1083 (1939). (8) A. Elliott and B. Malcolm, Nature, 178, 912 (1956). (9) M. L. Huggina, J . Am Chem. Soc., 74, 3963 (1952). (Second note b y Huggins on thia page.) (10) D.Fitts and J . G . Kirkwood, Proc. Natl. Acad. Sci., 43, 1046 (1957).
‘
STABI.LITY OF HYDROGEN-BONDED POLYPEPTIDE STRUCTURES IN SOLUTION
Dee., 1958
1487
positions in the chain that the single hydrogen bond can be located (assuming that only the 13 atom rings characteristic of the a-helix are formed) and Q is a normalizing factor which will be shown to be equal to the configurational partition function. When we consider a polypeptide with two hydrogen bonds a problem arises, since these could form adjacent to one another or form independently in different parts of the molecule. The relative probabilities of these two configurations is given by p1+1 P2
-E ) 2 (n -
e24F,/RT e-f3AS/R
5)c24F/RT e-34S/R
(5)
where (n - 8)(n - 9)/2 is the number of ways two hydrogen bonds can be formed in a polypeptide of n residues without being involved in the same helix. This relation is derived in the Appendix. Equation 5 is an expression of the entropy effect which opposes the formation of a new helix because of the large number of groups which must be frozen into the helical configuration to form the first bond. It is seen that the formation of two incipient helices increases in importance as chain length increases, as would be expected. It should be noted that this ratio is not explicitly dependent on temperature and is also relatively independent of solvent since the term A S is primarily configurational entropy with a relatively small solvent dependence. (See next section.) Values of A S will be in the neighborhood of 4-5 e.u. and it is shown easily that the relative probabilities are in the neighborhood of 2% for a chain 50 units long and reach 50% only for chains which are 1000 units long. We shall assume for our purposes that the chains considered are sufficiently short that only one helix need be considered as existing in the chain. With this assumption the calculations become quite easy. The method of generalization is discussed in the Appendix and for the treatment of very long chains where this assumption cannot be made the reader is referred to the work of Peller who treats the problem in a general way." The probability for the formation of m adjacent hydrogen bonds is given by P, =
(n - 3
- m ) ernAFpeD/'RT e - 3 4 S ~ e p / R Q
(6)
and since the sum of all probabilities must be unity, we shall have = 1
+
(n - 3
- rn)emAF/RT
~ - ~ A S W D / R( 7 )
where the factor of 1 arises from the probability of the completely unfolded state which, since we have chosen it as the reference state, has a probability of l/Q. Putting e A F / R T = p , we have n-4
= 1
5 6
0, respectively. P(n')is the normalized probability of a helix containing n' units.
which can be summed by the usual formula; the second term is the derivative of a geometric series Zmpm = pZrnn"-'
+ e-3~~vtD/m
(n
-3
- m)p"
(8)
m=l
The first term in the brackets is a geometric series (11) L. Peller, Thesis, Princeton University, 19.57.
=
p
d
Zp"
and its value can be found simply by differentiating the summation formula with respect to p. The result is Q
1
+ e-34S/R
cpn-z
(n
-
+
- 3)pZ (n (P
- 1)2
*)PI
(9)
It should be noted that Q =
W(m)e-Fm/RT
(10)
m
where W ( m ) is the number of ways m hydrogen bonds can be formed and F m is the free energy of formation of a molecule with m hydrogen bonds relative to the random chain. Since by statistical mechanics e-AFm/RT
71-4
m=l
Q
8 10 12 14 16 18 20 n'. Figs. l-4.-The distribution of helical lengths for a polypeptide twenty units long with AF,,,/RT = 0.4, 0.2, 0.1 and 0
e-AAm/RT
= Zwie-ei/kT
the summation being over-all configurations compatible with the formation of m hydrogen bonds, including those of the solvent, then W(m)
Q = m
zuie-ei/kT
(11)
i
is the complete configurational partition function (apart from the translational term) with the zero of energy so chosen that the free energy of the unfolded form vanishes. Accordingly we may use the usual formulas to determine the configurational entropy, heat capacity, enthalpy, etc. It is interesting to note that we have deriyed a statistical mechanical quantity starting wlth thermo-
JOHN A. SCHELLMAN
1488
dynamic assumptions. This is the opposite of the usual procedure.12 An expression also can be derived for , the mean value of m. This average is defined by = zmP,
(12)
which from eq. 5 and 6 is seen to be equivalent to
(”)
m P= d paln = Q
Q
’ -bP- -
(13)
and substituting the above explicit expression for Q we obtain pe-3ASndR
= 1
Fig. 3 the number of molecules with no hydrogen bonds is beginning to pile up, and finally in Fig. 4 the transition is very near to completion. When p = 1, there is no thermodynamic driving force for the formation of hydrogen bonds, the relative probabilities being determined entirely by the combinatorial term of eq. G. The distribution curves demonstrate a lack of symmetry in the transition which is missed entirely in the “all or none” treatment. In addition, it is seen that the intermediate configurations outweigh the completely helical state except in systems of
(n - , 4 ) ~ ” - ~(n - 2)pm-s
+ e-SASpeJR
pn-2
Vol. 62
-
+ (n - 2 ) p - ( n - 4)
( P - 1)s (n - 3)PZ (P
-
+ (n - 4 ) P
(14)
Equations 6 and 8 can be combined to give the great stability (p>>l). Nevertheless the calprobability P(m) as a function of p and ASpep. culated transition curves do not differ too greatly Figures 1-4 show the form of this function for four from the simple case, as may be seen in Fig. 5 which values of AFIRT. I n order to emphasize the dis- was constructed on the basis of the same assumed continuity involved in the formation of the first values for n and AXpep as the previous figure. Two hydrogen bond, n‘ = m 4 has been plotted rather features are of note: (1) the curve for the general than m itself. Decreasing p implies decreasing case is not symmetrical, turning abruptly a t low stability and can be regarded as resulting from an AFpep/RT (unfolded form) but very slowly at increase in temperature, addition of a hydrogen- high A F p e p I R T . This behavior has been observed bond-breaking solute, or the transfer of attention in at least one instance-the thermal unfolding from a more to a less stable helical polypeptide. In of ribonuclease. The slow, continued variation the construction of the figures a polypeptide con- off with AFpep/RTis probably the explanation of taining twenty units has been assumed and A S p e p the positive temperature coefficient of the levorotation of native proteins. Disordering fluctuations has been taken to be 4.3 e.u.14 I n Fig. 1 the molecule is in its normal folded increase with temperature and the disordered state, the breadth of the peak representing the forms have a high levorotation. (2) The transifluctuations from the completely folded form, tion occurs a t lower AFpep(higher T, more conLe., the occasional fraying off of one or two peptide centrated denaturant, etc.) than the “all or none” units from either end, although the completely case. This means that the existence of fluctuations hydrogen-bonded form has the highest probability. actually protects the helix against a total unfolding Figure 2 shows the helix in a state of transition. because the free energy of the group of fluctuant The most probable helical length is now 18 and states is lower than that of the totally hydrogenmolecules with no hydrogen bonds are beginning bonded helix. to appear. It should be noted that the state with The transition in the general case is an incomno hydrogen bonds is favored more than the states plete one which introduces a certain amount of with m = 5-14. This is a direct result of the con- complexity in the interpretation of data. For cept of nucleation; larger helices grow a t the ex- example, the slope of the fh vs.p curve is less steep pense of smaller ones. It also should be noted than for the idealized case, so that if the formula that in this particular state no configuration has a relating the sharpness of a transition to the number high probability and yet in spite of this the average of participating units (derived for the idealized helical length is quite high. This arises because case in ref. 5) is used, the number of units calcuthe helices which do exist tend to be long. I n lated will be less than the number of units involved. Indeed, the number of units participating in the (12) B y eq. 3, F is the Gibbs free energy. In the transformation from eq. 10 t o eq. 11 we have used relationships valid only for the transition is not clearly defined. This point is Helmholtz free energy, the reasons for this being that this developdiscussed by Peller, who introduces operational ment is by far the most familiar and that this difference between thermodynamic functions for such a transition. AEpep and AHpsp is of the order of 0.3cal. at one atmosphere pressure A further complexity arises in the interpretation (see below); L e . , it is negligible. At high pressure we would have instead of data obtained by a given experimental method. Unless there is a simple, preferably linear, correlae -A F m / R T = zwie -(ei +p V i ) / k T tion between the physical property being measured where p is the external pressure and VI is the volume contribution of and the average number of hydrogen bonds, a the ith eonfiguration. Instead of Q the isothermal-isobaric partition funotion A would be the normalizing factor in eq. 11 and subsequent strict dependence of the physical property on the equations. 1: degree of unfolding will be destroyed because of (13) T. L. Hill, “Statiatiaal Mechanics,” McGraw-Hill Book Co., the large number of species present. The simNew York, N. Y., 1956. plicity of the formulation given above results from (14) There is no real significance to this choice of entropy change
I(
+
except that it appears to be reasonably close to the truth theoretically and experimentally. 4.3 e.u. is the value which would obtain for an infinite helix which unfolds at 350OK. if AHpep 1500 cal., which is an approximate value of the heat of breaking a peptide hydrogen bond in aqueous solution.‘
-
(15) See, for example, ref. 16. The experimental part of this research will be submitted in a separate article with Dr. John Foss. (16) W. Harrington and J. Schellman, Conspt. rend. trau. Lob. Carlabare, Ser. chim., SO, No. 3 (1956).
..
Dec., 1958
STABILITY OF HYDROGEN-BONDED POLYPEPTIDE STRUCTURES IN SOLUTION
the assumptions (1) that only helices of the aform (13 atom H bond loop) exist, (2) that these are all of the same sense, and (3) that the linear equation 1, holds for the structures under consideration. All these approximations break down for short helices but, as is shown in Figs. 1 4 , short helices always have a low probability. The distribution function for n' equal to 5 to 8-10 should be regarded as less accurate than the other cases. Setting up the problem to include any number of a, T, y, etc., helices of both senses is straightforward. It is found that the partition function is given by &=I+ e-iAS/R[pn
- i + 1 - (n - i ) p 2 + (n - i
1489
1.o
0.8
0.6 *r: 0.4
-1)pJ
i
(P -
112
where i is the number of complete amino acids in the hydrogen bond loop characteristic of the helix under consideration. (See definition above.) It is t o be noted that the larger i is, the greater is the entropic disadvantage in helix formation. It can be shown that the combination of this entropy effect and the 0.5 kcal. of strain energy calculated by Donahuel' for the rr-helix favors the a-helix over the latter by a factor of about 50:l for a helix containing one hydrogen bond and by a rapidly rising amount for longer helices. This is the reason that such helices were neglected. It should be emphasized that this argument does not apply to chains with configurational constraints (as in proteins) or when side chain interactions are such as t o compensate for the extra instability of the Ir-helix. On the other hand, helices with i equal to one or two are entropically favored over the a-helix. Calculations on the basis of Donahue's estimates show that for helices containing only one or two hydrogen bonds, the concentration of the varient forms actually exceeds that of the a-type. However, as mentioned above, short helices play a very small part in the over-all picture of the distribution function. The Dependence of p and AF,,, on Experimental Parameters The discussion above has been concerned with transitions governed by variations in the somewhat abstract quantity p , rather than the usual parameters which are subject to precise experimental variation and control, such as temperature, solvent, pH, etc. This abstraction is very convenient for the purpose of general discussion because, regardless of the complexity of the system, the appearance of polypeptide transitions will always depend in a straightforward manner on AFpep. On the other hand, the dependence of the latter on temperature, pressure, amino acid sequence, tertiary interactions, chain length, pH, ionic strength, etc., can be exceedingly complex and in many instances is not as yet understood. The factors which influence AFpep can be divided into two types: those such as peptide hydrogen bonds, electrostatic effects, hydrophobic bonds, etc., which are indigenous to the molecule under consideration, and those which can be controlled (171 J. Donahue, Proc. N d . Acad. Sci., $9, 470 (1953).
0.2
/ 2
1
3
P. Fig, 5.-The helix coil transition for: A, a polypeptide twenty units long which can exist in intermediate co&gurations; B, a polypeptide of the same length which can exist in only the completely helical or random form; C, an infinitely long helix either case. f h is the fraction of reaidues in the helical conkguration.
externally, such as temperature, pressure and solvent composition, which we shall construe quite generally as including the effects of pH, ionic strength, etc., as well as the presence of inorganic and organic reagents. Naturally all the intramolecular interactions and forces are dependent on the external parameters. It is important t o remember that the individuality of the transitions of polypeptides and proteins arises from these intramolecular interactions which are different for all proteins and polypeptides. It is possible t o make quite general statements about the effect of external parameters such as T , P, pH, etc., on the stability of proteins taken as a whole, though the manner in which a given protein reacts to them will depend on its own individual pattern of stabilization. It should not be regarded as surprising that some proteins unfold a t high temperatures whereas others do not, that poly DL-alanine and cytochrome c do not unfold in urea whereas most proteins do, that some large proteins are very unstable while many sma,llproteins are quite stable, that some proteins undergo an acid denaturation whereas most do not, etc. These differences give us, in fact, some notion of the prominent stabilizing forces in the molecule under consideration. Polypeptides in Pure Solvents.-If there is only one solvent species present, it is possible to make use of the usual theory of high polymer solutions except that the problem at hand is especially simple because (1) we are not interested in the complete thermodynamics of the system but only in the changes which occur during the transformation from a hydrogen-bonded to a disordered structure; and (2) the effect of concentration of polypeptide, so vexing in high polymer theory, can be ignored a t this stage of development, the polypeptides and
JOHN A. SCHELLMAN
1490
proteins being treated as isolated molecules in contact with solvent only. This approximation is justifiable on the grounds that the initial stages of most protein transitions are first-order in protein concentration, though to be sure subsequent steps, particularly those involving sulfhydryl groups, frequently involve intermolecular interactions. Association reactions often take place with synthetic polypeptides in non-aqueous solvents and in such cases the present discussion will only cover the case of extreme dilution. On the basis of the quasi-crystalline model of polymer solutions, the molar free energy of a polypeptide can be written in form1*(15) where the symF = -RT In W F(C,Co) (15) bo1 i2 represents the number of configurations available to the polypeptide, W represents the total free energy of contact between the polypeptide and the solvent, andF(Co,C) is an additional free energy term which is dependent on the concentration of the polypeptide and the arbitrary choice of standard state. Since we are considering a system in which the polymer molecules are isolated, the last term will be identical for all structures and will cancel out of the equations for transitions. Thus if we are interested in the change in free energy in going from a folded to an unfolded structure we can write
+ +
which contains the ratio of the number of configurations available to the unfolded and folded states and the change in the solvent contact parameter, W , in going from one state to the other. Only those degrees of freedom of the molecule which are restricted in the folded state will contribute to the first term and only those portions of the molecule which change their contact with solvent will contribute to the latter. A rough analysis of the restrictions of configurational freedom of the polypeptide backbone in polyglycine already has been given.5 The result of this calculation tells us only that the disordering entropy forces in polypeptides oppose and match in order of magnitude the ordering, hydrogen-bonding forces in aqueous solution. I n the general case of complex polypeptides and proteins, the changes in configurational freedom of side chains (changes resulting from tertiary structure) also must be included. The second term presents many difficulties. The first approximation that usually is made is based on the theory of regular solutions and assumes that W is simply an energy term which is independent of temperature. This approximation was made in an earlier calculation in which it was assumed that W consisted largely of the heat of disruption of the peptide hydrogen bonds in the solvent under consideration, which was water.6 It has become increasingly apparent during the past ten years that this assumption is not valid in many cases and that the term W contains a contact entropy as well as a contact energy between solute and solvent. The work of Prigogine,l9 (18) E. A. Guggenheim, “Mixtures,” Oxford University Press, London, 1952. (19) I. Prigogine and V. Mathot, J . Chem. Phys., 20, 49 (1952).
Vol. 62
using cell theories of solution, has revealed how such entropy changes arise from the changes in the range and frequency of motion of molecules which change their immediate environment. PopleZ0 has carried the problem further by showing that marked contact entropy changes can result with molecules with permanent dipole moments. However, the state of development of this field is so little advanced a t the present time that i t is very often difficult to ascribe even the sign of the contact entropy in an ordered solvent such as water.21 Consequently observed entropy changes are comprised of contributions from the polypeptide backbone, the side chains and the entropy of contact, all of which are uncertain a t the present time. One hopeful feature is that the observed entropies are usually of the order of magnitude as the entropies to be anticipated from the peptide backbone alone so that perhaps the other contributions are smaller or cancel one another to a certain extent. The manner in which a polypeptide governed by eq. 16 will undergo transitions as the temperature and length is varied has been discussed already. The effect of pressure on the transformation has, however, received little theoretical attention. Assuming that the number of available configurations is not greatly affected by pressure, we have
I
where AVcont is the localized change in volume which results from the difference in contact between the polypeptide and the solvent.2s This change should be a very complicated one since it will depend in general on the numerous interactions of side chains with the solvent and, if ionizable groups are present, on the electrostrictive effects which will accompany changes in ionization during the transition. The contribution of such “tertiary” factors is difficult to assess a t the present time, but it is possible to make a crude estimate of the contribution of the change in solvent contact of the polypeptide backbone from the careful studies of the density of urea solutions of Gucker, Gage and M o ~ e r . It ~ ~is easy to show that if one assumes that urea molecules associate in aqueous solution, the heat of formation of the urea dimer AH can be obtained from the relation25 @I,
= AH(1
-
$9)
(18)
a t low urea concentration where @L is the relative molar heat content and cp is the osmotic coefficient. (20) J. A. Pople, Disc. Faraday Soc.. 16, 35 (1953). (21) Calculations based on the properties of the urea dimer in aqueous solution indicate that AScon& is of the order of $1 e.u. in thia cme. The degree to which this can be applied to peptide hydrogen bonds in water ia uncertain but it seems definite that the contact entropy change is much smaller than the configurational entropy. The sign of the 4Xcont favors the unfolded form and is in direct opposition to the usual interpretation of the liberation of solvent molecules when solute-solute contact is made. W. Kauzmann has independently arrived at the same conclusions.%* (22) W. Kauzmann, private communication. (23) We here deviate dightly from the treatment in ref. 15. Equation 17 establishes A V as the change in local Gibbs free energy rather than Helmholtz free energy. (24) F. Guc$er, F. Gage and C. Moser, J . Am. Chem. Soc., 60, 2582 (1938). (25) J. A. Schellman, Compl. rend. tras. Carlsbsro Lab., Xer. chim., 29, No. 14 (1955).
.
Dec., 1958
STABILITY OF HYDROGEN-BONDED POLYPEPTIDE STRUCTURES I N SOLUTION
1491
Actually it may be seen from the derivation of this equation that it applies to any extensive thermodynamic property of the solute and may be converted directly t o an equation for the volume
unfold. This may be the origin of the inactivation of enzymes by high pressure.30 All these remarks apply to aqueous solution only. Polypeptides in Mixed Solvents.-We have so far been concerned with the case of a system which @v AV(1 - rp) (19) consists of two components only-polypeptide and where AV is the volume change accompanying the a pure solvent. This condition is occasionally formation of the dimer and @?vis the partial rela- realized in polypeptide systems but almost never tive volume. This equation is to be applied in the in protein chemistry where various counter ions, concentration region where @jvis actually a linear buffers, etc., are present. The presence of salts function of 1 - cp. Deviations from linearity a t will always diminish the electrostatic forces which higher concentrations are attributable to approxi- are present and so can lead to a stabilization or an mations made in the derivation. Values of @?v unstabilization of compact forms, depending on the were taken from the studies of Gucker, Gage and total charge on the polypeptide and the manner in M~ser~ those ~ ; for cp from Scatchard, Hamer and which the individual charged groups are distributed Wood.26 Gucker’s data had to be converted from in the molecule under consideration. The addition a molar to a molal basis for proper comparison. of acid or base will favor an expanded configuration A plot of @jvus. f, - cp proved to be quite linear and by building up charge on the protein, etc. The from the slope it was found that AV = 3.4 ml. influence of electrostatic forces on helical stability If we assume that this volume change arises en- has been considered in detail by Peller.’l tirely from the formation of CO H N hydrogen Rather than attempt to discuss the problem of bonds, then the volume change per hydrogen bond solvent, composition in general, which would be a is given by formidable task, we shall instead turn our attention AVE E 3.4 X 3/4 2.5 ml. (20) to one type of solvent effect, namely, the influence because there are roughly 4/3 hydrogen bonds in of a mixed solvent containing two substances, one of which is more firmly bound to exposed peptide the urea dimer.26 Recognizing that the solvent contact situation groups than the other. It will be assumed that the in the urea dimer is certainly different from that properties of these two substances are such that in the peptide hydrogen bond in a helix, we may, hydrogen-bonded configurations are stable in the nevertheless, assume that the volume changes are weakly binding solvent and unstable in the strongly similar in order of magnitude. (The establish- binding solvent and inquiries shall be made about ment of the sign is the important matter.) The the behavior of the system in the presence of mixmost carefully investigated volume change is that tures of the two, where the properties can be anof P-lactoglobulin. Linderstr@m-Langz7has esti- ticipated to be intermediate. A discussion of the mated that a contraction of 700 ml./mole is problem along the lines of the quasi-crystalline involved in the unfolding of this molecule which model would be quite impossible at present since is now known t o be a dimer.28 If we assume we are dealing with binary solvents, in general of that most of the 320 peptide residues in the dimer different molecular size, in contact with a macroare hydrogen-bonded in the helical state and take molecule with differing contact energies and enthe contraction per hydrogen bond broken as 2.5 tropies. Progress can be made, however, if we ml. as calculated above, a total contraction per regard the system as consisting of the weakly mole is obtained which is in quite good agreement bound material symbolized by [SI as being in with the experimental value. On the other hand, excess and regarded as a solvent in which the we have been informed by W. Kauzmann22 that polypeptide and the strongly bound substance, the changes in AVContand AXcont which accompany symbolized by [D], are associating chemically. the formation of a hydrophobic bond are so large This corresponds to common practice in protein that even the formation of a few of them in a chemistry where proteins and denaturating agents polypeptide or protein molecule can have a big are allowed to combine in a solvent, usually water, effect on the thermodynamics of the system. and we wish to extend it to synthetic polypeptides Changes in ionization also will have a large effe~t.~gin organic solvents where t h e strongly bound subAccordingly the agreement found above is probably stance is usually formic acid or an equivalent hyfortuitous and the real volume change is a com- drogen-bond breaker. In the case where water is posite of all three effects. It does appear, however, the “solvent,” S, we will have to contend with the that the breaking of peptide hydrogen bonds may fact that it is itself a very powerful breaker of hybe an important if not dominant cause of the con- drogen bonds so that the native structure is usually traction observed during the unfolding of proteins. only partially folded to begin with and the denaturaThe final result of the analysis is that if the vol- tion only takes place in somewhat limited regions ume changes arise from the breaking of hydrogen of the molecule. In the helix-coil transitions of bonds and hydrophobic bonds, increasing the pres- synthetic polypeptides, the entire molecule usually sure should cause proteins and polypeptides to participates. We consider the unfolding to occur in three (26) G. Scatchard, W. Hamer and S. Wood, J . Am. Chem. Soc., 60, 3061 (1938). steps: First, the molecule is transported from the (27) K. u. Linderstr@m-Lang, Cold Spring Harbor Symposia Quant. mixed solvent to pure 5. This will give rise to a B i d . . 14, 117 (1950).
...
-
(28) R. Townaend and S. Timasheff, J. Am. Chem. SOC., 79, .3614 (1957). (29) C. Tanford, ibid., 79, 3931 (1957).
(30) F. Johnson, H. Eyring and M. Polissar, “The Kinetics of Molecular Biology,” John Wiley and Sons, Inc., New Pork, N. Y., 19.54.
JOHN A. SCHELLMAN
1492
free energy change AFl. Next, the molecule is unfolded in pure S to its final state, accompanied by a free energy change AFu,f (S). The polypeptide is then transported back to the mixed solvent with a change in free energy, AF3. Since we are ignoring effects of concentration, AF1 will differ from AF3 only by the difference in solvent interaction of the folded and unfolded forms of the molecule. The total free energy change for these three steps, which corresponds to the real transition, already has been evaluated for the special case of the urea denaturation of a helix in which only the peptide hydrogen bonds are attacked by urea. The resulting equation is5 AFu,,f = AFu.f ( S )
- 2nRT
In (1
i
For urea and polyglycine < v > 2n, for urea and a polymer with extensive side-chain hydrogen bonds 3n, etc. With other reagents such as detergents which also form hydrophobic bonds with exposed groups, the situation is more complicated because one cannot count simply the number of hydrophobic "bonds" as one can with hydrogen bonds. Let us now proceed to an examination of the behavior predicted by eq. 23 when both the temperature and [D] are permitted t o vary. For this purpose it will be convenient to introduce the quantity
+ K [ U ] ) (21)
where AF,nf(S) is the free energy change of unfolding in pure solvent, n is the number of peptide links, K is the association constant of urea for the NH or the CO groups of the peptide, and [U] is the urea concentration. The factor 2 arises from the fact that there are two sites per peptide link where a hydrogen bond can be formed. This equation immediately can be generalized to fit our present problem AFunf = AFUndS) - C u , R T In ( 1
Vol. 62
+ K,[DI)
(22)
where the subscript i refers to the type of site which is brought into contact with [D] by the unfolding, vi is the number of sites of type i, Ki is the association constant between site type i and D. This is merely an obvious extension of eq. 21 to include the possibility that not all sites in a polypeptide which change their solvent environment are peptide bonds and that the association constants will vary with the type of group involved; for example, side-chain tertiary interactions with [D] of various kinds also must be considered. These will most certainly differ from those of the peptide backbone. There is unfortunately very little that can be done with eq. 22 because the number and nature of all the binding sites are not known. If we make the assumption that the sites of dominant importance are the exposed peptide links, then we shall have only two kinds of site, the exposed CO and NH groups, and the total number of sites will be twice the number of peptide units involved in the transition. The special assumptions which can be made for the binding of urea to peptide links have been given elsewhere.6 FoI the general case we must use the device, common in problems of multiple equilibria, of writing the equation in the form
d
which is defined as the sharpness of a thermal transition where Ttis the transition temperature. I n using this relation we are reverting to the simple model in which only the helix and random chain are considered. There is no difficulty in extending the present treatment since eq. 23 can be written in the form AFu,r = nAFpep(S) -I- C =
n (AFpep(S)
-
-
+
RT In (1 < K > [D]) R T l n (1 < K > [D])] C
+
so that
+
+
AFpep = AFpep(S) - RT In (1 < K > ID]) where AF,,,(S) is the value which obtains in the pure solvent S and < v,> is the number of sites per residue. Consequently, the entire development of the preceding section can be applied to this case. We are seeking only semi-quantitative results and use eq. 25 for simplicity. Since6in this case
fh
= (1
+ e-AFud/RZ')-l
(25)
a simple differentiation shows that
It is to be noted that
s gives a guide not only to the sharpness of a transition but also to its direction. For if s is negative, it means that the fraction of helices decreases with increasing temperature; and if it is positive, the opposite will occur. ASunf can be determined by a differentiation of eq. 22 with respect to temperature, giving ASunf = A&,of(S) + R In (1 + KD) +
1
[D] A [D] 7(27)
+
where we have taken the liberty of assuming that < v > does not vary too much with temperature so that a fourth term involving its changes can be AFunf = AF,,f(S) - RTln (1 + [D]) (23) dropped and assuming also that the Gibbswhere < v > and are the effective number of Helmholtz equation can be applied to the effective sites and the effective binding constant, respec- equilibrium constant to give an effective heat per tively. It can be shown that < v > is a reason- site, A. This will be all right provided the able estimate of the number of sites and sites are similar. Entropies of unfolding are a reasonable average of the association constants, normally expected to be positive but eq. 27 shows provided that, and only provided that, the Kl's that there is a negative component in the third are mostly of the same order of m a g n i t ~ d e . ~term, ~ because we are dealing here with associations In this case the number < v > will be equal to a driven by hydrogen bond formation and similar simple factor times the number of peptide units. interactions which have a negative heat of association. The behavior predicted by eq. 27 can be (31) J. Sohellman, R Luinry and L. T. Samuels, J . A m . Chem. visualized by examining its limiting forms Soc , 7 6 , 2508 (1954).
*,
Dee., 1958
STABILITY OF HYDROGEN-BONDED POLYPEPTIDE STRUCTURES IN SOLUTION
1493
able variation throughout the structure (not just a t the extremities as in a uniform helix). Nevertheless, there is some profit in comparing the results obtained above with protein transitions. The situation is somewhat equivalent to deriving the general properties of solids from arguments which apply only to a simple cubic lattice. Regarded from this restricted, semi-quantitative point of view, From eq. 28 we see that ASun* will decrease with the foregoing results have a considerable appliincreasing concentration of [D] provided that cation to the transitions of proteins. A denaturing A< H > / T is negative and greater than two calories/ agent is an agent which decreases AF,,,, now to be deg. We can expect this to be so since we have regarded as an average. This can be done by inexplicitly stated that [D] binds more strongly creasing the temperature, increasing the pressure than the solvent to exposed sites. For the rela- (eq. 17), by direct combination with exposed sites tively weak binding of urea in water A/TG (eq. 23), or the building up of an electrostatic -5 cal./deg. a t normal operating temperatures. charge.ll The sites involved need not be the pepAt large [D] on the other hand, the entropy tide hydrogen bonds but also can be side-chain of unfolding increases with increasing [D] as can groups, which presumably accounts for denaturabe seen in eq. 29. It is thus seen that ASunf goes tion by detergents and organic solvents. Acid through a minimum. If the decrease in L S u n f is and alkali denaturations are probably a combinasufficient to change its sign, the transition will not tion of the last two factors. All evidence so far has only become broadened but s will become positive, pointed toward a quite low value for AFpep in indicating that the transition is inverted, with the aqueous solution, indicating that fluctuations in ordered structure most stable a t high tempera- the structure of proteins are very extensive-so tures, Such inverted transitions are known from extensive, in fact, that it is probably meaningless the work of Jacobsen and Kor~gaard-Christensen~~ to refer to the structure of a protein. The charand of Doty and Yang.a3 In fact, it was the ob- acteristic of a protein of continually changing its servations of the latter workers of broadened and configuration recently has been given the name inverted transitions which prompted the present “motility,” and there is some evidence that this investigation on the effect of binary solvents since property is involved in the enzymatic process. inverted and broadened transitions invariably The process we have been discussing (the reoccur in mixtures of solvents only. versible unfolding of an isolated molecule) appears Another effect of solvent mixtures easily is to be very different from the classical denaturation observed and easily predicted. As would be an- reaction which is irreversible and results in the ticipated, the temperature of the transition is de- formation of a precipitate. Accordingly, it might creased as the concentration of the more strongly be maintained that the results have no application bound [D] is increased. Putting eq. 23 in the to the general case. This is not necessarily true. form Since it has been established that most irreversible denaturations are a transition from a hydrogenAFunf = AHunt(S) - TASurtf bonded to a more or less random p ~ l y p e p t i d e , ~ RT In (I [D]) (30) and putting AFUnfequal to zero, we can solve for T t . it is a reasonable hypothesis that the initial stages of these reactions consist of an unfolding of the kind described here and that the difference between reversible and irreversible phenomena arises showing that Tt decreases as [D] is increased since because of what happens after the initial unfolding both AHudand ASunf are normally positive. This has taken place. This would account for the very lowering in Tthas been observed for several pro- high entropies of activation calculated by Eyring teins and polypeptides. (See ref. 11 for a dis- and Stearn.34 For example, ribonuclease has no cussion of its relationship with the freezing point SH groupa and its unfolding is reversible. Ovaldepression of a crystal.) bumin, on the other hand, coagulates because of the formation of intermolecular cross-links. The The Unfolding of Proteins situation is perhaps analogous to the melting of The application of the foregoing results to pro- crystals of quartz or glycerol. One does not teins requires a considerable extrapolation since hesitate to think of these transitions thermodythey depend almost exclusively on the validity of namically, even though they are essentially irrevereq. 1 which implies a long polypeptide containing sible processes. The suggestion is made here only one kind of monomer. Proteins, on the other that this may be a profitable procedure for the irhand, are aperiodic in structure, may contain reversible transitions of proteins. hydrogen-bonded units which are not a-helices, The behavior of proteins should be intermediate and are cross-linked by S-S bonds. Unless we between the “all or none” transition governed assume that a protein consists of a number of by eq. 25 and the fully fluctuating helix governed relatively independent helices, the constant, C, by eq. 14, since the degree of molecular freedom is difficult to define. Moreover, ALF,,, is a local in a protein should be intermediate between property which will in general undergo consider- these extreme cases. (32) C. F. Jacobsen and K. Korsgaard-Christensen, Nature, 161, Acknowledgments.-I wish to express my grati-
+
30 (1948). (33) P. Doty and Y. Y a w , J. Am. Chern. Soc.. 1 8 , 498 (1956).
(34) H.Eyring and A. E. Stearn, Chenz. Rem., 24, 253 (1939).
WILLIAMDRINKARD AND DANIEL KIVELSON
1494
Vol. 62
tude to the United States Public Health Service for the partial support of this investigation.
= (n - 9)ec Appendix The probability of a break of five hydrogen bonds A helix with one hydrogen bond contains five peptide units. If there were two such helices, is (n - 8)(l/p)eC, and the total probability of a ten residues must be involved, and these must not break of any size is Pb = e C / R T [ ( n - 9 ) (n - 1O)p (n - l l ) p * . . . ] overlap since otherwise a single helix would be m formed. If the ten residues are adjacent, the num= ~-C/RT (n - i l p i - 9 ber of possible positions in the chain is (n - 9); i=9 if there is a free peptide unit between them, the number is (n 10). The total number is thus
+
+
PI1= [(n - 9) + (n - 10) + . . . . + (n - n ) ] e 2 A F / R T e-BAS/R =
[
(n - 9)n,
- (n -
-
5'i]
from the usual formulas for geometric series. In the above equations
e2AF/RT e - 6 A S / R
i=9
- 8,
2
eZAF/RT e-GAS/R
p
= l/p
=:
k
e-AFmp/RT
The geometric series was allowed to go to infinity This result leads directly to eq. 5 . in this summation but easily can be corrected to This type of calculation easily can be extended to any finite chain length. Ignoring the difference any distribution of interest though, as pointed out between n - 9 and n - 8, we obtain in the main text, distributions containing more than one helix will be unlikely except for long chains. Pb = (n - S ) e - C / R T (1 - P) I n this case the method of Peller," which gives the number of units in helices and in random re- If A F p e p l R T = l/2, Pb is roughly three times gions of the chain without specifying the distri- P b 4 . This means that once formed, a break tends bution, is to be preferred because of its simplicity. to propagate itself. Such breaks will have little A fluctuation of some interest is the breaking of effect on the thermodynamics of a helix but can a helix, i.e., the breaking of at least four hydrogen give rise t o considerable changes in hydrodynamic bonds to form two separate helices. The probabil- properties such as viscosity and diffusion and to ity of breaking four bonds is given by kinetic properties such as deuterium exchange.
NUCLEAR RESONANCE AND THERMAL STUDIES ON HYDROGEN BONDS IN SOLUTION BY WILLIAMDRINKARD AND DANIEL KIVELSON Contribution from the Department of Chemistry, University of California at Los Angeles, Cal. Receiaed April .%'4#1868
The proton resonance shifts of the OH proton in HzO and CHsOH have been measured as functions of concentration in acetone and dimethyl sulfoxide solutions. It was found in all cases that the OH proton resonance was shifted to higher fields with increasing dilution indicating a decrease in average hydrogen bonding of the OH. A shift of the OH stretching frequency in the infrared to hikh wave numbers on dilution agrees qualitatively with the decrease of hydrogen bond strength. The heats of mixing of these solutions, measured as a function of the concentration, are not readily interpreted in terms of the simple hydrogen bonding scheme used in the analysis of the proton resonance data. For both H20 and CHIOH in acetone and dimethyl sulfoxide it has been suggested that hydrogen bonds are formed, the order of increasing strength being
\
CO
/
H O C $0
/
-
/ \
+
HO
, HO
/
' HO.
The heat of mixing of the acetone-water system is quite anomalous, exothermic
at low acetone and endothermic at low water concentrations.
Introduction Nuclear magnetic resonance spectroscopy is a relatively new and unexplored tool for the study of hydrogen-bonding in solutions. The proton resonance appears to shift to lower fields as the hydrogen bonding is increased'-6 and a theoretical expla(1) H.S. Gutowsky and A. Saika, J . Chem. Phya., 21, 1688 (1953). ( 2 ) R.A. Ogg, ibid., S I , 560 (1954). (3) C. M. Huggins, G. C. Pimentel and J. N. Shoolery, ibid., 23, 1244 (1955). (4) B. N. Bhar and G. Lindstriim, ibid., 23, 1972 (1) (1955). (5) A. D.Cohen and C. Reid, dbid., IS, 790L (1956). (6) G. J. Korinek and W . G. Schneider, Can. J. Chem., 85, 1157 (1957).
nation of this phenomenon has been p r o p ~ s e d . ~ ~ ~ An exhaustive study of these spectral shifts and of the correlation of these shifts with other solution data, such as heats of mixing, has not yet been made. The present study is an attempt to correlate the proton resonance and heat of mixing data of a few solutions. Solutions of water, methyl alcohol and diethyl ether in acetone and dimethyl sulfoxide were investigated. Dimethyl sulfoxide differs from acetone (7) J. N.Shoolery and B. J. Alder, J. C'hsm. Phys., IS, 805 (1955). (8) H. S. Gutowsky and P. J. Frank, Abstracts of Papers 133rd Meeting, A.C.S., p. 15L, paper 35.
-
.
