REFUSLUMRYA N D ROBERTHON-SANG YUE
1162
Dielectric Dispersion of Protein Solutions Containing Small Zwitterions’
by Rufus Lumry and Robert Hon-Sang Yue Laboratory for Biophysical Chemistry, School of Chemistry, Unizersity of Minnesota, Minneapolis, Minnesota (Received September 28, 19/34)
-
The molecular mechanisni responsible for the dielectric-dispersion behavior of small globular proteins has been examined using measurements in the presence and absence of small dissolved zwitterions. The zwitterions eliminate coniplications due to electroviscous effects and dielectric effects of the ion atmosphere. They have relatively little influence on the Kirkwood-Shuniaker effect. The kinetics of the latter are analyzed on the basis of known rates for proton processes to show that in the range of pH from 6.5 to 9, little complication due to these effects is to be expected. The agreement between results obtained with and without zwitterions over a range of solvent conditions demonstrates that for sufficiently small proteins the solution dielectric properties are controlled by and measure the mean-square permanent dipole moment of the protein and the rotational relaxation hehavior of the protein. The dielectric parameters for metmyoglobin, carboxymyoglobin, oxidized and reduced cytochrome-c, and chymotrypsinogen A are given for a variety of solution conditions. Distinct anomalies were found only in the cytochrome-c forms where relaxation time and dielectric increment increase with decreasing concentration of protein and in mixtures of bovine serum albumin with oxidized cytochrome-c. S o differences were detected between oxidized and reduced cytochrome-c or between oxidized and reduced myoglobin. The results provide confidence in the use of the dielectric dispersion method for some protein solutions and suggest several uses of the method for the study of the kinetics and equilibria of protein aggregation reactions.
Introduction As a source of information about the physical parameters of protein molecules, the method of dielectric dispersion has much to recommend it. It is simple, capable of considerable precision, and under certain conditions can provide, in addition to dielectric relaxation times, the distribution of such times and information about the first moment of the charge distribution.2 These quantities cannot for the most part be intepreted in terms of specific protein models in any useful way a t the present any more than can data from any other hydrodynamic measurement. However, providing that the specific processes measured by the method are known, the quantities can be of considerable help to protein chemists especially in studying changes in conformation and charge distribution. They may also ultimately have some utility as a means of determining the rate constants for aggregation processes or some conformation changes. Some years ago, Oncley and Wyman and their respective co-workers3 The Journal of Physical Chemistry
developed the method as applied to proteins and collected considerable information about well-known proteins. In recent years Takashima4-lo has been the principal worker in this area. The method has not grown in popularity because the possibility of several kinds of complication has lead to uncertainty regarding the actual molecular processes which determine the measured dielectric parameters. In part,icular, the (1) This work was supported by t h e Office of Naval Research, Department of Defense, Contract No. N O N R 710 (55).
(2) C. P. Smyth, “Dielectric Behavior a n 8 Structure,” McGraw-Hill Book Co., Inc.. New York, N. Y.. 1955. (3) E. J. Cohn and J. T. Edsall, “Proteins, Amino Acids and Peptides,” Reinhold Publishing Carp.. New York, N. Y., 1943.
(4) S. Takashima. J . A m . Chem. SOC.,78, 541 (1956). ( 5 ) S. Takashima and R. Lumry, ibid., 80, 4238 (1958). (6) S. Takashima and R. Lumry, ibid., 80, 4244 (1958). (7) S. Takashima, ibid., 80, 4478 (1958). ( 8 ) S.Takashima, J . Polymer Sei.,56, 257 (1962). (9) S. Takashima, Arch. Biochem. Biophys., 77, 454 (1958). (10) S. Takashima, Biochim. Biophys. Acta, 79, 531 (1964).
DIELECTRIC DISPERSIOK OF PROTEIN SOLUTIONS
dielectric relaxation and measured dielectric moment may have more to do with the behavior of the ion atmosphere of the protein than with the rotational diffusion and permanent moment of the rigid protein dipole. There are three types of complication which may be important. 1. Because of high ionic conductance of singly charged ions it is not possible to work a t sufficiently high salt concentrations to damp out the electroviscous effects resulting essentially from the dragging of ion atmosphere ions and solvent by the moving protein groups. This effect is a minimum a t the prevailing isoelectric point but is still nonzero because the condition of zero net charge for the protein system is partially achieved by balancing molecules with instantaneous net positive charge against those with net negative charge. Effects can become quite large a t pH values some distance from the isoelectric point. 2. If a protein molecule is charged, its ion atmosphere has the opposite average charge and on the average a complementary electrical distribution. The ion atmosphere can thus respond to the applied electrical field to undergo periodic polarization. The problem has been discussed particularly by O'Konski. The characteristic relaxation time for this process has been estimated at 3f-l x sec.,12in Ghich M is the molar concentration of salt. Thus the ion-atmosphere relaxation time for M KC1 is 5 X lO-'sec. and close to the expected rotational relaxation time of a protein of medium size or medium asymmetry. Increasing salt concentration reduces electroviscous effects and decreases the time for ion-atmosphere relaxation but also reduces the precision of the method. The method beconies inapplicable at salt concentrations much above M. Decreasing salt concentration decreases the magnitude of the ion-atmosphere dispersion but increases the electroviscous effect and shifts the ion-atmosphere relaxation time right into the range of rotational relaxation times. It is clear that no suitable compromise of salt concentration exists which will eliminate the possibility of complications. The ion-atmosphere effect is an example of the Maxwell-Wagner effect13 and can be quite large if the ionic strength is appreciable. Because of technical limitations, most protein dielectric work is done a t such low ionic strength that there is usually only one small ion per protein molecule. Hence it can b t anticipated that the magnitude of the effect usually will be very small. 3. I n 1952. Kirkwood and Shumaker14 suggested that the probable process measured in dielectric dispersion measurements in protein solutions was the field-induced niigration of protons from acid groups on
1168
the surface of the proteins t,o unprotonated basic groups of the protein. Kinetically, the rate of such niigration will depend on the particular acidic and basic groups involved and their orientation and also weakly on the salt' concentration which, as it increases, will increasingly stabilize an existing charge distribut,ion on the protein relative t,o rapid changes in this dist'ribution. Kirkwood and Shuinaker did not estimate t'he relaxation times for the migratory process, but, suggest'ed that the apparent dielectric increment produced in this way might be of the order found in prot'ein studies. Since that, time t'here has been lit,tle further development' of t,his idea and until recent,ly lit,t,le clarificat'ion of the kinetic problems involved. Takashi~na~ has attempted with some success to show t,hat protein dielectric dispersion studies are to be iiit>erpretedin t e r m of classical Debye behavior of a rigid dipole. Very recently, Scheider15 has considered theoretically the kinetic problems and has concluded from theory and experiment that the Debye behavior is t'he principal cause of the dielectric relaxation in bovine serum albumin. The first two general classes of effect's include hlaxwell-Wagner effects dependent on t'he difference in dielectric constant and conductance across a phase boundary and are thus associated mit'h the prot8ein as a suspended particle and Debye-Falkenhagen effect's which are direct consequences of t'he structure of the ion atmosphere about a charge particle. O'Konski" has developed a more general approach to the problem in which he shows that both types of effects can be treated as manifestations of a common set of elect,rostatic conditions dependent on dielectric const,ant, surface conductivity, and part,icle volume conductivity. Schwarz16 and Eigen and S c h w a r ~ ,have ' ~ arrived a t somewhat, similar conclusions. These st,udies plus numerous investigations of large colloidal particles and particularly of large colloidal particles and polymer molecules with large axial ratios clearly deinonstrat'e that the static and dynamic propert'ies of such particles in their interaction with an electric field are dominat,ed by ion-at'mosphere eff ect,s n-hich completely
(11) C. T. O'Konski. J . Phys. Chem., 64, 605 (1960). (12) H.Fdkenhagen. "Electrolytes," T h e Clarendon Press, Oxford, 1934. (13) K.W. Wagner, Arch. Elektrotech., 2, 371 (1914.) (14) J. G. Kirkwood and J. B. Shumaker, Proc. S a t l . Acad. Sci. U . 8..38, 855 (1952). (15) W.Scheider, "Relaxation Spectra of Permanent and Fluctuating Dipole Moments," Thesis, Harvard University, 1962. (16) G. Schware. 2. Physik, 145, 563 (1956). (17) M. Eigen and G. Schwara. Z . physik. Chem. (Frankfurt). 4, 380 (1955);J . Colloid Sci., 12, 181 (1957).
Volume 69, Xumber 4
April 1865
1164
overshadow any influence of the permanent dipole moment. The discussion of Eigen and Schwarz’8 on these matters is particularly illuminating. The ion-atmosphere effects drop off very rapidly with decreasing particle length and axial ratio, so that a t some upper limit of size and shape of the charged particle the permanent-moment effects can be expected to assume a dominant role in the interaction with an applied field. The basic problem in the use of the dielectric dispersion method for globular proteins has been to decide whether the ion atmosphere or the permanent moment dominates or indeed what is the range of size, shape, charge density, and ionic strength in which effects attributable separately to ion atmosphere and to permanent moment appear. In view of the potential utility of the dielectric dispersion method and our own interest in interpreting a t the molecular level a series of experiments on hemoglobin made some years ago by Takashima and Lumry,5’6we have devoted some attention to method modifications which might allow unequivocal interpretation of the data. Of these, the most successful appears to be the use of relatively high concentrations of small zwitterions instead of singly charged ions. Zwitterions have been used effectively in dielectric experiments by Shack and c o - w ~ r k e r sand ~ ~ by Shaw, et al. ,20 studying /?-lactoglobulin. Their particular use was justified in the latter experiments by a need to salt-in the globulin. Since in the cases reported zwitterions act effectively to salt-in protein^,^ it appears probable that to a t least a first approximation insofar as the protein is concerned, an ion atmosphere of such ions is equivalent to an ion atmosphere of single-charged ions. The zwitterion atmosphere can be made concentrated without serious loss in method precision. As a result, electroviscous effects can be eliminated and the ion-atmosphere relaxation time made very small since aside from small effects due to the asymmetry of the protein field there is no net ion displacement and the relaxation process becomes rapid. In fact, the latter will not be very much longer than the characteristic rotational time of the small zwitterion. Heme, with such an atmosphere the only probable sources of dielectric relaxation in the radiofrequency range are Debye rotator behavior and perhaps the Kirkwood-Shuniaker effect. Although the change in type of ion atmosphere should alter the time constants of the latter effect, it is not obvious that this effect can be eliminated. However, a realistic look a t the Kirkwood-Shumaker effect in terms of rate constants for protonic processes suggests that in most instances the effect cannot practically alter relaxation times for protein solutions except under very limited conditions. The Journal of Physical Chemistry
RUFUSLUMRY AND ROBERT HON-SANG YUE
It may, however, make a small contribution to the observed moment. I n this paper are described results obtained with and without zwitterions on several small proteins of current interest. The Kirkwood-Shumaker effect will be considered a t the end.
Experimental Material. Sigma Chemical Co. Type I11 horseheart cytochrome-c (Lot No. 31B-647) was used. The material was 95% pure by carbon monoxide binding and autoxidation tests. About 10 mg. of reagent grade potassium ferricyanide was added to 5 ml. of cytochrome-c solution to ensure that all the cytochromec was in the ferric state. I n making reduced cytochrome-c solution, oxidized cytochrome-c solution was reduced using platinum black with hydrogen gas until there was no further change in the absorbancy. Worthington Biochemical Corp. chymotrypsinogen A Lot No. CG685-90 was used. Sperm whale metmyoglobin was kindly furnished by Dr. H. Mizukami. It was chromatographed on a carboxymethyl cellulose column using the pH and ionic gradient technique (0.01 M phosphate buffer pH 6.05 to 0.1 hi’ phosphate buffer pH 7.60). Only the main component was used for the experiment. Carboxyniyoglobin was made by reducing the metmyoglobin with a very small amount of sodium borohydride in the presence of carbon monoxide. Sigma Chemical Co. bovine plasma albumin Lot No. A42B-98 was used. Carboxypeptidase A was obtained from Worthington Biochemical Corp. Reagent grade glycine and p-alanine were recrystallized from ethyl alcohol-water a t the isoionic pH. All protein concentrations were measured spectrophotometrically with a Beckman DU spectrophotonieter. Optical densities for a 1% solution in a 1-cm. spectrophotonietric cell at the respective wave length are: oxidized cytochrome-c a t 530 nip, 7.66; reduced cytochrome-c a t 530 mp, 22.4; metinyoglobin at 408 mp, 87.0; carboxymyoglobin at 580 mp, 7.55; chymotrypsinogen A at 280 nip, 19.7. These values were determined by the dry-weight method. Viscosity was measured with Oswald viscometers at 15.00 f 0.02”. Instrumental. A standard replacement bridge (18) M.Eigen and G. Schwarz, “Symposium on Electrolytes-trieste 1959,” Pergamon Press, London, 1961. (19) J. Shack, “Dielectric Absorption of Protein Solutions,“ Ph.D. Dissertation, Harvard University, 1939. (20) T. N . Shaw, E. F. Jensen, and €I. Lineweaver, J . Chem. Phys., 12, 439 (1944).
DIELECTRIC DISPERSION OF PROTEIN SOLUTIONS
1165
dipole moment, M , of the protein was estimated from eq. 2 using Wynian’s value of 8.j3 for the constant h
method employing a Boonton Radio llodel 250A RX meter, essentially an admittance bridge, was used for all experiments. 21 The bridge conductance scale was shifted to a maximum value of 120 pf. by fixed parallel inductance coils a t each of the 14 frequencies in the range 0.5 to 20 AIc. The measuring cell has been described5 but was modified for this work by covering the brass electrode support for the upper electrode with a Teflon cylinder compressed against the top of this electrode. The Teflon was effective in eliminating contact between brass and the solution and thus removed previous drift problems even when using conductivity water. The cell was rigidly joined to the instrumeni input terminals by the shortest possible brass rods hollowed to receive banana plugs mounted on the cell. The equivalent circuit was determined from readings on known KC1 concentrations. The resistance and capacitance of the solution in the cell were coniputed froin eq. l a and lb.22 L,, the residual series inductance, was 0.75 x lo-’ henrys and R,, the residual series resistance was 0.40 ohms. Both quantities were constant independent of plate separation and cell contents.
where IC is the Boltzmann constant, N is Avogadro’s number, and M is the molecular weight of the protein. An interesting and useful by-product of the investigation was the finding that electrode polarization effects a t our lower frequencies are a consequence of the siniultaneous presence of zwitterions and singlecharged ions. I n the range 0-5 X M KC1 and as high as 2% @-alanine neither kind of ion by itself produced polarization a t our lowest frequency, 0.5 lIc. The polarization observed when both kinds of ion were present was roughly proportional to the product of their concentrations. Since the polarization contribution from the protein was negligible when 1 or 2% glycine or @-alaninewas present, it was possible to correct for polarization contributions by subtracting the values of e’ observed with a zwitterion solution containing no protein from the value of e’ -__
+ Rp2Cp2w2+ L X w y ( R, - Rx> 1 + Rp2Cp2w2 RP2C,w
R
= ( R2 - (1
1
+ Rp2Cp2w2- ’.> +
C = R P
1
+ Rp2Cp2w2-
Rx)2
+
(1
RP2C,w
CX
+ Rp2Cp2w2
observed when both zwitterion and protein were present and the zwitterion concentration was the same in both experiments. Some experimental results of the polarization effects in solutions containing singlyequivalent parallel resistance; and C, was the equivalent charged and zwitterions are given in Figure 1. Results parallel capacitance. Using different plate separations from a typical experiment on a protein solution showing and plotting C - C, us. the reciprocal of the distance the effect of the correction are given in Figure 2. In between electrodes, the intercept gave C, and it was view of the confusion attending any understanding of found to be 20.0 pf. The real and imaginary parts electrode-polarization processes, this result is of some of the complex dielectric constant of the cell were since in zwitterion-free solutions the protein as calculated23from e‘ = C/Co in which Co = C K C I / D , + . ~ ~ interest ~~; dipole may be contributing to electrode polarization. C K C l = capacitance of a dilute KC1 solution a t the Procedure. Just before use, the protein solutions same fixed plate separation; and Dw.ter= low frequency dielectric constant of water taken as 82.0 a t 15.0°.20 e” = G,/COw in which G, = (1/R - 1/Ro). Ro is (21) We are indebted to Dr. 0. Schmitt for mnking this instrument the low frequency resistance outside the dispersion available. frequency range. The dielectric constant of the solu(22) Dr. D. Britton provided valuable aid in programming and computing. tion a t high frequency well outside the dispersion range (23) “International Critical Tables of Numerical Data. Phvsics. ” . was taken as (82.0 - 0.060~)in which c is the concenChemistry and Technology,” McGraw-Hill Book Co., Inc., New tration of protein in grams/liter of ~ o l v e n t . The ~ York. N. Y . , 1929.
R was the resistance of the solution in the cell in ohms; C was the capacitance of the solution in the cell; C, was the residual capacitance of the cell; R, was the
Volume 69, Number 4
April 1.965
RUFUSLUMRYAND ROBERT HON-SANG YUE
1166
0
0
90 -
-ionsare shown in Figure 3. The dielect,ric paranleters become very large at 10" ronrentration. In general, at concentrations niuch The Journal of Physical Chemistry
PERCENT OF CYTOCUROME -C
Fs"
Figure 3. Dielectric increment and relaxation time of oxidized cytochrome-c at pH ,,oo.
above O.l%, the capacitance dispersion curves for the experiments of this figure follow the simple dispersion equation very closely. At 0.1% protein the data were not sufficiently precise to establish the form of the capacitance dispersion curves, yet the large increases in estimated 7 and AD there and the more precise value at high protein concentrations suggest an unfolding process or an aggregation. The occurrence of such processes on lowering the protein concentration would be remarkable but a t both high and low salt concentrations there are large increases in reduced viscosity of the protein solutions which follow the satne dependence on protein concentration. These were first observed by T a k a s h i i ~ i aand ~ ~ have since been found in our laboratory3* and in that of K o w a l ~ k y . ~ ~ This behavior is currently under investigation and we have deferred further dielectric studies until a clearer interpretation of the viscosity behavior beconies available. The matter is of some concern since oxidation-reduction potentials and spectra are usually deternlined in low protein concentration whereas physical properties are studied at high protein concentrations It is worth noting- that both Dintzis2*and Scheiderlj observed a large increase in dielectric iiicrenient with decreasing protein concentration for both bovine plastlla albunlin BPA and hutilatl plasma albutnin. There be a cOllllllOn cause for behavior ill t,heseproteins and in cytochrome-c. There is also a significant indication t,hat the relaxation time is longer at t,he isoionic point t,han at pH 7.0. Also t'he dielectric increnient is snialler as would be expected. The relaxation tiiiie is 2 X lop8 see. at (29) J. C . Kendrew, G. Bodo, H . 11.Diiitzis. 11. D. I'nrrish, and H . Wyckoff. .\ra:atccre. 181, 662 (1958). (30) D. Keilin and E. F. Hartree, Niochem. J . , 39, 289 (1945). (31) s, Takashimat unpublished results, (32) J. F.Sullivan and 11. Lumry, (33) A . Kowaisky. unpublished results.
results.
DIELECTRIC DISPERSION OF PROTEIN SOLUTIONS
1169
higher protein concentrations which, assuming spherical protein, corresponds Do a diameter of 36.5 A. The ratio of relaxation times for met-Mb and oxidized cytochrome-c a t pH 7 in the presence of zwitterions is 1.5. The ratio of molecular weights without attempting to correct for hydration is 1.4. A few results obtained with reduced cytochrome-c are given in Table 1V. Although the range of con-
Table IV : Reduced Cytochrome-c Salt
% protein
0.79 0.62 0.50
concn.,"
Amino acid
M
5 X 2 X 2%@-alanine 4 X
r
pH
x
108,
AD
6ec.
a
6.40 2 . 4 0 . 9 5 9.80 2.3 0.61 8 . 3 0 2.4d 0.76
... , . .
0.OSb
-OC -0'
'
Dia Calculated as KCl concentration from conductance. electric parameters from Cole-Cole plot. Capacitance dispersion only. K'ot corrected for solvent viscosity. See Table I, footnote e.
centrations examined was not adequate for an unequivocal interpretation, the relaxation time appears to be independent of pH and the dielectric increment strongly dependent on pI-1. The dielectric parameters also increase with decreasing protein concentration. Chymotrypsinogen A . The results of studies of chymotrypsinogen A, CGN, are given in Table V.
Table V : Chymotrypsinogen A" Salt
% protein
concn.,'
Amino acid
0.80 0.75 ... 0.97 ... 0.97 , . . 0.97 ... 0.97 ... 0.97 ... 0.88 ... 0.70" 2% @alanine 0.70" 2% j3-alanine .
.
I
M
1x 1x 1x 2.3 x
T
pH
10-4 9.00 10-4 8 . 5 0 10-4 8.70 10-4 8 . 0 0 4 x 10-4 6.90 5 x 10-4 6.00 5 . 5 X lo-' 5 . 0 0 7.50 1X 3 . 5 x 10-4 7.00 5 x 10-4 6.00
x
lee,
8ec.
AD
6.1 6.6 7.1 6.0 5.3 5.3 4.8 6.4 4.7d 4.7d
1.07 1.24 0.95 1.04 1.20 1.20 1.00 1.00 1.20 1.20
a
0.10 0.10 0.09 0.09 0.08 0.09 0.07 0.09
o o
Calculated Dielectric parameters from Cole-Cole plots. aa KC1 concentration from conductance. Abnormal proteinsee text. Not corrected for solvent viscosity. See Table I, footnote e.
Those obtained in 2% @-alaninesolutions are particularly noteworthy since the protein was found to have a very low sedimentation constant (Sza,,) of 1.8s in
these solutions compared with that of the essentially monomeric protein in solutions of singly-charged ions which is about 2.7s. Dreyer and c o - ~ o r k e r sob~~ served low sedimentation constants in glycine buffer a t pH 3.0 and 0.1 A4 ionic strength but in our work a t neutral pH values no such effects of glycine were observed. Diff erence-spectrum studies in @-alanine did not indicate denaturation and the Cole-Cole plot was a perfect half-circle with dielectric parameters in the range expected for native proteins of this size. Further work on this interesting effect of small zwitterions on the chymotrypsin family of proteins is in progress. For the present, the effect is important insofar as it shows that zwitterions must be used with caution in dielectric studies of some proteins. With no added zwitterions, a was significantly different from zero, suggesting more than one dispersion process. The dielectric increment did not change with pH, which is not surprising since there are only two histidine residues, but there is an indication that the relaxation time increases with increasing pH. Some aspects of this. behavior are associated with dinierization and higher aggregation reactions and will be discussed in another place. Bovine Plasma A l b u m i n Plus Oxidized Cytochrome-c. Figure 4 shows the typical results of an experiment in which two proteins different in dielectric properties are combined in the same zwitterion solution. By itself, a 1.50/, solution of BPA in parallel experiments was found to have T = 1.2 X lO-'sec. and AD = 0.20. The behavior of bovine serum albumin in dielectric dispersion experiments is variable and complicated , ' 5 * z 1 so no attempt was made to obtain other than relative information. The results of Figure 4 were obtained from a solution containing 1.50% BPA plus 0.50% oxidized cytochrome-c in 2% p-alanine solution. The
*E" 4 -
a -
0
94
.e
H
loo
IO*
\
$
IO*-
€'
Figure 4. Cole-Cole plot for 1.50% bovine eerum albumin and 0.50oJ, oxidized cytochrome-c i n 2% p-alanine and 2 X IO-' M salt a t p H 6.70.
(34) W. J. Dreyer, R. D. Wade. and H. Neurath, Arch. Biochem. Bwphys., 59, 145 (1955).
Volume 69, Number 4
April 19fi5
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RUFUSLUMRY A N D ROBERT HON-SANQ YUE
_ .
pH was 6.70 and thus between the isoelectric points of the two proteins in this solution. If no heterogeneous association of the two proteins occurs, the Cole-Cole plot should be resolvable into two circular arcs since the critical frequencies are well separated. Somewhat similar results would be expected in the case that an association occurs but has an association-dissociation relaxation time small with respect to the period of the experimental frequencies. However, it is to be expected that the relaxation time for the protein association-dissociation process would be long relative to 1 Illc. Then, if aggregation occurs with long time constants, a wide variety of different Cole-Cole plots is possible. As shown in Figure 4, a deceptively simple Cole-Cole plot consisting of a single arc is observed over most of the frequency range and only a t lowest frequencies is complex behavior suggested. Only a large a value of 0.22 reveals the complexity of the situation although the average dielectric increment of 0.78 is considerably higher than might have been expected even on the basis of extensive aggregation. The mean relaxation time from Figure 4 was 7.1 X lop8 sec. The sedimentation pattern for the solution consisted of a major coniponent with S20,w= 5.0s and a red component of a very low concentration with the sedimentation constant of monomeric cytochrome-c. The molar ratio of cytochrome-c to BPA was 2 : 1 and it was apparent from the color of the heavy peak that most of i,he cytochrome-c moves with bovine serum albumin. I t might be assumed on the basis of the sedimentation experiments that there were only two components: pure cytochronie-c and a 1 : 2 compound of bovine seruni albumin and cytochrome-c. If it then be assumed that the relaxation time for the formation equilibrium of the compound is long with respect to lov7 sec., the characteristic time of the experiment, the Cole-Cole plot would be expected to consist of two displaced semicircles. This behavior was not observed. The value of a observed is not remarkable bui the very large apparent dielectric increment is. There would appear to be three possible explanations for this peculiar behavior. In the first, we can suppose that the relaxation time for compound associationdissociation is the same order as lo-’ see. Then a t low frequencies the dielectric experiment ‘[sees” only a weighted average of all forms. At high frequencies it “sees” the separate compounds so that high and low frequency results cannot be expected to be compatible. In this case neither dielectric increment nor critical frequency has much immediate significance. I n the second case, it may be assumed that compound equilibrium is either slow or fast relative to sec., but that there is a wide variety of compounds conThe Journal of Physical Chemistry
sisting of different proportions of cytochrome-c to BPA and including some forms making very high contributions to the measured dielectric increment. An appropriate distribution of these compounds could explain the Cole-Cole plot. If this case is correct, it would seem possible to use dielectric dispersion studies to detect heterogeneity not indicated in sedimentation experiments as is the case here. The third possibility is more serious. I t is possible that the dielectric properties of a heterogeneous protein solution are not additive functions of the contributions from the several species. This situation would make useful analysis very difficult. Much more extensive dielectric dispersion studies of mixed protein systems are necessary and this form of study should provide not only a test of the last alternative but also murh unique information about the kinetics and thermodynamics of aggregation reactions.
Discussion Value of the Method. Although the results of this work are generally favorable to the method of dielectric dispersion for protein study, they also indicate possible complexities which must be understood before the full power of the method becomes available for general protein study. The effect of p-alanine on CGN demonstrates the need for caution in choosing “inert” small zwitterions for this work. At the same time it shows that the method can be used to study reactions of this type and particularly to characterize altered forms. In fact, the method is potentially the best way to detect the presence of unfolding in such reactions and has real advantages over viscosity. The experiments with the heterogeneous mixture of BPA and cytochrome-c indicate that the method is not yet applicable to a t least some heterogeneous systems. Similarly, the peculiar behavior of cytochrome-c as protein concentration is lowered must be further investigated. Studies with hemoglobin by Takashima and Lumry’ showed no such effect down to about 0.2% protein to suggest that the phenomenon is not general, and Scheider15has concluded that similar behavior of BPA on lowering concentration is probably an artifact. Aside from these reservations, our experiments provide considerable confidence in the method when used with caution. Of equal importance, they lead to two significant conclusions. The first of these is that small zwitterions can be used effectively in some protein solutions to correct for electrode polarizations, to eliminate electroviscous effects, and finally to eliminate complications due to polarization processes of the counterion atmosphere. The second conclusion is that dielectric parameters obtained in the absence of
DIELECTRIC DISPERSION OF PROTEIN SOLUTIONS
1171
zwitterions and a t low salt are generally free of ion Table VI atmosphere and electroviscous effects. This is apTwo-center, indirect acid parent from the siniilarity of results obtained with and &of k d Pi and P2 are equivalent HzO PiH +P I without zwitterions. Electroviscous effects are exHa0' + Pp- +P2H ks{ groups.' pected to be a minimum a t the isoelectric point but Two-center, indirect base even in experiinents carried out a t pH values some OH- f PIH +P I HzO k d ( PI and PZare equivalent distance from the isoelectric point, there is little eviPP- HzO +P2H OH- k,\ groups. dence in this work for the occurrence of new dispersion Two-center, direct processes in zwitterion-free solutions. This simple Pi and Pi are equivalent situation may not always exist, however, since S h ~ t t ~ ~PiH + PP- +Pi PzH groups-direct contact or conobserved a strong pH dependence of dispersion betact through organized havior in ovalbumin solutions which suggests the apwater bridge. pearance of some relaxation mechanism other than Four-center, indirect acid dipole rotation a t conditions of large net charge. On Hp0 + CiH +Ci- + are nonequivalent the other hand, Takashima3'j could not confirm these H30+ + Ai +AiH + H20 findings. AzH + Hp0 +A2- + H 3 0 + It niay also be noted that even though heterogeneity C2H + H20 Cp- + HaO++ as measured by QI was not large for most of our protein Four-center, direct solutions, the presence of zwitterions reduced QI to zero CiH + A i - + CiAiH A and C are not equivalent and considerably improved precision. No reason for A2H + Cz- +Az- + CIH groups. Direct contact or this behavior can be given. If a protein molecule has contact through organized high asymmetry, multiple relaxation processes should water bridge between C1 and A,, C2 and A2. in general be detectable. Our proteins with the exception of BPA may be too symmetrical to make mula Pi and P2, for example, do not need to be on the same moletiple relaxation detectable. On the other hand, some cule. At low ionic strength or low zwitterion concentration there of the multiple dispersion previously reported may be is a correlation among charges on a given protein molecule. incorrect or due in part to small ion-atmosphere effects These fluctuations are an important aspect of the KirkwoodShumaker theory but their times in low electric field are very eliminated by the use of zwitterions. similar to the values for the group-pair processes. The fluctuaAlthough new studies using Zwitterions of proteins tions can occur no faster than the appropriate transfer processes previously investigated are necessary to support any and the charge-charge correlation has only the effect of slightly general conclusion, our results indicate that the dimodifying the rates of these processes. electric relaxation processes of proteins thus far studied are primarily those of protein orientation. I t is interesting to note that insofar as dielectric disthe pseudo-first-order constant for association and is persion is coiwerned, within our experimental errors thus usually the reciprocal of the larger of these two. met-Mb and COBIb are identical. Similarly, insofar The magnitude of the effect depends on the product of as the dependence on protein concentrations allows the number of the proton-donating and proton-acceptcomparisons, oxidized and reduced cytochrome-c are ing sites on a single protein molecule, and thus this very similar. Hence any conformational differences magnitude is significant only near the pH equal to between iron(I1) and iron(II1) forms of these proteins the effective pK, or pKb of the group involved dependhave only a small effect on the charge distribution or ing on whether the mechanism is protolysis or hyshape. if there is a large number, say 100, drolysis. Thus The Kirkwood-Shumaker Effect. This effect can of a particular kind of group with essentially the same only be due to proton migration by two mechanisms. effective pK, at one pH unit from pK each protein (1) The Indirect-Transfer Process. The indirect molecule has about 10 sites of one kind and 90 of the process shown in Table VI occurs when a proton leaves other. A 10% perturbation of the charge distribution an actid group and migrates to a basic group on the same away from the purely random distribution and due to or a different protein molecule. At low hydrogen ion the applied electric field could thus produce a meanconcentrations and low protein concentrations the protein has a better chance of being picked up by the molecule from which it was released. The effective (35) W. J. Shutt, Trans. Faradav SOC.,30, 893 (1934). time constant is as usual the reciprocal of the sum of (36) S. Takashima. Abstracts, 148th National .Meeting of the Amerithe true first-order rate constant for dissociation plus can Chemical Society, Chicago, Ill., Aug. 31. 1964. p. 31C.
+
+
+
+ +
+
+
Volume 69, Sumber 4
April 1966
1172
square dipole moment of from 102d2 to 5 x 103d2 depending on the dimension of the protein. This is generally considerably less than is observed experimentally. The smaller the protein the smaller the contribution from this source. Since a 10% perturbation is much larger than can occur in this kind of experiment, these contributions are certainly an upper liinit of the effect. At 2 pH units from the pK of the group the effect would be negligible. However, if the characteristic time of the proton migration is of the same order as the rotational relaxation time and the experiments are carried out very near the pK of the responsible group, multiple or complex dispersion with a contribution from the Kirkwood-Shumaker effect niay be detected. This matter has also been discussed by Scheider.15 If the Kirkwood-Shumaker relaxation time is very short relative to the rotational relaxation time, the proton migration will always be in phase with the applied field in the frequency range of rotational relaxation and the dielectric increment will contain the contributions from both processes. The relaxation time in this frequency range will, however, be that for rotational relaxation. If em can be chosen experimentally in a frequency range below that of the dispersion of the Kirkwood-Shumaker process, eo - E , will be independent of the latter process and the correct dielectric increment due to the dipole moment of the protein can be determined. It is important to note that the perturbing field used in mosi dielectric dispersion experiments is much smaller than kT and cannot produce a net change in the number of different kinds of protein charged groups. The field can only influence normal charge fluctuations to occur in such a way as to lower the electrostatic potential energy. Thus in the presence of the applied field a proton will tend to migrate from one carboxyl group to a crirboxylate group lying close to the negative electrode. ’\ligration from ammonium to a carboxylate group can occur only if the reverse process also occurs at some other place in the protein system. Both migrations can be influenced by the field in this “four-center” process. Two-center and four-center indirect-transfer processes are shown in Table VI. It is to be noted, however, that insofar as relaxation times are concerned, all four-center indirect processes can be considered as a sum of two-center indirect processes. Four-center processes between groups with well-separated pK, values can make only minor contributions to the induced moment since the magnitude of the contribution depends on the product of concentrations 3f acidic and basic forms of both kinds of groups. This product can be large only if the The Journal of Physical Chemistry
RUFUSLUMRY AND ROBERT HOWSANG YUE
pK, values are close together and close to the pH of the experiment. (2) The Direct-Transfer Process. Proton migration can occur through direct contact of donating and accepting groups or through complete hydrogen-bonded water bridges between such groups. Once again the contribution to dielectric increment will be significant only at pH values such that appreciable concentrations of both donor and acceptor groups exist. Although there is no precise description of protein hydration, hydration water will be organized to favor charge hydration and in so doing may effect a slight increase in the average length of water bridges for proton migration. It is unlikely that the effect will be large and thus unlikely that two-center direct processes can occur at a sufficient distance to make an important contribution to the induced moment unless there are many of them. It is also unlikely that side chains of glutamic, aspartic, histidine, or lysine residues have the freedom of motion which would allow direct contact for proton transfer to occur over appreciable distances. This is a consequence of the hydrophobic character of their tethering chains which generally requires the hydrocarbon sections to be well surrounded by other hydrocarbon parts of the protein. In fact, these tethers probably make at least as large a contribution to hydrophobic binding as any other side chains. This situation also explains why it is unlikely that we need consider dielectric effects due to the perturbation of charge group positions by rotation along the tethering chains. The direct-transfer processes are diagrammed in Table VI. The times of these processes can be variously estimated from the protolysis, hydrolysis, and direct-transfer processes of small molecules when the rates for the appropriate models are known. The calculated times for the protein groups may be slightly different from true values since the local proton concentration about a protein which has just released a proton will be slightly higher than the bulk average. This effect has been discussed by Scheider.ls Furthermore, the times may be a bit smaller because of the solid-angle limitations on proton and water approach to protein charged groups in diffusion-controlled processes. The times estimated in Table VI1 are reasonable approximations and will have to be improved with direct studies of proteins. Table VII is incomplete since a few additional model reactions remain t o be studied. A consideration of Table VI1 shows that for rotational relaxation times in the range of lo-’ to 3 X l o p 9sec. in the pH range from 6.5 to 9, proton migration processes
DIELECTRIC DISPERSION OF PROTEIN SOLUTIONS
1173
Table VI1 Type of mechanism
Indirect Indirect Indirect Indirect Indirect Indirect Direct Direct
Direct
-coo -
acid acid acid base base base
-NHZ Imidazole
-coo -
-NH, Imidazole -COO-, -COOH -NHZ, -NHa+
+ -NHz { -COOH -NHs+ + -COO-
1
Model
Group
/-COOH Direct -ImH +
r is the larger of
Ref.
Two-center proton migration times Acetic acid 10-6 or 3 X 10-11[H+] -1 sec. Ammonia 5 X lo-* or 3 X lO-lI[H+]-l see. Imidazole 5 X or 5 X 10-ll[H+-]-l sec. Acetic acid 1 or 10-ll[OH-] sec. Methylamine 5 X 10-8 or 3 X lO-ll[OH-] --I sec. Imidazole 5 X l o w 4or 3 X 10-ll[OH-]-l sec. Acetic acid formate 2 X 10-8 see. Methylamine 10-9 see. (direct) 10-8sec. (through single water molecule bridge) Four-center proton migration times Acetic acid-hydrazine sec.
a
a a
b C
a
a d
a
+ -1m + -coo-
Acetic acid-imidazole
2 X
see. a
XI. Eigen and L. De Maeyer in “Technique of Organic Chemistry,’’ A. Weissberger, Ed., Vol. VIII, Part 11, Interscience Publishers, Inc., New York, N. Y., 1963, p. 1031. Z. Luz and S.Meiboom, J. Am. Chem. Soc., 85,3923 (1963). ‘ Calculated from data S.Meiboom, A. Loewenstein, and S. Alexander, J. Chem. Phys., 29, 969 (1958). in footnote a.
‘
are almost invariably too slow or too improbable because of the concentrations of donor and acceptor to make contributions to dielectric behavior. Only the direct reactions are sufficiently rapid to provide coniplications. With the information currently available, the only’ such process which might provide a detectable contribution would be the four-center direct exchange between iniidazolium and carboxylate groups. Concentration factors make its importance unlikely even a t pH 7, the pK, of the imidazole, and it is certainly unimportant at much higher pH values. I n general, the groups with proton relaxation times in the range of small globular protein rotational relaxation times have pK values outside the central pH range and thus are of little interest unless the protein average moment is very small. Experiments with large proteins and a t pH values near carboxyl and ammonium pK values may be complicated by Kirkwood-Shumaker effects. However, T a k a ~ h i n i ahas ~ ~reported dielectric dispersion experiments with ovalbumin and BPA over a wide range of hydrogen ion concentrations including pH values near the pK, values of these two important ionizing groups. and he has not found either new dispersion processes or changes in dielectric behavior which indicate any contribution for the KirkwoodShumaker effect. Except under the unusual circumstances in which the Kirkwood-Shumaker effect may still present complications, it would now appear possible to carry out meaningful dielectric dispersion experiments on globular proteins a t least aslarge as BPA. Each new pro-
tein should first be examined using a zwitterion atmosphere and probably any drastic change in solution composition such as a large change of pH should also be examined in this way. If the zwitterion-protein solution demonstrates normal dispersion, and especially if this dispersion is essentially identical with that obtained in the absence of zwitterions, the dispersion can be assumed to be due to rotational relaxation and the electrostatic interaction to be due to the rootmean-square moment of the protein itself. I t is to be noted that although the apparent value of this moment must depend on the ion atmosphere, the presence of large concentrations of zwitterions has little if any effect on it. This behavior is another manifestation of the absence of long-range order in the zwitterion atmosphere. I t may also be useful in some cases to apply a viscosity test for rigid rotator behavior in dielectric dispersion. Takashimas studied the effect of solvent viscosity on the dielectric dispersion of met-Mb, ovalbumin, hemoglobin, and catalase. The first three proteins manifested behavior expected for a rigid rotator and not that expected if the viscosity dependence were due to changes in the motional times of small ions. The viscosity test is not infallible but can be useful either to supplement the zwitterion test or to replace it when the latter is inapplicable. Takashima’s results with the first three proteins cannot be generalized to all globular proteins, since he found that the dispersion behavior of catalase was independent of solvent viscosity. This is a most interesting Volume 69,Number
4
April 1965
J. LEOXARD A N D H. DAOUST
1174
result and suggests that further dielectric studies of catalase may lead to a more profound understanding of the method. It is to be hoped that the combination of studies by Takashima, Scheider, and those reported here will help to restore confidence and interest in the dielectric dispersion niethod for protein study since the method is potentially a powerful tool for such investigations.
Much work is obviously required to make the method suitable for routine use. There is a variety of applications of the method which when developed will provide useful and often unique inforniation about globular proteins. In addition, it may be noted that the by-product investigations in the present work present new and interesting phenomena for investigation.
Variation of Chain Dimensions of Polystyrene with Concentration'
by J. Leonard and H. Daoust Department of Chemistry, Uniaersitd de Montrhal, Montrdal, Canada
(Received September 68, 1964)
Osmotic pressures of polystyrene solutions have been measured at 25 O for concentrations up to 17Oj,, using dioxane and chlorobenzene as solvents. The data have been analyzed according to three different theories which deal with the concentration dependence of polymer dimensions. The variational theory developed by Fixman seems to give a more complete picture of the behavior of polymer chains in solution. The theories of Eizner and Grimley are theoretically applicable only in the dilute concentration range.
Introduction According to Fixman,2 the apparent second virial coefficient, S,varies with concentration for polymers in good solvents; it has been shown that the same phenomenon exists also for intermediate solvents. In the model used by Fixman, it is assumed that the intermolecular potential of average force of the system of polymer molecules in solution is a sum of pair potentials. The radial distribution function is calculated on the basis of the model of Flory and Krigbaum, through a variational solution of the Born-GreenKirkwood equation. The concentration dependence of the excluded volume factor a has been obtained by the minimization of the free energy with respect to a. We here recall the most important relations needed in the application of Fixnian's treatment. The empirical definition of S is given by S = (E/c - RT/M)(RTc)-* (1) where E is the osmotic pressure; c , the polymer conThe Journal of Physical Chemistry
centration; and M , the molecular weight of the polymer. In Fixman's variational theory,2 S is defined by
RTS
=
2i"o~~o~B'-'''M-~No(l - 7)
(2)
where A. is the intermolecular potential parameter defined by
A0
=
5.63(ao2 - 1)
(3)
and where a. is the excluded volume expansion factor a t zero concentration. B' is given by
B'
=
9.61/Ro2
(4)
Ro2 being the mean square end-to-end distance. N o (1) This pnper has been presented in part a t the 145th National Meeting of the American Chemical Society, New York, N. Y., Sept. 8-13, 1963. (2) M. Fixman, J . Chem. Phys., 33, 370 (1960); Ann. N . Y . Acad. S c i , 89, 657 (1961); J . Polymer Sci., 47, 91 (1960). (3) J. Leonard and H. Daoust, ibid., 57, 53 (1962).
