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The Journal of Physical Chemistry, Vol. 83,
(3) A. L. Cummings and E. M. Eyring, “Chemical and Biological Appllcations of Relaxation Spectrometry”, E. WynnJones, Ed., D. Reidel, Boston, 1975. (4) A. L. Cummings and E. M. Eyring, Biopolymers, 14, 2107 (1975). (5) T. Yasunaga, T. Sam, K. Takahashi, H. Takenaka, and S. Ito, Cbem. Lett., 405 (1973). (6) T. Yasunaga, T. Yoshikuni, T. Sano, and H. Ushio, J . Am. Cbem. Soc., 98, 813 (1976). (7) T. Sano, T. Yasunaga, Y. Tsuji, and H. Takenaka, Cbem. Insfrum., 6, 285 (1975). (6) The value of 6 cm3/molat the midpoint of the h-c transition seems too high; the volume change associated wlth internal proton transfer Is approximateb equal to the difference of A V’s of the two elementaty steps in eq 2. The net volume change should be very small. The relaxation amplitude in eq 1 depends on (AV)2.
George W. Tin John E. Stuehr”
Department of Chemistry Case Western Reserve University Cleveland, Ohio 44 106
Communications to the Editor
No. 22, 1979
strength both the calculated volume change for charge transfer and the compensatory positive volume change due to the relatively smaller degree of ionization in helical segments decrease and their difference remains small. Thus, while the naive model of ref 2 is hardly a basis for the interpretation of the volumetric titration curve, at least its result is not wildly at variance with the experimental A V for the helix-coil transition. With respect to the choice of kinetic parameters, the broken curve in Figure l a of ref 1 ( T as a function of pH) is equivalent to, and in agreement with, the lower broken curve in Figure 2 of ref 2. More generally, it is clear that any assumed mechanism for internal proton transfer which proceeds by acidic or basic ionization k
A,H
Received March I, 1979
A,Response to Comments by G. W. Tin and J. E. Stuehr
Sir: The origin of such substantial differences as exist between the calculations and conclusions in the preceding comment by Tin and Stuehr (hereafter ref 1) and our results (ref 2) are implicit in footnote 8 of ref 1and in the choice of the bimolecular rate constant in eq 2 of ref 1. The proposition that the volume change for the ionization of helical and coil side chains depends on the geometrical parameters which characterize the two regions and on the state of ionization of neighboring residues, much as the volume change for the ionization of low-molecularweight dibasic or polyfunctional acids depends on the position and state of ionization of neighboring charged or polar groups,3 is of course central to our nomination of the perturbation of internal proton-transfer equilibria as a possible source of acoustic absorption in poly(g1utamic acid). We have sought to estimate the size of this effect with the aid of simple expressions for the electrostatic free energy of the two regions: the Hermans-Overbeek4 equation for the coil and Hill’s result for the uniformly charged cylinder5for the helix. The rationale for the choice of the geometrical parameters is given explicitly in ref 2. As in any order-of-magnitude calculation of this sort it will be possible to quarrel both with the model and the choice of parameters and it does not appear to us to be profitable to enlarge upon ref 2. However, electrostatically, the transfer of a proton from a helical to a random coil region is equivalent to the transition of a negatively charged carboxyl group from the coil to the helical conformation. It is thus perhaps desirable to reconcile the large negative electrostrictive volume change (-6 cm3/mol at the midpoint of the transition) estimated in ref 2 for proton transfer at constant geometry with the fact that the overall volume change for the helix-coil transition6 is small and relatively insensitive to ionic strength. The overall degree of dissociation ( a )of the helix is less than that of the coil and hence, near the midpoint of the transition, the conformational change is accompanied by protonation of the side chain. The volume change for this process is large and positive (-11.4 cm3/mol). There would be no net electrostatic contribution to AV for the were approximately 1 / 2 helix-coil transition if q,elix/acoil and this is rather close to the ratio implied by the titration curves7 a t low (0.02 M) ionic strength. At higher ionic 0022-3654/79/2083-2930$01 .OO/O
5A,- + H+ kb
k’
+ H 2 0& A,H + OHkbl
(la)
(W
with a pH independent bimolecular rate constant will predict a maximum relaxation time (in water near room temperature) near pH 7 as does the lower broken curve in Figure 2 of ref 2. However, in the case of the diffusioncontrolled bimolecular association reactions of spherically symmetric ions of opposite charge the effect of the ionic charge is to increase the reaction rate by a factor8 f = $(rd)/kT{exp[$(ra)/kT]- 1)where $(rd) is the potential energy of interaction of the ion pair at the effective radius for reaction. Both the pulsed electric field and the acoustic measurements are carried out at low ionic strength where the Debye length is considerably greater than the interresidue spacing. Qualitatively, one would expect that under these circumstances the bimolecular rate constant a t a given carboxylate group on the polypeptide anion will depend on the degree of titration of neighboring groups. The solid curve in Figure 2 of ref 2 represents an effort, based on the model and the parameters used to calculate AV, to estimate the electrostatic enhancement of the reaction rate for the conditions employed by Yasunaga and his co-worker~.~ Again, other models, other parameters, and other conclusions as to the magnitude of the effect are doubtless possible. In particular, the use of a result appropriate to spherical geometry may be questioned. We would argue that the calculation of the rate of intramolecular proton transfer from the rate constant appropriate to a single-charged carboxylate ion will generally underestimate that rate when the polyelectrolyte is highly charged and the ionic strength is low. The helix-coil transition in poly(g1utamic acid) occurs near neutral pH where intramolecular proton transfer based on acidic and basic ionization reactions is relatively slow. Rather faster acoustic relaxation times near the midpoint of the helix-coil transition have been reported by Parker, Slutsky, and Applegatelo in poly(L-lysine) (T = 4.3 X s a t pH 10.2 and an ionic strength of 0,6 M) and by Hammes and Roberts1’ in poly(ornithine) ( T = 1.7 x s at pH 11.2 and an ionic strength of 0.2 M in 15% methanol-85 % water). The relaxation frequency for the mode qualitatively describable as intramolecular proton exchange is roughly kb([H+]+ KI for eq l a and k{([OH-] K j for eq lb, where K and K’are respectively the equilibrium constants for eq l a and Ib. For simple carboxylic acids12k b N 5 X 1O1O M-l s-l, for protonated amine groups kb’ = 2 X 1O1O M-l s-l.13 The midpoint of the helix-coil transition occurs in the region where the side chains titrate, so, with no acceleration due to charged neighboring groups, the relaxation frequency at the midpoint is about loll. [H+Imids-l in acidic solution and 4 X 10’OIOH-],id s-l in
+
0 1979 American Chemical Society
The Journal of Physical Chemistry, Vol. 83, No. 22, 1979 2931
Communications to the Editor
basic solution. The relaxation times so estimated, 1.6 X lo4 s for PLGA, 1.6 X s for poly(lysine), and 1.6 X lo-* for poly(ornithine), are of the same order of magnitude as those which have been observed in each of these systems and attributed to perturbation of the helix-coil equilibrium. To summarize, we argue that the elementary theory of diffusion-controlled reactions predicts a relaxation time for intramolecular proton transfer near the midpoint of the helix-coil transition in several uniform polypeptides comparable with the time which has been attributed to the transition itself. A simple electrostatic model suggests that the volume change for intramolecular proton transfer near the midpoint of the transition may be appreciable. The effect at low ionic strength of the high charge on the coil side of the transition is to decrease the relaxation time and thus to produce a pH dependence of the relaxation time qualitatively more like that predicted for the transition than the result for independent ionizations. Thus, the perturbation of internal charge transfer equilibria is worth considering as a possible contributor to the acoustic relaxation spectrum of uniform polypeptides. We believe that it is an open question whether acoustic absorption due to perturbation of equilibria involving the growth and contraction of the helix, uncomplicated by other concomitant relaxation processes, has been observed.
References and Notes G. W. Tin and J. E. Stuehr, preceding article in this issue. L. Madsen and L. J. Slutsky, J. Phys. Chem., 81, 2264 (1977). W. Kauzmann, A. Bdanzsky, and J. Rasper, J. Am. Chem. Soc., 84, 1777 (1962). C. Tanford, "Physical Chemistry of Macromolecules", Wiley, New York, 1961, p 482. T. L. HIIi, Arch. Biochem. Blophys., 57, 229 (1955). H. Noguchi and J. T. Yang, Blopo/ymers, 1, 359 (1963). M. Nagasawa and A. Holtzer, J. Am. Chem. Soc., 86, 536 (1964). P. Debye, Trans. Electrochem. SOC.,02, 265 (1942). T. Yasunaga, Y. Tsuji, T. Sano, and H. Takenaka, J. Am. Chem. SOC.,98, 813 (1976). R. C. Parker, L. J. Slutsky, and K. R. Applegate, J. Phys. Chem., 72, 3177 (1968). G. G.Hamrnes and P. B. Roberts, J . Am. Chem. Soc., 91, 1812 (1969). M. Eigen and L. DeMaeyer, Tech. Org. Chem., 8, 1034 (1963). K. R. Applegate, R. C. Parker, and L. J. Slutsky, J. Am. Chem. Soc., 90, 6909 (1968).
Department of Chemistry B E 10 University of Washington Seattle, Washington 9 8 795
L. J. Slutsky' L. Madsen
Received June 4, 1979
Water Penetration into Surfactant Micelles Publication costs assisted by the Universky of Lund
Sir: One crucial problem concerning the aggregation of amphiphilic molecules in an aqueous medium refers to the hydration of micellar aggregates. The classical picture of a micelle, first clearly stated by.Hartley in the 1930's, as an essentially spherical aggregate formed to minimize the contact between hydrocarbon (or fluorocarbon) chains and water (reviewed in ref 1) has been questioned on several occasions, most recently by Menger.2-4 In this communication we want to point out some aspects of the micelle hydration problem for ionic amphiphiles which were not treated by Menger. Water-micelle interactions have been studied with thermodynamic, transport, and spectroscopic techniques. The two former techniques rather invariably support the idea of a limited water-amphiphile interaction. As an
TABLE I: Hydration Numbers of Micellar Aggregates Expressed as the Number of Water Molecules per Amphiphile Ion amphiphile"
hydration no.
C ,,OSO,Na
a C, is a n-alkyl chain with n carbon atoms. P. Mukerjee, J. Colloid Sci., 19,7 2 2 (1964). F. Tokiwa and K. Ohki, J. Phys. Chem., 71, 1343 (1967). P. Ekwall and P. Holmberg, Acta Chem. Scand., 19,455 (1965). e B. Lindman and B. Brun, J. Colloid Interface Sci., 42, 388 (1973). M:C. Puyal, private communication.
example some hydration numbers determined by viscosity and diffusion methods are summarized in Table I. These (global) hydration numbers correspond approximately to the hydration of the polar head groups and the counterions (cf. ref 5 and 6). As regards the spectroscopic techniques, the picture is more ambiguous and such studies have in certain cases given rise to suggestions about a more hydrated micellar interior. A typical spectroscopic experiment concerning the micelle-water contact is to observe a parameter of a probe, intrinsic or added, and to correlate the value with that found in a pure solvent of a given polarity. Such an approach is connected with two problems, the seriousness of which is too often overlooked. First, to be able to deduce anything about the molecular environment in the micelle one must independently determine how the probe is distributed in the micelle. Secondly, the relation between the spectroscopic parameter and the nature of the molecular environment of the probe should either have a sound theoretical basis or it should be carefully experimentally tested. Several studies of water and ion penetration into micelles are open to criticism on the basis of one or both of these points. For example, Menger et aL3 studied water penetration with the amphiphile CH3(CH2)7C(=O)(CH2)7N+(CH3)3Br-. These authors measured the I3C NMR chemical shift of the carbonyl carbon and correlated it with the shift obtained in a range of solvents. In this case, there is neither an interpretation of the molecular cause of the shift nor an independent investigation of the distribution of the carbonyl group in the micelle. The two other experimental investigations cited by Menger in favor of a strong water penetration is due to Muller and co-workers' and Svens and Rosenholm.8 Recently Muller has showng that his earlier interpretations of 19Fchemical shifts in fluorocarbon systems were too simplistic and lacked a sound theoretical basis. As regards the X-ray scattering results of Svens and Rosenholm? it is clear that the particular model applied in the analysis of the data gives unphysical results and the investigation is at present inconclusive. A t the concentrations of the actual measurements, the measured polar radius implies that approximately 150% of the total solution volume is occupied by the spherical micellar aggregates! In conclusion, it seems that at present there is little reason to abandon the classical picture of a micelle as an aggregate with an apolar interior and a polar surface in contact with the aqueous medium.lOJ1 The structural similarity between different surfactant-water phases being clearly established, it is also pertinent to note the small hydration numbers observed for lamellar mesophases,12
0022-3654/79/2083-2931$01.00/00 1979 American Chemical Soclety
