L. Madsen and L. J. Slutsky
2264
rR4,+(LAmode) is the ionic volume in the extended state, because the LA mode arises from the molecule taking the all-trans form. Therefore r R N+(LAmode) does not correspond to rR4N+(SPT),which represents the hard core volume of ions and it follows the general relation that rR4,+(LA mode) > ~R~N+(SPT). In conclusion, it was confirmed that the molecular structure of the tetraalkylammonium ion in aqueous solution is an extended one, such as the all-trans from in CH2 chains, especially for lower members of n-paraffins. The ionic radii of the tetraalkylammonium ion shown in the first to forth columns of Table 111, correspond to an effective radii in aqueous solution. In addition, n-paraffins containing long CHpchains are, on the average, not in such an extended state and therefore their effective ionic radius should be much smaller than that obtained from the LA band.
References and Notes (1) S . Mizushima and T. Shimanouchi, J . Am. Chem. Soc., 71, 1320 (1949). (2) R. F. Schaufele and T. Shimanouchi, J. Chem. phys., 47, 3605 (1967). (3) R. F. Shaufele, Macromol. Rev., 4, 67 (1970).
(4) I. Sakurada, T. Ito, and K. Nakamai, J. Polym. Sci., C15, 75 (1966). (5) T. Shimanouchi, M. Asahina, and S. Enomoto, J. Polym. Sc/., B9, 93 (1962). (6) W. L. Peticolas, G. W. Hibler, J. L. Lippert, A. Peterlln, and H. G. Olf, Appl. Phys. Lett., 10, 87 (1971). (7) M. J. Folkers, A. Keller, J. Stejny, P. L. Goggin, G. V. Fraser, and P. J. Hendra, Colloid Polym. Sci., 253, 354 (1975). (8) H. 0. Oif, A. Peterlin, and W. L. Peticolas, J. Polym. Sci., 12, 359 (1974). (9) J. L. Koenig and D. L. Tabb, J. h c r o m l . Sci'.,Phys., BO, 141 (1971). (10) H. Okabayashi, M. Okuyama, T. Kitagawa, and T. Miyazawa, Bull. Chem. SOC.Jpn., 47, 1075 (1974). (11) H. Nomura and Y. Miyahara, Bull. Chem. Soc. Jpn., 48,2779 (1975). (12) S. Mlzushlma and T. Shimanouchi, Nippon Kagaku Zasshi, 64, 1064 (1943). (13) R. F. Shaufele, J. Chem. Phys., 49,4168 (1968). (14) J. F. Connolly and G. A. Kandalic, Phys. Fluids, 3, 463 (1960). (15) N. C. Stephenson, Acta Crystallogr., Sect. B , 17, 587 (1964). (16) A. Zalkin, Acta Crystallogr., Sect. B , 10, 557 (1957). (17) J. T. Edsail, J. Chem. Phys., 5, 225 (1937). (18) T. Shlmanouchi and S. Mizushima, Nippon Kagaku Zasshi, 64, 1215 (1943). (19) J. Levkov, W. Kohr, and R. A. Mackay, J . Phys. Chem., 75, 2066 (197 1). (20) R. A. Robinson, and R. H. Stokes, "Electrolyte Solutions", 2nd ed, Butterworths, London, 1959. (21) W. L. Masterton, D.Bolocofsky,and T. P. Lee, J. Phys. Chem., 75, 2809 (1971). (22) A. Bondi, J. Phys. Chem., 66, 441 (1964).
Relaxation Times for Acid Ionization and Internal Proton Transfer in Polypeptides in the Neighborhood of the Helix-Coil Transition L. Madsen and L. J. Slutsky" Department of Chemistry, University of Washington, Seattle, Washington 98 I95 (Received January 14, 1977;Revised Manuscript Recelved Ju/y 22, 1977)
Relaxation frequencies and normal coordinates are calculated for the system of coupled reactions formed by the acid ionizations at glutamyl residues in poly-L-glutamic acid. The results suggest that the perturbation of overall and internal ionization equilibria constitute a reasonable alternate interpretation of relaxations which have been attributed to perturbation of the helix-coil equilibrium.
I. Relaxation Times for Intramolecular Proton Transfer The kinetics of the proton-transfer reactions involved in the relaxation of fluctuations in the overall and internal ionization equilibria in solutions of proteins and uniform The polypeptides has been the object of some role of perturbation of proton-transfer equilibria in determining, and perhaps in complicating, the interpretation of relaxation spectra in the neighborhood of a reversible pH-induced conformational transition is also an object of c ~ n c e r n . ~We - ~ wish here to discuss the normal modes of the system of acid ionization reactions (reaction 1)of side -AHi
kfi
F=-=
-Ai
+ H'
kbi
chains in the helical (i = 1) and coil (i = 2) regions of a uniform polypeptide coupled to the ionization of an indicator (i = 3), to consider the application of these results to the interpretation of spectra observed in solutions of poly-L-glutamic acid in the neighborhood of the helix-coil The Journal of Physical Chemistry, Vol. 81,No. 24, 1977
io - a kb~[-A;] kb3[-A3]
W2O-W
kbi [-A;] kbz[-A;]
kb3[-&1
03'-
&i
[-A;]
0
=0
(2)
2265
Helix-Coil Transitions in Polypeptides
4
\
0 x 4
\
1 \ 1 OO
fh
1
O'
5
I
I
I.o
I
1
7
6 PH
Figure 1. Calculated relaxation fre uencies for proton exchange between poly-bglutamic acid (6.3 X 10 reskiue molar) and bromocresol M) as a function of pH. Circles are the relaxation purple (3 X frequencies determined by Cummins and Eyring.
3
'r
5
8
7
6 PH
r
Figure 3. The chemical factor for intramolecular proton transfer (solid curve in lower panel) and indicator-proton exchange (broken curve in lower panel) and the contribution of acid dissociation ( b )to the normal mode qualitatively described as intramolecular exchange as a function of pH. The upper panel gives calculated and observed values of the absorption amplitude (C) as a function of the fraction of the polymer present in the helical state ( f,) under the experimental conditlons of ref 15. The bar in the upper panel represents the average experimental error.
partial dissociative character is plotted, as a function of pH, in Figure 3. In a spectrophotometric E-jump experiment peptideindicator exchange is the mode which would be observed. The computed relaxation frequencies for this mode plotted in Figure 1are in very good agreement with those observed by Cummings and Eyring. It does not appear necessary to include the helix-coil transition in the kinetic basis set to explain the results of ref 8. In basic solution, and close to neutrality, the basis set must be expanded to include reactions of the form kf'
Figure 2. Calculated relaxation frequencies for overall ionization (-- -) and internal proton transfer (solid and broken, ., curves). The solid curve is calculated with the aid of Hill's expression for the potential at distance r from a cylindrical polyelectrolyte'* with geometrical properties of the helix and coil from the work of Rinaudo and Domard." The lower broken curve (- --) is Calculated neglecting the electrostatic effect of neighboring residues on the ionization kinetics of a given protonated glutamyl residue. Circles are the relaxation frequencies observed in ref 9.
-- -
-
7.7 X 1O1O s-l M-l) for the indicator are those determined by Lentz, Hutchins, and Eyring.lo Values of pK, for helical and coil residues are derived from the titration curves of Rinaudo and Domardll and the bimolecular rate constant for recombination at the glutamyl residues is taken to be 5 X 1O1O M-l s-l by analogy with simple carboxylic acids.14 The normal modes correspond fairly closely to overall ionization, proton exchange between helical and coil residues, and proton exchange between peptide and indicator. However in a conductiometrically sensed E-jump experiment the admixture of overall dissociation in the mode qualitatively describable as intramolecular proton transfer is a matter of some importance. Hence, if the internal transfer is represented by AIH + (1- b)Az- = A i + (1- b)A2H + bH+,the coefficient (b)which specifies the
-A-+ H,O-
kb'
-AH + OH-
(3)
In Figure 2 the calculated relaxation frequencies for overall ionization and internal proton transfer for poly-L-glutamic acid with eq 1and 3 as a basis are compared with the times observed by Yasunaga, Tsuji, Sano, and Takenakag in a conductiometrically sensed pulsed E-field experiment. The relaxation frequencies are calculated for the polymer concentration residue molar) of ref 9 with values of pK,, as a function of pH, derived from the titration curve given in ref 9. The experimental relaxation) which is interpreted by Yasunaga et al. as being the time for the helix-coil transition) is found to be independent of polymer concentration as is the calculated time for internal proton transfer. The calculated relaxation frequencies exhibit a minimum slightly on the acid side of neutrality as do the experimental values of 7-l. However the calculated times for the internal proton transfer are an order of magnitude greater than the times observed in ref 9 and it might legitimately be concluded that internal proton transfer is not what is observed by Yasunaga et al. Alternately it might be argued that the local concentration of hydrogen ion in the immediate neighborhood of the polymer, and The Journal of Physlcal Chemistry, Vol. 8 1, No. 24, 1977
L. Madsen and L. J. Slutsky
2200
hence the rate of intramolecular proton transfer, is significantly higher than the bulk concentration. There does exist a plausibility argument in favor of the second of these propositions. In a medium of dielectric constant D and inverse Debye length K , the electrostatic free energy change associated with the transfer of a proton from an infinite distance to distance r from a uniformly charged cylinder of radius p with an average linear charge density Zell isI2
(4) where e is the charge of the electron, KO and K1 are modified Bessel functions of the second kind of order 0 and 1. Rinaudo and Domard" estimate p = 8 8, for the poly-L-glutamic acid extended coil, 16 8, for the helix. If f and f'are respectively the average degrees of ionization of coil and helical residues (in this case estimated from the titration curve in ref 9 then for the extended coil Z / l = f13.6 charges/A, for the helix 211 = f'11.5 8,. In order to estimate the rate of diffusion-controlled proton transfer one wishes to estimate Wand hence [H'] a t r = p + rd where r d is the effective radius for reaction in the Debye-Smolu~howski~~ theory. It has been found that r d = 2.7 8, gives a good account of the rates of the bimolecular step of the proton-transfer reactions of a number of amino acids and small peptidesI4and this value is used here. The choice of the rather large value of p suggested in ref 11rather than the smaller diameter of the PLGA helix (13.5 8,) estimated by Tanford15* will minimize the electrostatic enhancement of the local hydrogen ion concentration at the polymer surface and hence the calculated rate of intramolecular proton exchange. The relaxation frequencies calculated from the parameters given in the foregoing and displayed in Figure 2 are in fair agreement with the experimental results of Yasunaga and his coworkers. The agreement could no doubt be improved by ad hoc selection of the geometrical parameters. The difference between the average degree of ionization of residues in helical and coil segments of the molecule determines the dissociative component of both the helix-coil transition and intramolecular proton exchange. In the transition region where both helical and coil segments are large the average change in dipole moment associated with the transfer of a proton from a helical to a coil residue will be large. We would thus argue that one might reasonably expect to observe this mode in a conductiometrically sensed pulsed E-field experiment and thus that perturbation of internal proton-exchange equilibria is a reasonable alternate interpretation of the relaxation observed in ref 9 and that a strong case can be made for the proposition that proton transfer is responsible for the relaxation observed in ref 8. Less directly the change in local charge density or in molecular dipole moment associated with an intramolecular proton transfer is electrostrictively coupled to the pressure field of an ultrasonic wave even if, as is the case in a uniform polypeptide, the transfer is between chemically equivalent groups and there is no intrinsic volume change. If the proposition that the volume change for intramolecular proton transfer is sufficiently great to account for the observed relaxation amplitudes can be made plausible, then the possibility that the ultrasonic relaxation observed in solutions of uniform polypeptides16 in the neighborhood of the helix-coil transition is in significant part due to perturbation of internal chargetransfer equilibria is worthy of consideration as is the The Journal of Physlcal ChemMiy, Vol. 8 1, No. 24, 1977
possibility that this mechanism is responsible for the acoustic absorption observed in protein solution near neutral pH. In the following, an argument, albeit a t best semiquantitative, is presented in support of the proposition that the acoustic relaxation amplitudes associated with intramolecular proton transfer are comparable with those observed in PLGA. 11. Relaxation Amplitudes The degree of advancement, 6, of a single normal reaction represented by the equation &gi[Ai] = 0 may be defined in terms of the coefficient (gi) of any species (Ai) by A[AJ = gi6 where A[Ai] is the displacement of the concentration of the ith species from its equilibrium value. The initial value of 6 when a reaction, initially at equilibrium, is subjected to a step variation in some external parameter such as the electric field, E, is then cro = (a In K/aE)(ds/d In K ) ( h E ) = (a In K/aE)rAE. In the absence of significant dispersion the excess ultrasonic absorption a at frequency f associated with a single process with relaxation time r is" a = C r f / [ 1 + (27rrf)2]. In dilute aqueous solutions, where C, and C, are very nearly equal and the temperature fluctuation associated with the passage of an ultrasonic wave has a neglible effect on the equilibrium of a reaction with a moderate AH
C = 2nZcpI'(A V ) ' / v R T
(5)
where p is the density of the solution, c is the velocity of sound, AV is the volume change (per mole) for the process in question, and V is the volume per mole of the solvent. The value of r for any normal reaction is readily calculated from the equilibrium composition. To estimate the amplitude of ultrasonic absorption and response to a pulsed electric field associated with the normal modes of the system of reactions represented by eq 1,values of AV and a In K/aE must be approximated. Experimental amplitudes are available only for ultrasonic absorption. We will thus be principally concerned with the calculation of the volume change for transfer of a proton from a residue in a helical segment of the poly-L-glutamic acid helix to a residue in a random coil segment. The electrostrictive volume change associated with intramolecular proton transfer a t constant molecular geometry may be estimated by appropriate differentiation of an approximate expression for the electrostatic free energy change (AW). If a helical segment of length L and total charge 2' is modeled as a cylinder of radius p with exclusion radius a, the electrostatic free energy in a medium of dielectric constant D and Debye length K - ~i d 2
and for a random coil of total charge Z and radius of gyration R, fully penetrated by
W2e2g(K) 20 Thus, for the transfer of a single proton from a random - 3zg(~)] coil to a helical segment, AW = (e2/D)[2z!g'(~) and, since K is proportional to D-'I2, aK/aP = - ( K / ~ D ) . (aD/aP). Thus, differentiating at constant values of the geometrical parameters
Helix-Coil Transitions in Polypeptides
aAW Av=-=--
aP
2267
Ne2aD 2Z’g’(~) - 3Zg(~) o2 aP
+
where N is Avogadro’s number and AV is the volume change per mole of protons transferred. Introducing the abbreviations y’ = Ka and y = KR
j[
+--KOYY’) K,2(y’)
21n(i)]:t3
[(1--3Y)’t
NeZaD A
v = 02G-
2KdY’) y’K1(y’)
+
18
(1 t y)(2y2 + 3y + 3)e-2y Y2
(6d)
We wish to compare the calculated ultrasonic absorption with the measurements of Barksdale and Stuehr16in 0.03 M NaBr. In this case the radius of the counterion (Na+) is 0.98 A and K - ~is 18 A. Taking Tanford’s estimate of the diameter of the PLGA helix, p = 6.75 A and the exclusion radius a = 7.73 A. Nagasawa and Holtzerls have determined the titration curves of PLGA in 0.02 and 0.05 M NaC1. We have estimated the degrees of dissociation of residues in helical and coil segments (designated respectively f’and f , a t an ionic strength of 0.03 by interpolation between values estimated from their results a t higher and lower ionic strength. We take Z’/L equal to f’/1.5 A. An estimate of the charge density in the random coil is more difficult. Near a midpoint of the helix coil transition the length of a coil segment is approximately (u)-lI2 or 16 residues. The radius of gyration is then about (16/ 6)1/2Ap15b where p, the length of a “statistical segment’’, is about three times the interresidue spacing or 11.4 A. Brant and Florylg have determined the “expansion parameter” A for PLGA in 0.3 M sodium phosphate to be 1.34. The results of Ricemsuggest that A might be 10-15% greater a t the lower value of the ionic strength of interest here. We have thus taken R to be 28 A and Z / R to be (16f/28) A-1. As long as R is significantly larger than the Debye length the calculated volume change is not very sensitive to the parameters which characterize the relatively less densely-charged coil. The pressure derivate of the dielectric constant of water has been determined by Owen and Brinkley,21 (Ne2/ D2)(aD/aP)= 10.1 X cm3/mol. When the appropriate Debye length and the geometrical parameters are substituted eq 6d becomes AV = lO.l(l.3f’- O.lf, cm3/mol (eq 6e). At the midpoint of the transition f’ f 0.5 so AV 6 cm3/mol.
-
--
In Figure 3, the amplitude calculated from eq 5 and 6e and the pH dependence of f and f’ are compared with experiment. The computed amplitudes are systematically less than the observed, but the disagreement does not exceed experimental error. The electrostrictive volume change calculated in this work results from the coexistence of helical and coiled regions of the molecule with different equilibrium charge densities and, in the simple model advanced here, would be zero for pOly-DL-glUtamiC acid where there is no helix-coil transition. Thus the observed absence of pH dependent excess absorption22in the DL polymer cannot be taken as a demonstration that either the helix-coil transition or the mechanism proposed here is responsible for the observed excess acoustic attenuation. We would therefore conclude, as have Eggers and F ~ n c k that ,~~ neither the times observed in the E jump or in ultrasonic experiments can be unambiguously attributed to a conformational transition. We would further argue that, in proteins or other spatially nonuniform polyelectrolyte systems near neutral pH where the value of r, and hence the amplitude, of any process which is principally acid or basic dissociation is necessarily small, intramolecular charge exchange is likely to be a significant contributor to the observed relaxation spectrum.
References and Notes (1) J. G. Kirkwood and J. B. Shumaker, Proc. Natl. Acad. Sci. U.S.A., 38, 855, 863 (1952). (2) R. D. White and L. J. Slutsky,Biopo&mers, 11, 1973 (1972); J. ColM Interface Sci., 37, 727 (1972). (3) R. Zana and J. Lang, J. Phys. Chem., 74, 2735 (1970). (4) P. J. kywocd, J. F. Rassing, and E. WynnJones, Adv. Mol. Rehation Processes, 8, 95 (1976). (5) R. C. Parker, L. J. Slutsky, and K. Applegate, J. Phys. Chem., 72, 3177 (1968). (6) F. Dunn and L. W. Kessler, J . Phys. Chem., 74, 2736 (1970). (7) R. D. White, S. Pattison, and L. J. Slutsky, J. Phys. Chem., 75, 161 (1971). (8) A. L. Cummings and E. M. Eyring, “Chemical and Biological Applications of Relaxation Spectrometry”, E. WynnJones, Ed., Reidel Publishina Co.. Boston. Mass.. 1975. (9) T. YasuAga, Y . Tsuji,’T. Sano, and H. Takenaka, J. Am. Chem. Soc., 98, 813 (1976). (10) D. J. Lentz, J. E. C. Hutchins, and E. M. Eyrina, . - J . Phvs. Chem.. 78, 1021 (1974). (11) M. Rinaudo and A. Domard, Siopo/ymers, 12, 2211 (1973). (12) T. L. Hili, Arch. Biochem. Siophys., 57, 229 (1955). (13) P. Debye, Trans. Electrochem. Soc., 82, 265 (1942). (14) M. Elgen and L. De Maeyer, Tech. Org. Chem., 8, 1034 (1963). (15) (a) C. Tanford, “Physical Chemistry of Macromolecules”,Wiley, New YWk, N.Y., 1961, pp 468-500; (b) ;bas,150-170; (c) lbid., 509-512. (16) A. D. Barksdale and J. E. Stuehr, J. Am. Chem. SOC.,94, 3334 (1972). (17) K. F. Herzfeld and T. A. Litovitz, ”Absorption and Dispersion of Ultrasonic Waves”, Academlc Press, New York, N.Y., 1959. (18) M. Nagasawa and A. Hoitrer, J. Am. Chem. Soc., 86, 538 (1964). (19) D. A. Brant and P. J. Flory, J. Am. Chem. SOC.,87, 2771, 2788 (1965). (20) S. Rice in “Biophysical Science”, J. Oncley, Ed., Wiley, New York, N.Y., 1959, p 69. (21) B. B. Owen and S. R. Brinkley, Phys. Rev., 64, 32 (1943). (22) R. Zana, J . Am. Chem. Soc., 94, 3646 (1972). (23) F. Eggers and Th. Funck, Stud. Blophys., 57, 101 (1976).
The Journal of Physical Chemistry, Vol. 8 1, No. 24, 1977
