The Journal of Physical Chemistty, Vol. 83,No. 22, 1979 2929
Communications to the Editor
COMMUNICATIONS TO THE EDITOR I
Comments on: “Relaxation Times for Acid Ionlzatlon and Internal Proton Transfer In Polypeptides in the Nelghborhood of the Helix-Coll Transition”, by L. Madsen and L. J. Slutsky
and ) ~uI are ’ , the density and time, C = [ ~ T ~ ~ I I / ( R T ) ] ( Ap V sound velocity in the solution of interest, AV is the effective volume change, is the amplitude function, B is the background (nonrelaxing) absorption, and the other symbols have their usual meanings. For a 0.0027 M solution of PGA, they showed that both relaxation amplitude and relaxation time passed through a maximum at the mid point of the helix-coil transition with values of A = 100 X s2/cm and 7 = 1.0 ~ s respectively, , at pH 5.21 and 37 “C (I = 0.03). In a recent paper, Madsen and Slutsky2(MS) suggested that acid ionization and internal proton transfer in polypeptides in the neighborhood of the helix-coil transition could be the mechanism responsible for the relaxation times found in this as well as other studies involving PGA.3-7 The mechanism they used for the analysis is -AiH
& -A; kh
+ H+
(2)
where i = 1for a helical residue and i = 2 for a coil residue. This mechanism will give a secular equation, the solutions of which are the two relaxation times, roughly corresponding to overall ionization and internal proton transfer. MS found that calculated relaxation times for the peptide indicator exchange were in very good agreement with those observed in a spectrometric E-jump s t ~ d y .In~ addition, they suggested that the acoustic relaxation amplitudes predicted for intramolecular proton transfer were comparable with those observed in ref 1. The concept that the ultrasonic relaxation,l exhibiting a sharp maximum centered exactly about the helix-coil transition midpoint, could be due to internal proton transfer is both surprising and distressing. In order to clarify the situation, we carried out a detailed analysis of the predictions of eq 2 with regard to relaxation times and amplitudes and compared the predictions with the results reported in ref 1. Rate and other constants are those used by MS in their analysis. Figure l a shows the predicted dependence of the relaxation time upon pH. In the vicinity of the helix-coil transition, both the magnitude of the predicted 7’s and the shape of the pH curve are quite different from the experimental findings. Figure l b shows the ultrasonic amplitude as a function of pH. The solid line is that predicted for the helix-coil mechanism; the dashed line is the predicted 0022-3654f 79f 2083-2929$0 1.OO/O
’
I
m-
DO
a =A+B f 1+ 0272 where the relaxation amplitude A = C7,7 is the relaxation
I
10”~ sed/un
Publication costs assisted by the National Institutes of Health
Sir: In 1972, Barksdale and Stuehrl published a paper on the kinetics of the helix-coil transition in aqueous poly@glutamic acid) (PGA) by ultrasonic absorption techniques. The results were interpreted by the following equation for a single relaxation process:
‘
Ld
Lo
I
PH a
-
%!i
1.0
5.1
J
PH
b
Flgure 1. Ultrasonic relaxation data for 0.0027 M aqueous PGA from ref 1: (solid curve) calculated for heilx-coil process; (dashed curve) predicted dependence for internal proton exchange, eq 2. (a) Dependence of 7 upon pH. (b) Dependence of uitrasonic relaxation amplitude, A , on pH.
curve for internal proton transfer. The agreement of the former with the experimental data is excellent. The predictions for internal proton transfer, however, show a half-width of relaxation amplitude much wider than experiment, a maximum at the wrong pH, and smaller values of the relaxation amplitude, A , throughout. Furthermore, for smaller values of AV, the values of A decrease dramatically.* On the other hand, in ref 1, we found (1) the ultrasonic amplitude was just what one would expect for the helixcoil transition if AV,,,, = 1 cm3/mol; (2) the half-width of the amplitude is that predicted for the helix-coil transition; (3) location of pH maximums for both relaxation amplitude and relaxation time is at the transition midpoint; and (4) the shape of the pH dependence of 7 is that predicted for the helix-coil transition. We see another problem with the MS analysis of the ultrasonic work. Their amplitude analysis was based on C , not A in eq 1. The quantity C is mainly a function of the amplitude function I’. Since I’ for both internal proton transfer and helix-coil transition depends on the fraction of helicity, it is not surprising to find that the proton transfer C passes through a maximum at the midpoint of transition. Multiplication of C by 7 to yield A shifts the predicted maximum to higher pH (dashed line, Figure lb). Finally, it should be pointed out that the proton transfer rate constants employed in the MS analysis were those for simple amino acids, i.e., encounter controlled values. If the rate constants for proton transfer in the polyamino acid are substantially smaller than encounter controlled, then the predicted relaxation times will shift out of the time range accessible to the ultrasonic technique. Based on these considerations, we conclude that the helix-coil equilibrium is much more compatible with the ultrasonic data than is the internal proton transfer reaction. We do not dispute the possibility that internal proton transfer may be an alternate explanation for the relaxation effects in some of the other work reported in the literature. Acknowledgment. This work was supported in part by a National Institutes of Health Grant to J.E.S. (GM 13116).
References and Notes (1) A. D. Barksdale and J. E. Stuehr, J . Am. Chem. Soc., 94, 3334 (1972). (2) L. Madsen and L. J. Slutsky, J. Phys. Chem., 81, 2264 (1977).
0 1979 American Chemical Society
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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
