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[Mn'] E Ani2/(X2 ON THE PHENOMENOLOGICAL TREATMENTS OF OPTICAL ROTATORY DISPERSION OF POLYPEPTIDES AND PROTEINS* BY JEN Tsi YANG CARDIOVASCULAR RESEARCH INSTITUTE, AND DEPARTMENT OF BIOCHEMISTRY, UNIVERSITY OF CALIFORNIA (SAN FRANCISCO) Communicated by Paul J. Flory, December 8, 1964 Since the discovery of the anomalous dispersion of a-helical polypeptides1' 2 and the introduction of the Moffitt equationl 4 ORD has been applied extensively to conformational studies of proteins and polypeptides. Although the early theories3' I were unsatisfactory,6 we do not yet have a relationship as simple as Moffitt's to describe ORD in the visible region. Instrumental limitation confined almost all early experimental data to wavelengths above 300 mg. Now that measurements can be extended to about 185 m/i, we detect conformation-dependent Cotton effects7' 8 (and CD9) between 180 and 240 mj.. It seems desirable to re-evaluate ORD analysis in the visible region. All treatments proposed are phenomenological, and success should be tempered with caution in interpreting the experimental results quantitatively or semiquantitatively. The Moffitt Equation.-Moffitt's equationl 4 [m'] = aoX02/(X2-X62) + boXO4/(X2 - (1) greatly stimulated the study of ORD of proteins and polypeptides because of its theoretical overtones. In particular, the virtual constancy of the bo value in various solvent media as predicted by Moffitt and confirmed by experiments (about -630 deg. cm2 decimole-1 for a right-handed a-helix with Xo = 212 mu)4 is very useful in estimating the helicity of proteins. Although initially the equation gained wide acceptance, it has now come to be regarded as empirical,6 and not especially characteristic of helices.10 The Drude equation [mn'] = E a)2X2/(X2 - X,2) (2) can always be expanded in inverse powers of (X2 - X62) where the parameter Xo is to be determined: [mn'] E ani2/(X2 - X02) + E ajX2(Xf2 - X02)/(X2 - X02)2 + 0[(X2 - X02)-3] (3) which is converted into equation (1) provided that ao 2 = ajXj2 boX04 = E ajX2(XA2 - Xo2) Za6Xi2(X2 - X02)2 - 0. (4a-c) It will be valid whenever XoI, so determined, is positive and considerably less than X2. If ao0XO4 happens to be equal to E ajXj4 [eq. (4a, b) ], i.e., boX04 = 0, then equation (2) becomes a one-term Drude equation: [m'] = ao0X2/(X2 - Xo2) or [m'] = k/(X2 - X2) (5) (In this case Xc = Xon) Moffitt proposed that the amide absorption bands, which are responsible for the 438 Downloaded by guest on September 27, 2021 VOL. 53, 1965 CHEMISTRY: J. T. YANG 439 helical rotations of polypeptides, are split into perpendicular and parallel (to the helical axis) components. Accordingly, a rotation in the visible region can be written as4 Z E + 2) (6) [in'] i a1XX1,/(X'-X2) i a11iX1112/(X'- Further he noted that the Ri of each parallel component may be exceedingly large, but is almost exactly compensated by the opposing Ri of its perpendicular partner, i.e., a1l -alli(ai's being proportional to the Ri's). Under these conditions, we can let X = Xo(l- A/2), X11i = Xo(l + A/2). (7) According to equation (4b), boX04 for such a band pair will not vanish, since the two terms on the right-hand side would have the same sign. The form of equation (1) is easily obtained by substituting equation (7) into equation (3) with only one band pair and by defining Ai = (eVs- XI )/XOi, b0i = (all - a1I)Ai, and a0i = (aIi + ai) + b,, (8) provided that A << 1 (say, JX1lj - Xli < 10 mu) and o,<<X2. It is then a simple matter to summarize the rotations of all band pairs to yield the MIoffitt equation. Today we know Moffitt's treatment neglected several important terms, which must therefore be added to equation (6) to account for the rotations due to the helical conformation. But the form of equation (1) remains valid, as is evident from equations (3) and (4). The Tuo-Term Drude Equation.-The observation of a negative (actually two overlapping) and a positive Cotton effect (180-240 miA) for the helical polypeptides renews interest in the two-term Drude equation: [ml] = aX,2/(X2- X12) + a2X22/(X2 - X22) (9) since each Drude term would account for one Cotton effect if the rotations are measured away from the optically active absorption bands." The four parameters, X1, X2, a,, and a2, of equation (9) can be evaluated graphically by plotting [m']. (X2/X,2- 1) against 1/(X2/X22- 1) or (X2/X,2- 1)/(X2/X22-1) for trial XA and X2 values until a straight line is obtained. This procedure is reminiscent of the plot now used for equation (1),4 but it can now be simplified with the aid of a computer. Equation (3) implies that a Drude equation with one or more terms can always be converted to equation (1), provided that the conditions of equation (4) are satisfied. The graphical solution of equation (1) yields a unique straight line. In contrast, the precision of contemporary experimental data is so limited that equation (9) may give more than one solution each time a different Xi or X2 is chosen. Even with computerized goodness of fit to experimental curves the significant figures obtained experimentally (usually 3 or 4) may not warrant a unique solution. Equation (9) was used by Imahori in describing the ORD of helical PBG with Xi = 190 mu and X2 = 220 mI.2 Recently, Yamaokal3 reintroduced equation (9) using Xi = 193 muA and X2 = 226 my. Shechter and Blout'4y 1" and Shechter et al.'6 of the same laboratory next proposed a four-term Drude equation for polypeptides and proteins (two terms for the coiled form). However, what they actually applied was Downloaded by guest on September 27, 2021 440 CHEMISTRY: J. T. YANG PROC. N. A. S. [Xi'] = A193X1932/(X2 - X1932) + A225X2252/(X2 - X2252), (10) an equation similar to equation (9). The A-parameters were determined for both the helical and coiled conformations; these reference values were then used to estimate the helicity in proteins through simple interpolation. A linear relation was also obtained by plotting A225 against A193 for various helical contents, but the intercept varied with the solvents used. It was further stipulated that the two measures from experimental A193 and A225 agreed with each other if only a-helices and random conformations are present in proteins. (Actually, in an [M'](X2/ 1932-1) versus 1/(X2/X2252 - 1) plot, A225 is derived from the slope A225(X2252/X1932 - 1), but, in contrast, A193 is computed from the intercept (A193 + A225X2252/X1932). Thus, the two A-parameters cannot be independently obtained by this graphical pro- cedure. In particular, since the intercept is usually the difference between two large numbers (e.g., A193 = +2900 and A225 = -2050 for PGA at pH 414) and therefore a small number, A193, so determined, largely reflects the numerical value of A225 over most of the helix-coil transition region, except for low helical content. It is not too surprising that an A193-A225 plot gives very nearly a straight line.17 Theoretically, equation (2) must satisfy the condition (noting aiaRi) :18 Eaj=o° (11) Therefore, a one-term Drude equation [equation (5) ] is empirical; the same is true for a two-term Drude equation unless the two equal but opposite Ri's and thereby the two aiX,2/(X2 - X,2) terms are so strong that they overwhelm the contributions of the remaining terms in equation (2). (Of course, under special circumstances the sum. of the Drude terms [equation (2)], excluding the two terms under con- sideration, could happen to be close to zero, even though their corresponding Re's are large. Then the sum of R1 and R2 may not be zero.) Experimentally, CD measurements of the helical form reveal one positive and two (not one) negative dichroic bands at 190, 206, and 222 muA with Ri's of +81 X 10-4,-29 X 10-40, and -22 X 10-40 erg cnm3 rad.9 These must be attributed to the 7r-7r* transitions and the n-7r* transition (the latter was not considered by i\Ioffitt). Obviously, the sum of R.'s in the present case does not satisfy equation (11); thus, the ORD of the helical conformation in the visible region can be represented by: [I'] = a1X19o2/(X2 - X1902) + a2X2o62/(X2 - X2062) + a3X2222/(X2 -222') n + E aj2Xj/(X2 - X22). (12) i = 4 The last term on the right-hand side may be approximated as another Drude term. If the X193 term in equation (10) is identified with the 190-m/A dichroic band, then the 206- and-222 muA Drude terms in equation (12) must, be combined into the X225 term of equation (10). However, it can easily be shown'9 that such a combined term must have a X2 in between, not greater than, 206 and 222 mu, since the numer- ators of the two terms have the same sign (both negative in this case). Thus, at least one positive Drude term must be present in order to reach a compromise X225 term in equation (10). We are, however, not even convinced that the X190 term in equation (12) would remain unaffected by such arbitrary manipulations; indeed, part of this positive term could be combined with the two negative terms. To Downloaded by guest on September 27, 2021 VOL. 53, 1965 CHEMISTRY: J. T. YANG 441 force a multi-term (at least four) Drude equation into two terms is purely an ex- pediency for graphical treatment; consequently, any attempt to calculate the apparent R,'s from equation (10) and identify them with the experimental values obtained directly from the CD measurements is very dubious.
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