Local Amide I Mode Frequencies and Coupling Constants in Multiple

J. Phys. Chem. B 2004, 108, 20397-20407

20397

Local Amide I Mode Frequencies and Coupling Constants in Multiple-Stranded Antiparallel β-Sheet Polypeptides Chewook Lee and Minhaeng Cho* Department of Chemistry and Center for Multidimensional Spectroscopy, DiVision of Chemistry and Molecular Engineering, Korea UniVersity, Seoul 136-701, Korea ReceiVed: July 1, 2004; In Final Form: August 31, 2004

Amide I local mode frequencies and vibrational coupling constants in various multiple-stranded antiparallel β-sheet polyalanines are calculated by using the semiempirical calculation method and Hessian matrix reconstruction methods. The amide I local mode frequency is strongly dependent on the nature and number of hydrogen bonds. Vibrational couplings among amide I local modes in the multiple-stranded β-sheets are shown to be fully characterized by eight different coupling constants. The intrastrand coupling constants are found to be much smaller than the interstrand ones. Introducing newly defined inverse participation ratios and phase-correlation factors, the extent of two-dimensional delocalization and the vibrational phase relationship of amide I normal modes are elucidated. The A-E1 frequency splitting magnitude is shown to be strongly dependent on the number of strands but not on the length of each strand. A reduced one-dimensional Frenkel exciton model is used to describe the observed A-E1 frequency splitting phenomenon.

I. Introduction The β-sheet polypeptide is known to be one of the most frequently observed secondary structures of proteins. Although a variety of spectroscopic methods have been used to determine various secondary structure elements, vibrational spectroscopies have been found to be exceptionally useful.1-14 One of the empirical methods that have been used to determine whether a given protein contains β-sheet polypeptide segments is to examine the line shape of the amide I IR band. A characteristic intense low-frequency peak around 1620-1630 cm-1 and, in some cases, a weaker high-frequency peak at 1680-1690 cm-1 have been considered to be the signatures of β-sheet polypeptides.4,15-17 Recently, Tokmakoff and co-workers carried out two-dimensional (2D) IR photon echo experiments for various proteins with antiparallel β-sheets and found a characteristic “Z”-shaped pattern for the amide I region of the 2D IR spectrum.18,19 Often, this split pattern in the IR absorption spectrum distinguishes it from other secondary structures of which the amide I band consists of overlapped components forming single bands which are typically at higher frequencies than the intense low-frequency β-sheet peak. To theoretically describe the line shapes of IR absorption and vibrational circular dichroism spectra of extended β-sheet polypeptides, Keiderling and co-workers carried out interesting simulation studies.12,20 As a minimal unit for extended multiplestranded β-sheets, a triple-stranded β-sheet with each strand having three peptide bonds was considered and its force field, dipole strength, and rotational strength were calculated by employing a density function theory calculation method, and those properties were directly transferred to simulate various spectra of large multiple-stranded β-sheet polypeptides. Their simulated IR and VCD spectra are in good agreement with experimental observations.5,21,22 Furthermore, the anomalous intensity enhancement of the alternately doubly isotopically * Address correspondence to this author. E-mail: [email protected].

labeled multiple-stranded antiparallel β-sheet was successfully explained.12 They found that the low-frequency modes are IR intense and that any two nearest neighboring local amide I mode vibrations are out-of-phase whereas those between two peptides belonging to neighboring strands and forming a hydrogen bond to each other are in-phase. This low-frequency mode was technically assigned to be the E1-mode. On the other hand, the high-frequency weak amide I normal modes show the opposite trend in terms of phase correlations between neighboring peptides and this high-frequency band was assigned to be the A-mode. The A-E1 frequency splitting magnitude was theoretically predicted and found to be a function of the number of strands.12 Despite that their theoretical approaches to calculating various spectra of extended multiple-stranded β-sheets were found to be very successful in quantitatively describing IR and VCD spectra, their simulation method can be useful only when a given protein contains a quasiregular structural pattern like multiple-stranded β-sheets. In other words, if it becomes desirable to calculate amide I normal-mode frequencies, eigenvectors, dipole strengths, and rotational strengths of the protein of which the 3D structure contains various secondary structure elements as well as nonrepeating segments, one might have to consider a large number of units to ultimately describe the force fields, atomic polar tensors, atomic axial tensors, etc. Therefore, a more desirable approach would be to develop a theoretical procedure to construct the Hessian matrix (in the local amide I mode subspace) of a given protein from the structural data by using a rather smaller unit peptide. To achieve this goal, we have carried out a series of quantum chemistry calculation studies of small polypeptides and developed an empirical correction method to predict diagonal Hessian matrix elements.23-26 Also, a recipe for determining off-diagonal Hessian matrix elements was presented and discussed in detail. Recently, these methods were used to numerically calculate one- and twodimensional IR spectra of an R-helical polyalanine in liquid water and the simulated spectra were found to be in excellent agreement with experiments.27

10.1021/jp0471204 CCC: $27.50 © 2004 American Chemical Society Published on Web 11/25/2004

20398 J. Phys. Chem. B, Vol. 108, No. 52, 2004 It was known that the amide I local mode frequency is strongly dependent on the local environment, which is largely determined by the relative distances and orientations of surrounding peptide groups and amino acid side chains as well as by the solvent molecules around the target peptide. Particularly, a single hydrogen bonding interaction of the peptide carbonyl group with either a water molecule or a neighboring peptide group can induce about a 20 cm-1 frequency red-shift. In the case when the peptide N-H proton forms a hydrogen bond with a water molecule, the amide I mode frequency is farther redshifted by about 10 cm-1.28-30 Therefore, for multiple-stranded β-sheet polypeptides, it is expected that there are different groups of peptides differing from one another by the number of hydrogen bonds. Furthermore, the hydrogen-bonding interaction will affect not only the amide I local mode frequencies but also the vibrational coupling constants between the associated two peptides. Once these amide I local mode frequencies and coupling constants, which correspond to the diagonal and offdiagonal Hessian matrix elements, respectively, are determined, the extents of delocalization and magnitudes of frequency-shift of amide I normal modes can be quantitatively described. Then, it will be possible to address the following questions: (1) How large is the amide I local mode frequency shift induced by a hydrogen bond in the cases of antiparallel β-sheet polypeptides? (2) how strongly are any two nearest neighboring peptides in a given strand coupled to each other? (3) Are the interstrand vibrational coupling constants larger than the intrastrand ones? (4) Are the amide I normal modes of antiparallel β-sheets delocalized on a single strand or on different strands? (5) Is it possible to theoretically predict the amide I local mode frequencies that are related to the diagonal Hessian matrix elements? (6) Does the transition dipole coupling model work for describing intra- and interstrand vibrational couplings? (7) What is the functional form of the A-E1 frequency splitting magnitude with respect to the number of strands? (8) Do the vibrational coupling constants depend on the chain length of each strand or on the number of strands? The main goals of this paper are to address these issues by carrying out semiempirical quantum chemistry calculations in combination with the Hessian matrix reconstruction method and to establish the structure-amide I mode frequency relationship. II. Semiempirical Quantum Chemistry Calculation Results Although there exist a number of different high-level quantum chemistry calculation methods, it is still very expensive to carry out ab initio geometry optimization and vibrational analysis of long polypeptides with a large basis set.31-40 Recently, we showed that the AM1 approximation is useful in calculating amide I mode frequencies and eigenvectors of lengthy R-helical polypeptides.27 The molecular structures, such as bond lengths and angles, and normal-mode frequencies obtained by using the AM1 method are known to be less reliable than other highlevel calculation methods with large basis sets. However, as long as the AM1 results are properly treated, it still can be reliable as shown below. For a series of R-helices, we found that the amide I local mode frequencies obtained by using the Hessian matrix reconstruction method and by employing AM1 approximation are linearly proportional to the corresponding CdO bond lengths. This linear relationship between amide I mode frequency and CdO bond length has been firmly established theoretically with various quantum chemistry calculation methods.23-28,39 Therefore, the linear relationship between them observed in the AM1 calculation suggests that,

Lee and Cho

Figure 1. The AM1 optimized structures of 210-510.

if the calculation results on the CdO bond lengths and amide I frequencies are properly rescaled, the semiempirical AM1 method can still be successfully used to calculate amide I mode frequencies of very large proteins. In the present paper, we will consider multiple-stranded antiparallel β-sheet polyalanines with 4 to 10 peptide bonds in a single strand. For each strand, the N-terminus is capped with an acetyl group and the C-terminus is blocked with a methylamine group. Hereafter, Mn (for M ) 1-8 and n ) 4-10) denotes the M-mer with n peptides in each strand. Among these various β-sheets, geometry optimized structures of four representative ones, 210, 310, 410, and 510, are shown in Figure 1. A. Amide I Local Mode Frequency. To determine a proper scaling scheme for the AM1 amide I frequency, we considered the fact that the CdO bond length is linearly proportional to the local amide I mode frequency regardless of the quantum chemistry calculation methods used. Carrying out ab initio geometry optimization and vibrational analysis of various polypeptides in different conformations with the HF/6311++G** calculation method, we already found that ν˜ HF ) HF HF where ν˜ HF and rCdO A - BrCdO are the amide I local mode -1 frequency (in cm ) and the CdO bond length (in Å) of a given peptide in polypeptide (see Figure 2a).23,25 The two constants, A and B, were determined to be 7009 and 4423.9 cm-1/Å, respectively. Collecting all AM1-calculated data including the

Multiple-Stranded Antiparallel β-Sheet Polypeptides

J. Phys. Chem. B, Vol. 108, No. 52, 2004 20399

Figure 2. (a) The amide I local mode frequencies for various polypeptides with two to five peptides, which were obtained by using the HF/6311++G** method with a proper scaling factor, are plotted with respect to the corresponding CdO bond lengths (see refs 23-25). (b) The amide I local mode frequencies of various R-helices,27 antiparallel β-sheets, and glycine dipeptides calculated with AM1 approximation.

present calculation results for various antiparallel β-sheets, glycine dipeptide analogue, and various R-helices,27 we also found that the unscaled AM1-calculated amide I local mode frequency is linearly proportional to the CdO bond length as AM1 ν˜ AM1 ) C - DrCdO with C and D being 7119.4 and 4108.8 -1 cm /Å, respectively (see Figure 2b). Now, the remaining task is to obtain a proper scaling procedure for converting the AM1 amide I frequency to that at the high level of HF/6-311++G**. This can be achieved by obtaining yet another relationship HF AM1 and rCdO . We have carried out complete AM1 between rCdO geometry optimizations and vibrational analyses of the glycine dipeptide analogue with varying dihedral angles φ and ψ for the entire Ramachandran spacesthe HF/6-311++G** results for the same set of dipeptide conformations were presented in HF AM1 and rCdO , we found ref 23. Comparing two data sets on rCdO that these two quantities are linearly proportional to each other HF AM1 as rCdO ) 1.6681rCdO - 0.88215. Now, combining these three linear relationships, we can finally determine a proper scaling equation, ν˜ scaled ) a + b(ν˜ AM1 - 1994.5) with a ) 1707.1 and b )1.8. Note that ν˜ AM1 on the right-hand side of this scaling equation is the unscaled AM1-calculated amide I frequency. By using the scaled amide I normal-mode frequencies and eigenvectors, the amide I local mode frequencies and coupling constants can be calculated by employing the Hessian matrix reconstruction method.24,25 For the sake of completeness, the Hessian matrix reconstruction method needs to be briefly outlined. It begins with an assumption that a given amide I normal coordinate, qj, can be written as a linear combination of the N amide I local coordinates, QR, as N

qj )

∑

QRURj

(1)

R)1

Here, {U1j, U2j, ..., UNj}is the eigenvector of the jth normal mode in the amide I subspace. Now, let us denote r0j to be the jth CdO bond length of the geometry-optimized polypeptide. Then, by using the quantum chemistry calculated jth eigenvector elements, the molecular structure can be properly distorted along the direction dictated by the eigenvector elements and the N CdO bond lengths denoted as r1j, r2j, ..., rNj are measured. Then, the difference between rRj and r0R is assumed to be directly proportional to URj as

URj ) Nj(rRj - r0R)

(2)

Although each one of the eigenvectors {URj} (R ) 1-N) can

Figure 3. Scaled amide I local mode frequencies of antiparallel β-sheets, Mn values with M ) 2-8 and n ) 4-10, are plotted with respect to the corresponding CdO bond lengths. The local mode frequencies of inner (two terminal) peptides in each strand are linearly fitted with line A (B). Depending on the number of hydrogen bonds and on the H-bonding site, there are three groups of amide I local modes (see the text for a more detailed description).

be properly normalized, there is no guarantee that they form a complete orthonormal set of eigenvectors. However, we confirmed that the reconstructed Hessian matrix is almost diagonally symmetric, as it should be, which suggests that those eigenvectors are fairly orthogonal to one another. Once the eigenvector matrix U is fully determined, the corresponding Hessian matrix H can be reconstructed as

H ) U-1ΛU

(3)

where Λ is the eigenvalue matrix obtained from the quantum chemistry vibrational analysis and the N diagonal matrix elements of Λ are, in the present case, the scaled AM1calculated force constants of the N amide I normal modes. In Figure 3, we plot the amide I local mode frequencies thus obtained with respect to the CdO bond lengths. The data points are not perfectly on a single line. In fact, the entire data set can be divided into two groups and each group is on line A or B in Figure 3. The data points on line A are associated with the inner peptide groups in a given strand, whereas the data points falling on line B are associated with the terminal peptide bonds. Note that the two terminal peptide bonds in each strand are connected to only one nearest peptide group. Due to the different chemical structures of terminal peptides in comparison to those inner peptides surrounded by two peptides, the amide I local mode frequencies cannot be directly compared with those of inner peptides. Therefore, we use two different linear lines to fit the

20400 J. Phys. Chem. B, Vol. 108, No. 52, 2004 calculation results and the resultant linear lines are given as

νInner ) 6646.0 - 4124.7rAM1 CO

(inner peptides)

νOuter ) 6126.6 - 3763.1rAM1 CO

(outer peptides) (4)

By using these two linear relationships between the amide I local mode frequency and the CdO bond length, as well as the scaling equation discussed before, it becomes possible to determine the amide I local mode frequencies of any arbitrary polypeptides once their AM1-optimized structures are determined. Even for the inner peptides in a given strand, there are three different types of peptides, which differ from one another by the number of hydrogen bonds with peptides in the neighboring strands. For double-stranded β-sheets, there are two different groups of peptides. The GH group contains peptides having a single hydrogen bond at the N-H site and the peptides in the GO group form one hydrogen bond at the CdO site. Now, in the cases of multiple-stranded β-sheets with more than two strands, the third type of peptide, the GOH group, exists and they involve two hydrogen-bonding interactions at CdO and N-H sites. The amide I local mode frequencies of these three peptide groups differ from one another due to the hydrogenbond-induced frequency red-shifts, and as can be seen in Figure 3 these four groups appear in distinctively different frequency regions. Upon a single hydrogen bond of the CdO group with the N-H proton in the neighboring strand, the amide I local mode frequency undergoes a red-shift of about 25 cm-1. On the other hand, the hydrogen bond at the N-H site induces a red-shift of about 10 cm-1. Note that the high-level calculation of the hydrogen-bond-induced frequency red-shift was also found to be quantitatively similar to these AM1-calculation results. The amide I local mode frequencies also depend on the interstrand distance because the hydrogen-bonding interaction strengths increase as the interstrand distance decreases. More specifically, as the number of strands increases, the amide I local mode frequency becomes farther red-shifted, though this effect induced by shortening of the hydrogen bond distance on the amide I mode frequency shifts is relatively small. B. Vibrational Coupling Constants. Vibrational coupling constants that are the off-diagonal Hessian matrix elements can be classified into two different types, i.e., intrastrand and interstrand coupling constants. We first consider the intrastrand coupling constants denoted as Fij. In Table 1, the average F12, F13, and F14 values for all Mn values obtained from the reconstructed Hessian matrixes are summarized. The average F12 value is found to be about 1.0 cm-1. The coupling constants between the jth and (j ( 2)th peptides, denoted as F13, are found to be quantitatively similar to F12 values (see Table 1). The coupling constants, F14, are small in comparison to F12, F13, or any other interstrand coupling constants. We next consider the interstrand coupling constants. For multiple-stranded β-sheets, the coupling constants between two peptides interacting with each other by directly forming a hydrogen bond are denoted as Fab (see Figure 4). The solid arrows in Figure 4 represent the transition dipole vectors associated with each amide I local vibration. Fac denotes the coupling constants between two peptides that are separated by one more strand. The average Fab and Fac values are -10.3 and -1.7 cm-1, respectively. In comparison to the intrastrand coupling constants such as F12, F13, etc., the interstrand coupling constant, Fab, is much larger. Therefore, it is expected that the amide I normal modes

Lee and Cho of multiple-stranded β-sheet polypeptides are strongly delocalized over different strands due to these strong interstrand coupling constants. The coupling constants Fad between two peptides that are separated by two strands are negligibly small. In addition to these interstrand coupling constants Fab and Fac, there are four more distinctively different types of interstrand coupling constants between two peptides that belong to two neighboring strands as shown in Figure 4. Fhh (Ftt) is the coupling constant when the two transition dipoles are aligned in a head-to-head (tail-to-tail) orientation. Because the two transition dipoles repulsively interact with each other in this case, the coupling constants of Fhh and Ftt are positive. The average Fhh and Ftt values are similar to each other in magnitude, 3.16 and 2.86 cm-1, respectively. In the cases of Fap1 and Fap2, the orientations of the two associated transition dipoles are antiparallel so that the two transition dipoles attractively interact with each other and these values are negative. The average Fap1 and Fap2 values are found to be -6.94 and -0.80 cm-1, respectively. Overall, the relative magnitudes of coupling constants can be summarized as

|Fab| > |Fap1| > |Fhh| ≈ |Ftt| > |Fap2| ≈ |Fac| ≈ |F12| ≈ |F13| > all other F’s Fab, Fap1, Fap2, and Fac < 0 Fhh, Ftt, F12, and F13 > 0

(5)

Although these vibrational coupling constants depend on the location of the peptide group in a given chain as well as on which strand the peptide group belongs to, the variances of these coupling constants are rather small so that the average coupling constants given above can be considered to be the representative values for each coupling. Recently, Moran and Mukamel reported vibrational coupling constants for the antiparallel β-sheet, where a dimer of dipeptide (22 in our notation) was considered.41 Since the size of their model system is very small, it is not possible to make a direct comparison of our results with theirs. III. Empirical Correction Method: Amide I Local Mode Frequencies Although we presented amide I local mode frequencies and coupling constants in Section II, it is still desirable to develop a theoretical model that can be used to quantitatively predict the diagonal and off-diagonal Hessian matrix elements because it is still a formidable task to carry out AM1 geometry optimizations and vibrational analysis of very large proteins. More specifically, for QM/MM calculations of proteins in solutions, it is necessary to carry out many million times of AM1 calculations to run a few nanosecond QM/MM simulation. Recently, we have developed an empirical correction method to predict the amide I local mode frequencies (diagonal Hessian matrix elements).25 By assuming that the peptide contains four sites, i.e., O(dC), C(dO), N(-H), and H(-N), and assigning properly adjusted partial charges, the mth amide I local mode frequency, ν˜ m, is expanded as a function of electrostatic potentials at these four sites as 4

ν˜ m ) ν˜ 0 +

lj(m)φj(m) ∑ j)1

(6)

where ν˜ 0 is the reference frequency, lj(m) is the expansion coefficient of the jth site of the mth peptide group, and φj(m) is



TABLE 1: Coupling Constants Calculated by Using the Hessian Matrix Reconstruction Method and Transition Dipole Coupling Modela M

N

F12

F13

F14

Fab

2

4

0.13 0.15/-0.23 0.83 -0.08/-0.68 0.64 -0.06/-0.45 0.91 -0.07/-0.68 0.78 -0.02/-0.53 0.98 -0.06/-0.68 0.86 0.00/-0.56 0.83 -0.04/-0.50 0.94 -0.04/-0.52 0.98 0.07/-0.48 1.02 0.08/-0.46 1.03 0.10/-0.44 1.05 -0.11/-0.43 1.06 0.11/-0.42 0.58 0.18/-0.25 1.09 0.07/-0.49 1.27 0.05/-0.54 1.13 0.13/-0.40 1.02 0.19/-0.30 1.16 0.15/-0.33 1.08 0.19/-0.30 0.95 0.10/-0.24 0.79 0.14/-0.39 1.28 0.18/-0.30 1.05 0.31/-0.19 1.11 0.10/-0.46 1.14 0.21/-0.36 1.31 0.45/-0.14 0.96 0.11/-0.41

1.40 0.72/1.00 0.93 0.88/0.77 1.08 0.93/0.86 0.93 0.89/0.77 1.02 0.92/0.83 0.91 0.90/0.76 0.99 0.92/0.81 1.33 0.94/0.85 1.30 0.95/0.85 1.36 0.97/0.87 1.34 0.98/0.87 1.36 0.99/0.88 1.36 0.99/0.88 1.36 1.00/0.88 1.66 1.00/0.97 1.41 0.96/0.85 1.40 0.97/0.83 1.48 1.00/0.86 1.54 1.02/0.92 1.49 1.01/0.89 1.53 1.02/0.92 1.53 0.96/0.92 1.68 0.99/0.88 1.64 1.02/0.90 1.59 0.82/0.74 1.66 0.98/0.85 1.53 0.81/0.69 1.52 0.87/0.90 1.37 0.94/0.86

-0.30 -0.19/-0.27 -0.38 -0.18/-0.28 -0.37 -0.18/-0.28 -0.39 -0.19/-0.29 -0.40 -0.19/-0.29 -0.41 -0.19/-0.29 -0.40 -0.19/-0.34 -0.47 -0.21/-0.31 -0.48 -0.21/-0.32 -0.52 -0.22/-0.33 -0.53 -0.23/-0.34 -0.54 -0.23/-0.35 -0.54 -0.23/-0.35 -0.54 -0.24/-0.35 -0.48 -0.22/-0.32 -0.53 -0.22/-0.33 -0.58 -0.23/-0.35 -0.57 -0.24/-0.35 -0.58 -0.24/-0.35 -0.58 -0.24/-0.36 -0.59 -0.24/-0.36 -0.52 -0.22/-0.32 -0.52 -0.23/-0.35 -0.61 -0.25/-0.36 -0.55 -0.22/-0.32 -0.59 -0.24/-0.35 -0.58 -0.22/-0.32 -0.61 -0.25/-0.35 -0.51 -0.22/-0.33

-9.13 -6.31/-8.02 -9.52 -6.31/-7.93 -9.30 -6.34/-8.04 -9.54 -6.32/-7.94 -9.37 -6.34/-8.01 -9.55 -6.33/-7.95 -9.40 -6.34/-8.00 -9.77 -6.35/-8.00 -10.31 -6.36/-8.01 -10.29 -6.37/-8.02 -10.27 -6.37/-8.02 -10.26 -6.37/-8.03 -10.26 -6.37/-8.03 -10.27 -6.37/-8.03 -10.52 -6.37/-8.05 -10.69 -6.42/-8.09 -10.70 -6.38/-8.03 -10.60 -6.39/-8.06 -10.48 -6.40/-8.08 -10.56 -6.40/-8.07 -10.51 -6.40/-8.08 -10.82 -6.34/-7.84 -10.82 -6.39/-8.06 -11.17 -6.40/-8.07 -10.97 -6.38/-8.06 -10.85 -6.39/-8.05 -10.75 -6.37/-8.03 -10.92 -6.27/-7.77 -10.27 -6.36/-8.01

5 6 7 8 9 10 3

4 5 6 7 8 9 10

4

4 5 6 7 8 9 10

5

4 6 10

6

4 6

7

4

8

4

av

Fac

-1.64 -0.78/-0.91 -1.60 -0.77/-0.90 -1.58 -0.77/-0.90 -1.56 -0.77/-0.89 -1.56 -0.77/-0.89 -1.54 -0.77/-0.89 -1.55 -0.77/-0.89 -1.81 -0.79/-0.92 -1.76 -0.78/-0.90 -1.73 -0.77/-0.89 -1.71 -0.77/-0.89 -1.69 -0.78/-0.90 -1.68 -0.77/-0.89 -1.68 -0.77/-0.90 -1.88 -0.78/-0.91 -1.90 -0.77/-0.90 -1.83 -0.77/-0.89 -1.93 -0.78/-0.91 -1.92 -0.77/-0.89 -1.80 -0.78/-0.09 -1.85 -0.76/-0.89 -1.72 -0.77/-0.90

Fad

Ftt

Fhh

Fap1

Fap2

-0.60 -0.24/-0.30 -0.62 -0.24/-0.30 -0.60 -0.24/-0.29 -0.60 -0.24/-0.30 -0.60 -0.24/-0.30 -0.59 -0.23/-0.30 -0.60 -0.24/-0.30 -0.65 -0.24/-0.30 -0.66 -0.24/-0.30 -0.67 -0.24/-0.30 -0.68 -0.24/-0.30 -0.68 -0.24/-0.30 -0.72 -0.24/-0.30 -0.72 -0.23/-0.29 -0.64 -0.24/-0.30

1.63 0.29/4.13 2.45 1.32/5.74 2.07 0.82/4.96 2.67 1.53/6.07 2.30 1.13/5.43 2.76 1.62/6.19 2.48 1.31/5.71 2.48 1.03/5.28 2.79 1.26/5.68 2.93 1.30/5.76 3.03 1.37/5.87 3.09 1.39/5.19 3.15 1.42/5.97 3.20 1.44/6.00 2.46 0.81/4.96 2.93 1.29/5.74 3.32 1.55/6.17 3.21 1.36/5.89 3.14 1.23/5.70 3.33 1.39/5.95 3.26 1.29/5.80 1.80 0.81/4.21 3.67 1.35/5.87 3.63 1.40/6.00 3.29 1.21/5.46 2.92 1.51/6.11 3.24 1.15/5.16 3.39 1.21/5.32 2.86 1.24/5.61

2.70 1.20/1.82 2.78 1.23/1.83 2.83 1.24/1.84 2.78 1.23/1.82 2.83 1.24/1.84 2.81 1.23/1.81 2.82 1.24/1.83 3.17 1.23/1.83 3.15 1.22/1.81 3.13 1.21/1.80 3.13 1.04/1.56 3.13 1.21/1.80 3.13 1.21/1.79 3.13 1.20/1.79 3.20 1.21/1.82 3.36 1.23/1.82 3.34 1.21/1.80 3.28 1.20/1.79 3.25 1.10/1.65 3.27 1.20/1.78 3.24 1.19/1.78 3.40 1.18/1.70 3.39 1.20/1.79 3.49 1.19/1.78 3.51 1.21/1.82 3.44 1.21/1.80 3.37 1.22/1.82 3.46 1.24/1.80 3.16 1.20/1.79

-6.50 -0.87/-5.53 -6.95 -2.00/-5.42 -6.35 -1.85/-5.56 -6.70 -1.96/-5.47 -6.33 -1.86/-5.56 -6.60 -1.93/-5.50 -6.35 -1.86/-5.55 -7.20 -2.16/-5.59 -7.06 -2.15/-5.63 -6.97 -2.16/-5.69 -6.93 -2.16/-5.71 -6.87 -2.16/-5.75 -6.83 -2.16/-5.76 -6.83 -2.16/-5.77 -7.30 -2.24/-5.73 -7.14 -2.21/-5.70 -7.09 -2.21/-5.71 -6.99 -2.23/-5.79 -6.90 -2.25/-5.86 -6.91 -2.25/-5.84 -6.85 -2.26/-5.89 -7.32 -1.87/-4.84 -7.34 -2.26/-5.80 -7.23 -2.30/-5.91 -7.36 -2.09/-5.00 -7.18 -2.28/-5.78 -7.10 -2.09/-5.09 -7.17 -2.54/-5.24 -6.94 -2.12/-5.60

-0.85 0.80/0.53 -0.95 0.78/0.49 -0.90 0.78/0.50 -0.97 0.78/0.49 -0.98 0.78/0.50 -1.00 0.78/0.49 -1.00 0.78/0.50 -0.85 0.76/0.45 -0.85 0.75/0.45 -0.84 0.75/0.45 -0.84 0.75/0.45 -0.82 0.75/0.45 -0.81 0.75/0.45 -0.82 0.75/0.45 -0.73 0.76/0.46 -0.79 0.75/0.44 -0.80 0.74/0.43 -0.76 0.74/0.44 -0.72 0.74/0.45 -0.75 0.74/0.44 -0.72 0.74/0.44 -0.74 0.79/0.56 -0.66 0.74/0.44 -0.74 0.73/0.43 -0.69 0.75/0.45 -0.49 0.74/0.43 -0.74 0.75/0.44 -0.71 0.77/0.56 -0.80 0.76/0.47

a Here two different transition dipoles are used. The TDC values are written in the second line for each case. The origin of the local mode transition dipole is assumed to be on the CdO axis and 0.868 Å away from the carbonyl carbon atom (see ref 46). For the two cases, the angles of the transition dipole vector are assumed to be 10° and 23° for the two cases, and the magnitudes of the transition dipole are 2.73 and 3.135 D/Å amu-1/2, respectively.

the electrostatic potential at the jth site of the mth peptide group. We have already determined the best sets of partial charges and lj(m) values for the four sites (see ref 24). The amide I local mode frequencies calculated with eq 6 are compared with those obtained from the scaled AM1 local mode frequencies. The empirical correction method, eq 6, works very well in this case. In Figure 5, we specifically plot the cases of 210-510. Note that the hydrogen-bond-induced local mode frequency shifts are

correctly predicted by this model. To specifically discuss the hydrogen-bond-induced effects on the local mode frequencies, let us consider 310 β-sheet in more detail (see Figure 5). The local mode frequencies of the first 10 peptides, denoted as p1-1 to p1-10, that belong to the first outer strand exhibit an alternating behavior. This can be easily understood by noting that the hydrogen-bonding interactions occur for every two peptides. Similarly, the local mode frequencies of the last 10

20402 J. Phys. Chem. B, Vol. 108, No. 52, 2004

Lee and Cho IV. Transition Dipole Coupling Theory: Comparison The inter- and intrastrand vibrational coupling constants play a critical role in the delocalization of each amide I normal mode. One of the most successful theories used to quantitatively determine the vibrational coupling constants between two amide I local modes is the transition dipole coupling theory developed by Krimm.1,42-45 The TDC theory is based on the assumption that a pair of peptides interact with each other via the dipoledipole interaction potential as

VDD(Q1,Q2) ) µ1(Q1)‚T˜ 12(φ,ψ)‚µ2(Q2) T˜ 12(φ,ψ) )

Figure 4. Notations of various coupling constants.

peptides in the third strand also exhibit the same alternating behavior. On the other hand, the 10 peptides p2-1 to p2-10 in the middle strand are capable of forming two hydrogen bonds at CdO and N-H sites. Therefore, the local mode frequencies of the peptides in the middle strand do not show any notable site dependence at all as can be seen in Figure 5. The cases of 410 and 510 can be understood similarly.

1 [I˜ - 3rˆrˆ] 4π0|r|3

(7)

The dipole moments of the two peptides are denoted as µ1 and µ2, respectively. The mass-weighted amide I local vibrational coordinates are denoted as Q1 and Q2. T˜ 12 is the dipole-dipole interaction tensor, and r ) r1 - r2 where ri (i ) 1, 2) denotes the position vectors of the two dipoles, and rˆ denotes the unit vector along the direction of r. Now, the dipole-dipole interaction potential is Taylor expanded with respect to the two local coordinates, Q1 and Q2, up to the second-order terms. Then, the coupling force constant F is given as

F12 ) (∂µ1/∂Q1)0‚T˜ 12‚(∂µ2/∂Q2)0

(8)

where the first dipole derivative is the transition dipole of the amide I local mode. Despite that this TDC theory has been extensively studied and widely applied to the simulations and

Figure 5. For 210-510, the amide I local mode frequencies obtained by using the Hessian matrix reconstruction and linear scaling methods are compared with those obtained with the empirical correction formula given in eq 6. Also, for the sake of comparison the corresponding CdO bond lengths are plotted. Here, p2-5 for example denotes the fifth peptide in the second strand.



Figure 6. (a) Simulated IR absorption spectra of 210-510. (b) The inverse participation ratios of each normal mode in 210-510. (c) The F(M) R spectrum, which is approximately a measure of the number of strands participating in a given normal mode, is plotted.

interpretations of amide I bands and frequency splitting behaviors, it was found that a single set of transition dipole coupling parameters such as magnitude, direction, and origin of the transition dipole is not enough to quantitatively determine the vibrational coupling constant for any arbitrary conformation of polypeptide. In this section, we will compare TDC constants for all β-sheets considered in this paper with semiempirical calculation results. Particularly, two different sets of TDC parameters are used. The first set of parameters used was given in ref 46. The magnitude of the transition dipole is 2.73 D/Å amu-1/2 and the angle between the CdO axis and the transition dipole vector is assumed to be 10°. The origin of this point transition dipole is on the CdO axis and is at 0.868 Å from the carbonyl carbon atom. The second set of transition dipole coupling parameters is obtained from the AM1 calculation of an NMA molecule. The angle between the CdO axis and the transition dipole vector is 23° and the magnitude of the transition dipole is 3.135 D/Å amu-1/2, but the origin of the transition dipole is assumed to be the same. In Table 1, the average TDC constants calculated are summarized. In general, the TDC theory works reasonably well in these cases of multiple-stranded β-sheet polyalanines. Particularly, the interstrand vibrational couplings are found to be well described by the TDC mechanism, though the TDC theory with parameters given above slightly underestimates the coupling constants in comparison to the quantum chemistry calculation results. V. IR Spectrum, Delocalization, and Phase Correlation A. IR Absorption Spectrum. By using the numerically calculated transition dipole strengths and normal-mode frequencies, the IR absorption spectra of four representative multiplestranded antiparallel β-sheets, 210, 310, 410, and 510, are simulated and plotted in Figure 6a. In addition to the bar spectra, the Lorentzian-broadened absorption spectra are shown in the same figuresthe dephasing constant was assumed to be 6 cm-1. The high-frequency amide I band is weak in comparison to the lowfrequency band, which is in good agreement with experiments.9,12,21 In convention, the weak high-frequency band was assigned to the A-mode because the amide I local mode

vibrations in a given strand are in-phase with each other whereas the strong low-frequency band was assigned to the E1-mode because the vibrations of two neighboring peptides are out-ofphase. The high-frequency (A-mode) band position is found to be independent of the number of strands whereas the lowfrequency E1-mode band position is strongly red-shifted as the number of strands increase. We will present a detailed discussion on the origin of the A-E1 frequency splitting in Section V.D below. Although the IR intensity distribution will be quantitatively described in Section V.C by calculating the so-called weighted phase-correlation factors introduced in ref 27, a brief discussion on the origins of a weak A band and a strong E1 band is first presented by considering the symmetric natures of antiparallel β-sheets. Let us consider Mn values for M being an even number, such as 210 and 410. From Figure 1, these two antiparallel β-sheets have C2, i, and σh symmetry elements so that they belong to the C2h point group. Now, let us consider the highfrequency modes. On the basis of the corresponding eigenvector obtained from the Hessian matrix reconstruction method, the high-frequency vibration can be assigned to the totally symmetric representation, Ag, so that it is IR inactive. However, other high-frequency modes that are localized on the GH group peptides are not perfectly Ag so that they are weakly IR active. In contrast, we found that the low-frequency modes are antisymmetric with respect to both the inversion and C2 rotation so that they are in the Bu symmetry representation, which is strongly IR active. Note that, however, the same symmetry analysis method cannot be used to describe the IR intensity distribution of Mn values having an odd number of strands because of broken symmetry. However, even for them, if all CdO bonds are considered, assumed to be aligned perpendicularly to each strand, and are on the same β-sheet plane, still the same symmetry arguments are acceptable and the reason the high (low) frequency peak in the IR absorption spectrum is weak (strong) can be similarly understood. Perhaps an even more general argument can be made by considering an infinite β-sheet. Then, the IR active and inactive modes can be identified and they will appear in the low- and high-frequency regions, respectively.


Lee and Cho

B. Delocalization of Amide I Normal Modes. Due to the intra- and interstrand vibration couplings, each individual amide I normal mode is delocalized over the entire β-sheet. However, the extent of delocalization depends not only on those coupling constants but also on local mode frequencies. To study delocalization behaviors of normal modes in multiple-stranded antiparallel β-sheets, we first calculated the inverse participation ratio (IPR)27,37 defined as, for the R’th amide I normal mode,

IPRR ≡ (

∑j UR,j4 )-1

(9)

where the summation is over all N peptides in the entire multiple-stranded β-sheets. Note that the IPR number approximately equals the number of amide I local modes involved in the Rth normal mode. Among various β-sheets, we will focus on the cases when each single strand contains 10 peptide bonds, N i.e., 210 to 510. In Figure 6b, the IPR spectra, defined as ∑R)1 4 -1 (∑j UR,j) δ(ω - ωR), are plotted. As the number of strands increases, the magnitude of the average IPR number increases from 7.3 for 210 to 13.8 for 510. Regardless of the number of strands, the normal modes are found to be extensively delocalized over the entire β-sheet. Although the inverse participation ratio defined in eq 9 is a good measure of the number of participated local modes in the Rth normal mode, it does not provide specific information about how may strands are involved in each amide I normal mode. To address this issue, we newly define a modified inverse participation ratio as follows for the Mn system

F(M) R

≡

1 M

n

(10)

2 2 (∑UR,j ) ∑ s)1 j∈s

The first summation over s is over the M strands and the second summation is over jth local modes belonging to the sth strand. Note that the range of F(M) R is from 1 to M, depending on the extent of delocalization over different strands. To provide a physical picture of F(M) R , let us consider two limiting cases. Suppose that the Rth normal mode is completely localized on just one strand. In this case, F(M) R is a unity. On the other hand, in the case when the Rth normal mode is evenly delocalized n over the M strands so that ∑j)1 U2R,j ) 1/M, F(M) R equals M. In (M) Figure 6c, the corresponding FR spectra for 210 to 510 are presented. The average F(M) R values for M ) 2-5 are 2, 2.3, 3.2, and 3.5, and the normalized values defined as 〈F(M) R 〉/M are 1, 0.77, 0.80, and 0.70. A slight deviation of 〈F(M) R 〉/M from 1 for M ) 3-5 can be understood by noting that the inner strand and outer strand are different from one another so that the diagonal Hessian matrix elements associated with the peptides in the inner strands differ from those with peptides in the outer strands. This broken symmetry causes a slightly uneven distribution of the eigenvector elements among the strands and deviations of 〈F(M) R 〉/M values, for M ) 3-5, from 1. Nevertheless, 〈F(M) 〉/M values are close to 1 and this result suggests R that most of normal modes are delocalized over all strands almost eVenly. This observation is in good agreement with the recent experimental finding by Tokmakoff and co-workers, who measured both IR absorption and two-dimensional IR photon echo spectra of various proteins containing a large portion of antiparallel β-sheets.18 C. Phase Correlation between Neighboring Local Modes. Although the extent of delocalization of each individual normal

mode could be quantitatively estimated by examining the IPR spectrum in Figure 6b, there is another critical issue on the phase relationship between different amide I local mode vibrations participated in a given normal mode. As briefly mentioned in the Introduction, Kubelka and Keiderling studied the phase correlations between neighboring local modes by directly examining the eigenvector elements of a few selected amide I normal modes.12,20 To quantitatively address this issue we introduced the socalled weighted phase-correlation factor (WPCF)27 that can be considered to be a measure of how the vibrational phases of two neighboring peptides are related to each other. For the cases of β-sheets, the WPCF of the Rth normal mode is defined as n-1

P(s) R

≡

sign(UR,jUR,j+1)|UR,jUR,j+1| ∑ j∈s

(11)

where s denotes the index of strand and the summation is over all (n - 1) sites in the sth strand. Although P(s) R is a measure of phase-correlation of local modes in a given strand, it is necessary to introduce another phase-correlation factor describing that between two peptides belonging to the nearest neighboring strands and forming a hydrogen bond to each other. To this end, let us define P[l] R for l ) 1 to n, as M-1

P[l] R ≡

sign(UR,lUR,l+1)|UR,lUR,l+1| ∑ s)1

(12)

In parts a and b of Figure 7, we plotted P(s) R spectra of 310 and 410. At first sight, the weighted phase correlation does not appear to exist because P(s) R values do not monotonically depend on the corresponding Rth normal-mode frequencies. However, considering two IR active modes in the high- and low-frequency regions (see the two star-marked peaks in Figure 6(a)), we find that P(s) R values of the IR-active high-frequency normal mode, indicated by a asterisk in parts a and b of Figure 7, are all positive, indicating that the local mode vibrations within a given strand are in-phase. This result explains why the high-frequency peak in the IR spectrum is comparatively weak. Noting that the transition dipoles of each local mode in a given strand are aligned to be in up-down-up sequence, the local mode transition dipoles are largely canceled out with one another in this case so that the corresponding high-frequency normal mode becomes weakly IR active. On the other hand, P(s) R values of the strongly IR-active low-frequency normal mode is negative, regardless of the number of strands in the antiparallel β-sheet, which shows that the local mode vibrations in a given strand are out-of-phase. Consequently, the transition dipoles associated with each local mode add up to produce a large transition dipole for the corresponding normal mode in the low-frequency region of the IR spectrum. This observation is in good agreement with Keiderling et al.’s finding in ref 20. Second, we calculated P[l)1] spectra for 210 to 510 and plotted R values in the highthem in Figure 7c. Interestingly, the P[l)1] R frequency normal modes are all negative, indicating that the two local mode vibrations in the neighboring strands are outof-phase. This observation is also found to be inconsistent with the fact that the high-frequency normal modes are relatively weakly IR active. The transition dipole of the jth peptide in the mth strand is approximately speaking in parallel with the jth peptide in the (m ( 1)th strand. Therefore, the out-of-phase vibrations of these two peptides in the neighboring strands produce a small transition dipole of the corresponding normal



Figure 7. Modified weighted phase-correlation factor, P(s) R , for each strand in 310 and 410 is plotted in parts a and b, respectively. For 210-510, the phase-correlation factor, P[l] R , for the case of l ) 1, is plotted in part c.

Figure 8. (a) For all antiparallel β-sheets considered in the present paper, the A-E1 frequency splitting magnitude ∆(M) is plotted with respect to the number of strands, M. ∆(M) is found to be independent of the length of each strand. (b) ∆(M) is calculated by considering the one-dimensional Frenkel exciton model.

mode. In strong contrast, the P[l)1] values of the low-frequency R normal modes are positive so that the transition dipoles of local modes are constructively interfering to make the corresponding low-frequency normal mode strongly IR active. Although only the case of l ) 1 was discussed in the present paper, the same trend was observed for all l ) 1 to n. Overall, the quantitative calculations of various WPCF’s presented in this subsection clearly showed that the distribution of amide I normal mode IR intensities is closely related to the relative vibrational phase relationships among local mode vibrations and to the orientations of local mode transition dipoles. D. A-E Frequency Splitting. One of the distinctive features that depend on the number of strands, M, is the A-E1 frequency splitting denoted as ∆n(M). For M varying from 2 to 8, ∆n(M) values are estimated from the simulated IR absorption spectra and plotted in Figure 8a. We found that the A-E1 frequency splitting does not depend on the number of peptides in a giVen strand. Thus, the A-E1 frequency splitting will be simply denoted as ∆(M) by ignoring its n-dependence. Now, from the data plotted in Figure 8a it becomes possible to address the following question: what is the functional form of the A-E1 frequency splitting magnitude with respect to the number of strands? Denoting the asymptotic value of ∆(M) to be ∆(∞)

and the initial value for the case of M ) 2 to be ∆(2), it is found that ∆(M) in Figure 8a can be fitted by using the following stretched-exponentially rising function,

∆(M) ) ∆(∞) - (∆(∞) - ∆(2)) exp{- k(M - 2)R}

(13)

In our case, the corresponding values in eq 13 are ∆(∞) ) 57 cm-1, ∆(2) ) 20 cm-1, k ) 0.47, and R ) 1.17. Perhaps eq 13 can be used to determine the number of strands of an arbitrary protein having a significant portion of antiparallel β-sheet polypeptides, once the corresponding parameters in eq 13 are predetermined experimentally in the future. However, it should be emphasized that the structures of natural β-sheets in proteins are likely to be strongly distorted due to interactions with either solvent or nearby polypeptide segments. Therefore, eq 13 should be carefully used and needs to be tested for various β-sheets in real proteins. The strong dependences of the E1-mode frequency and the A-E1 frequency splitting on M can be theoretically described by using a simplified one-dimensional Frenkel exciton model47-51snote that Cheatum et al.19 recently presented a more elaborate exciton model study of amide I IR and twodimensional vibrational spectra of an ideal antiparallel β-sheet


Lee and Cho shown in this paper the simulated IR absorption spectra for 210 to 510 by using the above one-dimensional exciton model were found to be in quantitative agreement with those in Figure 6a. This suggests that the red-shifting behavior of the E1-band in the multiple-stranded antiparallel β-sheets can be essentially modeled by this one-dimensional Frenkel exciton model with properly estimated coupling constant and local mode frequencies. VI. Summary

Figure 9. An interstrand hydrogen-bonded linear chain can be considered as a simple model for constructing the one-dimensional Frenkel exciton Hamiltonian. The relative energy levels and couplings are schematically shown.

polypeptides. The fact that the A-E1 frequency splitting ∆(M) was found to be rather insensitive to the chain length suggests that the intramolecular neighboring coupling within a single strand does not play a significant role in the A-E1 frequency splitting behavior. Thus, even though the β-sheet polypeptides should be better described in terms of spatially two-dimensionally delocalized excitons, the dimensionality can be effectively reduced down to one because of the above-mentioned insensitivity of ∆n(M) with respect to n. Therefore, a linear chain of M coupled oscillators can be effectively used to describe ∆(M) (see the linear hydrogen-bond chain in the box of Figure 9). The energy-level scheme for the one-dimensional Frenkel exciton model is shown in Figure 9. The peptide at site 1 belongs to group GH because it has one hydrogen bond at the N-H site, whereas the last peptide (site 5 in this specific case of 5n β-sheet) belongs to the GO group. The inner peptides all belong to the GOH group. The site energy differences between GH and GOH peptides and between GO and GOH peptides were estimated to be about 25 and 10 cm-1, respectively. The relevant interstrand coupling constant, Fab, was calculated to be -10.3 cm-1. Then, the corresponding Frenkel exciton Hamiltonian is simply given as M

H)

∑ j)1

νj a+ j aj +

M-1

+ Fab{a+ ∑ j aj+1 + aj+1 aj} j)1

(14)

where νj was denoted as the jth local mode frequency and the creation and annihilation operators for the jth local mode excitation were denoted as a+ j and aj, respectively. After diagonalizing the Frenkel Hamiltonian, one can obtain a set of eigenvalues and the maximum and minimum frequencies will be denoted as νmax(M) and νmin(M). In Figure 8b, νmin(M), νmax(M), and ∆(M) ) νmax(M) - νmin(M) are plotted with respect to M. It is found that the frequency difference ∆(M) obtained from the above simple one-dimension exciton model is qualitatively in good agreement with that in Figure 8asthough not

Carrying out a series of AM1 calculations for multiplestranded antiparallel β-sheet polyalanines and using the Hessian matrix reconstruction method, we investigated interstrand interaction-induced amide I local mode frequency shifts and their site dependencies and determined various vibrational coupling constants. The diagonal Hessian matrix elements or amide I local mode frequencies obtained from the AM1 calculations were directly compared with those predicted by using the empirical correction method developed recently. Also, it was shown that the transition dipole coupling theory works well in the cases of antiparallel β-sheet polyalanines. From the comparative investigation in the present paper, it is believed that the principal goal of addressing the eight questions given in the Introduction was achieved. The corresponding answers for each question can be summarized as follows: (1) depending on the hydrogenbonding site, the amide I local mode frequency shift is about 25 or 10 cm-1, (2) the intrastrand coupling constant between any two neighboring peptides is as small as about 1 cm-1, (3) the interstrand coupling constants are significantly larger than the intrastrand ones, (4) due to the large interstrand coupling constants, the amide I normal modes are strongly delocalized over the peptides in the other strands, (5) the empirical correction method in eq 6 can be used to quantitatively determine the amide I local mode frequencies, (6) the transition dipole coupling model works well in these cases of multiple-stranded antiparallel β-sheets, (7) the functional form of the A-E1 frequency splitting magnitude is approximately given as eq 13, and (8) both intraand interstrand coupling constants do not strongly depend on the chain length, though they are weakly dependent on the number of strands because the average interstrand distance slightly decreases with respect to the increasing number of strands. In addition, by investigating the natures of A- and E1bands with the reduced 1D Frenkel exciton model, the A-band at the high-frequency region is produced by the high-frequency modes that are relatively localized on the GH peptides, of which frequencies are close to 1690 cm-1, and the E1-band for Mn originates from the M coupled oscillators along the linear chain of peptides as shown in Figure 9. This latter finding explains why only the low-frequency E1 band becomes red-shifted as the number of strands increases, whereas the high-frequency A-band position does not. Overall, it is believed that the present model calculation results will be of use in describing various experimental observations. Acknowledgment. This work was supported by the creative research initiative program of KISTEP (MOST). We are grateful to Professor Tokmakoff for stimulating discussions and comments and for preprints of refs 18 and 19. References and Notes (1) Krimm. S.; Bandekar, J. AdV. Protein Chem. 1986, 38, 181. (2) Byler, D. M.; Susi, H. Biopolymers 1986, 25, 469. (3) Surewicz, W.; Mantsch, H. H.; Chapman, D. Biochemistry 1993, 32, 389.

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Local Amide I Mode Frequencies and Coupling Constants in Multiple

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