Amide Vibrations and Their Conformational Dependences in β-Peptide

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J. Phys. Chem. B 2010, 114, 16011–16019

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Amide Vibrations and Their Conformational Dependences in β-Peptide Juan Zhao and Jianping Wang* Beijing National Laboratory for Molecular Sciences, State Key Laboratory of Molecular Reaction Dynamics, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190, P. R. China ReceiVed: September 1, 2010; ReVised Manuscript ReceiVed: October 25, 2010

The characteristics of the amide-A and amide-I modes in a β-homoalanine dipeptide (β-HADP) have been examined as a function of backbone dihedral angles. The harmonic frequencies were obtained using the density functional theory. The anharmonic frequencies and diagonal anharmonicities were obtained by using the Morse potential. Local-mode frequencies and intermode couplings were obtained using the computed normal-mode frequencies and eigenvectors. It was found that the vibrational frequencies for the two types of amide modes are both conformational-dependent. The inter-amide-A and inter-amide-I couplings in the β-peptides were predicted to be generally weaker than those in the R-peptides. Structural bases of the amide-A and amide-I local modes in the β-peptides are discussed. 1. Introduction It is well known that traditional vibrational techniques such as linear infrared (IR) spectroscopy are very sensitive to molecular structure and its local microenvironment. To be able to use this technique, it is necessary to understand the relationship between molecular structures and their IR signatures. The amide-I band (mainly the CdO stretching motion) is commonly used as an IR probe for peptide conformation.1 The relationship between peptide secondary structure and the amide-I band position has been extensively studied; for example, it is believed that in aqueous solution R-helices, β-sheets, and random-coils have characteristic IR bands at 1638, 1610, and 1644 cm-1, respectively.2 However, such a picture seems to be quite naive. Recent IR studies have shown that the amide-I modes are exciton-like.3-6 Vibrational transition-intensity transfer occurs among the amide-I vibrators as a result of vibrational coupling. Frequency shift also occurs as a result of the coupling as well as solvent influence. The consequence is that a secondary structure of peptide does have a unique IR absorption band, but the band profile could also be influenced by solvent. Because of the presence of the coupling, a shifted IR band as a result of site-specific isotopic labeling does not reflect a purely labeled site state but rather a mixed state. Furthermore, recent studies of linear IR and femtosecond 2D IR experiments and simulations have shown that vibrational parameters such as diagonal anharmonicities, mixed-mode off-diagonal anharmonicities, as well as vibrational couplings of the amide-I modes are very sensitive to peptide structures.7-20 These are the studies mainly carried out for the R-peptides. For the β-peptides, however, the IR signatures of their secondary structures have not been systematically studied. The β-peptides, which are the oligomers of the β-amino acid residues, are very important unnatural foldamers owing to their biological functions.21,22 A β-amino acid contains one additional carbon atom in the backbone in comparison with an R-amino acid. Despite their seemingly higher conformational flexibilities, the β-peptides have been shown to display high folding propensities.23 The conformation of a β-peptide is mainly * Corresponding author. Tel: (+86)-010-62656806. Fax: (+86)-01062563167. E-mail: [email protected].

determined by its sequence. About six to seven β-amino acid residues are capable to form stable secondary structures if the residues are properly selected,24-26 whereas for the R-peptides, some 10-20 amino acid residues are usually necessary to form stable secondary structures.27 The β-peptides also form a larger variety of secondary structures than do the R-peptides. Up to now, apart from turns and hairpins,28-31 various β-helices with 14-,32-37 12-,38-41 10-,42,43 8-,44,45 and 12-/10-46,47 membered intramolecular H-bonding rings were identified. 1D and 2D IR studies have been carried out to examine the structures and structural dynamics of the β-peptides using the amide-A28,29 as well as the amide-I modes.47,48 The amide-A mode (mainly the N-H stretching vibration) is localized and can thus be used to probe local structures and local structural distributions. Theoretical and experimental studies have indicated that local chemical environment, hydrogenbonding conditions, and nearby solvent configurations all play a role in determining the frequency and line shape of the amide-A mode.49-53 Very recently, the vibrational properties of the amide-A and amide-I modes, including the harmonic and anharmonic properties of the R-peptide oligomers, have been studied by ab initio calculation,14,54 molecular dynamics simulations,55 and IR experiments.56 In the present work, using a β-homoalanine dipeptide (βHADP, CH3CONHCβHCH3CRH2CONHCH3, Figure 1) as a model molecule, we examine the harmonic and anharmonic vibrational parameters of the amide-A and amide-I modes and their dependences on the backbone conformations. Because of the additional β-carbon atom in the β-HADP with respect to the R-alanine dipeptide (ADP, CH3CONHCRHCH3CONHCH3), the β-HADP has one additional dihedral angle, θ (Figure 1). In this work, by fixing θ at 180°, we examine the vibrational properties of the two pairs of the amide-A and amide-I modes in the β-HADP along the φ and ψ dihedral angles by ab initio calculation. Full anharmonic frequency computations would be permitted if the systems were smaller, such as the R-aminoacid-based ADP.57 For the β-HADP, we turn to use Morse potential to evaluate the anharmonic vibrational parameters. We also examine the strength of the vibrational coupling between two amide-A modes as well as that between two amide-I modes.

10.1021/jp108324p  2010 American Chemical Society Published on Web 11/11/2010

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Figure 2. Typical potential energy curve for the amide-A mode of the β-HADP. The solid dots indicate the ab initio result using the DFT theory, fitted using the Morse potential function.

V(δq) ) D(1 - e-R(δq))2

Figure 1. Top: Molecular structure of the β-homoalanine dipeptide (β-HADP) with three backbone dihedral angles defined as φ (∠CNCβCR), θ (∠NCβCRC), and ψ (∠CβCRCN). Hydrogen atoms on the CR and Cβ atoms are not shown. Bottom: The potential energy surface as obtained by scanning the φ and ψ angles at θ ) 180°. The transition dipoles of the amide-A and amide-I modes are illustrated as arrows.

We focus on revealing and understanding the sensitivities of thesevibrationalpropertiestotheβ-peptidebackboneconformation. 2. Calculation Methods 2.1. Harmonic Frequency. Harmonic frequencies of the fundamental transitions for the 3N-6 normal modes of the β-HADP were obtained by ab initio computations at the B3LYP density functional level with the 6-31G** basis set using Gaussian 03.58 This combination of theory and basis set has been used to examine various conformations of the β-peptide in a recent study.59 If the vibrational properties were evaluated in 3D space (φ, θ, ψ) as defined in Figure 1, then the computations involved would be very expensive. In this work, by fixing θ at 180°, a series of the β-HADP structures with ∆φ ) ∆ψ ) 15° in the range of (-180 e φ e 180°, 180 e ψ e 180°) were partially optimized. In total, there are 576 independent dipeptide structures to be examined. Trans amide unit (-CONH-) was assumed in the β-HADP. Fixing θ at 180° corresponds to a case in which two amide unites are largely separated, and thus intermode coupling is expected to be relatively small. The conformational dependence of the vibrational properties of the two amide-A modes and the two amide-I modes in such a restricted space were examined. Furthermore, several typical β-peptide conformations with θ * 180° were also examined in the end. 2.2. Anharmonic Frequency. The anharmonic frequencies of the two amide-A modes and the two amide-I modes of the 576 β-HADP structures were obtained by fitting the Morse potential function.60-62 For a diatomic molecule, the Morse potential is the function of the distance between two nuclei. In this work, for any vibrational mode, we express the Morse potential on the basis of the normal-mode coordinates. Shown in Figure 2 is the case of the amide-A mode. The solid dots represent the ab initio energy of the β-HADP molecule computed as the N-H bond is stretched (δq). However, the displacements of other atoms during the vibration of the amide-A mode were also taken into account during the energy calculation. These energies were then used to fit to the Morse potential energy function

(1)

where D is the so-called bond dissociation energy (in diatomic molecule) and R is a parameter that sets the range of the potential. D and R can be obtained from the fitting. The parameters D and R are connected via µ and ω

 2Dµ

R ) 2πω

(2)

where µ and ω are the reduced mass and harmonic frequencies for the mode. The vibrational energies were then calculated according to

(

E(ν) ) hω ν +

1 1 - ∆hω ν + 2 2

)

(

2

)

(3)

where ν is vibrational quantum number (ν ) 0, 1, ...). ∆ is the overtone diagonal anharmonicity (or simply the diagonal anharmonicity) of the vibrational mode, which is given by

∆)

hω 4D

(4)

A similar Morse potential protocol has been used to study the anharmonic properties of N3-.63 Furthermore, the 13C-isotopic effect on the anharmonic frequency and anharmonicity of the amide-I mode for several typical β-peptide conformations is evaluated using a similar procedure. To do this, we label the two amide units (-CONH-) by 13C at the same time. 2.3. Vibrational Coupling. The ab initio calculations provide the eigenvalues (frequencies) and the eigenvectors for each normal mode. The information of local-mode frequencies and the intermode couplings associated with a group of normal modes can be retrieved by using a wave function demixing approach, as previously described.64 We apply the approach to the two amide-A modes and two amide-I modes of β-HADP in various conformations. 3. Results and Discussion 3.1. Potential Energy Surface of the β-HADP. The potential energy surface (PES) of the β-HADP obtained at the B3LYP/6-31G** level of theory is given in Figure 1. The lower energy conformation is located around (φ ) -75°, ψ ) -90°), whereas the most unstable structure appears around (0°, 30°). The PES is somewhat similar to that of MeCO-β-Ala-NHMe, which was reported very recently.65 However, because of the

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Figure 3. Normal-mode vibrational frequencies of the two amide-A modes. (a,c,e) Harmonic, anharmonic fundamental and anharmonic overtone transition frequencies for the amide-A mode on the Ac side. (b,d,f) Harmonic, anharmonic fundamental, and anharmonic overtone transition frequencies for the amide-A mode on the NMe side.

difference in their side chains, slight difference exists in the two PES profiles. 3.2. Harmonic Frequencies of the Amide-A Modes. The harmonic frequencies of the fundamental transitions of the two amide-A modes in the φ-ψ dihedral angle space (φ-ψ map) are shown in Figure 3a,b. The frequencies are found to be very sensitive to the φ and ψ dihedral angles, indicating that they are chemical environment-sensitive. However, the conformational dependences of the two modes are quite different. The harmonic frequency of the amide-A mode on the acetyl side (denoted as Ac), ωa, is generally lower than the harmonic frequency of that on the NH-methyl side (denoted as NMe), ωn. It is seen that ωa changes more significantly along φ than along ψ, whereas ωn changes more significantly along ψ. It should be noted that the two CdO groups and two N-H groups can hardly form an intramolecular hydrogen bond (IHB) in the β-HADP. The closest H · · · O distance is found to be ca. 3.69 Å for the conformation with φ ) -75° and ψ ) -75°. Therefore the IHB yields insignificant influence on frequencies. It is also clear that there is no symmetry in the φ-ψ map of the harmonic frequencies of the two N-H modes in Figure 3a,b. The N-H bond length as a function of the φ and ψ dihedral angles is given in Figure 4. It indicates that the two φ-ψ maps of the N-H bond length are quite similar to the corresponding harmonic frequencies of the amide-A modes but with opposite trend. Higher frequencies correspond to shorter bond lengths and vice versa. These results reveal that the amide-A mode is indeed highly localized on the N-H stretching motion in the β-HADP. 3.3. Anharmonic Frequencies of the Amide-A Modes. The anharmonic frequencies of the fundamental and overtone

Figure 4. Conformational dependence of the N-H bond length in angstroms: (a) on the Ac side and (b) on the NMe side.

transitions for the two amide-A modes, obtained by fitting the potential curve to the Morse potential, were plotted as a function of the φ and ψ angles in Figure 3c,d. It is found that the conformational dependences of the anharmonic fundamental frequencies, νa and νn, are quite similar to those of ωa and ωn, respectively, however, with a considerable red shift in each. Statistical distributions of ωa, ωn, νa, and νn, are given in Figure 5a,b. The mean values of the four frequencies (ωa, ωn, νa, and νn) are found to be 3638.2, 3659.2, 3442.1, and 3464.3 cm-1, respectively, showing that the anharmonic decreases for the two modes are both on the order of 195 cm-1. The anharmonic overtone transition frequencies (ν2a and ν2n) of the two amide-A modes are also found to exhibit conformational sensitivity (Figure 3e,f). Furthermore, the patterns of the overtone frequencies of the two modes are quite similar to those of the corresponding fundamental frequencies. There is a relationship between the fundamental frequency and the overtone frequency for the ith anharmonic oscillator; that is, ν2i ) 2νi + ∆i, where ∆i is the anharmonic correction term (the pure-mode

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Figure 6. Conformational dependence of the harmonic frequencies for the two amide-I modes.

Figure 5. Statistics of the vibrational frequencies (a,b) and the anharmonicities (c) for the amide-A modes on the Ac side (black) and on the NMe side (red).

anharmonicity). Therefore, the similarity between the fundamental frequency and the overtone frequency indicates that the pure-mode anharmonicities for the two amide-A modes, which are ∆a ) 2νa - ν2a and ∆n ) 2νn - ν2n, respectively, are less conformational-dependent. A similar conclusion was also drawn for the R-amino-acid-based glycine dipeptide,53 showing that the amide-A modes in both the R- and β-peptides share similar properties. We perform statistical analysis on the anharmonicities, and the results are given in Figure 5c. The mean value of the two diagonal anharmonicities (∆a and ∆n) is ca. 195.5 cm-1, which is generally equal to the anharmonic decreases mentioned above. This can be simply explained by the fact that the Morse potential fitting is based on a local-mode picture, in which the anharmonic interactions with the remaining 3N-7 modes are neglected. Under such circumstances, the anharmonic drop is the pure-mode anharmonicity. This is a quite different picture from the normal-mode-based anharmonic treatment, where all intermode interactions are considered. The approach used in this study allows one to examine the pure-mode contribution to the anharmonic decrease in the potential energy of a given mode. 3.4. Harmonic Frequencies of the Amide-I Modes. Unlike the amide-A mode, which is localized on the N-H stretching motion, the amide-I mode in the R-peptides is known to be substantially delocalized. Here the conformational dependences of the harmonic frequencies for the two amide-I modes in the β-HADP are given in Figure 6. Clearly, the harmonic frequencies of the two amide-I modes are conformational-sensitive, showing somewhat different φ-ψ dependences between the two modes. The low-frequency mode (ωl), which is usually the asymmetric CdO stretching motion, changes more dramatically along ψ than along φ (Figure 6a), whereas the high-frequency component (ωh), which is usually the symmetric CdO stretching motion, changes more dramatically along φ (Figure 6b).

Furthermore, it is seen that along φ, ωl changes even more dramatically at certain ψ value, for example, at (60°, 60°). Similar situation occurs for ωh on the ψ axis. This demonstrates the extreme conformational sensitivity of the two amide-I frequencies, which is mainly due to the normal-mode nature of the amide-I mode. The mixing degree of the two local modes can be revealed by the potential energy distributions (PEDs). Table 1 lists the PED value for the β-HADP structure with the φ dihedral angle changing from 15 to 105° at ψ ) 60°, which is computed mainly using the CdO stretching motion for the low-frequency amide-I mode. Table 1 also lists the PED of highfrequency amide-I mode with ψ dihedral angle changing from 45 to 135° at φ ) 60°. Clearly, when φ changes from 15 to 105°, ωl decreases as the PED value of the CdO on the NMe side (C10dO12) decreases and increases again as the PED of C10dO12 increases. It is also clearly seen that when the φ angle changes from 45 to 60° and from 75 to 90°, the PED of C10dO12 changes dramatically (from 7.9 to 1.6% and from 1.9 to 11.7%, respectively). Meanwhile, ωl changes from 1773.3 to 1769.1 cm-1 and from 1769.8 to 1776.0 cm-1, respectively. Therefore, it is clear that the value of ωl depends on the PED composition of the mode. The situation for ωh is somewhat different from that of ωl: the PED values and the frequencies do not change so dramatically (Table 1, right side). The statistical distributions of the two amide-I harmonic frequencies are given in Figure ˜ l ) 1775.9 7a,b. The mean values of ωl and ωh are found to be ω ˜ h ) 1783.8 cm-1; the difference is caused by cm-1 and ω chemical environment and coupling effects. 3.5. Anharmonic Frequencies of the Amide-I Modes. The φ-ψ dependence of the anharmonic frequencies of the two amide-I modes has been examined. Figure 8 shows the harmonic and anharmonic frequencies of the two amide-I modes along the dihedral angle φ at ψ ) 0° and along ψ at φ ) 30°. The angular dependence of the anharmonic frequencies is quite similar to that of the harmonic frequencies. This reveals that the harmonic and anharmonic frequencies of the amide-I mode have similar conformational dependences. It is also found in Figure 8 that the mean value of the transition frequency decreases by ∼19 cm-1 when the anharmonic correction is considered. This value is smaller than that found for the amide-I mode in the R-peptide by ab initio calculation (30 cm-1).64 The reason for this has been explained in section 3.3 for the amide-A mode. Here even though the amide-I mode is normal mode in nature, we treat it on the local-mode basis. A similar assumption was also made for the amide-I mode in the R-peptide.66 The statistics of the anharmonicities of the two amide-I modes obtained in the whole φ-ψ space are given in Figure 7e,f. It is found that the mean values of ∆l and ∆h are 13.7 and 16.7 cm-1, respectively. It should be mentioned that there are some singularities in the entire φ-ψ map of the anharmonic frequencies for the two amide-I modes. For these ill-behaved structures, we find that the potential curves of the asymmetric CdO

J. Phys. Chem. B, Vol. 114, No. 48, 2010 16015 TABLE 1: Harmonic Frequencies (cm-1) of the Two Amide-I Modes and Their Potential Energy Distribution (PED, in percent) for Some Structures PED

PED

φ, ψa

ωl

C6dO5 (Ac)

C10dO12 (NMe)

15, 60 30, 60 45, 60 60, 60 75, 60 90, 60 105, 60

1775.3 1774.1 1773.3 1769.1 1769.8 1776.0 1776.2

0.4 1.3 8.2 14.7 14.4 3.9 0.7

14.2 13.5 7.9 1.6 1.9 11.7 14.0

a

φ, ψ 60, 60, 60, 60, 60, 60, 60,

45 60 75 90 105 120 135

ωh

C6dO5 (Ac)

C10dO12 (NMe)

1771.2 1775.0 1778.4 1779.0 1776.6 1773.5 1771.4

6.4 1.4 1.2 1.3 1.7 2.5 4.5

9.9 13.5 13.6 13.5 13.3 12.9 11.7

Dihedral angles φ and ψ in degrees.

Figure 8. Dihedral angle-dependent harmonic and anharmonic frequencies of the two amide-I modes. (a) ψ dependence at φ ) 0° and (b) φ dependence at ψ ) 30°. Figure 7. Statistics of the harmonic normal-mode and local-mode frequencies and anharmonicities for the two amide-I modes.

stretching motions, in particular, are very much like those of harmonic oscillators. Therefore, the Morse potential fitting does not yield satisfactory D values, which leads to the unsatisfactory anharmonic frequencies as well as the broader distribution of the anharmonicities shown in Figure 7e,f. 3.6. Local-Mode Frequencies and Intermode Couplings. The local-mode frequencies of the two amide-A modes were examined by decoupling the two normal modes. Because of its high localization nature, the local-mode frequencies and the conformational dependence of the amide-A mode are quite similar to those of the normal-mode frequencies, and thus no further discussion is given here. The local-mode frequencies of the two amide-I modes and the conformational dependences were obtained; the results are given in Figure 9a,b. After decoupling, the amide-I mode on the NMe side usually has a lower local-mode frequency, ω0l , whereas that on the Ac side generally has a higher local-mode frequency, ωh0. The conformational dependences of ω0l and ωh0 are clearly different; this is because they are site frequencies belonging to different amide groups and thus have their own structural basis. In other words, local chemical environment, for example, types of the bonding and atoms in the neighborhood of the amide unit, plays a major role here. However, one notices

Figure 9. Conformational dependence of the harmonic local-mode frequencies and the CdO bond length of the two amide-I modes: (a) ωl0 for the low-frequency mode, (b) ω0h for the high-frequency mode, (c) rCdO on the NMe side, and (d) rCdO on the Ac side.

that the profile of ω0l is similar to that of ωl (Figure 6a) to some extent and that of ωh0 is similar to that of ωh (Figure 6b). This suggests that a generally weak coupling existed between the two normal modes. (See below.) The statistics of ω0l and ωh0 are shown in Figure 7c,d. Two mean values of the local-mode frequencies are ω ˜ 0l ) 1777.9

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Figure 10. Statistics of the vibrational coupling between the two amide-A modes (a) and that between the two amide-I modes (b).

cm-1 and ω ˜ h0 ) 1781.9 cm-1, which are also very close to the corresponding the most probable values. It is found that ω ˜l ˜h - ω ˜ h0 ) 1.9 cm-1. The average ω ˜ 0l ) -2.0 cm-1 and ω frequency splitting between the two normal modes due to coupling is 3.9 cm-1 () 2.0 + 1.9), and the average splitting ˜ 0l ) is 4.0 cm-1. The latter of the harmonic local mode (ω ˜ h0 - ω represents the extent of average local-mode nondegeneracy. In a previous work,14 it was found that the average frequency splitting of the two amide-I local modes is 9.5 cm-1 in glycine dipeptide. This indicates the two amide-I local modes are less nondegenerate in the β-HADP. To assess the strength of anharmonic interactions between the two amide units in the β-HADP, we obtained the coupling between the two amide-A modes and that between the two amide-I modes. Statistics of the coupling constants on the φ-ψ space for the two amide-A modes and two amide-I modes are shown in Figure 10. The value of the coupling for amide-A modes (Figure 10a) ranges from -0.32 to +0.17 cm-1, indicating a weak coupling case. In a previous work,53 it was found that the anharmonic coupling of the two amide-A modes in glycine dipeptide is mostly within in (1 cm-1. This indicates that the inter-amide-A mode coupling becomes even weaker in the β-HADP. The statistics of the amide-I mode coupling is given in Figure 10b. It shows that the coupling varies from -5.7 to +5.6 cm-1. These values are also smaller than that of the glycine dipeptide, which range from -13.2 to +17.8 cm-1.14 These generally smaller couplings are mainly caused by the additional methylene in the backbone of the β-peptides. In addition, we found that both the inter-amide-A and inter-amide-I mode couplings are sensitive to the φ-ψ angles (data not shown); however, because of their small values, the sensitivity is not as significant as those found in the R-peptides.14,53,64,67 The CdO bond length (rCO) as a function of the φ and ψ dihedral angles is also shown in Figure 9c,d. It is shown that the φ-ψ map of rCO is quite similar to that of the corresponding local-mode frequency but with opposite trend. Higher frequencies correspond to shorter bond lengths and vice versa. This indicates that the amide-I local mode is mainly composed of the CdO bond stretching motion, which is quite similar to the case of the amide-I mode in the R-peptide. To demonstrate this further, we analyze the correlation between vibrational frequency and CdO bond length. Figure 11a,b gives the correlation between ωl and rCO on the NMe side and that between ωh and rCO on the Ac side, respectively. In each case, the frequency and bond length are roughly anticorrelated, with correlation coefficients determined to be -0.81 for Figure 11a and -0.92 for Figure 11b. The degree of correlation improves when local modes are considered because the two local modes represent two decoupled site states. This is clearly shown in Figure 11c,d. The correlation coefficient of ω0l and rCO on the NMe side, and that of ωh0 and rCO on the Ac side are found to be -0.83 and -0.96, respectively. This

Figure 11. Correlations between the amide-I mode frequency and CdO bond length for the β-HADP. (a) ωl versus the rCdO on the NMe side; (b) ωh versus the rCdO on the Ac side; (c) ωl0versus the rCdO on the NMe side; and (d) ω0h versus the rCdO on the Ac side.

confirms that in the entire φ-ψ map, the low-frequency localmode is dominated by the CdO stretching motion on the NMe side, and the high-frequency local-mode is dominated by the CdO stretching motion on the Ac side. 3.7. Amide Vibrational Properties of the β-HADP in Typical Conformations. The calculations discussed above were carried out for the β-HADP with a fixed dihedral angle θ (θ ) 180°). Actually, the backbone conformational space for the β-peptides has three dimensions (φ, θ, ψ). A typical β-polypeptide conformation has a set of characteristic dihedral angles, for example, the so-called C8 at (-66, -45, 95°), C6 at (110, 60, 180°), 14-helix populates at (-155, 60, -135°), 12-helix at (-90, 90, -110°), and β-sheet at (180, 180, 180°).68 In the C8 and C6 conformations, an intramolecular H-bonding ring is formed by eight and six atoms, as illustrated in Figure 12. The other three conformations are also given in Figure 12 for comparison. The vibrational properties of the β-HADP in these structures were examined. Table 2 gives their harmonic and anharmonic frequencies of the two amide-A modes, and those for the two amide-I modes are given in Table 3. Their total energies with respect to that of the C8 structure are also listed in Table 2. It is found from Table 2 that the energies of the C8 and C6 structure are much lower than those of other conformations because of the IHB stabilization. The energy in β-sheet conformation is found to be the highest. However, a previous work68 showed that for longer peptide chain such as β3oligopeptide (containing 12 β-amino acids), because of the formation of IHB, the 14-helix becomes the lowest-energy conformation. In the β-peptide we considered here, there is no IHB in 14- and 12-helix conformations. For the C8 structure, the IHB is formed between the NH group on the NMe side and the CO group on the Ac side; both the harmonic and anharmonic frequencies of the amide-A mode on the NMe side, ωn and νn, are much lower than those of other structures. The red shift of the vibrational frequencies is clearly due to the IHB, whereas for the C6 structure, the IHB is formed between the hydrogen atom of the NH group on the Ac side and the oxygen atom of the CO group on the NMe side. In this case, the harmonic and anharmonic frequencies of the amide-A mode on the Ac side, ωa and νa, become significantly lower. The IHB effect on the amide-I modes are seen in Table 3: each has a low-frequency component whose frequency is much lower

J. Phys. Chem. B, Vol. 114, No. 48, 2010 16017 modes for these structures range from 2.8 to 19.6 cm-1, which also fall into the distribution of the anharmonicities shown in Figure 7. These results indicate that the vibrational parameters for the amide-A and amide-I modes for typical β-HADP conformations show conformational sensitivities. The isotopic effect on the harmonic and anharmonic frequencies and the anharmonicities of the two amide-I modes for these conformations were examined, and the results are listed in Table 3. It is found that on average both the harmonic and anharmonic frequencies are lowered by ca. 45 cm-1 from the unlabeled case. This is fairly close to the value of the red shift caused by 13C/ 12 C substitution of the CdO stretching mode measured from FTIR experiment (37 cm-1)69 and much closer to the value predicted by ab initio calculations (43 cm-1)16 in the R-peptide oligomers. Table 3 also shows that except for the low-frequency mode of the β-sheet conformation, the pure-mode anharmonicity decreases by 0.4 to 0.9 cm-1 after 13C labeling. Such a decrease can be explained mainly as the mass effect. This is in agreement with the belief that lower-frequency vibrational mode tends to be less anharmonic. In addition, Table 3 shows that upon 13Clabeling the inter-amide-I coupling also decrease slightly in the selected conformations except C8. 4. Conclusions

Figure 12. β-HADP in the C8, C6, 14-helix, 12-helix and β-sheet conformations.

In this work, using the β-homoalanine dipeptide as a model of the β-peptides, a set of vibrational parameters of the amide-A and amide-I modes was examined in a restricted backbone conformational space (-180 e φ e 180°, 180 e ψ e 180°, θ ) 180°). The obtained parameters include harmonic and anharmonic normal-mode frequencies, diagonal anharmonicity, vibrational coupling, and local-mode frequencies. The conformational sensitivities of these parameters were examined to explore their structural basis in the restricted conformational space. Typical β-peptide conformations (C8, C6, 14- and 12helices, and β-sheet) were also examined. Results were compared with previously published results of the R-peptides. Both the harmonic and anharmonic frequencies of the amide-A mode in β-HADP are conformational-dependent and exhibit similar conformational sensitivities, even though the

than the case of β-sheet. We also can see that the couplings between the two amide-A modes for the C8 and C6 conformations are much stronger than those of other conformations. This is probably due to the presence of the two H-bonded amide units that are closer in distance because of the formation of IHB. It also depends how the two amide-A modes are oriented in space. For structures having no IHB, such as the 14-helix conformation, the coupling of the two amide-I modes becomes somewhat weaker. In addition, the anharmonicities of the amide-A modes for these structures, shown in Table 2, even though their θ value is quite different from 180°, fall into the most probable distribution of the anharmonicities shown in Figure 5c. The diagonal anharmonicities of the two amide-I

TABLE 2: Harmonic Frequency (ω, in cm-1), Anharmonic Frequency (ν, in cm-1), and Coupling Constant (β, in cm-1) of the Two Amide-A modes for the Selected Structures and Their Energies (E, in kilojoules per mole) C8 C6 14-helix 12-helix β-sheet a

φ, θ, ψa

ωab

ωn

νa

νn

∆a

∆n

β

E

-66, -45, 95 110, 60, 180 -155, 60, -135 -90, 90, -110 180, 180, 180

3643.8 3621.8 3642.7 3634.0 3640.9

3618.5 3642.8 3651.4 3658.2 3643.8

3451.7 3426.8 3446.0 3437.7 3444.7

3423.1 3447.7 3456.0 3462.9 3448.3

192.1 195.0 196.7 196.3 196.2

195.4 195.1 195.4 195.3 195.5

0.6 0.34 -0.04 -0.01 -0.05

0 8.21 9.81 16.51 21.42

Dihedral angles φ, θ, and ψ in degrees. b Subscript: a ) Ac side; n ) NMe side.

TABLE 3: Harmonic Normal-Mode and Local-Mode Frequencies (ω, in cm-1), Anharmonic Frequency (ν, in cm-1), Diagonal Anharmonicity (∆, in cm-1), and Coupling Constant (β, in cm-1) for the Two Amide-I Modes and Their 13C Isotopic Effects φ, θ, ψa C8

-66, -45, 95

C6

110, 60, 180

14-helix

-155, 60, -135

12-helix

-90, 90, -110

β-sheet

180, 180, 180

a

12

C dO C13dO C12dO C13dO C12dO C13dO C12dO C13dO C12dO C13dO

Dihedral angles φ, θ, and ψ in degrees.

ωl

ωh

ω0h

ωl0

∆ω0

β

νl

νh

∆l

∆h

1756.1 1711.8 1762.1 1718.0 1781.0 1736.3 1782.3 1737.5 1780.0 1735.4

1773.0 1728.6 1780.6 1735.6 1788.3 1743.5 1786.4 1741.4 1788.0 1743.1

1757.2 1712.8 1780.5 1735.6 1781.1 1736.4 1783.2 1738.4 1782.7 1737.9

1772.0 1727.5 1762.1 1718.0 1788.3 1743.4 1785.5 1740.5 1785.3 1740.6

-14.8 -14.7 18.4 17.6 -7.2 -7.0 -2.3 -2.1 -2.6 -2.7

4.08 4.13 -0.67 -0.31 0.79 0.66 1.69 1.63 3.78 3.64

1738.9 1695.3 1744.3 1701.2 1761.4 1717.3 1775.3 1731.0 1777.2 1732.5

1759.0 1715.4 1761.4 1717.1 1769.0 1724.8 1771.8 1727.4 1776.3 1731.8

17.2 16.4 17.8 16.9 19.6 19 7.0 6.5 2.8 2.9

14.0 13.3 19.2 18.5 19.3 18.7 14.6 14 11.7 11.3

16018

J. Phys. Chem. B, Vol. 114, No. 48, 2010

mode is highly localized on the N-H stretching, as is seen in the case of the R-peptides. The structural basis of the amide-A mode lies in the conformational dependence of the N-H bond length. Similar conclusions were also previously drawn for the amide-A mode in the R-peptides.53 Such a conformationdependent high-frequency stretching mode was also seen for the -CRD species in deuterated R-peptide.57,70,71 Such structural sensitivities were also shown for typical β-HADP conformations with θ * 180°. However the diagonal anharmonicity of this mode is predicted to be less structural-sensitive. Furthermore, generally weaker inter-amide-A interaction is found in the β-peptide than in the R-peptide, which is obviously due to a longer inter-amide unit distance in the former. Nevertheless, our results suggest that with the aid of these parameters, the amide-A modes shall be useful in reporting the local conformation and local microenvironment of the β-peptides. For the two amide-I modes, the anharmonic treatment using the Morse potential approach seems to be quite successful. This approach allows one to evaluate the pure-mode contribution to the anharmonic properties for a given mode. The conformational dependences of the obtained anharmonic frequencies are similar to those of the corresponding harmonic frequencies, and both are sensitive to the β-peptide structures. The average frequency difference between the two local modes is smaller than that found in the R-peptide, indicating less nondegeneracy in the β-peptide. A simple explanation is that the two amide units are further separated in the β-HADP, leading to a less significant difference in their chemical environments in terms of the electrostatic influences exerted on the amide-I vibrations. Furthermore, even though the coupling between the two amide-I modes is also conformational-dependent in β-HADP, the coupling is generally weaker than that found in the R-dipeptides. Again, this is mainly because of the additional carbon atom in the β-amino acid backbone. The vibrational frequencies, anharmonicities, and inter-mode couplings for the selected conformations with θ * 180° are found to fall into the distributions of those in the restricted conformational space (θ ) 180°). Therefore, the conclusion drawn above is generally valid. The diagonal anharmonicity for the amide-I mode in β-HADP is found to be similar to that in the R-peptides. A slightly smaller diagonal anharmonicity is predicted for the 13C-labeled amide-I mode in the β-peptide for the first time. In summary, the computational results in this work allow us to gain a deep understanding of the structural bases of the vibrational properties of the amide-A and amide-I modes in the β-peptides, even though the conclusions were drawn from simple dipeptides. The vibrational properties of the two types of modes in the β-peptide oligomers shall be examined next by experimental linear and nonlinear IR techniques. Acknowledgment. This work was supported by the National Nature Science Foundation of China (20773136, 30870591), by the National High-Tech Research Program of China (863, 2008AA02Z114), by the National Basic Research Program of China (973, 2007CB815205), and by the Chinese Academy of Sciences (Hundred Talent Fund). References and Notes (1) Krimm, S.; Bandekar, J. AdV. Protein Chem. 1986, 38, 181. (2) Jackson, M.; Haris, P. I.; Chapman, D. Biochim. Biophys. Acta 1989, 998, 75. (3) Torii, H.; Tasumi, M. J. Chem. Phys. 1992, 96, 3379. (4) Hamm, P.; Lim, M.; Hochstrasser, R. M. J. Phys. Chem. B 1998, 102, 6123. (5) Mukamel, S.; Abramavicius, D. Chem. ReV. 2004, 104, 2073. (6) Wang, J.; Hochstrasser, R. M. Chem. Phys. 2004, 297, 195.

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