Article pubs.acs.org/JPCB

Dissecting Amide‑I Vibration in β‑Peptide Helices Juan Zhao and Jianping Wang* Beijing National Laboratory for Molecular Sciences; Laboratory of Molecular Reaction Dynamics, Institute of Chemistry, Chinese Academy of Sciences, Beijing, 100190, P. R. China S Supporting Information *

ABSTRACT: The vibrational properties of the amide-I modes of β-peptides in five helical conformations (8-helix, 10-helix, 12-helix, 14-helix, and 10/12-helix) from tetramer to heptamer were examined by ab initio calculations. The normal modes have been first decoupled into local modes, whose transition energies are found to be intrinsically sensitive to peptide structure and intramolecular hydrogen bonding interactions. By further removing the intramolecular hydrogen bonding interactions, pure local modes are obtained, whose transition energies still exhibit some conformational dependence in 8-helix and 10/12 hybrid helix, but not much in homogeneous 10-, 12-, and 14-helical conformations. This suggests that a set of nearly degenerated pure local-mode transitions can be specified when excitonic modeling the amide-I vibration in latter cases. The work provides important benchmark measurements for understanding the complexity of the amide-I absorption spectra of β-polypeptides.

I. INTRODUCTION β-Peptides, the bio-oligomers composed of β-amino acid residues, are known to be important unnatural peptides because they can be biologically as active as the α-peptides that are composed of naturally occurring α-amino acid residues.1,2 β-Peptides are very important model systems to address fundamental questions of protein folding, biomolecular interaction, as well as molecular recognition.3 A β-amino acid contains one additional carbon atom in its backbone in comparison to an α-amino acid, making β-peptides structurally more flexible, and allowing more choices for side chain connections. Thus, the structures of the β-peptides are more abundant than those of the α-peptides. A β-peptide oligomer can adopt several unique helical conformations that were never seen in α-peptides. Representative structures of such reported in the past decade are 8-helix,4,5 10-helix,6 12helix,7−9 14-helix,10−15 and 10/12-helix,16,17 which are termed according to the number of atoms participating in the intramolecular hydrogen bonds. Understanding the structure of β-peptides is of great importance. The amide unit (−CONH−) that connects a chain of amino acid residues into polypeptides has several IRactive vibrations. Among them, the amide-I mode, which is mainly the CO stretching vibration, is known to be extremely sensitive to the molecular structure and local microenvironment,18−23 and is usually used as structure marker of peptides. For example, α-peptide in α-helix has the amide-I band at about 1640 cm−1, while for the β-sheet conformation, there are two peaks in the amide-I region, positioned at about 1610 and 1680 cm−1.24 However, only a limited number of studies of the spectrum−structure relationship of β-peptide have been reported in the literature. For example, the structural dynamics of a β-peptide in 12/10-helical conformation have been © XXXX American Chemical Society

examined using femtosecond pump−probe two-dimensional infrared (2D IR) spectroscopy,16 and the nanosecond temperature jump-induced folding of a 15mer, 14-helical β-peptide has been reported very recently.25 Conformation of β-peptide oligomers has also been studied by matrix isolated26,27 and gasphase28,29 spectroscopic techniques. Because of the extra backbone carbon atoms periodically present in the backbone of β-peptides, the amide units are spatially distributed in a way quite different from those in αpeptides. Thus, it is expected that the amide-I vibrational properties, including local-mode frequencies, mode delocalizations, pure-mode diagonal anharmonicities, mixed-mode offdiagonal anharmonicities, as well as anharmonic vibrational couplings, in β-peptides, are quite different from those in αpeptides. For example, in our recent work, vibrational properties of the amide-I band in a β-dipeptide were examined by ab initio calculation.30 The vibrational transition frequencies of the amide-I modes were found to be conformation dependent, and the interamide-I couplings are found to be generally weaker than those in α-peptides.20,31−33 Very recently, the amide-I bands characteristic of β-peptides in the 14- and 12/10-helical conformations were examined using linear IR spectroscopy and computations.34 In the same work, the vibrational spectra of the two conformations were simulated using frequency−frequency time-correlation function. In this work, we aim to explore the structure−spectrum relationship of β-peptide oligomers, particularly in helical conformations. For this purpose, several well-known β-helices, namely, 8-, 10-, 12-, and 14-helices, and 10/12-hybrid helix, Received: November 18, 2014 Revised: December 28, 2014

A

DOI: 10.1021/jp5115288 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

Geometry optimization and normal-mode vibrational frequency analysis of these peptide oligomers were carried out in gas phase using the density functional theory (DFT) at the level of B3LYP/6-31+G*. The backbone dihedral angles (ϕ, θ, ψ) were fixed during optimizations. Amide-I vibrational absorption spectra of these peptides were simulated by applying Gaussian line shape function with full width at half-maximum (fwhm) of 16 cm−1 to the computed intensities, against the computed normal-mode frequencies. Such a fwhm is in general agreement with experimentally determined ones reported earlier. 16 All the calculations were carried out using Gaussian09.37 No scaling factor was used for the calculated vibrational frequencies, and the obtained vibrational frequencies are overestimated in comparison with typically reported solution-phase results of β-peptides.25,34 In addition, it is noted that with the initial dihedral angles obtained at the HF/631G* level of theory as input, a full structural optimization at the level of B3LYP/6-31+G* may yield somewhat different dihedral angles (see the Supporting Information, Table S2). However, we find that the averaged angles after the full optimization are similar to those obtained at the level of HF/631G*. Also, the computed IR spectra obtained by the full structural optimization are similar to those obtained by the constrained geometry optimization at the level of B3LYP/631+G* (see the Supporting Information, Figures S3 and S4, where only the results for the 12H conformation are given). This justifies the method used in the present study. The potential energy distribution (PED) analysis has been known to be very effective in describing relative contributions of normal-mode coordinates to the total change in the potential energy for a given vibration mode.38 In this work, the PED values were computed39 and were used to characterize the extent of mode localization. Further, the participation ratio40,41 was also used to simultaneously characterize the extent of mode delocalization of all the amide-I modes in a given peptide. Here, the participation ratio was computed using the eigenvectors for each amide-I normal mode;42 thus, it is on the basis of Cartesian coordinates. The PED method, however, is on the basis of internal coordinates. The participation ratio, denoted as P, measures the amide-I mode delocalization degree by P = 1 for a completely localized state and by P = 1/N for a completely delocalized state, where N is the total number of amide units in a peptide. B. Local Mode Frequency and Vibrational Coupling. The normal-mode calculations also provide the normal-mode eigenvalues (vibrational transition frequencies or energies from v = 0 to v = 1 with v being vibrational quantum number) and eigenvectors (normal coordinates) for the amide-I modes of each peptide. They were used to decouple the computed amide-I normal modes. Approximations are made to construct the N-by-N amide-I wave functions, i.e., only pairwise coupling in the N subspace of the amide-I modes is considered, and the amide-I mode is localized on the CO stretch. By mathematically demixing the wave functions, i.e., the matrix composed of N-by-N normal coordinates of the CO species only, one is able to obtain N uncoupled local-mode frequencies and an Nby-N coupling matrix.30,43 This method is referred to as the wave function demixing (WFD) hereafter, and can be summarized in the following equation:

each having four to seven amide units, are chosen. Normalmode and local-mode frequencies, vibrational coupling, potential distribution function, and participation ratio, are computed and their conformational sensitivities are analyzed. The influence of the intramolecular hydrogen bonding interactions on the local-mode frequency is also examined.

II. CALCULATION METHODS A. Ab Initio Calculations. β-Peptides containing four to seven amide units in five representative helical conformations, namely, 8-helix (denoted as 8H, ϕ = −111.5°, θ = 68.6°, ψ = 13.9°, defined in Figure 1), 10-helix (10H, 73.5°, 51.3°, 73.6°),

Figure 1. Typical β-peptide heptamers in five different helical conformations (8H, 10H, 12H, 14H, 10/12H). The dihedral angles are shown in the 8H conformation. Dashed lines indicate intramolecular hydrogen bonds. The amide units are indexed from the Nterminus to the C-terminus.

12-helix (12H, −88.5°, 89.3°, −111.4°), 14-helix (14H, −141.6°, 59.9°, −133.3°), 10/12-helix (10/12H, 89.5°, 65.9°, −110.6° for the first, third, and fifth amino acid, and −99.3°, 61.3°, 89.9° for the second, fourth, and sixth amino acid), were chosen to examine the vibrational characteristics of amide-I modes in β-peptides. These dihedral angles were taken from previous works35,36 obtained at the HF/6-31G* level of theory with additional constraints on individual residues within the structure without further optimization. These helices are denoted as 8H, 10H, 12H, 14H, and 10/2H in this work. Figure 1 also shows intramolecular hydrogen bonds formed between various amide groups in each case. Optimized threedimensional structures of the β-heptamer in five helical conformations are shown as Figure S1 in the Supporting Information.

νLM = νNM − [βi , j] B

(1) DOI: 10.1021/jp5115288 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B Here, νLM denotes local-mode frequencies, νNM denotes normal-mode frequencies, and [βi,j] denotes the coupling matrix. The obtained coupling matrix is more or less symmetric, with the matrix elements in the lower-left triangle almost identical to those in the upper-right triangle, from which averaged values of pair wise couplings are obtained. In addition, the WFD coupling constants were compared with those obtained by using the transition dipole coupling (TDC) model.38 In the TDC model, the coupling constants were calculated as βij =

1 μi⃗ ·μj⃗ − 3(nij⃗ ·μi⃗ )(nij⃗ ·μj⃗ ) · 4π ·ε0 rij3

(2)

Here, ε0 is the dielectric constant, μ⃗i is the transition dipole of the ith amide-I mode, n⃗ij is the unit vector connecting the ith and jth transition dipoles, and rij is the distance between the ith and jth transition dipole centers. In this work, μ⃗i of the amide-I mode in β-peptides was set to 2.5 D Å−1 amu−1/2 in magnitude and 20° orientation with respect to the CO bond direction (pointing from C to O). Those parameters were obtained by a DFT calculation at the same level on an isolated single βpeptide (CH3CH2CONHCH2CH3, N-ethylpropionamide, denoted as NEPA henceforth).

III. RESULTS AND DISCUSSION A. IR Signatures of the β-Helices and Their Underlying Vibrational Transitions. Amide-I Absorption Signature. The vibrational transition frequencies and relative transition intensities of the normal amide-I modes for the 20 peptides in helical structures (8H, 10H, 12H, 14H, and 10/2H, with number of peptide units ranging from four to seven in each case) were obtained by ab initio calculation. The computed normal-mode spectra, obtained by plotting the normal-mode transition intensities broadened using Gaussian functions against their transition frequencies, are shown in Figure 2. The results show that absorption bands are both conformation and chain-length dependent. For the five different helical conformations, a general observation is that as the chain length increases, the main peak position gradually shifts to the low frequency side. This is because the averaged local-mode frequencies decrease at longer chains (see section B). In the 8H conformation (Figure 2a), two clearly separated absorption bands are formed and the peak separation increases slightly as the chain length increases. For the cases of 10H, 12H, and 14H (Figure 2b−d), two major components are seen, and the peak separation also increases as the chain length increases, even though the peak separations are much smaller than that in the case of 8H. In addition, the overall amide-I band positions in these three cases are much higher in frequency than those in 8H, because the local-mode frequencies of these three conformations are generally higher than those in 8H (see section B). In the 10/12H conformation (Figure 2e), the spectral band is broad and contains more than three peaks, and the overall band position is red-shifted from those of both 10H and 12H conformations. In other words, the spectral signature of a 10/12-helix is not a simple average of 10H and 12H. This is true both in peak position and in band profile. The structural reason for the 10/12-helix being different from both the 10-helix and 12-helix lies in their slightly different dihedral angles (see Section A). These results indicate the chain-length and conformational sensitivities of the amide-I absorption spectra of the β-peptides. In addition, the two-component

Figure 2. Simulated amide-I IR spectra of five helical conformations as a function of peptide number N. The total area in each case is set to be proportional to N.

feature with finite peak separation in the 8H, 10H, 12H, and 14H conformations seems to somewhat resemble the case of the α-helix formed by the α-peptide, whose amide-I band also contains two peaks with limited frequency separations.43,44 For the 8H, 10H, 12H, 14H and 10/12H conformations with reasonable length of backbone chain, for example, heptamer, the main peak is positioned at about 1692, 1717, 1720, 1741, and 1698 cm−1, respectively, all of which are lower than that in a single peptide case: the amide-I band of NEPA is found to be 1746.7 cm−1, computed at the same level of theory. In peptide oligomers, the intramolecular hydrogen bonding interactions and interamide interactions (through bond and through space) are important factors affecting the vibrational modes. These are the two reasons that the amide-I mode in an isolated β-peptide differs from those in oligomeric peptides. Further, by examining the results shown in Figure 2, for peptides in homogeneous helical conformation (8H, 10H, 12H, and 14H), it seems that the smaller the intramolecular hydrogen bond ring is, the lower the amide-I peak position. Key normal-mode frequencies are listed in Table 1, and a complete set of frequencies and other vibrational parameters is given in Table S1 of the Supporting Information. For peptides in hybrid helical conformations, the normal-mode features are examined below. Origin of the Amide-I Spectra. To further reveal the origin of the computed amide-I spectra, normal-mode transition frequencies and relative transition intensities are plotted as stick spectra in Figure 3. The broadened spectra are also shown in Figure 3, which can be compared either vertically for the same helix with varying chain length or horizontally for different C

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The Journal of Physical Chemistry B Table 1. Frequencies (ν, in cm−1) of the Selected Amide-I Normal Modes Marked by Asterisk in Figure 3 for the Five Helical Conformationsa 8H

10H

12H

N

ν

PED

ν

PED

ν

PED

4

1694.0

2(16), 3(55)

1731.7

1730.3

1726.1

4(77)

1743.9

1691.7

2(11), 3(27), 4(33) 5(78)

1718.6

3(29), 4(45) 1(55), 2(20) 3(70)

1733.0

2(7), 3(23), 4(25), 5(15) 6(77)

1719.4

1(12), 2(63) 3(61), 4(7)

1(24), 2(19), 3(24), 4(10) 1(25), 2(23), 3(7), 4(23) 1(14), 3(46), 5(7) 2(38), 3(5), 4(32) 1(9), 3(25), 4(31), 6(6)

5

1724.9 6

1689.9

1725.1 7

1688.4

1724.9

3(13), 4(23), 5(15), 6(15) 7(77)

1742.6

1740.7 1717.7

1739.9

1(35), 2(42) 4(66)

1(36), 2(41)

14H N

ν

4

1738.5 1760.9 1741.0

1(19), 4(59) 2(42), 3(40) 1(38), 5(40)

1715.0 1742.8 1709.5

1759.6 1734.2 1743.2 1726.6 1742.4

2(10), 3(69) 1(22), 4(26), 5(16), 6(11) 2(29), 4(6), 5(22), 6(23) 1(10), 4(56), 7(8) 2(8), 3(15), 5(11), 6(35), 7(8)

1739.8 1698.7 1740.9 1698.0 1740.3

5

6 7

PED

set to 0.8 for better viewing. Clearly the transition frequencies are distributed in groups of sticks with a unique pattern for each helix. For the 8H conformation, the sticks are generally divided into two separated distributions, which is why two absorption peaks are seen in Figure 2a. The high-frequency group always has two transitions and the number of amide-I modes in the low frequency region increases as the chain length increases. For the case of 10H, with increasing chain length, three groups of normal modes begin to show. At N ≥ 5, the high-frequency group always contains two sticks. The number of transitions of the low-frequency group increases at longer chain length. However, there is no obvious grouping in the cases of 12H, 14H, or 10/12H, especially for longer chains. In addition, for the 10/12H conformation, the transition frequencies are more widely distributed than those in the remainder conformations, which is the reason for their broad IR spectra. B. Local-Mode Properties. Dissecting the Normal Modes. Local-mode frequencies are intrinsic properties of peptide local structures and local structural distributions, which is particularly true in the gas phase, as we discussed here. In this work, two sets of local modes are defined: the first is a set of local modes that are decoupled from the normal modes by the WFD method. Because there are intramolecular hydrogen bonds involved in the β-peptides, these local modes also inevitably reflect the consequences of the hydrogen bonding interaction. As we show here, the intramolecular hydrogen bonding interaction plays a significant role in determining the amide-I local-mode frequency distributions. There is a general rule for the local-mode frequency: if the amide NH group and/ or CO group participates in forming hydrogen bond, its vibrational frequency will be lowered; otherwise, its vibrational frequency will be purely local-conformation dependent. The estimated red shift in amide-I frequency in an N−H hydrogenbonded amide is ca. 13 cm−1, while that in a CO hydrogenbonded amide is ca. 20 cm−1. These values are obtained from the DFT computations, at the level of B3LYP functional with the 6-31+G* basis set, for NEPA−NEPA dimer (Figure 4 and Table 2). During the calculation, the amide group of interest is

ν

1736.4 1720.9

1722.9

1733.4

1(21), 2(28), 5(21), 6(6) 1719.3 2(8), 3(8), 4(45), 5(6), 6(5) 1739.9 4(7), 6(23), 7(40) 10/12H PED 2(66), 4(9) 1(69), 3(7) 2(5), 3(42), 4(5), 5(19) 1(72) 2(42), 4(28) 1(73) 3(10), 4(5), 5(47) 1(73)

a

PED analysis is shown in the form of local modes and their contributions listed in parentheses (in %). Only the contributions ≥5% are listed.

helices with the same chain length. The peak intensity of each spectrum is normalized, while the highest stick in each case is

Figure 3. Simulated amide-I IR spectra of the five helical conformations with increasing chain length. The distributions of the transition frequencies are also shown in the stick spectra. The main peak of each is normalized for better viewing. The highest stick is scaled to 0.8 in each case. The PEDs of the modes labeled by red asterisk are listed in Table 1. D

DOI: 10.1021/jp5115288 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

intramolecular hydrogen bond in a β-peptide oligomer; and second, the difference in the strengths of intramolecular hydrogen bonds in various helical conformations is not considered. Local-Mode Frequencies. It is interesting to see how νLM and ν0LM behave, because they are crucial for vibrational excitonic modeling of the amide-I mode in β-peptides. Figure 5

Figure 4. Hydrogen bonding in NEPA dimer: (A) carbonyl group forming hydrogen bond, (B) NH group forming hydrogen bond. Isotopic label (13C18O) is used to decouple the amide-I interaction with the unlabeled amide unit of interest.

Table 2. Ab Initio Calculated Normal-Mode Frequencies (in cm−1) of NEPA Dimer, with 13C18O Labeled NEPA Binding to a Nonlabeled NEPA in Position A or B, Respectively (Figure 4), and Resulting Frequency Differences C ONH amide-I frequency/cm−1

12 16

NEPA monomer HN12C16O···HN18O13C 12 16 C ONH··· 18O13CNH ΔνCO ΔνNH

1746.7 1726.3 1733.6 20.4 13.1

Figure 5. Local-mode frequencies of the five helical conformations with different chain lengths in the presence of local hydrogen-bonding interaction (left column, νLM), and in the absence of local hydrogenbonding interaction (right column, ν0LM).

in natural abundance of isotopes (12C16O), while its hydrogen-bonding partner is isotopically labeled (13C18O). This decouples the two amide-I modes so that only the hydrogen-bonding effect can be examined. A similar calculation is also carried out for N-methylacetamide (NMA)-NMA dimer, where NMA is a common model compound for a single αpeptide. In this case, the hydrogen-bonding on the CO or NH side causes 21.5 or 13.3 cm−1 red shift, respectively (data not shown). In addition, a similar calculation is carried out on the NEPA-2D2O cluster, where the NH and CO groups each form a hydrogen bond with a water molecule. A total red shift in the amide-I frequency is calculated to be ca. 30 cm−1, quite close to the sum of two single hydrogen-bonded cases. This suggests that the hydrogen bonding effect on the amide-I vibration frequency shift is cumulative in this case. Thus, another set of local modes can be obtained by removing the frequency shift due to hydrogen bonding interactions. Such dehydrogen bond local modes will only be sensitive to local structural environment and covalent bond, and thus can be considered as a set of “pure” local modes. The following equation describes the relationship between the two types of local modes and the hydrogen-bond effects: ν 0 LM = νLM + ΔνCO + ΔνNH

compares the values of νLM (left panel) and ν0LM (right panel) for these β-polypeptides. It can be seen that without removing the hydrogen bonding effect, νLM exhibits significant site dependence. There is a dramatic difference between terminal groups and those in the middle, which is true almost for all the helices. With the 8H conformation as one example (Figure 5a), the amide-I local-mode frequency of the amide unit at the acetyl end (ca. 1721 cm−1) is slightly lower than that at the NH-methyl end (ca. 1725 cm−1), while those amide units in the inner region have quite lower values (with average of ca. 1695 cm−1). The reason lies in the structure. Figure 1 shows that, for the first amide unit on the acetyl side (left end, also called the N-terminus), only the carbonyl group participates in hydrogen bonding; while for the last amide unit on the NH-methyl side (right end, also called the C-terminus), only the NH group participates in hydrogen bonding. However, for the remainder of the amide units, both CO and NH groups participate in forming hydrogen bonds. Thus, the local modes of the amide units in the inner region are generally lower in frequency. Another example worth mentioning is 10/12H (Figure 5e): there are two types of hydrogen bonds involved in this hybrid structure. The local-mode frequencies of the amide units forming the 12-membered hydrogen bond ring are generally lower than those of amide units in the 10-membered ring. For example, in heptamer, the local-mode frequency of the fourth amide unit (forming 12-membered ring) is lower than those of the third and fifth amide units (forming 10-membered ring). The local-mode frequency distributions differ from those of pure 10H and 12H conformations (Figure 5b,c). Thus, it must be due to the unique structure of the hybrid 10/12H conformation.

(3)

where νLM is the decoupled local-mode frequency using the WFD method, ΔνCO is the frequency red shift when amide carbonyl forms a hydrogen bond (ΔνCO > 0), ΔνNH is the frequency red shift when amide N−H group forms a hydrogen bond (ΔνNH > 0), and ν0LM is the pure local-mode frequency. Using eq 3, a set of pure local-mode frequencies due to helical conformation and local chemical environment can be estimated by subtracting the assumed red shifts due to NH hydrogen bond and/or CO hydrogen bond. However, two assumptions are made here: first, the strength of the intermolecular hydrogen bond between NEPA dimer is similar to that of the E

DOI: 10.1021/jp5115288 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

To further test the chain-length dependence of the frequency shift resulting from intermolecular hydrogen bonding, we also examined the frequency shift for a hydrogen-bonded NEPA−βglycine dipeptide dimer with selective 13C18O substitutions in a similar way as the NEPA−NEPA dimer described above. The computations were also carried out at the level of B3LYP/ 6-31+G*. It was found that the shift is 20.2 cm−1 for the CO hydrogen-bonded β-GDP in comparison to a non-hydrogenbonded β-GDP, and 12.5 cm−1 for the N−H hydrogen bonded β-GDP in comparison to a non-hydrogen-bonded β-GDP, indicating insignificant chain length dependence. This justifies the empirical model used to decouple the hydrogen bonding influences from the local modes to yield pure local modes. C. Mode Delocalization. Amide-I modes in α-peptides are known to be delocalized onto nearby amide units, through covalent bonds and even across hydrogen bonds.45,46 In this section we examine the delocalization extent of the computed amide-I normal modes in these β-peptides. There are two metrics for this purpose, as we mentioned earlier: the potential energy distribution functions on the basis of internal coordinates, and the participation ratio on the basis of Cartesian coordinates. In particular, the largest PED value (PEDmax) can be used to quantify the localization degree of a given vibrational mode, as has been suggested in our recent work.42 The participation ratios of all the normal modes are plotted in Figure 7. The PED analysis of selected normal modes

However, these site dependences diminish substantially after removing the intramolecular hydrogen bonding effect. The results of pure local-mode frequencies are shown in the right panel of Figure 5. The results suggest that for the 10H, 12H, and 14H (Figure 5g−i) conformations, the site-dependence of the pure local-mode frequencies becomes much weaker, while for the 8H (Figure 5f) and 10/12H conformations (Figure 5j), the pure local modes still bear some characteristics of their counterparts on the left side. Figure 6 shows the averaged local-mode frequencies (with or without hydrogen bonding influences) for these β-peptides.

Figure 6. Averaged local-mode frequencies of the five helical conformations with different chain lengths: (a) in the presence of local hydrogen bonding interaction (νLM); (b) in the absence of local hydrogen bonding interaction (ν0LM). N is the number of the amide units in each peptide.

The local-mode frequencies in 8H are generally the lowest, and those in 14H are generally the highest, indicating the conformational dependence. Also, under the influence of hydrogen bonding interactions (Figure 6a), the local-mode frequencies are dependent on the chain length: they decrease as the chain length increases. This is because more and more amide units are involved in hydrogen bonding interactions at longer chains, and, at the same time, the weight of nonhydrogen-bonded terminal amide units decreases. The pure local-mode frequencies (Figure 6b), interestingly enough, do not vary much as the chain length changes, reflecting their insensitivities to the size of peptides. In our previous work,30 the local amide-I mode characteristics using a β-dipeptide were examined. The local-mode frequencies were found to be dependent on the backbone dihedral angles (ϕ, θ, ψ). Because there is no intramolecular hydrogen bond (except for the C8 structure, which is similar to 8H in this work), the dipeptide results are already pure local modes. Additionally, it is noted that most of the obtained pure localmode amide-I frequencies, either the individual site frequencies or the averaged ones, differ from that of the β-peptide monomer (NEPA). The difference should be attributed to the structural aspect of the peptide oligomer, rather than the inaccuracy of eq 3 in evaluating the hydrogen-bonding effect.

Figure 7. Participation ratio of the amide-I mode for the five helical conformations with different chain lengths. The normal modes from left to right are arranged from low to high frequency.

with significant transition intensities (marked in Figure 3) is given in Table 1, and the PEDs for all the amide-I modes are given in Table S1 of the Supporting Information. In Figure 7, for the 8H conformation (Figure 7a), as the chain length increases, the participation ratios of the two highfrequency normal modes are always close to 1, indicating a highly localized nature of the two modes; while for the remaining modes their participation ratios range between 0.2 to F

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The Journal of Physical Chemistry B 0.6, indicating delocalized states. On the other hand, the PED analysis in Table 1 indicates that the low-frequency mode with the highest intensity contains significant contributions from the amide units in the inner region, while the highest-frequency mode, with PEDmax = 77% for N = 4, is localized on the amide unit at C-terminus. This indicates that PEDmax is indeed a good measure for a localized mode. The other high-frequency normal mode is localized on the amide unit at the N-terminus (see Supporting Information Table S1). For the 10H conformation (Figure 7b), the participation ratios are generally between 0.4 and 0.9. The average P values fluctuate dramatically at varying chain length: 0.53 at N = 4, 0.80 at N = 5, 0.57 at N = 6, and 0.56 at N = 7. This indicates that the delocalization is sensitive to the chain length. The PED analysis indicates the low-frequency modes are mainly attributed to the vibration of the amide units in the inner region, while the highest-frequency mode is mainly due to the interaction of the first two amide units at the acetyl side. The PED analysis for all the modes (Supporting Information Table S1) indicates that the other highest-frequency normal-mode has a similar nature. The last two amide units at the NH-methyl side mainly contribute to the mid frequency modes (Supporting Information Table S1). For the 12H conformation (Figure 7c), the participation ratios of the normal-modes are between 0.2 and 0.4 in this conformation, indicating the amide-I normal modes are always largely delocalized. The PED values listed in Table 1 indicate that the normal modes are always composed of several amide units, agreeing with the participation ratio results. For the 14H conformation (Figure 7d), the participation ratios indicate the vibrational modes in this conformation are also generally delocalized; however, their degree of delocalization, as reflected in PED (Table 1), tends to increase for longer chain. This differs from the case of 12H. For the 10/12H conformation (Figure 7e), participation ratio values are generally higher for the two high-frequency modes particularly for longer chains, indicating the highly localized states of the two modes. This is confirmed by the PED analysis shown in Table 1. Further, PEDs in Table 1 show that the highest-frequency mode is always due to the first amide unit, whose amide hydrogen atom participates in forming a 10membered hydrogen bond ring; while the other high-frequency mode is due to the amide unit near the C-terminal end where only the amide hydrogen atom participates in a 12-membered hydrogen bond ring (Supporting Information Table S1). D. Vibrational Interaction. To assess the strength of vibrational interaction between pairwise amide-I modes in these β-helices, we examine the coupling constants (generally denoted as βi, j for the ith and jth amide units). The results of the WFD coupling and TDC evaluated for the five heptameric helical structures are given in Figure 8 for comparison. The results of shorter peptides are not listed because we find, for a given conformation and a given coupling computing method, the couplings between two similarly spaced amide units are quantitatively similar. From Figure 8 one can see that overall the sign and magnitude of the coupling are conformational dependent. In particular, the vibrational interactions between the local amide-I modes in 10H and 8H are generally weaker. Couplings in Hydrogen-Bonded Amides. Clearly the coupling between the two amide units forming a hydrogen bond is always negative and significant. As shown in Figure 1, for the 8H conformation (Figure 8a), the hydrogen bonds

Figure 8. Comparison of vibrational coupling of the amide-I modes for the five helical heptamers obtained by using the WFD method (filled circle) and TDC method (open circle). βi,i+n indicates the coupling between the ith and (i+n)th amide groups.

formed between the ith and (i+1)th amide units; for the 10H, 12H, and 10/12H (Figure 8b,c,e), the hydrogen bonding are formed between the ith and (i+2)th amide units; while for the 14H (Figure 8d), the hydrogen bonds formed between the ith and (i+3)th amide units. The averaged coupling constants, βi, i+1 in 8H; βi, i+2 in 10H, 12H, 10/12H; and βi, i+3 in 14H, are found to be −2.4 cm−1, −2.0 cm−1, −6.6 cm−1, −5.5 cm−1, and −4.4 cm−1, respectively, by the WFD method. Nearest Neighbor Couplings. We also can see that the couplings between the two nearest amide units (βi, i+1) are positive in 10H, 12H, and 14H, with averaged values obtained by the WFD method to be about +0.6, +2.2, and +3.3 cm−1, respectively. However, in 10/12H the backbone dihedral angles vary periodically for the two nearest amide groups, so the couplings between every two amide pairs also vary in a similar way. For example, the coupling constant between the first and second amide unit is about the same as that between the third and fourth amide units (ca. 2 cm−1), and the coupling constant between the second and third amide units is similar to that between the fourth and fifth amide units (ca. −0.5 cm−1). Further, the couplings between amide groups spaced at longer distance become weak, which is true for all helices. Comparison with the TDC Results. We also calculated the couplings by using the TDC model to compare with those obtained using the WFD method (Figure 8). It is noted that the magnitude and sign of the coupling constants obtained by the two methods are in a similar trend. Large differences are mostly seen for the amide units over very short distances, for example, βi, i+1 and βi, i+2. This is because, if the two amide units are spatially close, the charge distribution overlap occurs, and the transition dipole approximation starts to fail.38 Under such circumstances, the results obtained by the WFD method are more reliable. G

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Figure 9. Change of transition energy from local modes to normal modes. Only heptameric β-peptides in the 8H (left) and 10H (right) conformations are given. LM0: pure local mode; IHB: intramolecular hydrogen bond; LM: local mode; NM: normal mode. The assignment of amide unit is given in the LM0 and LM states.

Figure 9 shows the transition energies for the ab initio computed normal modes (NM), WFD decoupled local modes (LM), and hydrogen-bond removed local modes (LM0, i.e., the so-called pure local modes) in the cases of heptameric 8H and 10H conformations. The drawings for the five heptameric helices are given in Figure S2 of the Supporting Information. Figure 9 summarizes the relationship among the three sets of amide-I mode frequencies in β-peptides, which can be expressed as

Here, the 10/12H hybrid conformation is an interesting case, where the difference of coupling between the nearest two amide groups is quite small for β1, 2, β3, 4, β5, 6 (ca. 0.2 cm−1), but large for β2, 3, β4, 5, β6, 7 (ca. 2.2 cm−1). This can be simply explained as the effect of the interamide distance: the averaged distance between two carbonyl bond central point between two amide units (1−2, 3−4, and 5−6, respectively) are ca. 4.8 Å, which is longer than that of the 2−3, 4−5, and 6−7 pair, i.e., ca. 3.3 Å. A similar phenomenon can also be seen for the coupling constants βi, i+2, βi, i+3 in this hybrid helix. As for the amide pair that is hydrogen bonded, the difference between the couplings from the two methods is generally larger. In this case, the formation of a hydrogen bond changes the charge distribution, and also changes the transition charge distribution. A typical example of such is the 8H conformation, where the hydrogen bond occurs between the nearest neighboring amide units. Figure 8a shows that for the two amide units in nearest neighbor, the TDC results differ from the WFD results in both magnitude and sign. The comparison of the TDC and WFD results present in this study is very important for the amide-I modes in β-peptides. The TDC method is known to work well for long-distance interamide coupling in α-peptides in various conformations. In β-peptides, the coupling constants, even though being seemingly small for the amide chromophore separated at long distance, are very important, because their collective effect will make an impact on linear IR absorption band structure and transition intensity distribution. Thus, in principle, none of the couplings should be left out in modeling the linear IR or 2D IR spectra of the amide-I modes in β-peptides. However, it should be mentioned that the WFD method, even though being more reliable, becomes computationally expensive for large molecules. One the contrary, the TDC method is relatively cheap but effective at longer distance, for any sized molecules. Once parametrized, the TDC method can be applied to any peptide structures, optimized or not. This is the advantage of the TDC model. In addition, for near neighbors where the breakdown of the TDC model occurs, the WFD method should be used instead. This combination of coupling scheme has also been very popular in modeling the amide-I modes of the α-peptides (for example, ref 19). Evaluation of the couplings for short peptides in various conformations using the WFD method would provide a coupling map for this purpose. From Local Modes to Normal Modes. To have a clear picture about the relationship between the local modes and normal modes and the roles played by the intramolecular hydrogen bond and vibrational couplings, we examine their relative energy levels of the first vibrational excited states.

νNM = ν 0 LM − ΔνCO − ΔνNH + [βi , j]

(4)

Note that we also have νNM = νLM + [βi,j] as described eq 1. For a given β-peptide in certain helical conformation, there is a set of pure local-mode transition energies (i.e., transition frequencies), each corresponding to an amide unit. From Figure 9, one sees that the LM0 band is the narrowest. Intramolecular hydrogen bonding interaction alters (lowers) some of the relative transition energies, and also causes a broadening in the band structure (becoming the LM states); however, this does not change the origin of each vibrational state. Second, the normal-mode bandwidth becomes more widely spread than the local-mode bandwidth as a result of a network of interamide coupling [βi,j]. This is generally true because the interamide couplings are generally nonzero, which tends to further spread the transition band and scramble the identities of the LM states (becoming the NM states). A strict one-to-one relationship between the LM and NM states no longer exists, even for highly localized modes. The largest difference between the bandwidth of the normal modes and that of the local modes is found in the case of the 12H conformation (see Supporting Information Figure S2), suggesting the strongest coupling overall in this case. This is in agreement with the results shown in Figure 8c (filled circles). Quite the opposite, the smallest difference between the LM and NM states is shown for the 10-helical structure (Supporting Information Figure S2), indicating the weakest coupling overall (Figure 8b, filled circles). Finally the weak site-dependence in the pure local-mode frequencies (except the terminal units) predicted for the 10H, 12H, and 14H conformations (Figure 5 right column) indicates that a set of nearly degenerated local-mode transitions can be assumed, as a very good starting point, for vibrational excitonic modeling of the amide-I vibration in these helical β-peptides. Vibrational excitonic modeling has been widely studied for the amide-I modes in polypeptides consisting α-amino acid residues.45,47 Various methods have been proposed to model the pure local-mode frequencies and hydrogen-bonding H

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main focus of this work is on the influence of intramolecular force fields on the β-helical structures and their amide-I properties; the effect of solvent is not considered. Nevertheless, it provides us a benchmark measurement to understand the structure−spectrum relationship. Further studies may include implicit or explicit solvent effects on peptide backbone and side chain, which will result in altered conformations and thus altered amide-I band signatures. Molecular dynamics simulations can be used to sample structural distributions, from which vibrational coupling distributions can also be examined. The 2D IR spectroscopy can also be used to further reveal the vibrational signature underneath the complicated amide-I absorption spectra.16,25,34

influences for vibrational excitonic modeling of the amide-I mode.48,49 Thus, a quick and reliable approach to obtain a set of pure local-mode frequencies for β-peptides would be extremely useful. The results presented in this work thus provide important benchmark measurements for β-peptides for such purpose.

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SUMMARY In this work, the vibrational properties of the amide-I band in several typical β-helical oligomers (8-, 10-, 12-, 14-, and 10/12helices) were examined by using quantum-chemical computations and post analyses. Normal-mode and local-mode frequencies and their distributions are obtained. The delocalization of the normal modes is examined using the potential energy distribution and participation ratio. Interamide vibrational couplings are evaluated by demixing the amide-I wave functions, and by using transition dipole approximation. The infrared absorption spectra of the β-peptide oligomers in the amide-I region exhibit conformational and chain-length sensitivities. As the chain length grows, grouping of the normalmode transition energies occurs differently for different helices. In addition, for longer chain, the overall normal-mode frequencies and the strongest peak position are generally redshifted for all the conformations. The local-mode frequencies obtained by decoupling the normal modes are intrinsically conformation and chain-length dependent. However, such local modes are also influenced by intramolecular hydrogen bonding interactions for the βpeptides considered in this work. For a given amide unit, if its CO and/or NH group participate in hydrogen bond, its amide-I vibrational frequency will be lowered. Because of the unique hydrogen bonding interactions in each of the five helices, the local-mode frequencies are affected differently. However, a set of “pure” local-mode frequencies for each helical β-peptide can be obtained approximately by removing the hydrogen bonding influences. The resultant pure local-mode frequencies are thus only sensitive to peptide local chemical environment and covalent bonding interactions; and naturally weak site-dependence is found in the homogeneous 10-, 12-, and 14-helical conformations. These pure local-mode frequencies are very important parameters for modeling the amide-I vibrational spectra of β-peptides. The coupling constants are another set of parameters having conformation sensitivities. Different coupling patterns exist for different conformations. Couplings between the amide units forming hydrogen bonding are found to be the strongest for a given conformation. The magnitude and sign of the coupling constants obtained by the transition dipole interaction method are generally similar to those obtained by the wave function demixing method. However, similar to the case of α-peptides, both the covalent bond and hydrogen bond affect the charge distribution of nearby amide units; thus, the transition dipole approximation only works well for amide units that are spatially far apart. A typical case is the 8H conformation, where the nearest neighboring amide units are predicted to be weakly coupled by the TDC scheme, but are found to be more strongly coupled by the WFD method. These predictions about helical β-peptides await experimental verifications, for example, by 2D IR spectroscopy.16,22,50−53 In summary, the results in this work allow us to gain deeper insight into the amide-I vibrational signature of representative helical β-peptides, and provide theoretical support for analyzing the structure of β-peptide by using the IR spectroscopy. The

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ASSOCIATED CONTENT

* Supporting Information S

The amide-I normal-mode frequencies and potential energy distributions of β-peptides in five helical conformations; a comparison of average dihedral angles obtained for the 12H conformations at the levels of HF/6-31G* and B3LYP/631+G*; optimized three-dimensional structures of the βheptamer in five helical conformations with backbone atoms shown only; change of transition energy from local modes to normal modes in β-peptide heptamers in different helical conformations; computed IR spectra and local-mode frequencies for the 12H conformations with different chain lengths obtained by complete optimization and constrained geometry optimization. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel: (+86)-010-62656806, Fax: (+86)-010-62563167, E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Nature Science Foundation of China (21173231 and 91121021), and by the Chinese Academy of Sciences (Hundred Talent Program).

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REFERENCES

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