Structural Dependence of Extended Amide III Vibrations in Two-Dimensional Infrared Spectra

Two-dimensional infrared (2D-IR) spectroscopy is a powerful experimental method for probing the structure and dynamics of proteins in aqueous solution. So far, most experimental studies have focused on the amide I vibrations, for which empirical vibrational exciton models provide a means of interpreting such experiments. However, such models are largely lacking for other regions of the vibrational spectrum. To close this gap, we employ an efficient quantum-chemical methodology for the calculation of 2D-IR spectra, which is based on anharmonic theoretical vibrational spectroscopy with localized modes. We apply this approach to explore the potential of 2D-IR spectroscopy in the extended amide III region. Using calculations for a dipeptide model as well as alanine polypeptides, we show that distinct 2D-IR cross-peaks in the extended amide III region can potentially be used to distinguish α-helix and β-strand structures. We propose that the extended amide III region could be a promising target for future 2D-IR experiments.


S1.1 Molecular structures
For the dipeptide N -acetyl-l-alanine-N -methylamide, molecular structures with the chosen backbone dihedral angles were constructed.Subsequently, these structures were each optimized while keeping the dihedral angles ϕ and ψ fixed at the chosen values.
The polyalanine-10 models consist of ten alanine units, where the terminal carboxylic acid and amine groups are capped by a methyl group.Here, idealized structures were generated using Avogadro V1.2.0.were adapted from the crystalline structure of the antiparallel poly-l-alanine β-sheet 3 for two strands placed in vicinity of each other.These idealized structures were then optimized with fixed backbone dihedral angles.The lengths of the hydrogen bonds between the two strands of the idealized β-sheet are between 192.2 and 208.1 pm.
To construct the polyalanine-15/16 models, we performed molecular dynamics simulations, starting at idealized α-helix, β-strand, and β-hairpin structures, respectively.Again, the termini are capped by methyl groups.These initial structures were generated using the molfacture function featured in VMD. 4 The topology was created using the AMBER99SB-ILDN force field. 5The molecular dynamics (MD) simulations were done with Gromacs, 6 using the water model tip3p 7 and standard equilibration methods at room temperature and normal pressure.As the MD simulations show, both systems start do deform.We took a snapshot from the early phase of each of the MD simulations.Results for additional snapshots from later points in the same trajectories are shown in Figs.S15-S17 below.All these snapshots were then optimized quantum-chemically with fixed backbone-dihedral angles.
Coordinates of all optimized molecular structures are provided in the data set at Ref. 8.

S1.2 Localization of normal modes
After performing a harmonic frequency calculation, we identify the modes belonging to the extended amide III region.For the dipeptides, these can be identified straightforwardly based on their wavenumbers.For the polyalanine-10 and polyalanine-15 models, the assignment used here is listed in Tables S10 to S18.
For the subset of these extended amide III modes, a unitary transformation U is performed, such that the resulting modes, are maximally localized. 9 This transformation provides the Hessian in the basis of the localized-mode coordinates, in which the harmonic potential energy surface can be expressed as, where ωi = Hii are the localized-mode frequencies and Hij are the off-diagonal elements of the Hessian with respect to localized modes.

S1.3 L-VSCF/L-VCI
We approximate the anharmonic potential energy surface in localized-mode coordinates where the anharmonic one-mode potentials Ṽ (1) i (q i ) are calculated on a grid of 16 points per mode 12 and Hij are the off-diagonal elements of the localized-mode Hessian, which describe the harmonic coupling between localized modes.Now, the vibrational Schrödinger equation can be solved using the L-VSCF/L-VCI ansatz. 12,13By diagonalizing the L-VCI matrix (including the ground state as well as singly and doubly excited states, i.e., L-VCISD), With this, the anharmonic excitation frequencies, and transition dipole moments (between ground state and singly excited states as well as between singly and doubly excited states), can be calculated.
To illustrate the methodology, the (harmonic) coupling matrices and anharmonic L-VSCF and L-VCI frequencies as well as the corresponding anharmonic shifts for two structures of the dipeptide are listed in Sections S2.1 and S2.2 (Tables S1-S6).
We note that the harmonic coupling can introduce errors in L-VSCF/L-VCISD already at the level of a harmonic potential energy surface [i.e., using Eq.(S3)]. 14While for the dipeptides, these errors are negligible (below 0.05 cm −1 in all cases), for polyalanine-10 and polyalanine-15/16 these errors accumulate and become noticeable.Fig. S18 compares the calculated one-dimensional infrared spectra of the polyalanine-10 and polyalanine-15 models.
In particular, there is a shift between the harmonic spectra calculated by diagonalizing the Hessian and those obtained with L-VCIS using the harmonic potential energy surface.This shift is due to the error introduced in L-VCIS by the harmonic coupling as discussed in Ref.
14.This error is also present in the anharmonic spectra.However, it mainly constitutes a constant shift of the vibrational frequencies and should not affect the intensity patterns in the calculated 2D-IR spectra.
Table S3: Anharmonic shifts ∆ 2i and ∆ ij from the 2D spectrum for each pair of double peaks for the dipeptide molecule in the α-helix region with the angles ϕ α = −80, ψ α = −20 in cm −1 .The ordering of the table correlates to the location of peaks as seen in the spectrum.Table S6: Anharmonic shifts ∆ 2i and ∆ ij from the 2D spectrum for each pair of double peaks for the dipeptide molecule in the β-strand region with the angles ϕ β = −120, ψ β = 140.The ordering of the table correlates to the location of peaks as seen in the spectrum.

S2.4 Mean values of frequencies and coupling matrices for the dipeptide
Table S7: Mean frequency values and standard deviations of the dipeptide molecules.

Figure S1 :
Figure S1: Vibrational normal modes of the dipeptide N -acetyl-l-alanine-N -methylamide in the α-helix region with the angles ϕ α = −80 • and ψ α = −20 • .(a) shows the bending vibration of C α -H(1) along the N-C α bond, (b) shows the bending vibration of C α -H(1) perpendicular to the N-C α bond and (c) and (d) show the classical amide III vibrations, that combine N-H bending and C-N stretching vibrations.

S2. 2
Figure S2: Vibrational normal modes of the dipeptide N -acetyl-l-alanine-N -methylamide in the β-strand region with the angles ϕ β = −120 • and ψ β = 140 • .(a) shows the bending vibration of C α -H(1) along the N-C α bond, (b) shows the bending vibration of C α -H(1) perpendicular to the N-C α bond and (c) and (d) show the classical amide III vibrations, that combine N-H bending and C-N stretching vibrations.

Figure S3 :
Figure S3: 1D spectra of the N -acetyl-l-alanine-N -methylamide molecule in the α-helix region within the range of 1170 cm −1 to 1370 cm −1 .Each plot shows the different spectra for varying ψ α with fixed ϕ α values.The solid lines are anharmonic spectra, the dashed lines are harmonic spectra.Additionally, vertical lines indicate the frequency of the anharmonic peaks.

Figure S4 :
Figure S4: 1D spectra of the N -acetyl-l-alanine-N -methylamide molecule in the α-helix region within the range of 1170 cm −1 to 1370 cm −1 .Each plot shows the different spectra for varying ϕ α with fixed ψ α values.The solid lines are anharmonic spectra, the dashed lines are harmonic spectra.Additionally, vertical lines indicate the frequency of the anharmonic peaks.

Figure S5 :
Figure S5: 1D spectra of the N -acetyl-l-alanine-N -methylamide molecule in the β-strand region within the range of 1170 cm −1 to 1370 cm −1 .Each plot shows the different spectra for varying ψ β with fixed ϕ β values.The solid lines are anharmonic spectra, the dashed lines are harmonic spectra.Additionally, vertical lines indicate the frequency of the anharmonic peaks.

Figure S6 :
Figure S6: 1D spectra of the N -acetyl-l-alanine-N -methylamide molecule in the β-strand region within the range of 1170 cm −1 to 1370 cm −1 .Each plot shows the different spectra for varying ϕ β with fixed ψ β values.The solid lines are anharmonic spectra, the dashed lines are harmonic spectra.Additionally, vertical lines indicate the frequency of the anharmonic peaks.

Figure S7 :
Figure S7: Mean frequency values and standard deviations in cm −1 of the dipeptide molecules as shown in TableS7.

Figure S10 :
Figure S10: Ramachandran plot with all found backbone angles of the polyalanine molecules in the α-helix region.The round dots show the angles of the polyalanine-10 molecules in the α-helix region (red) and the β-strand region (blue).The red crosses (0) show the backbone angles of the polyalanine-15 molecule shown in figure 3(b).The other crosses show the backbone angles of the polyalaine-15 molecules shown in figure S15.

Figure S11 :
Figure S11: Ramachandran plot with all found backbone angles of the polyalanine molecules in the β-strand region.The round dots show the angles of the polyalanine-10 molecules in the α-helix region (red) and the β-strand region (blue).The red crosses (0) show the backbone angles of the polyalanine-15 molecule shown in Fig. 3b.The other crosses show the backbone angles of the polyalaine-15 molecules shown in figure S16.

Figure S12 :
Figure S12: Ramachandran plot with all found backbone angles of the polyalanine molecules in the β-hairpin region.The round dots show the angles of the polyalanine-10 molecules in the α-helix region (red) and the β-strand region (blue).The blue crosses (0) show the backbone angles of the polyalanine-16 molecule shown in Fig. 3b.The other crosses show the backbone angles of the polyalaine-16 molecules shown in figure S17.

Figure S13 :
Figure S13: Calculated 2D-IR spectra (L-VSCF/L-VCI) of (a) polyalanine-10 in an idealized α-helical and an idealized β-strand conformation and of (b) a snapshot of polyalanine-15 extracted from a molecular dynamics simulation of an α-helix and a β-strand for the ⟨ZZXX⟩ polarization condition.In the insets above the 2D-IR spectra, the calculated harmonic and anharmonic infrared spectra are shown for comparison.Horizontal and vertical lines indicate the five largest peaks of the 1D spectra.

Figure S14 :
Figure S14: Calculated 2D-IR spectra (L-VSCF/L-VCI) of (a) polyalanine-10 in an idealized α-helical and an idealized β-strand conformation and of (b) a snapshot of polyalanine-15 extracted from a molecular dynamics simulation of an α-helix and a β-strand for the ⟨ZXXZ⟩ polarization condition.In the insets above the 2D-IR spectra, the calculated harmonic and anharmonic infrared spectra are shown for comparison.Horizontal and vertical lines indicate the five largest peaks of the 1D spectra.

Figure S15 :
Figure S15: Calculated 2D-IR spectra (L-VSCF/L-VCI) of four additional snapshots of polyalanine-15 extracted from a molecular dynamics simulation of an α-helix.In the insets above the 2D-IR spectra, the calculated harmonic and anharmonic infrared spectra are shown for comparison.In each plot the maximum is set to one.

Figure S16 :
Figure S16: Calculated 2D-IR spectra (L-VSCF/L-VCI) of four additional snapshots of polyalanine-15 extracted from a molecular dynamics simulation of an β-strand.In the insets above the 2D-IR spectra, the calculated harmonic and anharmonic infrared spectra are shown for comparison.In each plot the maximum is set to one.

Figure S17 :
Figure S17: Calculated 2D-IR spectra (L-VSCF/L-VCI) of three additional snapshots of polyalanine-15 extracted from a molecular dynamics simulation of an β-turn.In the insets above the 2D-IR spectra, the calculated harmonic and anharmonic infrared spectra are shown for comparison.In each plot the maximum is set to one.

Figure S18 :
Figure S18: Comparison of 1D spectra of the polyalanine molecules in the range of 1110 cm −1 to 1440 cm −1 .In the left column the spectra for polyalanine-10 in the α-helix region (a), β-strand region (c) and the β-hairpin (e) are shown, in the right column the spectra for polyalanine-15 in the α-helix region (b), β-strand region (d) and the β-hairpin (f) are shown.The solid colored lines show the results from the L-VSCF/L-VCI calculations, the dashed colored lines show the results from the L-VSCF/L-VCI calculations with harmonic potentials and the dashed/dotted gray lines show the results from the harmonic calculations using Turbomole and SNF.

Table S8 :
Mean coupling matrix of the dipeptide in the α-helix region with corresponding standard deviations in cm −1 .

Table S9 :
Mean coupling matrix of the dipeptide in the β-strand region with corresponding standard deviations in cm −1 .

Table S10 :
Mode contributions for polyalanine-10 in the α-helix configuration in percent for each vibrational mode n mode and their corresponding frequencies in cm −1 and intensities in km/mol.As a subset we chose the modes 184-214.The modes 184-194 include the classical amide III vibrations, the modes 195-204 are the C α -H(2) vibrations and the modes 205-214 are the C α -H(1) vibrations.

Table S11 :
Mode contribution for polyalanine-10 in the β-strand configuration in percent for each vibrational mode n mode and their corresponding frequencies in cm −1 and intensities in km/mol.As a subset we chose the modes 184-214.The modes 184-194 include the classical amide III vibrations, the modes 195-204 are the C α -H(2) vibrations and the modes 205-214 are the C α -H(1) vibrations.

Table S12 :
Mode contributions for polyalanine-10 in the β-hairpin configuration in percent for each vibrational mode n mode and their corresponding frequencies in cm −1 and intensities in km/mol.As a subset we chose the modes 204-235.The modes 204-219 include the classical amide III vibrations, the modes 220-227 are the C α -H(2) vibrations and the modes 228-235 are the C α -H(1) vibrations.

Table S13 :
Mode contributions for polyalanine-15 in the α-helix configuration in percent for each vibrational mode n mode and their corresponding frequencies in cm −1 and intensities in km/mol.As a subset we chose the modes 263-314.The modes 263-284 include the classical amide III vibrations, the modes 285-299 are the C α -H(2) vibrations and the modes 300-314 are the C α -H(1) vibrations.

Table S14 :
Mode contributions for polyalanine-15 in the α-helix configuration in percent for each vibrational mode n mode and their corresponding frequencies in cm −1 and intensities in km/mol (continued).

Table S15 :
Mode contribution for polyalanine-15 in the β-strand configuration in percent for each vibrational mode n mode and their corresponding frequencies in cm −1 and intensities in km/mol.As a subset we chose the modes 263-314.The modes 263-284 include the classical amide III vibrations, the modes 285-299 are the C α -H(2) vibrations and the modes 300-314 are the C α -H(1) vibrations.

Table S16 :
Mode contribution for polyalanine-15 in the β-strand configuration in percent for each vibrational mode n mode and their corresponding frequencies in cm −1 and intensities in km/mol (continued).

Table S17 :
Mode contribution for polyalanine-16 in the β-hairpin configuration in percent for each vibrational mode n mode and their corresponding frequencies in cm −1 and intensities in km/mol.As a subset we chose the modes 286-334.The modes 263-284 include the classical amide III vibrations, the modes 285-299 are the C α -H(2) vibrations and the modes 300-314 are the C α -H(1) vibrations.

Table S18 :
Mode contribution for polyalanine-16 in the β-hairpin configuration in percent for each vibrational mode n mode and their corresponding frequencies in cm −1 and intensities in km/mol (continued).