Coherent 2D-IR Spectroscopy of a Cyclic Decapeptide Antamanide. A Simulation Study of the Amide-I and A Bands

Abstract

The two dimensional infrared photon echo spectrum of Antamanide (-¹Val-²Pro-³Pro-⁴Ala-⁵Phe-⁶Phe-⁷Pro-⁸Pro-⁹Phe-¹⁰Pro-) in chloroform is calculated using an explicit solvent MD simulation combined with a DFT map for the effective vibrational Hamiltonian. Evidence for a strong intramolecular hydrogen bonding network is found. Comparison with experimental absorption allows to identify the dominant conformation. Multidimensional spectroscopy reveals intramolecular couplings and gives information on its dynamics. A two color amide-I and amide-A cross peak is predicted and analyzed in term of local structure.

1 Introduction

Understanding the structure and folding dynamics of peptides in solution is an important challenge. 2D-NMR techniques introduced in the 1970s are an important structure determination tool with a μs time resolution [1, 2]. A large number of experimental and theoretical studies have been devoted to the development of femtosecond multidimensional techniques in the infra-red, visible and ultra-violet regions [3, 4, 5, 6, 7, 8, 9, 10, 11]. In a 2D-IR photon echo experiment, three femtosecond pulses with wavevectors k₁, k₂ and k₃ interact with the molecule to generate a coherent signal which is heterodyne detected in the direction k_I = −k₁ + k₂ + k₃ (see Fig.1). The signal is recorded as a function of the three time delays between pulses [11].

Figure 1.

a coherent four wave mixing 2D-IR experiment

2D-IR spectroscopy probes high frequency vibrations. The theoretical description of vibrations in peptides is much more complex than that of spin dynamics probed by NMR and extracting structural information requires extensive modeling. IR techniques can determine the vibrational dynamics on much faster time scales than NMR. Numerous theoretical studies of 2D-IR have been conducted [12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. A high level electronic structure simulation is necessary in order to accurately model the vibrational dynamics and spectra. Most theoretical approaches use a semi-classical treatment, where the high frequency optically active vibrations are treated quantum mechanically and all other degrees of freedom are treated as a classical bath giving rise to a fluctuating Hamiltonian. The determination of this Hamiltonian has been a topic of interest during the last few years [16, 17, 18, 19, 20]. Several theoretical tools which combine electronic structure calculations, molecular dynamics and excitonic Hamiltonian description have been developed [18, 19, 20, 22, 23]. Peptides have four amide vibrational modes (amide I, II, III and A) localized on the backbone peptide bonds and will be denoted local amide modes(LAMs). The vibrational Hamiltonian consists of the local amide Hamiltonian expanded in LAMs and their couplings between the different units. Most studies had focused on the amide-I band [19, 22, 23]. Its frequency in solvent was calculated using a map, that connects it to the local electrostatic field. Such maps allow long time (ns) molecular dynamics (MD) simulation.

We had developed an electrostatic DFT Map [18] of all amide states in N-methylacetamide, a model of the peptide bond. The map provides a first-principles effective vibrational Hamiltonian which includes the fundamental, overtone, combination frequencies and transition moments of the amide III, II, I and A. It is based on vibrational eigenstate calculations of a 6th order anharmonic DFT potential in the presence of up to 3rd rank multipole (octapole) electric field. The map was first applied to the amide I band combined with the ab-initio map of Torii and Tasumi [14] for the covalent bonded nearest neighbor couplings and transition dipole coupling model (TDC) for non bonded units [12, 13]. We denote it MAP1 [19].

A extension of this map, denoted here MAP2, has been constructed in order to describe the amide-I band as well as other amide vibrational modes, including the vibration amide-A, amide-II and amide-III [18] using improved coupling models. For nearest covalently bonded units the ab-initio map of Torii and Tasumi has been replaced by a higher level DFT map and for non bonded units higher multipoles of the couping have been included [20].

Highly accurate electrostatic maps are required in order to reproduce the variation of the vibration with its environment (molecule, solvent and intramolecular hydrogen bonds dynamics). The hydrogen bonding network is crucial for determining the structure and dynamics of peptide in solution. The sensitivity of vibrational spectra to hydrogen bonding has been studied in several systems [24, 25, 26, 27] including the decapeptide Antamanide [26].

Antamanide is a cyclic decapeptide (see Fig. 2) which has been extensively studied by 2D-NMR spectroscopy and molecular dynamics simulations [28, 29, 30, 31]. It has been isolated from extracts of the poisonous Amanita phalloides and inhibits the toxic principles of this mushroom. In mammals, phalloidins block the depolymerization of F-actin into G-actin in the liver cell membrane. This process can be prevented by the presence of Antamanide [32, 33]. The determination of the possible conformations in solution have been the subject of a very large number of studies since this discovery [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. Yet the stable conformations have not been fully determined [37, 38, 39]. Antamanide was first studied by linear IR spectroscopy and electronic circular dichroism in different solvents [34, 35]. A 1D-NMR experiment suggested that Antamanide exists in solution in a equilibrium of two conformers with a 1 μs exchange time [36]. One corresponds to a fully intramolecular hydrogen bonded structure and the other allows for hydrogen bonding with the solvent. A subsequent NMR study showed some inconsistencies between structurally dependent NMR parameters and a single stable conformation [37]. For this reason, and the fact that this peptide is reasonably small, Antamanide has served as a benchmark for a multiconformational search algorithms based on NMR data [38, 39]. Using this algorithm Blackledge et al. [39] have identified four conformations called A128, E116, G129 and G193, each configuration corresponding to a specific hydrogen-bonding network. Based on these conformations Moran et al. [26] conducted a study using both linear IR and non-linear IR response. This study did not include the solvent dynamics, only considered frozen configurations and used a crude model to determine the frequency of the amide vibrations.

Figure 2.

The cyclic decapeptide Antamanide. The six amide group containing one C=O vibration and one N–H vibration are highlighted in red and the four isolated CO vibrations are highlighted in blue.

The present study have several goals. First, we describe the dynamics of Antamanide in an explicit solvent to reproduce the complex inhomogeneous broadening of the vibrational lineshapes of the amide-I and amide-A absorption band. Second, we test the ability of the two maps (MAP1 and MAP2) to reproduce these lineshapes. Antamanide is a perfect molecule for this purpose since its size is at the crossover between the small molecule with well resolved spectra and long polypeptides with a very large broadening. It also shows a strong intramolecular hydrogen bond with typical infrared signatures. Finally we propose a two-color experiment for probing vibrational dynamics in greater detail. In section II we describe the MD simulations which were started from several conformations suggested by NMR spectroscopy and an X-ray structure. The simulation of the infrared absorption and comparison with experiment is described in section III. Section IV presents the 2D-IR one color response. In the last section, we discuss the amide-I amide-A crosspeak.

2 Molecular Dynamics Simulations

2.1 Simulation protocol

MD simulations were performed using the NAMD 2.6 program [41] with the CHARMM27 force field [42]. Chloroform was modeled using an all-atom model with force field parameters taken from Ref. [43]. All simulations used periodic boundary conditions with a 2 fs time step. Long-range electrostatic interactions were computed using particle-mesh Ewald (PME) [44, 45] and a real space cutoff of 12 Å was used for nonbonded interactions. Langevin dynamics with a 1 ps⁻¹ damping coeffcient were used to achieve a constant temperature of 250 K. The 1 atm constant pressure was maintained using a Nose-Hoover Langevin piston [46, 47] with a decay period of 200 fs and a 100 fs damping time, when pressure regulation was employed.

Our simulations start from the five configurations of Antamanide suggested by NMR and X-ray experiments: XRAY corresponds to the structure identified by Karle et al. [40] through an X-ray study of Antamanide in crystals. The other four conformations (A128, E116, G129 and G193) were identified by Blackledge et al. [39] in solution. The dihedral angles for these structures are reported in Table 1.

Table 1.

Dihedral angles (in degrees) of the five conformations used as a starting point for the MD simulations. XRAY denotes the structure of Antamanide in a crystal determined by X-ray crystallography [40]. A128, E116, G129 and G193 correspond to the four structures suggested by Blackledge et al. [39].

	XRAY		A128		E116		G129		G193
	Φ	Ψ	Φ	Ψ	Φ	Ψ	Φ	Ψ	Φ	Ψ
1Val	−113	158	−96	105	−79	127	−97	146	−105	162
2Pro	−64	161	−70	133	−72	143	−76	133	−80	148
3Pro	−80	−21	−93	17	−96	20	−106	43	−95	24
4Ala	−103	−22	−98	83	−93	−45	−134	67	−93	−49
5Phe	70	30	−87	40	67	35	−79	63	53	27
6Phe	−78	161	−88	130	−89	126	−96	135	−140	135
7Pro	−62	160	−68	144	−76	142	−80	140	−72	138
8Pro	−92	−4	−92	22	−93	17	−95	18	−86	5
9Phe	−101	−22	−96	−54	−95	−42	−112	54	−100	57
10Phe	56	48	72	25	71	19	−58	71	−53	−4

The Antamanide molecule was embedded in a pre-equilibrated orthorhombic 52 Å chloroform box containing between 1102 and 1105 chloroform molecules. To equilibrate the solvent around the peptide a 10000 steps energy minimization was performed using a harmonic constraint on the polypeptide with the force constant 2500 kcal mol⁻¹ Å⁻¹. In a second step, we have performed a 200 ps equilibration in the NPT ensemble using a reduced force constant for the harmonic constraint of 200 kcal mol⁻¹ Å⁻¹ to equilibrate the pressure and the box size. Finally, we performed a 15 ns NVT simulation without any restraints, recording snapshots every 500 fs. We observed a relaxation of all conformations during the first nanosecond . All calculations are based on the 14 ns trajectory which follows the first nanosecond equilibration period.

2.2 MD simulations results

The average dihedral angles with the corresponding standard deviations starting from the XRAY and E116 conformations are reported in Table 2. The two configurations clearly relax to the same structure. The standard deviation of all dihedral angles is less than 11°, indicating that this configuration is stable. Most dihedral angles change from their initial value, but we highlight Ψ₅ and Ψ₁₀ which starting from respectively 30° and 48° for XRAY conformation and 35° and 19° for E116 conformation relax to respectively −51° and −49°. The average conformation has approximately a cyclic invariance Φ_n ≈ Φ_n+5 and Φ_n ≈ Φ_n+5. This follows directly from the location of the proline group (see Fig. 2).

Table 2.

Average dihedral angles and corresponding standard deviation (in degrees) of the XRAY and E116 conformations. Calculations are made during the last 14 ns.

	XRAY				E116
	〈Φ〉	σΦ	〈Ψ〉	σΨ	〈Φ〉	σΦ	〈Ψ〉	σΨ
1Val	−70.4	10.0	151.7	10.5	−71.5	10.0	154.9	8.0
2Pro	−68.1	6.9	157.1	8.0	−68.7	6.8	157.9	7.6
3Pro	−80.3	7.7	11.3	8.3	−80.0	7.8	10.9	8.9
4Ala	−80.9	9.0	−48.2	7.4	−81.2	9.1	−48.5	7.4
5Phe	72.5	6.6	−51.1	8.6	72.5	6.7	−50.5	8.6
6Phe	−72.0	10.2	156.3	8.3	−71.8	10.3	156.3	8.3
7Pro	−67.5	6.7	157.6	7.5	−67.7	6.8	157.7	7.5
8Pro	−79.3	7.4	8.1	8.4	−79.3	7.5	8.3	8.5
9Phe	−80.3	9.0	−50.8	7.1	−80.7	9.0	−50.4	7.1
10Phe	72.1	6.4	−48.5	9.0	72.3	6.4	−48.5	8.9

Using these average angles we built an average conformation used to determine the hydrogen bond network of that configuration. We assume the existence of a hydrogen bond if the distance H···O is less than 3.2 Å and the angle N–H···O less than 60°. This network is essential for identifying the possible conformations. As shown in Fig. 2, Antamanide contains four proline groups, thereby only six peptide groups contain an NH group. This allows for at most six intramolecular hydrogen bonds. The molecule and the hydrogen bond network fluctuate around this average structure and H bond breaking occurs during the simulation. The calculated hydrogen bonding networks for the average conformations are shown in Fig. 3. For the XRAY/E116 conformation, we observe six hydrogen bonds, which is the maximum possible number. All six N–H vibrations are hydrogen bonded, two C=O group are doubly bonded, two C=O vibration are singly bonded and the other four C=O are not bonded. This hydrogen bonding network has been pointed out previously by Moran et al. [26]. They performed electronic structure calculation to determine the different possible structures and the Kabsch-Sander criterion was used to determine the hydrogen bond network [48].

Figure 3.

Hydrogen bonding network of the different average conformations. Each arrow link an N–H group to a C=O group. An hydrogen bond is defined if the distance H···O is smaller than 3.2 Å and the angle N–H···O smaller than 60°

The average dihedral angles and associated standard deviations of A128 and G193 are given in Table 3. As in XRAY/E116, the angles Ψ₅ and Ψ₁₀ show a significant deviation from their initial values. For both A128 and G193, Φ₆ also experiences large relaxation. Starting from −88° and −140° respectively the angle relaxes to −144° and 71° for the conformations A128 and G193. For A128 we observe a non negligible deviation for the angle Ψ₁ (from 105° to 151°). As a result, the average configurations A128 and G193 can be approximately deduced from each other by the symmetry ${\Phi }_n^{A128}\approx {\Phi }_{n+5}^{G193}$ and ${\Psi }_n^{A128}\approx {\Psi }_{n+5}^{G193}$. Both A128 and G193 experience relatively small fluctuations around their average structure (less than 12° for all angles except for two angles with large deviations > 15° (in bold in Tab 3)) indicating that these configurations are less stable. The trajectories of the angles with large deviations, Φ₆ and Φ₅ for A128 and Φ₁ and Φ₁₀ for G193, are displayed in Fig. 4.

Table 3.

Average dihedral angles and corresponding standard deviation (in Degrees) of the A128 and G193 conformations. Calculations are made during the last 14 ns. In bold are highlighted the angles which experience larger deviations

	A128				G193
	〈Φ〉	σΦ	〈Ψ〉	σΨ	〈Φ〉	σΦ	〈Ψ〉	σΨ
1Val	−71.6	10.2	150.9	9.2	−121.7	24.1	154.6	7.3
2Pro	−68.5	7.0	152.9	7.6	−71.8	6.8	158.8	7.0
3Pro	−80.6	7.9	5.1	8.9	−80.4	7.6	12.9	9.6
4Ala	−90.0	9.1	70.1	8.5	−78.7	10.1	−57.5	7.9
5Phe	−75.5	7.4	54.6	16.9	70.3	7.3	−45.6	11.6
6Phe	−144.0	18.2	153.6	7.9	−71.2	10.7	143.3	11.4
7Pro	−70.3	6.7	158.2	7.1	−67.1	7.1	152.2	7.9
8Pro	−80.1	7.5	8.9	8.8	−80.2	7.7	4.8	8.4
9Phe	−78.4	9.3	−58.1	7.5	−90.1	8.5	69.4	8.3
10Phe	70.5	7.1	−45.6	10.9	−73.9	8.8	38.8	27.2

Figure 4.

Time evolution of dihedral angles. (a) evolution of Φ₆ and Ψ₅ for the A128 conformation. (b) evolution of Φ₁ and Ψ₁₀

Fig. 3 also shows the hydrogen bond networks of A128 and G193. Unlike XRAY/E116, the hydrogen bond network of A128 and G193 does not have the symmetry of the molecule. However the network of one conformation can be deduced from the other by a n to n + 5 permutation of the amide groups. For G193 we recover the network observed by Moran et al. [26], but this is not the case for A128. This previous study missed the symmetry A128/G193 observed here. In that study the configurations were optimized using electronic structure calculations but the solvent was not explicitly included. As in XRAY/E116, all N–H modes are hydrogen bonded. However, for the C=O we observe some differences. In both A128 and G193 configuration one C=O is doubly bonded, four are singly bonded and three are not bonded. These difference have clear signatures in the linear and non linear IR spectra.

The average and corresponding standard deviation of the dihedral angles of G129 are reported on the Table 4. As in XRAY/E116, G129 has approximately the Φ_n ≈ Φ_n+5 and Ψ_n ≈ Ψ_n+5 symmetry. However this configuration experiences much larger deviations for several angles. As shown in Fig. 5, the time evolution of the angles Ψ₁ Ψ₄ and Φ₄ shows that the fluctuations correspond to jumps between different states.

Table 4.

Average dihedral angles and corresponding standard deviation (in Degrees) of the G129 conformation. Calculations are made during the last 14 ns. In bold are highlighted the angles which experience larger deviations

	G129
	〈Φ〉	σΦ	〈Ψ〉	σΨ
1Val	−100.8	14.8	134.5	18.9
2Pro	−72.9	7.9	152.6	8.8
3Pro	−80.8	7.9	1.6	9.4
4Ala	−95.2	18.2	76.9	19.1
5Phe	−77.4	8.5	76.3	18.6
6Phe	−103.6	17.0	125.9	18.7
7Pro	−71.1	8.3	152.0	9.9
8Pro	−80.4	7.9	0.7	10.0
9Phe	−90.8	10.4	75.2	12.1
10Phe	−77.5	8.6	74.3	14.8

Figure 5.

Time evolution of dihedral angles for G129 conformation. (a) Trajectory of Φ₆ and Ψ₅. (b) Ψ₁ Ψ₄ and Φ₄.

The hydrogen bond network of the average conformation G129 is sketched in the Fig. 3. Like the dihedral angles, the n → n + 5 symmetry is observed. Unlike the other configurations where all N–H modes where hydrogen bonded, using our hydrogen bond criterion we found that G129 has only four intramolecular hydrogen bonds and this configuration is thus less stable. The less restrictive criterion used in Ref. [39] yielded a fully hydrogen bonded structure. However, the two additional hydrogen bonds are weaker.

3 The effective Hamiltonian and absorption spectrum

We had used the simulation protocol of Ref. [18, 19] to generate a set of eigenenergies. As shown on Fig. 2 because of the presence of four proline groups, four amide CO-NH out of ten units only contains isolated CO groups. Assuming that each unit contains one or two high frequency vibrations (amide-I only or both amide-I and amide-A), we write the following effective vibrational exciton Hamiltonian

$$\begin{array}{cc}\hfill H_S& =\hslash \sum _{n=1}^{N_1}\left({\omega }_{1n}{B}_{1n}^{†}{B}_{1n}-A_{1n}{B}_{1n}^{†2}{B}_{1n}^2\right)+\hslash \sum _{n\ne m=1}^{N_1}{J}_{1nm}{B}_{1n}^{†}{B}_{1m}\hfill \\ \hfill & +\hslash \sum _{n=1}^{N_2}\left({\omega }_{2n}{B}_{2n}^{†}{B}_{2n}-A_{2n}{B}_{2n}^{†2}{B}_{2n}^2\right)+\hslash \sum _{n\ne m}^{N_2}{J}_{2nm}{B}_{2n}^{†}{B}_{2m}\hfill \\ \hfill & -\hslash \sum _{n=1}^{N_1}\sum _{m=1}^{N_2}{K}_{12nm}{B}_{1n}^{†}{B}_{1n}{B}_{2m}^{†}{B}_{2m}\hfill \end{array}$$ (1)

where $B_{1n}^{†}$, $B_{1n}$, $B_{2n}^{†}$ and B_2n are boson creation and annihilation operators corresponding to amide-I and amide-A frequency vibration located on the nth local unit. Ω_1n, A_1n, Ω_2n and A_2n are respectively the frequency and anharmonicity of the amide-I and amide-A vibration on the nth unit. We have N₁ = 10 amide-I local vibrations (Ω₁ ≈ 1650 cm⁻¹) and N₂ = 6 amide-A local vibrations (Ω₂ ≈ 3300 cm⁻¹). The two bands are well separated. The intermode couplings are represented by the constants J_1nm and J_2nm which modify the eigenstates by inducing a delocalization of the vibrations. We have neglected the linear J coupling between the amide-I and the amide-A vibrations. Given the frequency mismatch Ω₂ – Ω₁ ≈ 1650 cm⁻¹ its effect on the eigenstates is small. On the other hand we consider quartic coupling between the amide-I and amide-A bands (K coupling) given by the constants K_12nm. This coupling is essentially local so that we simply set K_12nm = δ_nmK_12nn.

The dipole interaction with the optical field is given by

$$H_{int}=-E\left(t\right)\left[\sum _{n=1}^{N_1}{\mu }^{\left(1\right)}\left(B_{1n}^{†}+B_{1n}\right)+\sum _{n=1}^{N_2}{\mu }^{\left(2\right)}\left(B_{2n}^{†}+B_{2n}\right)\right]$$ (2)

where μ⁽¹⁾ and μ⁽²⁾ are respectively the transition dipole of the amide-I and amide-A vibrations. The eigenvalues of the Hamiltonian H_S (Eq. 1) are sketched in Fig. 6. Since we have neglected all interband relaxation processes and in particular the coupling between the amide-A and the amide-I band, we have three blocks: the ground state, the one, and the two excitons blocks.

Figure 6.

Energy scheme of the amide-A and amide-I bands

Our MD simulations have generated an ensemble of configurations. For each configuration we have calculated the Hamiltonian parameters and the corresponding eigenvalues and eigenvectors. The parameters were calculated using electronic structure calculations of MAP1 [19] and MAP2 [18, 20]. MAP1 provides the parameters for the amide-I Hamiltonian only. The local amide-I frequencies (Ω_1n) and its anharmonicities (A_1n) are parametrized with the multipole electric field on the local amide unit generated by the environment. The parametrization is based on the 6th order anharmonic potential in 5 normal coordinates (including all 4 amide modes) of the model system (N-methylacetamide) constructed with DFT (BPW91/6−31G(d,p)) calculations. The linear amide I couplings between the neighboring amide unit (J_1nm) are calculated by the Torii and Tasumi map [14], which is based on the Hartree-Fock ab-initio calculations of the glycin dipeptide (GLDP) for different Ramachandran angles. Non-neighbor electrostatic couplings are given by the transition dipole coupling model [12, 13]. MAP2 provides the complete set of parameters for the peptide Hamiltonian of all 4 amide modes (I, II, III and A) in the same way as MAP1, and also provides higher level neighbor and non-neighbor couplings. The linear neighboring couplings (J_1nm, J_2nm) and the nonlinear neighbor couplings (K_12nn) are parametrized by the Ramachandran angles based on the anharmonic vibrational potential of the GLDP constructed at the BPW91/6−31G(d,p) level. Non-neighbor electrostatic couplings are calculated by the transition multipole coupling mechanism including dipole-dipole (∼ R⁻³), dipole-quadrupole (∼ R⁻⁴), quadrupole-quadrupole and dipole-octapole (∼ R⁻⁶) interactions. These higher multipole interactions were found crucial for the amide-II, III and A interactions [20].

Using our four MD trajectories (A128, E116, G129 and G193), we have performed the calculation of the vibrational absorption spectra of the amide-I and the amide-A region. The fluctuations of the eigenenergies and the eigenvectors at the femtosecond time scale are responsible for homogeneous and inhomogeneous broadening [49, 50, 11]. In this study, slow inhomogeneous dephasing is included microscopicaly. Homogeneous dephasing (fast fluctuations) is added by a dephasing rate [49, 11]. Γ = 5.5 cm⁻¹ for the amide-I band [19, 5].Γ = 22 cm⁻¹ for amide-A (assuming that the dephasing is scaled as the square of the energy Γ ∝ ε²). We assume the same Γ for all transitions in each band. Our classical simulations generate an ensemble and the final signals are given by averaging over the various snapshots.

The linear absorption is given by

$$I\left(\omega \right)=\langle \sum _e\frac{{\Gamma }_e{\mid {\mu }_{ge}\mid }^2}{{(\omega -{\omega }_e)}^2+{\Gamma }_e^2}\rangle$$ (3)

The average 〈···〉 is over the distribution ω_ge and μ_ge in the various snapshots, ω_e is the vibrational frequency of the excited state e, Γ_e is the homogeneous dephasing and μ_ge is the transition dipole from the ground state to the excited state e.

The simulated linear absorption using the two maps are displayed in Fig. 7. For comparison, we show the experimental spectrum from Ref. [35]. A 55 cm⁻¹ redshift for MAP1 and a 58 cm⁻¹ redshift for MAP2 was introduced to match theory and experiment.

Figure 7.

Linear absorption spectra of the four conformations. Solid line (MAP2), dashed line (MAP1), open circle (experiment [35]). To match with experiment a 55 cm⁻¹ redshift for MAP1 and a 58 cm⁻¹ redshift for MAP2 are introduced.

Each configuration shows a specific absorption profile. MAP1 and MAP2 give similar lineshapes for all configurations except for G129. E116 shows two main peaks. For MAP2 the high frequency peak is located around 1670 cm⁻¹ and the low frequency peak is located around 1629 cm⁻¹. The corresponding MAP1 frequencies are 1627 cm⁻¹ and 1671 cm⁻¹. Eigenvector analysis shows that the low frequency peak corresponds mainly to vibration located on the hydrogen bonded CO whereas the high frequency peak represents unbounded CO modes. G129 for MAP1 has one strong peak at 1646 cm⁻¹ with a smaller one near 1670 cm⁻¹ whereas for MAP2, two peaks with egal intensity, located respectively at 1656 cm⁻¹ and 1669 cm⁻¹ are visible. A128 and G193 have three peaks. For MAP1 and MAP2 the low frequency peak of both configurations is located respectively near 1631 cm⁻¹. For A128, the central peak is located near 1649 cm⁻¹. For G193, this peak is only well resolved for MAP1 at 1649 cm⁻¹, however for MAP2 a shoulder is visible near 1656 cm⁻¹. For G193, the last peak is located near 1666 cm⁻¹ for both MAP1 and MAP2. For A128, the position of this peak depend on the map. It is located around 1663 cm⁻¹ for MAP1 and near 1670 cm⁻¹ for MAP2.

The sensitivy of the linear absorption in the map can be directly related to the localization length of the system. Using MAP2 we have calculated the amide-I average localization length defined as

$$L=\langle \frac{1}{N_1}\sum _{e=1}^{N_1}\frac{1}{{\Sigma }_m{\mid {\Psi }_e\left(m\right)\mid }^4}\rangle$$ (4)

where Ψ_e(m) is the probablity amplitude to find an amide-I exciton on the site m. For E116 we find L = 2.0, for G193 L = 2.1, for A128 L = 2.4, and for G129 L = 2.8. Consequently the amide-I exciton is much more delocalized in G129 then in the other conformations and the corresponding eigenstates are more sensitive in the couplings between the vibrational groups which is the major difference between MAP1 and MAP2. In the same spirit E116 is the less delocalized configuration and also the less sensitive to the coupling.

The absorption of Antamanide in chloroform has been measured in the amide-I and amide-A region by Ivanov et al. [34, 35]. The amide-I shows one main peak at 1668 cm⁻¹ and a small peak at 1630 cm⁻¹. The experimental spectrum is very similar to our simulated E116, suggesting that this configuration is dominant. As was pointed out previously [36], the conformation of this molecule strongly depends on the solvent and in particular in its ability to form hydrogen bonds with the molecule. Since MAP2 is based on a higher level of theory than MAP1, in the reminder of this study we will focus on MAP2.

The linear absorption spectra of the amide-A band for all configurations are shown in Fig. 8a. A 150 cm⁻¹ redshift was introduced to match with experiment. For all configurations the amide-A band is much broader than the amide-I (∼ 140 cm⁻¹ compared to ∼ 70 cm⁻¹). A128 and G193 have a similar shape characterized by an asymmetric peak near 3340 cm⁻¹ with longer tail to the red. G129 is broader with a maximum at 3330 cm⁻¹ but the asymmetric shape is similar. E116 is narrower than G129 but has a more symmetric band with a main peak at 3330 cm⁻¹ and a shoulder at 3300 cm⁻¹. The overall spectrum is redshifted compared to the other configurations. Due to both their strong asymmetric shape and relatively small width, the configurations A128 and G193 fail to reproduce the experimental spectrum (see Fig. 8a). G129 has the broadest absorption, close from the experimental (∼ 130 cm⁻¹ compared to 140 cm⁻¹), E116 is narrower (∼ 105 cm⁻¹) but has a more symmetric shape, in better agreement with the experimental absorption. From the amide-A absorption, both G129, E116 seem to play a role in the dynamics.

Figure 8.

a) Linear absorption spectrum of all possible configurations in the amide-A region, open circle: experimental data. A redshift of 150 cm⁻¹ has been introduced to correct our theoretical data. b) Linear absorption spectra of a mixture E116/G129 ranging from 0% of G129 to 50% of G129, open circle: experimental data.

No jump between the different configurations was observed during our MD simulations, however such transitions could occur on a much longer time scale which has been suggeted by NMR. In that case two or more conformer could give a contribution to the spectrum. We have investigate the linear absorption using a distribution of different conformers. However none of them improve drastically the shape of the spectrum. In Fig. 8b is displayed the linear absorption in the amide-I band using a mixture of E116 and G129 conformers with a proportion ranging from 0% to 50% of G129. All spectra have been normalized to the low frequency peak. As increasing the G129 proportion the high frequency peak becomes broader. Indeed we increase the inhomogeneous fluctuations. Since G129 doesn't have significant optical density in the 1625 cm⁻¹ frequency range, the ratio between the intensity of the low frequency peak and the intensity of the high frequency peak decreases. However even for pure E116 this ratio is smaller then the experimental one (respectively ∼ 0.5 and ∼ 0.7). The differences between experimental spectrum and simulations are most likely to come from the limitations of our theoretical model than from a superposition of different conformer absorptions.

4 One color 2D-IR photon echo. Simulation of the amide-I and A bands

We now turn to the simulation of the 2D-IR photon echo signal in the amide-I and amide-A regions. Three quantum pathways contribute to the signal, as shown by the Feynmann diagrams (Fig. 9) : the ground state bleaching (GSB), the excited state emission (ESE) and the excited state absorption (ESA). The GSB and the ESE both give a negative contribution to the spectrum and probe the one-exciton block. The ESA contribution is positive and probes the two-exciton block. The signal, recorded as a function of the three times intervals between the pulses (t₁,t₂ and t₃), is given by the response function of the third order S(t₁, t₂, t₃) (see Ref. [49]). A double Fourier transform is applied over the times t₁ and t₃ [10] and the signal is finally displayed as

$$S({\Omega }_1,t_2,{\Omega }_3)={\int }_0^{\infty }\mathrm{d}{t}_1{\int }_0^{\infty }\mathrm{d}{t}_{3}S(t_1,t_2,t_3)e^{i{\Omega }_1{t}_1+i{\Omega }_2{t}_2}$$ (5)

The third order response function S(t₁, t₂, t₃) was calculated using the sum-over-states (SOS) expressions Eq. (5.19) of Ref. [11] as implemented in the Spectron code [19].

Figure 9.

Feynman diagrams characterizing the three elementary process of the photon echo: the excited state absorption (ESA), the excited state emission (ESE) and the ground state bleaching (GSB)

We assume pulses short enough to cover the entire amide-I or the amide-A bandwidth. The splitting of the two bands (∼ 1650cm⁻¹) is large and interband coherence cannot be created by a single color pulse. We used a rectangular pulse shape centered around 1650 cm⁻¹ with a spectral width of 250 cm⁻¹ for the amide-I region and centered around 3315 cm⁻¹ with a spectral width of 500 cm⁻¹ for the amide-A region. All transitions inside one band are excited equally and interband transitions are not possible.

2D-IR photon echo spectra of the amide-I region for all four configurations are displayed in Fig. 10, both imaginary part and absolute value of the signal are shown. In the imaginary part, pairs of negative and positive peaks are observed in all cases. GSB and ESE give negative contributions along the diagonal corresponding to transitions between the ground state and the first excited states. The ESA gives rise to positive peaks associated with the negative peaks but redshifted along ₃. This redshift comes directly from the anharmonicities 2A_1n of each vibrational mode. Measurement of the interval between the negative and positive peaks gives typical value of 15 cm⁻¹, in good agreement with measurement of the amide-I anharmonicity [51, 52].

Figure 10.

2D-IR photon echo amide-I spectra for all conformations. Left: Imaginary part of the signal. Right: Absolute value of the signal

For E116, we identify the two main peaks observed previously in the absorption spectrum. Due to the elimination of inhomogeneous broadening by the photon echo [49] these two peaks are well resolved. Neither the imaginary part nor the absolute value signals show cross peaks, however we can identify some off-diagonal tails which come from the weak coupling between the two types of vibrational modes. Both high frequency and low frequency peaks have a similar diagonal and off-diagonal width. For the absorption of G129 two overlapping peaks was observed. In the 2D-IR photon, these two peaks are well resolved with cross peaks. Note that one cross peak overlaps with the ESA contribution of the high frequency peak, reducting its intensity. For A128 we clearly observe three peaks with strong cross peaks. Theses cross peaks indicate coupling between the different modes and imply delocalization of exciton. For G193, only two peaks are well resolved, however a diagonal red tail is observed in the absolute value of the signal near the high frequency peak. By comparing the tails of the different peaks in all configurations we notice that E116 and G193 has the smallest tails whereas G129 and A128 have the strongest. As described previously, A128 and G129 have a greater delocalization length than the other two configurations, indicating a stronger influcence of the couplings between the local modes.

The 2D-IR photon echo spectra in the amide-A region are shown in Fig. 11. As noted for the linear absorption, all spectra are much broader than the amide-I region. By using larger homogeneous dephasing we have increased the bandwidth, however by looking at the diagonal bandwidth we also see that inhomogeneous dephasing is very large in the amide-A region. The negative to positive peak splitting gives a value of 35 cm⁻¹ for the anharmonicity of the NH vibration. This is smaller than previous measurement and calculation of the NH anharmonicity (∼ 60 cm⁻¹) [52], reflecting the accuracy of our map. As for the central frequency of both amide-I and amide-A bands the anharmonicity requires an overall redshift. However we believe that the fluctuations around this value are well represented by our simulation.

Figure 11.

2D-IR photon echo amide-A spectra for all conformations. Left: Imaginary part of the signal. Right: Absolute value of the signal

In the amide-A region, differences between spectra of the various configurations are not as clear as in the amide-I region due to the large homogeneous and inhomogeneous bandwidth. Unlike 1D-IR, in a 2D-IR spectrum, homogeneous and inhomogeneous dephasing are separated. The diagonal width of the spectrum indicates inhomogeneous dephasing and the off-diagonal width characterizes the homogeneous dephasing. All spectra show a similar off-diagonal width. Indeed, we used a single dephasing parameter to describe the fast dynamics in all conformations. However the diagonal width depend on the conformer. As noticed in the linear absorption G129 has the broadest lineshape among all configurations. Due to a lower number of intramolecular hydrogen bond the configuration G129 is less stable. This large inhomogeneous dephasing can be attributed to larger fluctuations of the molecular structure.

5 Two-color 2D-IR photon echo : simulation of the amide-I / amide-A cross peak

In a two-color experiment, the first two pulses span the Ω₁ ∼ 1650 cm⁻¹ amide-I band and the other two are tuned to the Ω₂ ∼ 3315 cm⁻¹ amide-A band. This technique measures the amide-I/amide-A crosspeak. With this pulse configuration, only two Feynamn diagrams, ESA and GSB, contribute to the signal. Indeed, for the third diagram, ESE occurs during the interaction with the third laser (frequency Ω₂), however the system has only been previously excited by the the first two lasers to states with energy around Ω₁. ESA probes the amide-I+amide-A states. Theses states represent one exciton amide-I and one exciton amide-A interacting through the quartic coupling K_12nn.

The two-color spectra (imaginary part and absolute value) of all conformations are displayed in Fig. 12. All spectra show a ensemble of pairs of negative and positive peaks. For two isolated vibrational modes with frequencies Ω₁ and Ω₂ coupled by a nonlinear coupling K₁₂ without fluctuations, the two-color spectrum is given by a negative peak at (Ω₁, Ω₂) and a positive peak at (Ω₁,Ω₂ – K₁₂). The intensity of the peaks is directly proportional to the nonlinear coupling. For E116, the spectrum shows two pairs of peaks along the amide-I band corresponding to the two absorption modes observed previously. Surprisingly, both pairs have a similar intensity, which wasn't the case for the one color experiment and the linear absorption. Since the coupling K_12nn between the amide-I and the amide-A band is local, only 6 amide group are sensitive to this spectroscopic technique, the amide groups number 3, 4, 5, 8, 9 and 10 (see Fig. 3). The high frequency band is mainly given by the amide groups 3,5,9 and 10 (unbonded CO groups) and the low frequency band is mainly given by the amide groups 4 and 8 (hydrogen bonded CO groups). Because both pair of peaks have a similar intensity, the K coupling of the low frequency modes should be approximately twice as strong as the K coupling of the high frequency mode. Indeed, by analysing our hamiltonian parameters we found that K_12nn ∼ 13 cm⁻¹ for the low frequency modes while K_12nn ∼ 6 cm⁻¹ for the high frequency modes. G129 has one pair of equal intensity peaks. Since those two peaks are strongly overlapping, no local information on the system can be extracted from this spectrum. In A128 and G193 situations, in contrast, the spectra contains some features which can be interpretted in terms of local structure. Indeed, since 4 of the 10 amide groups doesn't interact with the amide-A band, a part of the inhomogeneous fluctuations has been removed. Moreover, some amide-I modes are coupled to different frequency interval of the amide-A band. The optical response is given in terms of the eigenstates of the sytem and not in terms of local structure. However, in peptides the eigenvalues are overlapping and it is in general difficult to keep track of the eigenstates. Consequently, we tried to interpret the 2D-IR two-color spectra of A128 and G193 in terms of local frequencies. Of course, the coupling between the modes change the position of the absorption band and modify quantitatively our interpretation, however qualitatively we can detect some local information in those spectra. For A128 and G193 we have computed the 6 averaged local frequencies (both amide-I and amide-A) corresponding to the modes 3,4,5,8,9 and 10. Those frequencies has been reported in the spectra (Fig. 12). The average frequencies cannot account for the strong fluctuations occuring in the system and thus doen't account in general for the spectra. However we can see that for A128 the peak located around (−1630, 3180) cm⁻¹ is most likely due to the amide group 5 (see Fig. 3) with the average local frequency located at (−1637, 3321) cm⁻¹. The tail located near (−1660, 3375) cm⁻¹ due to the local mode 3 whose average frequency is (−1664, 3354) cm⁻¹. For G193, the same tail is observed and can be assigned to the local mode 8 with the average frequency (−1662, 3352) cm⁻¹.

Figure 12.

2D-IR photon echo two-color spectra of all conformations. Left: Imaginary part of the signal. Right: Absolute value of the signal. For A128 and G193 the 6 averaged local frequencies of amide groups 3, 4, 5, 8, 9 and 10 (see Fig. 3) are marked by a cross

6 Conclusions

We have used a combined MD, electronic structure protocol to study the nonlinear infrared response of Antamanide. MD simulations were performed in chloroform starting from four conformations suggested by NMR spectroscopy and the X-ray structure. These five trajectories yield only four different structures. We have compared two DFT maps for the effective vibrational Hamiltonian of the amide-I and amide-A bands. By comparing the absorption spectra using MAP1 and MAP2 with experiment, we found one conformation (E116) with a strong intramolecular hydrogen-bond network to be dominant. MAP2 based on a higher level of theory was then used to simulate the linear absorption of the amide-A and showed a good agreement with experiment. We then used MAP2 to predict the 2D-IR photon echo spectra of the amide-I and amide-A bands. We show that the small couplings between vibrational groups induce crosspeaks in the spectra. We have observed that when the excitons are more delocalized, the crosspeaks were more intense. We also notice that conformation G129 which experiences larger fluctuations yields a broader spectrum. Finally, we propose a two-color experiment in order to probe the amide-I/amide-A crosspeak. This spectrum reflects the nonlinear K coupling between the amide-I and the amide-A band. Since only some of the amide modes are sensitive to this coupling some inhomogeneous fluctuations are removed. For two configurations some spectral feature has been also interpreted in term of local structure.

Extract local information from IR spectroscopy (linear and nonlinear) is always a difficult task. 2D-IR, in general does not probe structure but the vibrational dynamics governed by local anharmonicity and couplings between modes (J and K). Moreover, the interplay of different phenomena increases also the difficulty of the interpretation. However, in some situations, in particular in medium sized molecule like Antamanide, IR spectroscopy, being sensitive to the intramolecular hydrogen bonding network, can be used as a complementary tool to NMR spectroscopy. Moreover, 2D spectroscopic techniques such as photon echo, double quantum coherence, chirality [21], has the power to extract the specificity of each molecule. For a molecule with a strong intramolecular hydrogen bonding network similar to Antamanide, two-color experiment, reavealing the amide-I/amide-A crosspeak can be an e cient technique. However, for larger molecules, this technique will not be effective since by increasing the inhomogeneous fluctuations we increase the overlapping between ESA and GSB contributions resulting in a vanishing spectrum.