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C. Two-Dimensional IR Spectroscopy on the Amide I Band

C. Two-Dimensional IR Spectroscopy on the Amide I Band

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Proteins and Peptides



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Figure 14 A model calculation of the 2D-IR spectra of a idealized system of two

coupled vibrators. The frequencies of these transitions were chosen as 1615 cm 1

and 1650 cm 1 , the anharmonicity as  D 16 cm 1 , the coupling as ˇ12 D 7 cm 1 ,

and the homogeneous dephasing rate as T2 D 2 ps. The direction of both transitions

as well as the polarization of the pump and the probe pulse were set perpendicular.

The spectral width of the pump pulses was assumed 5 cm 1 . The figure shows

(a) the linear absorption spectrum and (b) the nonlinear 2D spectrum. In the 2D

spectra, light gray colors and solid contour lines symbolize regions with a positive

response, while negative signals are depicted in dark gray colors and with dashed

contour lines.



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Hamm and Hochstrasser



2 ps in order to make the features as distinct as possible. In Fig. 14b, where

the coupling is switched on, pairs of a positive and a negative peak appear

along the diagonal of the 2D spectrum at positions corresponding to those of

the related fundamental lines. In addition, off-diagonal peaks show up, each

consisting of a positive and a negative contribution. When the coupling is

switched off, the off-diagonal peaks would disappear, emphasizing that this

technique is analogous to 2D NMR (COSY) spectroscopy. Of course in this

IR experiment the pump pulse is not transferring coherence. A fixed delay

time of approximately 0.6 ps was introduced and the narrow frequency

band of the pump field restricted the range of coherences that are initially

excited.

Experimental 2D spectra were obtained from three different peptide

samples, whose structures are shown in Fig. 15. The first sample, a de novo



Figure 15 Structures of the three peptide samples investigated by two-dimensional IR spectroscopy: a de novo cyclic penta peptide (cyclo-Mamb-Abu-Arg-GlyAsp), apamin, and scyllatoxin.



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cyclic penta peptide (cyclo-Mamb-Abu-Arg-Gly-Asp), is small enough so

that all amide I transitions are spectrally resolved in the linear absorption

spectrum (see Fig. 16a), allowing the properties of the nonlinear 2D-IR

spectra and the validity of the excitonic coupling model to be examined in

detail. Both its NMR and x-ray structures are known (133). The peptide is

stabilized by a hydrogen bond between the Mamb1-Abu2 peptide bond and

the Arg3-Gly4 peptide bond and was specifically designed to form a single



Figure 16 Absorption spectrum of the cyclo-Mamb-Abu-Arg-Gly-Asp in D2 O.

The dashed line shows a representative spectrum of the pump pulses (width

¾12 cm 1 ) utilized to generate the 2D-IR spectra. (b,c) 2D pump-probe spectra of

the same sample measured with the polarization of the probe pulse perpendicular

and parallel to the polarization of the pump pulse, respectively. The dashed contour

lines mark regions where the difference signal is negative (bleach and stimulated

emission), while the solid contours lines mark regions where the response is positive

(excited state absorption). The most prominent off-diagonal bands are marked by

arrows. (d,e,f) A global least-squares fit of the experimental data, used to refine the

coupling Hamiltonian in Equation (29c). (From Ref. 42.)



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Hamm and Hochstrasser



well-defined conformation in solution with an almost ideal type II’ ˇ-turn

centered at the Abu-Arg residues (133). The other samples investigated are

apamin (134), a small neurotoxic peptide component found in the honeybee

venom and scyllatoxin (135,136), a scorpion toxin with a high affinity for

apamin-sensitive potassium channels. Both peptides were chosen for their

variety of structural motifs, which are stabilized in the solution phase by

disulfide bridges. Apamin (18 amino acids) is one of the smallest globular

peptides known and has a short ˛ helix and a ˇ turn (134). Scyllatoxin,

with 31 amino acids, is the smallest known natural peptide containing both

an ˛ helix and a ˇ sheet (135,136).

Figure 16 shows the nonlinear 2D-IR spectra of the cyclic pentapeptide for perpendicular (Fig. 16b) and parallel (Fig. 16c) pump and probe

polarizations, respectively. In both cases, the dominating signals are found

along the diagonal of the 2D-IR spectrum, where the spectra are significantly better resolved than the linear absorption spectrum (Fig. 16a). More

germane for this discussion, however, is the off-diagonal region, where

cross-peaks appear (see arrows in Fig. 16b), the strongest of which was for

the 1610–1584 cm 1 level pair. The other cross-peaks are weaker but are

easily verified in cuts through the 2D-IR spectra along the probe axis for

certain pump frequencies, where they can be identified by their dispersive

shapes (see, for example, arrow in Fig. 17).

The intensity of the cross-peak, relative to the intensity of the diagonal

contribution, is larger when the spectrum is measured with the polarization

of the pumped and probed beams perpendicular (Fig. 16b). The anisotropy

rkl of each peak measures the angle between the pumped and the probed

transitions through rkl D 15 3 cos2 ϕkl 1 so that along the diagonal of the

2D spectra values close to 0.4 are observed as expected, since here the

same transition is pumped as is probed. Anisotropies smaller than 0.4 are

observed in the off-diagonal region since pumped and probed transitions

are in general not parallel, explaining the higher contrast of the cross-peaks

in the perpendicular spectrum.

The ultimate goal is to deduce from these experiments the resonance

couplings ˇij since these are the numbers that may be directly related to the

structure of the peptide, given that one has a reliable model for computing

the coupling Hamiltonian from the structure. The off-diagonal anharmonicities εkl can be deduced from the intensities of the off-diagonal peaks

(42), so that the resonance couplings ˇij could be computed according

to Equation (24) in the weak coupling limit (valid for the cyclic penta

peptide) or with the help of a diagonalization of the two-excitonic matrix.

However, additional ambiguity arises since the zero-order energies εi , on



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Figure 17 Cuts through the 2D-IR pump probe spectra shown in Fig. 16b and

c along the direction of the probe axes for selected pump frequencies which were

chosen to match the peaks in the linear absorption spectrum (squares: 1648 cm 1 ;

circles: 1620 cm 1 ; triangles: 1610 cm 1 ; diamonds: 1584 cm 1 ). The frequency

positions of the pump pulses are marked by the vertical dotted lines. (From Ref. 42.)



which the nonlinear response of the system depends very critically, are a

priori not known. In the localization limit, where each one-excitonic state

is predominantly localized on one individual molecular site, their eigenenergies (observed in the absorption spectrum, Fig. 16a) are essentially the

same as the zero-order energies εi . Nevertheless, there are still 5! D 120

permutations of how to distribute these frequencies to the five peptide

groups. In Ref. 42 it was shown that this ambiguity can be resolved with

the additional information obtained from the measured anisotropies and

with empirical rules for the amide I frequencies.

In the simplest model, as proposed by Krimm et al. (118) and Tasumi

et al. (123), the coupling Hamiltonian is given by a simple dipole-dipole

coupling term:

ˇij D



Ei Ð Ej

r3ij



3



Erij Ð E i Erij Ð E j

r5ij



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(28)



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Hamm and Hochstrasser



where the directions of the transition dipoles E i and the vectors connecting

two sites Erij relate the coupling Hamiltonian to the structure of the peptide.

The position and the direction of each dipole vector with respect to the

peptide bond have been assigned as depicted in Fig. 18a (118,123). Using

this formalism and the known x-ray structure of the peptide, one would

obtain for the coupling Hamiltonian (in cm 1 )





ÐÐÐ

6

8

8

1

8

4

1

 6 ÐÐÐ





8 ÐÐÐ

0

1

ˇkl D  8

(29a)





8

4

0 ÐÐÐ

4

1

1

1

4 ÐÐÐ

(where numbering starts from the Mamb-Abu-peptide bond). However, the

fact that the position of the dipole with respect to the peptide group has to

be specified clearly stresses that the dipole approximation is not appropriate

to describe the coupling (137). In other words the size of the peptide group

whose distributed charges give rise to the transition dipole is of the same

order as the separation between pairs of peptide groups. Nevertheless,

it would be very valuable to find an effective Hamiltonian based on

Coulomb interactions in order that the coupling matrix elements can be

related directly to structure. In the present case we have performed density



Figure 18 Models from which the excitonic coupling between pairs of peptide

groups were calculated: (a) The direction and location of the transition dipole of

the amide I mode (118,123) from which the coupling between two peptide groups

is calculated according to a dipole-dipole interaction term [Eqaution (28)] (b) The

nuclear displacements, partial charges, and charge flow of the amide I normal mode

obtained from a DFT calculation on deuterated N -methylacetamide (all experiments

were performed in D2 O) (42). With this set of transition charges, the multipole

interaction is computed, avoiding the limitations of the dipole approximation.



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functional theory (DFT) calculation on model compounds chosen to mimic

the local electronic structure of individual peptide bonds (such as deuterated

N-methylbenzamide, C6 H5 –COND–CH3 , and N ,N -dimethylacetamide,

and NMA) in order to calculate the nuclear displacements, partial charges,

and charge flows during amide I vibration (see Fig. 18b). With this set

of transition charges, the multipole electrostatic interaction has been

calculated, yielding as coupling Hamiltonian:





ÐÐÐ

10

7

1

0

ÐÐÐ

4

6

2

 10





ˇkl D  7

(29b)

4 ÐÐÐ

4

1





1

3

4

ÐÐÐ

11

0

2

1

11

ÐÐÐ

whose patterns are somewhat similar to the dipole-dipole Hamiltonian in

Equation (29a), but which in detail differs considerably. The Hamiltonian

in Equation (29b), together with an assignment of the observed absorption frequencies to the five peptide groups, was used as a starting point

of a least-square Levenberg Marquardt algorithm to globally fit all the

experimental information available (i.e., parallel and perpendicular 2D-IR

spectrum and linear absorption spectrum; see Fig. 16a,b,c) simultaneously.

The perpendicular 2D-IR spectrum was weighted most in this global fit

since it is believed that it carries the most significant structural content. The

modeling included homogeneous and inhomogeneous broadening mechanisms with widths of 12 and 10 cm 1 , respectively (30,42). The resulting

fits are shown in Fig. 16d,e,f, yielding as a refined Hamiltonian:





1618

11

7

1

0

4

6

2

 11 1588





D

(29c)

Hrefine

7

4

1671

6

1



kl





1

6

6 1648

1

0

2

1

1 1610

Except for the coupling constant between the Gly4-Asp5 peptide group

and the Asp5-Mambl peptide group [the 4-5 element in Equation (29c)],

the refined Hamiltonian in Equation (29c) is almost identical to the Hamiltonian in Equation (29b). Apparently electrostatic interaction describes the

coupling between two peptide groups reasonably well when the groups

are not neighboring in the peptide chain. However, it appears as if the

through-bond effect cannot be neglected for chemically bonded pairs of

peptide groups. Our ab initio calculations on model compounds (NMA)

clearly showed that the amide I normal modes are accompanied by a flow of

charge to the methyl groups (corresponding to the C˛ atoms in the peptide



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Hamm and Hochstrasser



chain) and also a bending of the methyl C–H bonds. Both effects will

couple neighboring groups by through bond interactions involving the C˛

atom, and the amide I mode cannot be viewed as entirely localized on the

peptide group (as assumed in Fig. 18b). Quantum chemistry calculations

on dipeptides and tripeptides, which are definitely feasible with presentday computer technology, will enable a better description of the coupling

between adjacent peptide bonds.

Figure 19 shows the results on apamin and scyllatoxin (30). Owing

to the larger size of these peptides, the different amide I states underneath

the amide I band are no longer spectrally resolved, so that the absorption

spectrum appears as a broad band (width 30–40 cm 1 ) with only very



Figure 19 Absorption spectrum, measured 2D-IR spectrum, and modeled 2D-IR

spectrum (from top to bottom) of apamin (a, b, c) and scyllatoxin (d, e, f) in D2 O.

The dashed contour lines mark regions where the difference signal is negative

(bleach and stimulated emission), while the solid contour lines mark regions where

the response is positive (excited state absorption). The isosbestic line is marked by

a dashed-dotted line. The thick lines mark the position of the local maxima (thick

solid lines) and minima (thick broken line) of the probe spectra as a function of

the pump frequency.



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weak substructure (Fig. 19a,d). Nevertheless, in this case it can be unambiguously concluded from the 2D-IR spectra (Fig. 19b,e) that the amide I

states are delocalized vibrational excitons. The variation of the response

with pump frequency is enhanced by the extra lines in the contour plots

in Fig. 19, which mark the positions of the local minima (thick dashed

lines) and maxima (thick solid lines) of the transient probe spectra as

a function of pump frequency. If the states excited by the narrowband

pump pulse were localized on individual sites, one would observe only

their anharmonic response so that these lines would directly follow the

diagonal of the 2D graph. This is clearly is not what is observed. Model

calculations of these spectra, based on the known structure of the peptides

(see Fig. 17c,f), feature an excellent qualitative agreement with the experimental results. In contrast to the experiments described before, no detailed

information on the zero-order frequencies εi is available so that they were

divided into two groups: those peptide groups that are hydrogen bonded

within the macro molecule and those that are not (30). The dependence of

the transient response on parameters for homogeneous and inhomogeneous

broadening permitted an estimate of their values (12 cm 1 and 24 cm 1 ,

respectively). Homogeneous broadening is essentially determined by the

width of the transient holes, while inhomogeneity controls how much the

transient response deviates from the diagonal (30). In addition, since the

excitonic wave functions are known from the fit, the degree of delocal˚ This corresponds to

ization could be estimated to be approximately 8 A.

approximately 11/2 helix turns in an ˛ helix. An upper limit is thereby

set on how far coherent transport of vibrational energy can take place,

such as would be required for the propagation of the so-called Davidov

soliton (138,139). Furthermore, if experiments can establish the coherence

length at 300 K for excitations in the various secondary structure motifs,

the computational effort needed to search for structures that match spectra

might be considerably deduced.

Coherent transport of vibrational energy is further limited by vibrational energy relaxation. Experiments on the amide I band of different

peptides (NMA, apamin, scyllatoxin BPTI, and the cyclic pentapeptide)

revealed a vibrational relaxation rate of approximately T1 D 1.2 ps, which

is essentially independent of the particular peptide (30,53). A similar value

has recently been reported for myoglobin at room temperature, with only a

weak dependence of the relaxation rate on temperature down to cryogenic

temperatures (140). In other words, vibrational relaxation of the amide I

mode reflects an intrinsic property of the peptide group itself rather than a

specific characteristic of the primary or secondary structural motifs of the



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Hamm and Hochstrasser



peptide. It is significantly faster than that of other C O modes, such as in

acetylbromide CH3 BrC O (65), heme groups (67,98,141,142), or metalcarbonyls (143). The relaxation rate of 15 N–NMA is essentially the same

as that of 14 N–NMA (95), suggesting that the Fermi resonances responsible for the fast relaxation rate of the amide I mode do not involve much

motion of the N atom. On the other hand, vibrational relaxation limits

the maximum time window in which spectral diffusion processes can be

observed by nonlinear IR techniques, so that knowledge and hence control

of the mechanism of vibrational relaxation of the amide I band might help

to extend this observation limit.

D. Spectral Diffusion of the Amide I Band



We have presented two types of nonlinear IR spectroscopic techniques

sensitive to the structure and dynamics of peptides and proteins. While the

2D-IR spectra described in this section have been interpreted in terms of the

static structure of the peptide, the first approach (i.e., the stimulated photon

echo experiments of test molecules bound to enzymes) is less direct in that

it measures the influence of the fluctuating surroundings (i.e., the peptide)

on the vibrational frequency of a test molecule, rather than the fluctuations of the peptide backbone itself. Ultimately, one would like to combine

both concepts and measure spectral diffusion processes of the amide I band

directly. Since it is the geometry of the peptide groups with respect to each

other that is responsible for the formation of the amide I excitation band, its

spectral diffusion is directly related to structural fluctuations of the peptide

backbone itself. A first step to measuring the structural dynamics of the

peptide backbone is to measure stimulated photon echoes experiments on

the amide I band (51).

The result of such an experiment on the de novo cyclic penta peptide,

which has been introduced previously in this paragraph, is shown in Fig. 20.

Qualitatively, the results are very similar to the results of the stimulated

photon echo system on isolated test molecules embedded to proteins. As a

function of T, the signal decays on a time scale corresponding to vibrational

relaxation of the amide I states T1 /2 D 600 fs. As a function of , on the

other hand, a significant peak shift is again obtained. As in the previous

case, the peak shift, represented in Fig. 20 by the normalized first moment

M1 T , slightly decays within the first ps, which is the time window accessible to these experiments in the moment. Similar results are obtained for

apamin (51).

However, the interpretation of these results is considerably more

complex since one has to deal with spectral diffusion of coupled states,



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Figure 20 The stimulated (three-pulse) photon echo signal of the amide I band

of cyclo-Mamb-Abu-Arg-Gly-Asp as function of delay time and T (see Fig. 3)

and its normalized first moment. The first moment decays with time (T) due to

conformational fluctuations of the peptide backbone.



rather than that of isolated vibrations. At this stage, we model the experiment within a simple Bloch picture, which assumes a strict separation of

time scales of dephasing that implies a homogeneous bandwidth in a fixed

inhomogeneous distribution of transitions. The relevant Feynman diagrams

are depicted in Fig. 21. In the weak coupling limit (see Section IV.A), we



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