C. Two-Dimensional IR Spectroscopy on the Amide I Band
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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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315
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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317
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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319
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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