D. Spectral Diffusion of the Amide I Band
Tải bản đầy đủ - 0trang
Proteins and Peptides
323
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
Copyright © 2001 by Taylor & Francis Group, LLC
324
Hamm and Hochstrasser
Figure 21 The Feynman diagrams, which have to be taken into account to model
the rephasing part of the stimulated photon echo system of a excitonically coupled
system of vibrations. The fjiig and fjk C lig are states of the one-exciton and
two-exciton manifold, respectively.
obtain for the total response function (51):
3
Rl D
lD1
h
2
0,i
2
i εi
0,j ie
εj t
Ð [1 C ei εi
εj T
] Ð [1
eiεij t ]
i,j
ðe
Ct /T2
e
T/T1
30
which, in the presence of inhomogeneity, has to be convoluted with distribution functions for the excitonic energies εj . The response in Equation (30) is
exactly zero in the harmonic limit εij D 0 , as expected. A model calculation of Equation (30), based on the known structures of these peptides, the
coupling constants determined from the 2D-IR experiment [Equation (29c)]
and parameters for vibrational relaxation (T1 D 1.2 ps) and homogeneous
broadening (T2 D 0.7 ps), but neglecting inhomogeneous broadening, is
shown in Fig. 22b. A sharp coherence spike at T D D 0 is obtained, in
clear contrast to the experimental results. This coherence spike is due to
the interstate coherences jjihij, that form after the second light interaction,
which are expressed by the term 1 C ei εi εj T in Equation (30) and which
are all in phase only at delay zero T D D 0. As time T increases, these
terms dephase rapidly owing to their different beating frequencies. The
square law detector in this particular experiment, which measures the tintegrated intensity of the third-order polarization [Equation (17)], strongly
averages out the beatings in the third polarization, so that only weak structures remains in the signal (see arrow in Fig. 22b).
In order to obtain a model that indeed reproduces the experimental
results, inhomogeneous broadening (modeled by a diagonal disorder of
Copyright © 2001 by Taylor & Francis Group, LLC
Proteins and Peptides
325
Figure 22 (a) Model calculation for the stimulated photon echo signal of the
cyclic model peptide (cyclo-Mamb-Abu-Arg-Gly-Asp) based on its known structures. The same coupling constants were employed as in the model simulations
of the 2D-IR spectrum in Fig. 16. The parameters for homogeneous broadening
(T2 D 0.7 ps), vibrational relaxation (T1 D 1.2 ps), and inhomogeneous broadening
(diagonal disorder 20 cm 1 ) were also the same. The pulse duration of the laser
pulses was set to 120 fs. (b) The same calculation as in (a), but with υ-shaped laser
pulses and neglecting the inhomogeneous broadening. A sharp coherence spike now
occurs at T D D 0, which is not seen experimentally.
20 cm 1 ) and the ﬁnite duration of the laser pulses (120 fs) had to be taken
into account, giving the result shown in Fig. 22a, which features an excellent qualitative ﬁt to the experiments. Inhomogeneity almost completely
suppresses the coherence spike and gives rise to a large peak shift. Somewhat surprisingly, the main content of a stimulated photon echo experiment
on a system of coupled states is essentially the same as in the case of
an isolated transition, despite the apparent complication of the vibrational
exciton coherence terms, namely that the peak shift is a qualitative measure
of the inhomogeneity of the transitions.
In agreement with this conclusion, the Bloch picture applied here to
derive Equation (30) does not predict a decay of the ﬁrst moment, since
the Bloch description omits spectral diffusion processes. Nevertheless, it
is possible to understand the existence of a peak shift within the Bloch
description, and this suggests a qualitative interpretation for its decay.
As we have seen from photon echo experiments on spectroscopic probes
Copyright © 2001 by Taylor & Francis Group, LLC
326
Hamm and Hochstrasser
embedded within proteins, the peptide is ﬂuctuating on a wide range of
time scales starting from the subpicosecond regime. The coupling scheme
in the excitonic system of the amide I band must therefore be continuously
rearranging on these time scales. The decay of the ﬁrst moment found in the
experimentally obtained stimulated photon echo signal is considered to be
a direct experimental manifestation of ﬂuctuations of the peptide backbone.
A more involved theory of spectral diffusion processes in coupled excitonic
systems has been worked out recently by Mukamel et al. (48,49) and will
help to better place these experiments on a more quantitative basis. The
photon echo experiment presented here is not mode selective and measures
spectral diffusion processes that represent an average of all peptide groups.
Nevertheless, the results clearly prove that the peptide backbone is ﬂuctuating on a very fast picosecond time scale.
Additional evidence for structural ﬂexibility of small peptides on very
fast time scales can be obtained from dynamical hole-burning experiments.
An example is shown in Fig. 23, (30) which reﬂects a cut through the 2D
Figure 23 The response of a hole burned into the amide I band of scyllatoxin
by a narrowband pump pulse (width ¾10 cm 1 , the spectrum of the pump pulse
shown as thin solid line) as a function of the delay time between pump and probe
pulse (thicker solid line: 0 ps; thick dashed line: 2.4). Vibrational T1 relaxation was
compensated for in this plot by scaling the 2.4 ps spectrum in order to facilitate a
direct comparison with the time zero signal.
Copyright © 2001 by Taylor & Francis Group, LLC
Proteins and Peptides
327
spectrum of Fig. 19e with the time separation of pump and probe pulse
being varied between 0 and 2.4 ps. After selectively exciting some of the
excitonic states of the amide I band of scyllatoxin with the narrowband
pump pulse (width ca. 10 cm 1 ), a spectral hole and an anharmonically
shifted excited state absorption are observed. As time goes by, the hole
broadens and the signal starts to wash out. Even though this process is
slower than T1 relaxation of 1.2 ps, it is clearly observable within the
available time window. In the more conventional case of an inhomogeneously broadened band, this process is usually referred to as spectral
diffusion. However, the situation can be given a more quantitative basis in
this example since the energies of the different transitions underneath the
amide I band are strongly correlated. Structural ﬂuctuations of the peptide
backbone continuously rearrange the excitonic coupling scheme so that
population can ﬂow between the different excitonic states. Two contributions to the spectral diffusion process might be distinguished: that of the
diagonal and that of the off-diagonal elements of the coupling Hamiltonian.
The former might reﬂect, for example, ﬂuctuations of the structures of the
hydrogen bonds formed by the peptides that have a major effect on the diagonal energies. On the other hand, spectral diffusion from ﬂuctuation of the
off-diagonal elements of the exciton matrix would directly relate to structural ﬂuctuations of the peptide backbone since the coupling constants are a
sensitive measure of the orientations and separations between neighboring
peptide groups. The results discussed in the previous paragraph suggest
that electrostatic Coulomb interaction can account for this interaction so
that an extremely simple geometric expression, rather than sophisticated
quantum chemistry calculations, would be required to calculate the correlation functions of the coupling (48,49). In addition, the technique introduced
in Fig. 23 is potentially mode selective both in the diagonal and the offdiagonal region. Consequently, one might be able to deduce from 2D-IR
spectroscopy a complete set of diagonal and off-diagonal ﬂuctuation correlation functions, where the time separation of pump and probe is introduced
as a third dimension (3D spectroscopy). The off-diagonal ﬂuctuation correlation functions also might be obtainable from MD simulations so that a
direct link between theory and experiment could be established.
E. 2D-IR Spectroscopy Using Semi-Impulsive Methods
The IR-2D spectroscopic technique applied in Section IV.C (30,42) utilized
the frequency domain: after selectively bleaching individual one-excitonic
states using a narrowband intense pump pulse, a broadband probe pulse
Copyright © 2001 by Taylor & Francis Group, LLC
328
Hamm and Hochstrasser
recorded the nonlinear response of the bleached transition, as well as that
of other transitions coupled to the bleached transition. By continuously
tuning the frequency of the pump pulse, 2D spectra were constructed. One
frequency dimension is the center frequency of the pump pulses, and the
other comes from dispersing the probe pulse. It has been shown that cross
peaks in those 2D spectra are related to the strength of coupling between
pairs of peptide units (42).
Mukamel and others have proposed time domain 2D spectroscopic
techniques on vibrational transitions that utilize impulsive excitation
through a ﬁfth-order Raman effect of low-frequency modes (33,34,48).
Lately this method has been demonstrated experimentally on neat liquids
(36,37,40). More recently, Mukamel and coworkers (47) have described
third-order nonlinear coherent experiments on excitonically coupled twoand three-level systems, in which electronic transitions are excited with
three laser pulses. The pulses are chosen short compared with relaxation
and coupling mechanisms, but long compared with the transition frequency,
corresponding to the so-called semi-impulsive limit. Model spectra on
coupled two-level systems comparing various time orderings (photon
echo, reverse photon echo, transient grating, reverse transient grating)
illustrate that the excitonic coupling gives rise to cross peaks, from which
the strength of coupling between individual pairs of transitions can be
determined.
From the experimental viewpoint, these concepts require a measurement of the complete third-order ﬁeld generated by the interaction of the
sample with three incident ﬁelds. Such a measurement requires a heterodyne detection scheme using phase locked laser ﬁelds for the pump pulses
and a local oscillator pulse with which to perform spectral interferometry.
We have recently presented a much simpler semi-impulsive scheme (53),
which, in terms of the underlying nonlinear response functions, resembles
the transient grating experiment discussed by Mukamel et al. (47). Such
a transient grating experiment can be thought of as a ﬁeld from each of
the ﬁrst and second pulses (wavevectors k1 and k2 ), which arrive at the
sample simultaneously, forming a grating that scatters a ﬁeld from the third
pulse (wavevectors k3 ) into the direction ks D k1 k2 C k3 . Our experiments also work in the time domain in the semi-impulsive limit. In the
proposed simpliﬁed scheme, the ﬁrst and second light ﬁeld interactions
corresponding to wavevectors k1 and k1 originate from one laser pulse,
while both the third ﬁeld and the local oscillator ﬁeld originate from the
a second laser pulse with wavevector k2 , so that the scattered ﬁeld has
a wavevector ks D k1 k1 C k2 D Ck2 . In other words, the scheme is a
Copyright © 2001 by Taylor & Francis Group, LLC