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D. Spectral Diffusion of the Amide I Band

D. Spectral Diffusion of the Amide I Band

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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



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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



eiεij 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



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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



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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.



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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



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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



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