Tải bản đầy đủ - 0 (trang)
E. 2D-IR Spectroscopy Using Semi-Impulsive Methods

E. 2D-IR Spectroscopy Using Semi-Impulsive Methods

Tải bản đầy đủ - 0trang

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



Proteins and Peptides



329



conventional pump-probe configuration, where the third-order polarization

generated by the combined pump and probe pulses generates an electric

field propagating in the direction k2 of the probe pulse. The probe field

acts as the local oscillator in heterodyning the generated field. Therefore

we define these as pump/probe self-heterodyne experiments, rather than

pump-probe experiments, in order to emphasize its relationship to semiimpulsive methods and that interstate coherences of multilevel systems,

rather than incoherent population states, are being probed.

The relevant Feynman diagrams to describe such experiments are

essentially the same as those depicted in Fig. 21, with appropriately altered

labeling of the wave vectors and time coordinates. In the weak coupling

limit, we obtain for the total response function (53):

Rl D 2

l



2

0,i



2

0,j



Ð e



iεj t2



i εj εij t2



e



e



t2 /T2



i,j



ð 1 C ei εi



εj t1



e



t1 /T1



31



The complex third-order generated field, which could be obtained experimentally from a phase locked heterodyne configuration, is proportional to

the two-dimensional Fourier transform of Equation (31):

1



E 3 ω1 , ω2 D

0



D2



1



dt1 eiω1 t1



2

0,i

i,j



2

0,j



0







Ð





1

1

iω1 C

T1



Rl t1 , t2

lD1







1



1

i εj



ω2 C







ð





8



dt2 eiω2 t2



1

T2



1



C

i εj



εi



ω1



i εj



εij









1 

C

T1



ω2







1 

C

T2



32



Along the ω2 D 0 axis, the 2D spectrum exhibits resonance peaks

at the fundamental frequencies of the exciton states εj together with resonance peaks at frequencies εj εjj shifted by diagonal anharmonicity and

having opposite phase. In the cross peak region ω1 6D 0 , pairs of resonance

peaks show up (again having opposite phase), which are now separated by

off-diagonal anharmonicity εij . These cross peaks can be unambiguously

identified by their ω1 -frequency, which is equal to the separations between

the corresponding fundamental frequencies εj εi (which can be either



Copyright © 2001 by Taylor & Francis Group, LLC



330



Hamm and Hochstrasser



positive or negative). The off-diagonal anharmonicity is related to the resonance couplings ˇij [Equation (24)], so that this method is another means

of two-dimensional spectroscopy from which structural information can be

deduced.

In a self-heterodyne experiment, however, there is no independent

control over the phase of the local oscillator field, so that the complete information on the complex third-order polarization of Equation (32) cannot be

obtained. It is necessary to analyze in more detail the measurement process

in order to determine the accessible information. In the actual experiment

the spectrometer performs the Fourier transform of the generated thirdorder field of Equation (31) with respect to time coordinate t2 , generating

the field components of E 3 t1 ; ω2 given by:



E 3 t1 ; ω2 D 2



2

0,i



2

0,j



i,j





Ð





1

i εj



εij



1

i εj



1

ω2 C

T2







 Ð [1 C ei εi

1 

ω2 C

T2



εj t1



]e



t1 /T1



33



The square-law detector then measures the total intensity at ω2 , which

includes the probe electric field E2 ω2 :

jE2 ω2



E 3 t1 ; ω2 j2 ¾

D jE2 ω2 j2



2 Re EŁ2 ω2 E 3 t1 ; ω2



(34)



so for the υ-function pulses, the difference signal has the form:



A t1 ; ω2 D



2

0,i



4 Re



2

0,j



i,j





Ð





1

i εj



ω2 C





1

i εj



εij



ω2



1

T2





 Ð [1 C ei εi

1 

C

T2



εj t1



]e



t1 /T1



35



This expression describes the difference signal in the time domain taken in

the manner of the experimental data shown in Fig. 25a. A spectral analysis

of this signal is obtained utilizing a cosine-Fourier transform of the signal



Copyright © 2001 by Taylor & Francis Group, LLC



Proteins and Peptides



331



along the t1 axis:

S 3 ω1 , ω2 D



1



2



dt1 cos ω1 t1 Re E 3 t1 ; ω2



(36)



0



An analytical form for Equation (36) can be obtained in a straightforward

manner, but it does not make the important features of the signal

transparent. Instead, the signal is illustrated in Fig. 24 from a model



Figure 24 Self-heterodyned 2D-IR signal simulation: A model calculation for

an idealized system of two coupled vibrators with frequencies 1615 cm 1

and 1650 cm 1 , diagonal anharmonicity ε11 D ε22 D 16 cm 1 , off-diagonal

anharmonicity ε12 D 2.5 cm 1 , homogeneous dephasing rate T2 D 2 ps, and a

population relaxation time T1 D 4 ps. The figure shows (a) the linear absorption

spectrum (b) the semi-impulsive self-heterodyne 2D spectrum. For the sake of

clarity, υ-shaped laser pulses are assumed, in which case the response function Rl

directly corresponds to the third-order polarization P 3 t1 , t2 . In the 2D spectra,

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

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

contour lines. The cross peaks in (b) have different shapes, allowing them to be

assigned unambiguously.



Copyright © 2001 by Taylor & Francis Group, LLC



332



Hamm and Hochstrasser



calculation for an idealized system of two coupled vibrators. The

same parameters as in Fig. 14 were chosen: frequencies 1615 cm 1

and 1650 cm 1 , diagonal anharmonicity  D 16 cm 1 , coupling ˇ12 D

7 cm 1 resulting in a off-diagonal anharmonicity of ε12 D 2.5 cm 1

[Equation (24)]. We used a homogeneous dephasing rate of T2 D 2 ps and

a population relaxation time T1 D 4 ps in order to make the features as

distinct as possible. The 2D spectrum is shown in Fig. 14b together with the

linear absorption spectrum in Fig. 14a. Two cross peaks emerge at positions

where the vertical lines, originating from the peaks of the linear absorption

spectrum, cross the unit slope diagonals originating from the corresponding

other absorption peak. Since only the real part of E 3 t1 ; ω2 participates

in the self-heterodyne experiment, positive and negative ω1 frequencies

cannot be distinguished. As a consequence, the cosine Fourier transform of

Equation (36) exhibits cross peaks in the ω1 > 0 region, which originate

both from jεi εj j and from jεj εi j. This is the reason why diagonals in

both directions are drawn in the 2D spectra of Fig. 24b. In effect the loss of

information about the imaginary part of the third-order polarization causes

the negative half side of the ω1 -axis to be flipped onto the positive side, and

each cross peak term then appears twice in a 2D spectrum, which is defined

only for the positive half of the ω1 -axis. This superposition of cross peaks

will introduce a loss of resolution and make it more difficult to analyze

congested spectra. However, as also seen from Fig. 24, the duplicated cross

peaks have different shapes and therefore may be distinguished by analyzing

both their positions and shapes.

Figure 25a shows the response of the cyclic penta peptide introduced

in Section IV.C after semi-impulsive excitation with an intense, ultrashort

pump pulse, in the time domain. Pronounced beatings, originating from



Figure 25 Self-heterodyned 2D-IR semi-impulsive 2D-IR spectra: (a) The

frequency resolved signal of the cyclic pentapeptide as a function of probe frequency

ω2 and delay time t1 between pump and probe pulse. Pronounced beatings are

observed in the signal, marked, for example, by the arrow. (b) The cosine-Fourier

transform with respect to delay time t1 of the data in (a). The data along ω1 D 0 are

suppressed in the 2D contour plot for a better distinctness of the cross peak region.

Light gray colors and solid contour lines symbolize regions with positive response,

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

lines. Cross peaks appear in the 2D spectrum where vertical lines, which mark the

peak position in the linear absorption spectrum, cross diagonal lines originating

from a coupled band. The most prominent cross peak is related to the absorption

lines at 1584 cm 1 and 1620 cm 1 (peak I and I’).



Copyright © 2001 by Taylor & Francis Group, LLC



Proteins and Peptides



Copyright © 2001 by Taylor & Francis Group, LLC



333



334



Hamm and Hochstrasser



interstate coherences between excitonic states, are observed in the signal

as a function of delay time t1 of the separation between pump and probe

pulse (marked, for example, by the arrow). The spectral analysis of these

data according to Equation (36) is shown in Fig. 25b together with the

linear absorption spectrum of the peptide. Prominent positive and negative peaks are observed along the ω2 -axis for ω1 D 0, which essentially

reflect the bleach, stimulated emission, and excited state absorption of each

individual linear absorption bands. Furthermore, cross peaks appear in the

ω1 > 0 region (some of which marked by arrows) at positions where the

vertical lines marking the peak positions in the linear absorption spectrum

cross-diagonal lines originating from related absorption lines. The most

prominent cross peak is related to the 1584 cm 1 and 1620 cm 1 transition (peak I and I’), in perfect agreement with the frequency domain 2D-IR

spectra shown in Fig. 16b and Fig. 16c, where the strongest cross peak was

observed between the same set of transitions. A few other distinct cross

peaks are marked as well. The elongated shape of the cross peak connecting

the absorption line at 1673 cm 1 with that at 1648 cm 1 (peaks II) emphasizes the higher spectral resolution of this method in the ω1 direction, which

is limited by T1 vibrational relaxation. In addition, there is inhomogeneous

broadening because the high flexibility of the corresponding Arg3-Gly4Asp5 peptide groups permits a range of structures to exist in equilibrium

(42,133). However, the cross peaks are inhomogeneously broadened only

along the ω2 -axis and remain sharp in the ω1 direction. This is a consequence of the inhomogeneous distributions being correlated so that they

affect neither the level spacings nor the interstate coherences. Clearly this

may not always be the case, and methods to evaluate the degree of correlation will be needed.

For uncoupled systems, where all groups are independent of each

other, off-diagonal anharmonicity would vanish, εij D 0, i 6D j, and only

diagonal anharmonicity would remain. In that case, Equation (35) would

not exhibit oscillations as a function of t1 so that the oscillations observed

in the experimental signal are an independent proof of the existence of offdiagonal anharmonicity and hence of exciton coupling between the amide I

states.

In conclusion, the rather straightforward one-color pump-probe

scheme of the self-heterodyne method seems particularly appropriate for

smaller coupled systems where the vibrational spectra are less congested.

The main content of the present approach and the frequency domain

method described in Section IV.C are the same, namely the existence and

magnitudes of cross peaks and their relationship to couplings between



Copyright © 2001 by Taylor & Francis Group, LLC



Proteins and Peptides



335



sites. However, they are complementary in terms of spectral resolution and

broadening mechanisms. Fully heterodyned echo experiments in which all

the Bohr frequencies are examined separately were recently reported (146).

Can Peptide Structures Be Determined by Nonlinear 2D-IR

Spectroscopy?



We shall conclude this chapter with a few speculative remarks on possible

future developments of nonlinear IR spectroscopy on peptides and proteins.

Up to now, we have demonstrated a detailed relationship between the

known structure of a few model peptides and the excitonic system of

coupled amide I vibrations and have proven the correctness of the excitonic

coupling model (at least in principle). We have demonstrated two realizations of 2D-IR spectroscopy: a frequency domain (incoherent) technique

(Section IV.C) and a form of semi-impulsive method (Section IV.E), which

from the experimental viewpoint is extremely simple. Other 2D methods,

proposed recently by Mukamel and coworkers (47), would not pose any

additional experimental difficulty. In the case of NMR, time domain Fourier

transform (FT) methods have proven to be more sensitive by far as a result

of the multiplex advantage, which compensates for the small population

differences of spin transitions at room temperature. It was recently demonstrated that FT methods are just as advantageous in the infrared regime,

although one has to measure electric fields rather than intensities, which

cannot be done directly by an electric field detector but requires heterodyned echoes or spectral interferometry (146). Future work will have to

explore which experimental technique is most powerful and reliable.

We have investigated peptides whose structures were known beforehand from NMR or x-ray spectroscopy and related these structures to

2D-IR spectroscopy. Ultimately, one would like to deduce the structure

of an unknown sample from a 2D-IR spectrum. In the case of 2D NMR

spectroscopy, two different phenomena are actually needed to determine

peptide structures. Essentially, correlation spectroscopy (COSY) is utilized

in a first step to assign protons that are adjacent in the chemical structure

of the peptide so that J coupling gives rise to cross peaks in these 2D

spectra. However, this through-bond effect cannot be directly related to the

three-dimensional structure of the sample, since that would require quantum

chemistry calculations, which presently cannot be performed with sufficient

accuracy. The nuclear Overhauser effect (NOE), which is an incoherent

population transfer process and has a simple distance dependence, is used

as an additional piece of information in order to measure the distance in



Copyright © 2001 by Taylor & Francis Group, LLC



336



Hamm and Hochstrasser



space between assigned protons. These constraints, together with molecular

mechanics (MM) calculations, are generally sufficient to unambiguously

determine the 3D structure of a peptide.

Similar procedures might turn out to be necessary in the case of 2D

vibrational spectroscopy. The spectroscopy demonstrated so far essentially

reflects the first step of the procedure in NMR spectroscopy (COSY). So

far, we have investigated only the amide I subspace. However, all amide

vibrations (N–H, amide II, etc.) might turn out to be equally important

in revealing the required information, each of them giving different

and hopefully complementary pieces of information. Couplings between

different amide subspaces need to be explored. Incoherent population

transfer out of the amide I transition appears to be very efficient (T1 D

1.2 ps), and the mechanism and the course of this transfer is completely

unknown.

One reason for our hesitancy in predicting the structure of the cyclic

pentapeptide from the 2D spectra was the inaccuracy of the coupling Hamiltonian. If one wants to calculate couplings from a structure, one will have to

separate the problem into as small as possible units, since a complete and

sufficiently accurate quantum chemistry calculation of the entire peptide

will not be feasible. We used, as a first step, the –CO–NH– group as such

a unit, which turned out to be too small, at least for peptide groups adjacent in the amino acid sequence. However, even larger units such as di- and

tripeptides are still within the scope of state-of-the-art computer technology.

Quantum chemistry calculations may be sufficiently accurate to calculate

couplings (i.e. the off-diagonal elements of the coupling Hamiltonian), even

though they might fail to predict the transition frequencies (i.e., the diagonal elements). However, since both quantities are measured independently

in 2D-IR spectroscopy, in contrast to conventional 1D spectroscopy, the

former information might be sufficient.

In conclusion, what we have summarized in this article can be viewed

as a very first step in the direction of a novel structural analysis method,

which opens a vast new field of experimental research where much work

remains to be done. Any 2D-IR spectroscopy certainly will be limited to

molecular sizes smaller than are tractable by 2D NMR and x-ray spectroscopy because of the intrinsically poor spectral resolution of vibrational

transitions. However, the one tremendous prospect that makes this research

worthwhile is the inherent high time resolution of IR spectroscopy, which

covers all biologically relevant time scales, starting from the subpicosecond

time regime. The IR photon echo experiments have proven experimentally

for the first time that the peptide backbone itself indeed fluctuates on these



Copyright © 2001 by Taylor & Francis Group, LLC



Proteins and Peptides



337



fast time scales. If one succeeds to predict structures from 2D-IR spectroscopy one will also be able to follow structural modifications on the

same ultrafast 1 ps time scale. These could be either equilibrium fluctuation of the peptide backbone or concerted structural rearrangements after

triggering, for example, with the help of photochemical reactions (144,145)

or temperature jumps.



APPENDIX: DIAGONAL AND OFF-DIAGONAL

ANHARMONICITY IN THE WEAK COUPLING LIMIT



The one-exciton Hamiltonian H1 mixes the monomeric site states jii to

create the one-excitonic states j k i D k qki jii (see Fig. 13). When the

system is harmonic  D 0 , the two-excitonic eigenstates of the twoexcitonic Hamiltonian H20 are simply Boson product states of the one

excitonic states: faij j k ij l i C j l ij k i g, p

l Ä k, where the normalization

1

factors are aii D 2 for l D k and aij D 1/ 2 for l 6D k, respectively. The

corresponding eigenvalues are εk C εl . The transformation matrix between

the two-exciton basis and the site basis f˛ij jiijji C jjijii g, i Ä j, is

Qkl,ij D 2aij akl qki qlj C qli qki



(37)



Anharmonicity is introduced by reducing the site energies of only the

double excited monomeric site states jiijii by an energy  (see Fig. 13).

The perturbed Hamiltonian H2 D H20 C V consists of a harmonic part H20

and an anharmonicity term V, which mixes the harmonic two excitonic

states. The matrix elements of V in the site basis are:

Vij,i0 j0 D



 Ð υij υi0 j0 υii0



(38)



The matrix elements of V in the excitonic basis are:

Vkl,mn D







1

Qkl,ij υij Qij,mn

D







ij



Qkl,ii Qmn,ii



(39)



i



Up to this point, this expression is exact. The problem can be evaluated

assuming that the coupling is weak so that each one-excitonic state is

predominantly localized on an individual monomeric site:

qij D υij C q0ij D υij C



ˇij

εj



εi



Copyright © 2001 by Taylor & Francis Group, LLC



with



ˇij

εj



εi



<1



(40)



338



Hamm and Hochstrasser



where ˇij are the off-diagonal terms of the one-exciton Hamiltonian H1 in

the site basis. Then we obtain for the diagonal anharmonicities in first order

of V and lowest order in ˇij :

εkk D Vkk,kk D







(41)



and for the off-diagonal anharmonicities

εkl D Vkl,kl D



q2ki q2li D



2

i



4



2

ˇkl

εk εl



2



(42)



ACKNOWLEDGMENTS



We would like to thank Manho Lim for the significant contribution in setting

up the instrumentation and taking the data presented here and William

DeGrado for introducing us to the cyclic model penta-peptide. We are

indebted to Dr. Nien-Hui Ge for her careful contributions regarding some of

the key formulas in this paper. Dr. Matthew Asplund gratefully contributed

the spectrally resolved echoes prior to their publication. The research was

supported by NIH and NSF with instrumentation developed under NIH

RR13456. Peter Hamm is grateful to the DFG for a postdoctoral fellowship.

REFERENCES

1.

2.

3.



4.

5.

6.



7.



Wăuthrich K. NMR in Biological Research: Peptides and Proteins. New York:

American Elsevier, 1976.

Bax A. Two-Dimensional Nuclear Magnetic Resonance in Liquids. Boston:

Kluwer, 1982.

Ernst RR, Bodenhausen G, Wokaun A. Principles of Nuclear Magnetic

Resonance in One and Two Dimensions. Oxford: Oxford University Press,

1987.

Sanders JK, Hunter BK. Modern NMR Spectroscopy. New York: Oxford

University Press, 1993.

Karplus M, Petsko GA. Molecular dynamics simulations in biology. Nature

1990; 347:631–639.

Elber R, Karplus M. Enhanced sampling in molecular dynamics: use of the

time-dependent hartree approximation for a simulation of carbon monoxide

diffusion through myoglobin. J Am Chem Soc 1990; 112:9161–9175.

Zhou HX, Wlodek S, McCommon JA. Conformation gating as a mechanism

for enzyme specificity. Proc Natl Acad Sci USA 1998; 95:9280–9283.



Copyright © 2001 by Taylor & Francis Group, LLC



Tài liệu bạn tìm kiếm đã sẵn sàng tải về

E. 2D-IR Spectroscopy Using Semi-Impulsive Methods

Tải bản đầy đủ ngay(0 tr)

×