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