IV. VIBRATIONAL EXCITONS IN CYCLIC PENTAPEPTIDE
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IR Spectroscopy in Polypeptides
359
structures (42). The structure is not unique, and several conformations are
compatible with the NMR and x-ray measurements. In our study we used
the crystallographic structure to obtain the atomic coordinates of cyclo(AbuArg-Gly-Asp-Mamb) shown in Fig. 2. The backbone conformation traces
out a rectangular shape with a ˇ-turn centered at the Abu-Arg bond.
Figure 2
3D structure of the pentapeptide. (From Ref. 42.)
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360
Piryatinski et al.
To obtain the one-exciton Hamiltonian we used the central frequencies
for the peptide CO vibrations reported in Ref. 41 and assigned 1 D
1588 cm 1 to Abu-Arg, 2 D 1671 cm 1 to Arg-Gly, 3 D 1648 cm 1
to Gly-Asp, 4 D 1610 cm 1 to Asp-Mamb, and 5 D 1618 cm 1 to
Mamb-Abu. The dipole-dipole couplings among CO vibrations were
calculated from
Jmn D
m
Ð
n
ˆ Ð
3m
jRmn j3
ˆ Ð
m
m
n
m, n D 1, . . . , 5
(24)
˚ from the carbon atom
by assigning each dipole on a CO bond 0.868 A
°
and forming 25 angle with respect to the bond. The absolute value of
each dipole moment is 0.37 D (1,2,15,41). Using these parameters, the
one-exciton Hamiltonian (in cm 1 ) assumes the form
1588
7.2
5.7
1.7
7.0
0.7
0.6
7.6
7.2 1671
(25)
h D 5.7
0.7 1648
2.2
6.2
1.7
0.6
2.2 1610
0.3
7.0
7.6
6.2
0.3 1618
Since j n
m j > Jnm , each one-exciton state is a weakly perturbed
localized CO vibration. The one-exciton eigenstates are
je1 i D
je2 i D
je3 i D
je4 i D
je5 i D
0.98j1i 0.07j2i
0.05j1i 0.03j2i
0.16j1i 0.15j2i
0.10j1i C 0.07j2i
0.10j1i 0.98j2i
0.07j3i C 0.08j4i C 0.18j5i
0.08j3i C 0.98j4i 0.17j5i
0.21j3i 0.17j4i 0.94j5i
0.97j3i 0.05j4i C 0.19j5i
0.03j3i 0.01j4i C 0.15j5i
26
where jni, n D 1, . . . , 5 represents ﬁrst excited vibrational state of the nth
CO vibration. The corresponding one-exciton energies are ε1 D 1586 cm 1 ,
ε2 D 1610 cm 1 , ε3 D 1617 cm 1 , ε4 D 1650 cm 1 , and ε5 D 1673 cm 1 .
The two-exciton manifold consists of two types of doubly excited
vibrational states. The ﬁrst are overtones (local), where a single bond is
doubly excited. The other are collective (nonlocal), where two bonds are
simultaneously excited (43,50). We denote the former OTE (overtone twoexcitation) and the latter CTE (collective two-excitation). A pentapeptide
has 5 OTE and 10 CTE. The two-exciton energies are determined by the
parameters gn in the Hamiltonian [Equation (17)], which in turn depend
on the peptide group energies n , the anharmonicity n , and dipole
moment ratio Än , n D 1, . . . , 5. We set them equal for all CO units
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IR Spectroscopy in Polypeptides
361
p
and D 16 cm 1 adopted from the experiment (15,41) and Ä D 2.
The two-exciton energies are obtained from the poles of the exciton
scattering matrix (rather than a direct diagnolization of the 15 ð 15 twoexciton Hamiltonian, which is much more expensive). They are ε1 D
3157 cm 1 , ε2 D 3195 cm 1 , ε3 D 3200 cm 1 , ε4 D 3205 cm 1 , ε5 D
3220 cm 1 , ε6 D 3227 cm 1 , ε7 D 3235 cm 1 , ε8 D 3258 cm 1 , ε9 D
3259 cm 1 , ε10 D 3264 cm 1 , ε11 D 3283 cm 1 , ε12 D 3287 cm 1 , ε13 D
3288 cm 1 , ε14 D 3322 cm 1 , and ε15 D 3331 cm 1 . In general the twoexciton eigenstates are linear combinations of the OTE and the CTE.
However, since j 0m 2 n j > Ä2 Jnm , most two-exciton states can be
classiﬁed as weakly perturbed OTE or CTE type.
Having introduced the one- and two-exciton states, we next turn to
the line broadening. We denote the dephasing rate of the ﬁrst vibrational
transition by and the overtone by 2 . In all calculations and 2 are
set identical for all peptide groups. The anharmonicity D 16 cm 1 is
ﬁxed and independent of disorder. We have employed six models:
A.
B.
C.
Small homogeneous dephasing rates D 0.2 cm 1 and 2 D
0.4 cm 1 .
Large homogeneous dephasing rates D 5 cm 1 and 2 D
10 cm 1 , which correspond typically to experimental values
(15,41).
Static diagonal disorder. The n’th peptide energy is
represented as
n
D.
E.
D
n
C
n
n D 1, . . . , 5
(27)
where n is average energy of the n’th peptide group set
to the central frequencies in the one-exciton Hamiltonian
[Equation (25)]. The random variables n , representing energy
disorder, are assumed to be uncorrelated random Gaussian
variables with variance d D 12 cm 1 equal for all peptide
groups: D 0.2 cm 1 , 2 D 0.4 cm 1 .
Same as model 3 except that the homogeneous dephasing rates
of each peptide are adopted from experiment and set to D
5 cm 1 , 2 D 10 cm 1 .
Static off-diagonal disorder. The exciton coupling is given by
Jmn D Jmn C
mn
n 6D m;
n, m D 1, . . . , 5
(28)
where Jmn is average coupling energy between the m’th and the
nth peptide groups, whose values are given in Equation (25). mn
are uncorrelated Gaussian random variables with equal variances
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362
Piryatinski et al.
D 12 cm 1 for all mn , (n 6D m; n, m D 1, . . . , 5). The homogeneous dephasing rates are D 0.2 cm 1 , 2 D 0.4 cm 1 .
Same as model 5 except that the homogeneous dephasing rates
are taken from experiment and set to D 5 cm 1 , 2 D
10 cm 1 .
od
F.
Different peptide groups in the pentapeptide have inhomogeneous
broadening, which varies in the range of ¾ 3–12 cm 1 (41). Some of
the experimentally observed lines are therefore dominated by homogeneous
and others by inhomogeneous broadening. We introduced these different
models in order to study the signature homogeneous and inhomogeneous
broadening on the 2D PE spectra. In models (A), (C), and (E) we used
small homogeneous broadening in order to resolve all resonances. Some of
these resonances are not resolved in models (B), (D), and (F) which use
larger, more realistic, homogeneous broadening.
The degree of one-exciton state localization can be described by the
inverse participation ratio (52–54):
1
5
Pε D
j
ε
nj
4
(29)
nD1
where ε n is the n’th component of one-exciton wavefunction with energy
in the interval [ε, ε C dε]. For our model P ε may vary between P D
1 (localized state) and P D 5 (delocalized state). The participation ratio
distribution as well as the density of states are shown in Fig. 3. For models
A and B the exciton states are well localized, since j n
m j < Jnm . In
models C and D diagonal disorder slightly increases the one-exciton state
delocalization, resulting in the participation ratio ¾1.3 in the maximum
of the density of states (dotted line in the plot), and in models E and F
the off-diagonal disorder corresponds to the state delocalization within ¾2
peptide groups near the density of states (dotted line) maxima.
The linear (1D) absorption spectra of all models are presented in
Fig. 4. Model A shows ﬁve well-resolved one-exciton lines. In model B
the lines ε2 and ε3 are poorly resolved due to the increased homogeneous
broadening. Diagonal disorder in models C and D further broadens the
spectra. Since off-diagonal disorder induces state delocalization, the oneexciton resonances shift for models E and F and become ε01 D 1578 cm 1 ,
ε02 D 1605 cm 1 , ε03 D 1618 cm 1 , ε04 D 1652 cm 1 , and ε05 D 1679 cm 1 .
Additional information related to the one- and two-exciton dynamics
can be obtained from the 2D spectra, as will be shown in the following
section.
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IR Spectroscopy in Polypeptides
363
Figure 3 Solid line: Inverse participation ratio of one-exciton states for models
A–F. Dotted line: Density of one-exciton states for models C–F.
Copyright © 2001 by Taylor & Francis Group, LLC
364
Figure 4
Piryatinski et al.
Infrared absorption (1D) spectra for models A–F.
V. 2D PHOTON ECHOES OF A CYCLIC PENTAPEPTIDE
In a 2D three-pulse spectroscopy, two of the three pulses are time-coincident
and differ only by their wave vector. The system thus interacts once with a
single pulse and twice with a pulse pair. In the 2D photon echo technique
we set t2 D 0. We consider the heterodyne signal [Equation (10)]. This
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IR Spectroscopy in Polypeptides
365
signal can be measured by mixing the third-order signal with the heterodyne
pulse arriving with delay time t3 in the direction determined by the phase
matching conditions ks D k3 C k2 k1 . We shall display the 2D PE signal
in the frequency domain by performing a double Fourier transform:
S
2,
1
1
D
1
dt3
dt1 exp i
0
2 t3
Ci
1 t1
S t3 , 0, t1 .
(30)
0
The 2D PE signal is computed using the response function given by
Equations (21)–(23). Since we consider the 2D response on the time
scale smaller than the dephasing times, only the coherent component of
the response function [Equation (21)] contributes to the signal. The 2D
Fourier transform PE signal determined by Equations (22) and (30) has the
following form (17):
S
2,
1
DS1
2,
1
CS2
2,
(31)
1
The ﬁrst component,
2,
S1
1
D
i
2
ð
a
b
c
d
2
abcd
1
εd C i
1
1 C εc C i
cd,ab εc C εd
εc C εd
εa C εb C 2i
32
represents correlations between one-exciton states shown by the Feynman
diagram (Fig. 5(1)). The second component,
2,
S2
1
D
a
b
c
d
abcd
ð
ð
1
1
1 C εc C i 2
1
ω
ω
εc
1
εc C ε d
2
i C
i
0
ω
dωcd,ab ω
1
εa C εb C i 2
i
0
0
33
0
is induced by correlations between one- and two-exciton states and is
represented by the Feynman diagram in Fig. 5(2).Ł
Below we ﬁrst analyze the absolute value of the 2D PE signal and then
consider its phase by looking at the real and the imaginary parts separately.
Ł
The components of the signal calculated according to these diagrams, using the
sum-over-state approach presented in Appendix A, coincide with Equations (32)
and (33) in the narrow line limit − 0 , J
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