J. Phys. Chem. B 2004, 108, 10415-10427
10415
Two-Dimensional Infrared Spectroscopy of Isotopomers of an Alanine Rich r-Helix† C. Fang,‡ J. Wang,‡ Y. S. Kim,‡ A. K. Charnley,‡ W. Barber-Armstrong,§ A. B. Smith III,‡ S. M. Decatur,§ and R. M. Hochstrasser*,‡ Department of Chemistry, UniVersity of PennsylVania, Philadelphia, PennsylVania 19104, and Department of Chemistry, Mount Holyoke College, South Hadley, Massachusetts 01075 ReceiVed: January 9, 2004; In Final Form: February 24, 2004
The two-dimensional infrared spectra of a series of doubly isotopically substituted 25-residue R-helices were measured with femtosecond three pulse infrared time domain interferometry. The insertion of 13Cd16O and 13 Cd18O labels at known residues on the helix permitted the vibrational couplings between different amide I′ modes separated by one, two, and three residues to be measured. The 2D IR signal of one residue in 25 was readily studied, confirming this approach is applicable to labeled proteins. We identified the couplings between each pair of isotopomer levels and between them and the helix exciton band states: the 2D IR spectra proved that the amide vibrations of the R-helix are delocalized. Cross-peaks, originating from the coupling of the isotopomer pairs, were systematically analyzed. Besides the separated pair modeling and second-order perturbation theory estimates, the experimental results were compared in detail with a full matrix diagonalization simulation based on averaged Hamiltonian matrices that represent the amide I′ vibrator’s one- and two-exciton states. The main features of the 2D IR spectra could be predicted by this modeling. The experimental results were in good agreement with a set of couplings that were derived from transition chargetransition charge interactions for all but the nearest neighbors, for which the coupling is more influenced by through-bond interactions between the adjacent amide groups. The possible ranges of the magnitudes of the three largest coupling constants β12, β13, and β14 were explored by various approaches to be within a few cm-1 accuracy of a preferred set of absolute values and their associated error bars: |β12| ) 8.5 ( 1.8, |β13| ) 5.4 ( 1.0, and |β14| ) 6.6 ( 0.8 cm-1. The signs were independently indicated to be β12 > 0, β13 < 0, and β14 < 0.
Introduction An important challenge for physical chemistry is the development of methods that measure the time dependences of structural changes in complex systems. Because of their intrinsically high time resolution, multidimensional infrared spectroscopies1-15 2D IR and 3D IR, can contribute significantly to this challenge, complementing the knowledge on average structures obtained by the established methods of structural biology and their timedependent variants. The concepts underlying the operational aspects of multidimensional infrared experiments arose from many years of theoretical and experimental research on laser technology, nonlinear optics, and infrared spectroscopy. However, the principles now used for the manipulation of the multidimensional data sets in time or frequency domains, phase cycling, projections, properties of multidimensional Fourier transforms and many other procedures, often of significant complexity, are also text book material in NMR,16 even though the practical aspects of the two experiments are very different and the methods arose quite independently. The earliest 2D IR experiments used the powerful pumpprobe method1 and the current style of 2D IR heterodyned echo results followed soon after.4 The spectral line narrowing and manifestations of couplings and correlations in 2D IR arise from the contributions of the multilevel photon echo portions of these †
Part of the special issue “Gerald Small Festschrift”. * Corresponding author. E-mail:
[email protected]. ‡ University of Pennsylvania. § Mount Holyoke College.
signals.17-19 Although the first photon echo experiments on molecules were in the infrared at 10.6 µm,20 and most of the coherent NMR responses were reproduced many years ago in the infrared by excitation of the vibrational modes of methyl fluoride at 3 µm,21 the main echo technique developments in molecular spectroscopy have been in the optical regime. In seminal molecular studies, Wiersma and Aartsma22 reported two pulse echoes with nanosecond time resolution. As more reliable, shorter laser pulses became available, these optical measurements were extended to liquids and glasses in which the motions were faster.23 Photon echo experiments began to incorporate heterodyning, gating, three pulse methods (see, for example,24 and references therein), and more recently, 2D displays25 and schemes for deducing the time correlations of the frequency fluctuations26 were introduced. With reference to applications in biology, femtosecond pulse IR experiments had been carried out on proteins and aqueous systems throughout this early period,27-32 but short, stable, and tunable IR pulses suitable for vibrational echoes were less available than in the optical region. Nevertheless the infrared two pulse echo method was extended to the picosecond regime by means of a free electron laser;33 then, based on advances in infrared materials and the titanium sapphire laser, femtosecond three pulse vibrational echoes in the infrared were accomplished for aqueous ions34 and later for peptides35 and small proteins.35,36 Heterodyning of three pulse phase-locked IR echoes enabled the assembly of multidimensional vibrational spectra4 enlarging the scope of 2D IR spectra from that based on the closely related,37 pump-probe methods.1 Recently, dual frequency phase-locked 2D IR of peptides have
10.1021/jp049893y CCC: $27.50 © 2004 American Chemical Society Published on Web 04/16/2004
10416 J. Phys. Chem. B, Vol. 108, No. 29, 2004 been reported.14,15 The field of multidimensional IR spectroscopy of vibrators has now become very active37,38 and has outstanding potential for the study of structure and molecular dynamics in liquids,39,40 glasses, and biological systems. The IR spectra of proteins and peptides are intimately connected with their complete three-dimensional structures on the length scale of chemical bonds, but in practice the vibrational transitions of proteins are not all spectrally resolved. Multidimensional nonlinear infrared methods provide additional information and can in principle be used to measure whether, and by how much, the modes are coupled. The structural constraints from these nonlinear experiments when combined with established concepts of chemical connectivity would be sufficient to establish features of three-dimensional structures.11 However, the intrinsic spectral widths of the vibrational transitions and the small frequency separations between many of the modes make the inversion of spectral data to structure a formidable challenge. The use of isotopic replacements has been essential in the interpretation of vibrational spectra and their relationship to structure. Isotopomers have frequencies, force fields, and anharmonicities that are different from one another. Of particular interest in applications of 2D IR is the use of isotopes to shift frequencies into regions where their couplings can be measured, free from interference by other modes of the system. For the amide I′ mode, which is mainly a CdO stretching vibration, the shifts by 13Cd16O and 13Cd18O substitution are large enough to displace the substituted amide group frequencies beyond the range of the natural distribution of frequencies found in most secondary structures. Our strategy41 is to insert both 13Cd16O and 13Cd18O labels into the helix. Because their isotope shifts are different, a pair of isotopic peaks will be created, separated by approximately 25 cm-1, and 2D IR spectroscopic methods can then be used to analyze the coupling between that specific pair of molecular transitions. If 13Cd16O is substituted for one of the residues of a 25-residue helix, a 13Cd16O diagonal peak will appear to the lower frequency of a much stronger and broad 12Cd16O diagonal peak, corresponding to the set of helical exciton states of the remaining 24 residues. The couplings within the band and between the 13Cd16O label and the band states will also show up in the off-diagonal regions of the 2D IR spectrum. If we insert an additional 13Cd18O label into the helix, another vibrator will be shifted out of the exciton band and appear at even lower frequency. The cross-peaks between these two isotopically labeled amide I′ modes assist in the measurement of the vibrational coupling between the two labeled vibrators. Because those labels can be inserted into various positions of the helix chain, the 3D structure of the molecule can be revealed, as reflected in the distances between the modes and relative orientations of the pairs of transition dipoles. In addition, the population relaxation times, the inhomogeneous distributions, and the correlation function of the fluctuations of the vibrational frequencies at various sites can be measured. However, the 2D IR signals from different levels interfere with one another. So we need to develop efficient methods to process and interpret the data. Our approach to simulate 2D IR spectra of peptides is based on considerable previous knowledge of the vibrational force fields,42-45 the polarized linear-IR spectra of helices,46,47 the 2D IR spectra of helical proteins based on calculated interaction energies,12,48 the vibrational circular dichroism of helices,49 and general effects of isotopic substitution on spectra. Comparisons of high-level computations of the amide I′ modes with estimates from through-space coupling models50 aptly demonstrate the
Fang et al. complexity of interamide coupling and the sensitivity of the coupling to the polypeptide conformation. The through-space coupling between modes is readily computed to all orders of the multipole expansion, but there are also through-bond interactions51-54 and polarizability effects need to be considered.55-61 A series of isotopically substituted 25-residue alanine rich helical polypeptides based on Ac-(A)4K(A)4K(A)4K(A)4K(A)4Y-NH2 have been investigated. The linear-IR and the 2D IR spectra of these structures provide quantitative information regarding the atomic-level interactions between nearby amide units in the R-helix, whereas the presence of the cross-peaks in the 2D IR evidences the coupling between the 13Cd16O and 13Cd18O vibrators. The resulting spectra also provide dynamical information on the amide I′ modes of these helices. A preliminary report of some aspects of this work has recently appeared.41 Methods and Experimental Results Materials. The isotopically labeled 25-residue alanine rich helical polypeptides of the sequence Ac-(A)4K(A)4K(A)4K(A)4K(A)4Y-NH2 containing the isotopic amide carbonyl labels 13Cd16O and 13Cd18O were obtained by automated peptide synthesis using Fmoc chemistry and purified by reversed-phase HPLC, as described previously.62 The identities of the peptides were confirmed via electrospray mass spectrometry (EMS).63 N-Fmoc-alanine-1-13C was purchased from Cambridge Isotopes; preparation of the Fmoc-protected 18O labeled alanine was required for solid-phase peptide synthesis. The L-alanine-1-13C and 18OH2 were purchased from Isotec and 18O-labeled-alanine1-13C was prepared by 18OH2 exchange following well-known procedures.64 The incorporation of 18O was monitored by mass spectrometry. After 4 days at 50 °C no unlabeled amino acid was observed, and the amino acid was isolated as its HCl salt. Protection by Fmoc was then accomplished using Fmoc-Cl and an in situ silylation procedure,65 providing a moderate yield (47%) of the protected amino acid after recrystallization. The product was confirmed by 1H and 13C NMR, IR, and mass spectrometry. As expected, the mono-18O product was present only as a minor component, whereas ∼75% of the product had two 18O equivalents (H3CCH(NH-Fmoc)13Cd18Os18OH) which assured that more than 87% of the final synthesized helices would contain the desired 13Cd18O isotope. We synthesized 250 mg of Fmocs18O-labeled alanine-1-13C in this manner for the subsequent solid-phase peptide synthesis. The monoamide model compound MC1 with the molecular formula H3CCH(NH2)13Cd18ONHCH2CH(CH3)2 was prepared to study the 13Cd18O amide I′ band independent of the coupling effects present in our peptides. The EDC mediated coupling of Fmocs18O-labeled alanine-1-13C with isobutylamine proceeded smoothly, and removal of the Fmoc protecting group with piperidine in DMF provided the desired amide model compound in 41% yield for the two steps. A portion of this material was transformed to its HCl salt (MC1 salt) by treatment with anhydrous HCl in methanol. The analogous unlabeled compound MC2 was prepared by the same method. The isotopomers contain none, one, or one each of the isotopic amide carbonyl labels 13Cd16O and 13Cd18O. We employ the following notation for these compounds: [alanine residue with 13Cd16O, alanine residue with 13Cd18O]. When there is no alanine isotopic substitution we insert a zero. We measured: [0,0] (data not shown), [0,11], [12,13], [11,13], and [11,14]. The effect of 13C in natural abundance is to cause 17.8% of the helices to have one 13Cd16O residue randomly distributed by residue number. Previous studies of the 25-residue peptide have
Isotopomers of an Alanine Rich R-Helix
J. Phys. Chem. B, Vol. 108, No. 29, 2004 10417
Figure 1. Carbonyl orientations in a perfect R-helical structure. The numbers inserted define the parent residues of the labeled carbonyl groups as used in the text. The intrachain hydrogen bond is between amide units 11/12 and 14/15, which corresponds to a carbonyl at residue 11 and an N-H group of residue 15.
shown that the central residues are nearly completely helical at 0 °C.62 Figure 1 shows a view of the residues that are the subject of this paper. The view is along the helix axis and the amide carbonyl oxygen atoms are colored in red. The numbering is as in the text and the figure captions. One notes that the carbonyl groups are all approximately parallel to the helix axis. Linear Spectroscopy. FTIR spectra (Perkin-Elmer Spectrum 2000) of 10 mg/mL peptide solutions (5-10 mM on a residue basis) were taken at 2 °C, in a temperature controllable cell with CaF2 windows and a path length of 25 µm (Teflon spacer, Harrick Scientific), at 2.0 cm-1 resolution. No concentrationdependent spectral changes were observed in this range. For background subtraction, spectra were obtained of the wellpurged chamber incorporating the 3 mm pinhole sample holder, the solvent (0.1% phosphoric acid-D2O buffer, pD ) 3), and the peptide solution (helix in the buffer). Figure 2 presents the linear-IR data for each of the compounds studied. The linear spectra were well reproduced by a simple curve fitting procedure in which the helix region was fitted by two Gaussian contributions and each of the isotopomer transitions by a single Gaussian. Another Gaussian centered at 1596 cm-1 accounted for the 13C in natural abundance. At 2 °C the fitting parameters for [0,11], [11,13], and [11,14] are similar but those of [12,13] are somewhat different. Figure 2 also shows simulations based on theoretical model that will be discussed in detail later in the paper. The main helix exciton band (the 23 or 24 unlabeled residues) of [0,11], [11,13], and [11,14] peaks at about 1631.7 ( 0.5 cm-1 and it has a full width at halfmaximum (fwhm) of 28.3 cm-1. The weaker, higher frequency component at ∼1658 cm-1 has a fwhm of 28.9 cm-1. The 13Cd16O isotopic label peaks at 1594.0 ( 0.3 cm-1, and the 13Cd18O isotopic label peaks at 1570.7 ( 0.8 cm-1. The separations between the two isotopic labels are 24.7 and 22.4 cm-1 in [11,13] and [11,14], respectively. The fwhm for the 13Cd18O label is 15.2 cm-1. Regarding the relative integrated intensities, the transitions of the 13Cd18O labels are similar to those of the 13Cd16O label in both [11,13] and [11,14]. Because of unavoidable uncertainties in baseline subtraction and the contributions of 13C isotopes in natural abundance, the ratios of the 13Cd18O to the 13Cd16O extinction coefficients from the linear spectra should be considered as minimum values. The FTIR spectrum of [12,13], which has its two isotopic labels adjacent to each other, shows significant differences compared with the other samples. A similar unusually large shift of the
Figure 2. Linear FTIR spectra of the isotopomers [0,11] (a), [12,13] (b), [11,13] (c), and [11,14] (d) with notation defined in the text. The thin lines are the simulations.
main helix band was seen for a 25-residue alanine rich R-helix having two adjacent 13Cd16O labels in the center of the sequence.66 The 13Cd18O mode shifts to 1568.3 cm-1 and the 13Cd16O mode to 1596.3 cm-1. The helix exciton transitions have their main peak at 1633 cm-1, which represents a significant blue shift from the other isotopomers. The integrated intensity of the 13Cd16O region is a factor 4.6 larger than that of the 13Cd18O band. The fwhm of the 13Cd18O label is now about 12.7 cm-1, whereas the main absorption band component has a relatively larger fwhm of 29.0 cm-1. The separation between the two isotopic labels of [12,13] is increased to 28.0 cm-1. Independent information regarding the frequency and intensity changes introduced by 13Cd18O substitution of a peptide group was obtained from the labeled model peptide MC1. Both the MC1 salt (in 99.9% D2O at ∼55 mM) and free amine showed a peak at 1597.6 cm-1 with a fwhm of 25.6 cm-1, and a peak extinction coefficient (for 13Cd18O) of ∼314 M-1 cm-1. The MC2 salt amide I′ peak at 1657.2 cm-1 has a fwhm of 26.8 cm-1, and an of ∼310 M-1 cm-1. Therefore, for amide CdO isotopomers (12Cd16O and 13Cd18O), the transition dipole moment magnitudes which are proportional to x are equal, and the isotope shift of the 12Cd16O and 13Cd18O amide I′ peaks is 59.6 cm-1, which agrees well with the former reported value of the peptide 13Cd18O isotope shift.67 The temperature-dependent difference FTIR in the amide I′ region on [11,14] when referenced to its 0 °C spectrum, showed two-state behavior with a melting temperature near 40 °C: the helix band was centered around 1628 cm-1, and the more random conformations peaked at 1658 cm-1. These results confirm that the peak in the helix region at 1658 cm-1 is in fact due to the presence of residual disordered structures. The experiments at 0 °C are clearly in the helical regime. The isotope transitions were not evident in linear spectra of the disordered state.
10418 J. Phys. Chem. B, Vol. 108, No. 29, 2004 Nonlinear Spectroscopic Methods. The nonlinear echo spectra were obtained using heterodyned time domain interferometry (heterodyned echo) in precisely the manner described in detail previously for experiments in the 6 µm region.4,7-9,13,37 Only information essential to the particular experiments reported here is given in this paper. The three infrared pulses incident on the sample, each of about 300 nJ, arrived with intervals τ, between the first (k1) and second (k2) pulses. In most of the present experiments the third pulse (k3) arrives time coincident with the second pulse (waiting time T ) 0 condition). The phasematched signal at wave vector -k1 + k2 + k3 was detected as a function of τ and t, the latter being the interval between the local oscillator and k3 fields. A double Fourier transform was applied in τ and t to generate the 2D IR spectrum along the frequency axes ωτ and ωt . To emphasize the isotopic peaks, we chose the center frequency of the pulses at ∼1580 cm-1, the mean of the 13Cd18O and 13Cd16O peaks. This meant that the 120 fs mid-IR laser pulses did not provide uniform coverage of the complete spectral range of the helix and its isotopomers. Therefore, the helix exciton 2D IR spectra in the region of the main strong band at ca. 1630 cm-1 are somewhat distorted in this presentation. By scanning the monochromator with a 5 nm step size, and continuously monitoring the k2 intensity on one of the channels of the 32-element MCT array detector (InfraRed Associates, Inc.), we obtained the pulse shape, and it showed a fwhm of ∼150 cm-1 with a moderately flat top region, having intensity changes less than 12% within the spectral range of 1560-1600 cm-1. Because this is the region where two isotopomer levels are excited, no convolution scheme with the laser pulse shape is needed to process the 2D IR spectra, although the main helical band region is obviously affected and modified by the finite pulse width. Pump-probe spectra were obtained by measuring the difference in transmission of the k2 beam when the sample was pumped by the k1 beam. The k2 probe beam intensity was reduced by a factor of ∼15 for this experiment, and the k1 pump beam was chopped at half of the repetition rate of the laser. The probe beam was continuously monitored through a monochromator and recorded by a 32-element MCT array detector having elements spaced in the spectrum by 14.5 nm. A finer sampling of the pump-probe spectrum was achieved by scanning the monochromator in 3 nm intervals and then averaging the spectral data seen by all the 28 active channels at each grating position. Because the quality of the spatial/ temporal overlap between the pump and probe beams along with the sample conditions varied from experiment to experiment, the magnitude of the transient absorbance varied from sample to sample. Pump-Probe Experimental Results. The broad-bandpump/broad-band-probe method1,5 was applied to 13Cd18O model compound MC1 to determine any significant isotope effect on the anharmonicity, and to the helix isotopomers to confirm their anharmonicities and for use in fixing the phase of 2D IR spectra.16 The pump-probe spectra for [12,13], [11,13], and [11,14] are shown in Figure 3 as the thicker lines having peak transient absorbances of 0.13, 0.67, and 0.50 mOD, respectively, in the main helical band region. The pump-probe spectra were fitted in the usual way1 by a bleaching component, the shape of which was obtained from the FTIR, and a new absorption shifted by the diagonal anharmonicity. Because the net pump-probe difference signal is close to zero, we conclude that the transition dipole for the v ) 1 f v ) 2 transition is nearly equal to twice that of the v ) 0 f v ) 1 transition but 1.23 times broader. The anhar-
Fang et al.
Figure 3. Broad-band-pump/broad-band-probe spectra of samples (b)(d) (in D2O-phosphoric acid buffer at 0 °C) and their corresponding rephasing 2D IR real part projections, shown as thin lines. The main helical band signal was normalized to 1, and the probe axis was split into two regions to be analyzed separately. The associated phase and normalization factors are described in the text.
monicity of the amide 13Cd18O group in the model compound was found to be 14.2 cm-1, smaller than the value of 16.0 cm-1 for the amide 12Cd16O group in N-methylacetamide.1 The least squares curve fitting of the isotopic regions of the pump-probe spectra of [12,13], [11,13], and [11,14] in Figure 3 provided in each case values for the anharmonicities for both isotopically substituted amides of (14.7, 13.7), (15.6,12.2), and (14.8,13.9) cm-1 respectively. We obtained the decay curves of the pump-probe signals by varying the time interval between the k1 pump and k2 probe beams. The experimental decay curves were globally fitted to a double exponential decay, where the much faster component representing 90% of the signal was assumed to be the T1 decay. The slow component is on the several picoseconds time scale. We found ca. 480 fs for the T1 relaxation time of both isotopomers. The population relaxation times of the overtone states of the isotopically replaced amide I′ modes of [11,13] and [11,14] were deduced to be ca. 320 fs by attributing the increased width of the v ) 1 f v ) 2 transition to a T1 process. 2D IR Spectroscopy Results. In the present work we focused mainly on the isotope region of the 2D IR spectrum. For [0,11], the exciton band states and the 13Cd18O labeled amide band clearly appear in the 2D spectrum (see Figure 4a), along with the cross-peaks between them. The signal from 13Cd16O modes in natural abundance is clearly present although it is not so evident in Figure 4. The other three isotopomers all clearly exhibit the spectral features of the labeled amides, as shown in Figure 4b-d. An effective way to analyze the data is to generate the absorptive spectra by carefully adding the rephasing (-k1 arrives first, R) and nonrephasing (k2 or k3 arrives first, NR) spectra.13,37,68 The frequency dependence of the signal phase was not measured, but the range of phase variations that could be introduced by the optical materials being used was obtained by calculations based on realistic quadratic and cubic phase parameters typical for the materials used in the optics of the experiment. We chose the phase factors to equalize the real parts of the R and NR projections (onto the ωt axis) and the corresponding pump-probe spectra. To account for the small
Isotopomers of an Alanine Rich R-Helix
Figure 4. 2D IR absolute magnitude rephasing spectra and their simulations. All pulses have the same polarization and the isotopomer notation is as in Figure 2 ([0,11] (a), [12,13] (b), [11,13] (c), and [11,14] (d)). Simulations are arranged on the right side of each spectrum (a′)(d′). Sixty equally spaced contour lines were drawn from zero to the maximum intensity of the main helical band in each case.
frequency-dependent phase associated with the photon echo spectra, we used different phase factors in the isotopic region (1540-1600 cm-1) and in the main helical band region (16101700 cm-1) to separately equalize the 2D IR projections and the pump-probe spectra. The projections are scaled using a normalization factor to match the signal strengths of the corresponding regions in the pump-probe spectra. It is found that each region can be matched well by using a specific constant phase factor. Because of the different noise levels associated with the photon echo and pump-probe measurements, and also the large difference in the IR absorbances of the helix band (∼0.2 OD) and the weak isotopomer levels (∼0.01 OD), the scale factors for the isotopic region and the main helical band were different. When a specific region of the real part of the 2D IR spectrum near the isotopomer bands was selected, the intensity ratio |R|/ |NR| integrated over this selected frequency box was about 2.3, but slightly different for each derivative. This ratio was used to scale the R and NR spectra. It varies with the size of the observation window and becomes close to unity when the whole spectral range is included, as expected from theory.13,37,38 The relevant frequency box is shown in Figure 5. Because of the spectral broadening along the diagonal (ωτ ) ωt) in R and the spectral broadening along the anti-diagonal (ωτ + ωt ) constant) in NR, the tail of the main exciton band along the diagonal contributes significantly to the R signal in the region of the
J. Phys. Chem. B, Vol. 108, No. 29, 2004 10419
Figure 5. Sum of the real parts of the 2D IR rephasing (R) and nonrephasing (NR) spectra of samples (a)-(d) and their corresponding simulations ([0,11] (a), [12,13] (b), [11,13] (c), and [11,14] (d)). The marked inner square indicates the finite piece of the spectrum on which the [R + NR × ratio] summation was performed as discussed in the text. Simulations are arranged on the right side of each spectrum (a′)(d′). The relative signal strengths were chosen for each spectrum to show the proper contrast between the negative (blue) and positive (red) features, and the contour lines were drawn in 1% intervals from the minimum to the maximum value of each region shown. The listed multiplication factors indicated the relative intensity ratios between each spectrum presented here in reference to their main helical band, which should have the similar signal amplitude in all four cases.
isotope transitions. To compensate for this, the absorptive spectra with the symmetric line shapes shown in Figure 5 were obtained by using the aforementioned scale factors. The absorptive 2D IR spectra of Figure 5 (left column, b-d) in the region of the 13Cd16O and 13Cd18O amide I′ transitions clearly show 8 resonant peaks. Besides the two pairs of strong diagonal peaks corresponding to the v ) 0 f v ) 1 and v ) 1 f v ) 2 transitions of each mode, a cross-peak pair, separated by the off-diagonal anharmonicities, appeared both above and below the diagonal. The 13Cd18O peaks are upright with their contour node separating the positive and negative parts almost parallel to the ωτ axis, whereas the 13Cd16O peaks are tilted and elongated along the diagonal with the node having a positive slope. On the basis of our previous work,15 these results suggest that the inhomogeneous distributions of the two isotopomer levels do not have the same correlation to the levels in the exciton band. Resolution enhancement of the spectra by means of window functions introduced distortions but the improved resolution of the spectral components confirmed that the two isotopic labels were split more in [11,13] than in [11,14], and that the crosspeak intensity is relatively smaller for [11,13], indicative of its smaller off-diagonal anharmonicity. For the quantitative analysis,
10420 J. Phys. Chem. B, Vol. 108, No. 29, 2004
Figure 6. Overview of the energy levels of the doubly tagged helix. The zero-order isotope shifts are δa (for 13Cd18O) and δb (for 13Cd16O) and ∆a and ∆b are their unperturbed diagonal anharmonicities. The shaded areas represent the helix one- and two-exciton bands that become perturbed by the two isotopomer levels.
the narrow-band-pump/broad-band-probe spectra were computed from the 2D IR data69 using a 10 cm-1 fwhm Gaussian distribution along the ωτ axis centered at either the 13Cd16O or 13Cd18O amide I′ resonance and then projecting its integrated real part onto the ωt axis. Curve fitting along the ωt axis based on the known parameters already derived from the linearIR spectroscopy, T1 measurements and broad-band-pump/broadband-probe experiments provided off-diagonal anharmonicity values for [12,13], [11,13], and [11,14] of ∆12,13 ) 9.0 ( 1.0, ∆11,13 ) 3.2 ( 0.8, and ∆11,14 ) 4.5 ( 0.6 cm-1, respectively. The relative values agree with the qualitative observation that in the 2D IR rephasing magnitude spectra, cross-peaks due to the coupling between the two isotopomer levels show the intensity trend [12,13] > [11,14] > [11,13]. The cross-peaks between the helix exciton states and the isotopomers are present in the spectra. In Figure 4 these peaks are at ωτ in the region 1600-1650 cm-1. They can be analyzed by plotting the magnitude spectra along ωτ for a particular ωt or by the integration of a range of Gaussian-weighted ωt spectral components centered at each isotopomer transition. These crosspeaks are spread vertically, providing the projection of the spread of the exciton states along the diagonal of the main peak region, onto the ωτ axis. The interaction of the 13Cd16O level with the exciton band is evident in Figure 4b-d, and a careful examination shows that the coupling of 13Cd18O to the band states is present in every case. Discussion Figure 6 shows a cartoon of the zero order and delocalized energy levels involved in the 2D IR experiment. The exciton bands of the tagged helix are shaded and the isotopomer levels
Fang et al. are located below these bands. The band states and the trap states are mixed and shifted from their zero-order positions by the couplings. The two isotopomer levels below the one-exciton band are those seen in the linear-IR spectrum. A typical pathway that gives rise to an infrared echo signal is indicated by the dashed and solid arrows.70 The first pulse represented by a solid up-arrow creates a coherence with one of the isotopomer transitions and the second two pulses represented by dashed lines transfer this coherence from one isotope to the other by means of transitions involving the two-quantum levels. The solid and dashed lines represent interactions on the bra and ket sides of the density operator. The five upper levels are the two isotopomer overtones and their combination band. In addition we have the combination bands of the isotopomers with the remaining helical band states. In Figure 6 the levels are identified by their zero-order energies. The signal is formed by the sum of contributions from many pathways involving all the one- and two-quantum states arising from the isotopomers and their couplings to the band states. The dynamical parameters of each of the isotopomers were found to be equal within experimental error, but the fitting procedure would not have allowed a 20% difference in the dephasing times to be confidently identified. Consistent with this conclusion regarding the similar dynamics obtained from fitting the spectral shapes are the plots of the absolute value squared of the 2D IR signal Fourier transformed only along the detection axis, to yield the ωt versus τ dependence. This procedure gave signals along τ that were similar for values of ωt corresponding to the different isotopic components. This specific signal is the zero waiting time response: the signal along the τ axis should peak at finite positive τ, indicating the inhomogeneous contribution to the transition. The peak shifts were not very well determined but are a few hundred femtoseconds for both isotopomers. From these results we conclude that the assumption of a fixed inhomogeneous distribution of frequencies is reasonable for the present analysis. The shapes of the 2D IR spectra, in particular the spread along the diagonal of the main helix part, indicate the presence of a broad distribution of states that we attribute to the helix excitons. Therefore, the experiment suggests that there are significant frequency fluctuations of the amide I′ mode along the helix; otherwise only a few transitions related to the helix A and E type states would be observed. The isotopomer transitions are significantly narrower than the band transitions. Furthermore, the spectra of the helix band transitions are significantly narrowed in the direction perpendicular to the diagonal, which is the result of the echo-like response of the 2D IR signals from the helix excitons. This picture is confirmed from the shapes of the isotopomer transitions, which are much more symmetric and whose inhomogeneous width contribution corresponds more closely to the fluctuations occurring at a single amide I′ site. The standard deviation of these fluctuations is ca. 4 cm-1, according to fits of the line shapes. The results also show clearly from the existence of the crosspeaks that the isotopomer residues are coupled in all cases: the vibrational state containing one excitation in each isotopomer is never at the same energy as the sum of the two isotopomer energies. We need a model to estimate the coupling constants from these measured off-diagonal anharmonicities. The coupling between the residues also influences the apparent diagonal anharmonicities, regarded as the differences in frequency between the v ) 0 f v ) 1 and the v ) 1 f v ) 2 transitions of the isotopomer. In linear spectroscopy it is usual, and often reasonable, to create models to explain splittings even without
Isotopomers of an Alanine Rich R-Helix direct experimental knowledge of the existence of any coupling or of the energies of the zero-order states. The existence of crosspeaks in 2D IR shows that there is coupling. The linear-IR spectra indicate that the helix exciton band states are influenced by the isotopic replacement particularly in the case of [12,13], where the exciton transitions are shifted significantly and the intensity distribution in the exciton region undergoes large changes upon isotopic substitution. The modeling of the spectra needs to accommodate all these observations and provide an explanation for them. Separated Pair Modeling. There are too many coupling constants involved in generating the helix states for them to be determined unambiguously from our limited set of data. Therefore, some models are required that allow particular couplings to be estimated. As a first simple approach to compute the combination and overtone energy shifts, we can consider the energies of a separated pair of tagged residues, 13Cd16O with frequency νj16 and 13Cd18O with frequency νj18 , having v ) 1 f v ) 2 transitions at νj16 - ∆16 and νj18 - ∆18 and a combination band frequency of νj16 + νj18 - ∆16,18 where ∆16,18 is the off-diagonal anharmonicity and ∆16 and ∆18 are the diagonal anharmonicities. We can then estimate ∆16,18 from a simple model if only two modes were to be involved.11,71 The unperturbed diagonal anharmonicities were measured from model compounds, as described above. For this estimate, the Hamiltonians Hs for the single excitations (2 × 2) and Hd for the double excitations (3 × 3) are written in the basis of the uncoupled isotopomers for which ∆16,18 ) 0; a harmonic approximation is used, in which the part of the bilinear transfer matrix element between v ) 1 and v ) 2 states of different groups is at x2 times those for v ) 0 to v ) 1. By diagonalization of Hs and Hd, sets of eigenvalues and eigenvectors are obtained. The observed linear-IR spectral frequencies are the eigenvalues of Hs and the differences between those of Hs and Hd predict the transition frequencies in the v ) 1 f v ) 2 region of the 2D IR spectra. It is a simple calculation to find the coupling constants, βij, needed to generate the observed two fundamentals, two overtones and one combination band, which are largely represented by the isotopomer band separation in the linear-IR and the off-diagonal anharmonicity measured by 2D IR. We find the values |β12,13| ) 11.3 ( 1.8 cm-1, |β11,13| ) 5.4 ( 1.0 cm-1, and |β11,14| ) 5.8 ( 0.8 cm-1. Comparisons with theory are given later in this paper. As discussed below, the signs of βij can be obtained by comparing the relative intensity ratio of the vibrational transitions associated with different isotopomer levels to the experimental results. In accordance with the observations that the 13Cd16O transition is significantly stronger than the 13Cd18O transition in [12,13] but smaller than the latter in both [11,13] and [11,14], we deduced the signs β12,13 > 0, β11,13 < 0, and β11,14 < 0. The relative intensity ratio of the 13Cd16O transition to the 13Cd18O transition after perturbation predicted by this two-level system analysis, incorporating the calculated angle between the two transition dipole moments in each case, is 3.2 ( 0.3, 0.69 ( 0.04, and 0.40 ( 0.04 for [12,13], [11,13], and [11,14] respectively. It is also manifest in this approach that the diagonal anharmonicities of the two isotopomer levels can be fitted simultaneously with a comparable magnitude (( 1.0 cm-1) to the experimental observables. Though the foregoing separated pair model is conceptually simple, it is oversimplified because the isotopomer levels interact not only with each other but also with the remaining 12Cd16O amides of the helix. This effect may be partially accommodated by incorporating the one- and two-exciton band influence into
J. Phys. Chem. B, Vol. 108, No. 29, 2004 10421 the two-level analysis. Using the simulation results of coupling constants and unperturbed energy separations, we derived the typical exciton band effect on the two isotopomer levels. Inclusion of these energy shifts from the isotope-band interactions into Hs and Hd before matrix diagonalization should generate a more reliable set of coupling constants. As expected, the resulting coupling magnitudes become relatively larger in all cases, with the mean values +12.2, -6.1, and -6.6 cm-1 for β12,13, β11,13, and β11,14 respectively. Because the 13Cd16O level is closer to the one-exciton helical band, it will be pushed down more than the deeper trap 13Cd18O level, resulting in a reduced energy separation between the two isotopic levels compared with the zero-order unperturbed levels. Perturbation Theory Estimates. Although we have carried out a full matrix diagonalization simulation described later, it seemed of interest to test how useful is a perturbation theory approach to understanding the linear and 2D IR spectra. The zero-order isotope shifts are significantly larger than the presumed couplings, so it would not be unreasonable to expect perturbation theory to give useful estimates of the repositioning of the energy levels due to coupling. Theories of isotopic trap spectra72 that incorporate the coupling to exciton bands are wellknown. If the unsubstituted sites are labeled by the index m and the isotopically substituted sites by i and j, the secondorder perturbation energy change ∆(2) i (i,j) in the one-exciton region of the isotopic site i when there is also an isotope substitution at site j is given by 2 ∆(2) i (i,j) ) -δi - 2B /δi +
δjβij2 δi(δj - δi)
(1)
where δi is the magnitude of the isotope downshift at site i. The result depends on the factor B2 ) (β122 + β132 + β142 + etc.) where β12 is the nearest neighbor coupling in a perfect helix, β13 is the next-nearest neighbor coupling, and so on, ranging over all distinct coupling constants of the helix. The parameters of the simulation (see below) yield a value of B2 ≈ 117 (cm-1)2, which causes a -3.49 cm-1 shift for all the 13Cd18O peaks and -5.44 cm-1 for all the 13Cd16O peaks through the second term in eq 1. On the basis of the generated set of parameters used for the simulations shown in Figures 2, 4, and 5, the perturbation theory results are in reasonable agreement with experiment for the various isotopomers. For instance, the 13Cd16O peak positions differ by 2.25 (2.1) cm-1 in [12,13] and [11,13], and 1.34 (2.6) cm-1 in [12,13] and [11,14], where the values calculated from perturbation theory are followed by experimental results in parentheses. Corresponding peak differences for 13Cd18O are -0.93 (-1.2) cm-1 and -0.56 (-3.0) cm-1. Additionally, the peak separations between the two isotopomer levels are 27.25 (28), 24.08 (24.7), and 25.35 (22.4) cm-1 for [12,13], [11,13], and [11,14] respectively. The [11,14] peak separation does not agree as well as the others. Second-order energy shifts for the two-exciton region can also be calculated in terms of the same coupling constants, isotope shifts and the unperturbed diagonal anharmonicities ∆i and ∆j. The perturbation energy change ∆(2) 2i (i,j) in the twoexciton overtone region of the isotopic site i when there is also an isotope substitution at site j is given by (2) (i,j) ) -2δi - ∆i ∆2i
2βij2δj 4B2 + δi + ∆i (δi + ∆i)(δj - δi - ∆i) (2)
10422 J. Phys. Chem. B, Vol. 108, No. 29, 2004
Fang et al. The simulation of the linear-IR and 2D IR spectra was based on the following one- and two-exciton Hamiltonians, H(n) 1 and H(n) , which describe the frequencies and delocalization of 2 amide I′ modes of a helix as 25 coupled oscillators: 25
H(n) 1 )
∑ m)1
25
(m + δm + ξ(n) m )|m〉〈m| +
β(n) ∑ ml |m〉〈l| m*l)1
(4)
H(n) 2 ) 25
∑ (m + l + δm + δl + ξ(n)m + ξ(n)l - ∆mδlm)|lm〉〈lm| + l,m)1 25
∑ l,m)1 Figure 7. Energy level diagram of the complete set of one- and twoexciton energy levels for the [11,13] isotopomer. The parameters used to generate these energy levels are discussed in text. This figure makes it clear that the isotopomer levels are well separated from the band states in the one-exciton region but their two overtones and the combination band are much closer to the two-exciton band states.
In the limit of second-order perturbation theory, the off-diagonal anharmonicity is independent of B2 and depends only on the coupling between the two isotopically substituted residues:
∆ij ) -
2βij2(∆i + ∆j) (δi - δj + ∆i)(δj - δi + ∆j)
(3)
We see that in this approximation the coupling to the band states influence the diagonal but makes no contribution to the offdiagonal anharmonicity. The full simulation will show that this simplification is robust. However, the deep trap limit perturbation theory should be inadequate because the diagonal anharmonicity is similar in magnitude to the separation between the isotopomer levels. One can see from Figure 7, which is a matrix diagonalization of the complete problem using the same parameters, that the combination band state comes very close to the two-exciton band states: the energy distribution of the band states, rather than just their zero-order positions must be important in determining the off-diagonal anharmonicity of the isotopomer pair. We now consider a model for the spectra that does incorporate these couplings in a more reasonable way through a more complete simulation. Numerical Simulations of the Helices. The computations are based on an empirical computational method48 for predicting the linear-IR and 2D IR spectra of amide I′ modes for R-helices of finite length. It incorporates transition charge-transition charge interactions between all except nearest neighbor amides. The nearest neighbor couplings are obtained from density functional theory calculations. The inhomogeneous broadening is incorporated by averaging over Hamiltonians that contain Gaussian random fluctuations of their diagonal frequencies. The input includes atomic coordinates for the helix. The pulse frequency and spectral width were chosen to match those used in the experiment. The homogeneous broadening is chosen to match observations on peptides. The method follows the earlier idea50 that the spectra of nearly degenerate sets of amide I′ modes can be considered without regard to their interactions with other types of modes.
′x2β(n) lm (|lm〉〈mm| + |mm〉〈lm|) +
25
′ β(n) ∑ mp|lm〉〈lp| l,m,p)1
(5)
The m is the vibrational frequency of the transition of the mth amide unit, ∆m its anharmonicity, which only appears in the overtone states |mm〉, and δm its zero-order isotope shift chosen to be zero for 12Cd16O residues, -43 cm-1 for 13Cd16O, and -67 cm-1 for 13Cd18O. The diagonal energy fluctuations {ξ(n) m }, for the nth helix of a distribution, are sampled from Gaussians with standard deviation σ. The diagonal elements of the interaction term, and the effects due to H-bonding that shift the amide I relative to amide I′, are assumed to be incorporated into the parameters. The ∑′ in eq 5 means omitting terms with equal indices. The β(n) ml are the intersite interaction terms. A perfect R-helix model with the dihedral angles of (φ ) -58°, ψ ) -47°, ω ) 180°) is used and fluctuations in the off-diagonal interaction terms are not included in the displayed simulations, although we could estimate the effect of such disorder (see below). The 13C natural abundance contribution is also included by randomly sampling the frequencies from a pool containing 1.1% of molecules with the 13Cd16O zero-order shift. For the nearest neighboring amide units, the coupling is evaluated by ab initio density functional theory (DFT) optimization and normal-mode analysis at the B3LYP/6-31+G** basis level on glycine dipeptide (Ac-Gly-NMe) in the configuration of the R-helix.48 The two computed symmetric (higher frequency) and antisymmetric amide I′ normal modes are linear combinations of two almost equally weighted local amide modes. Therefore, the intermode coupling is +7.5 cm-1. This result computes the total through-bond plus through-space contributions to the coupling. For the amide unit pairs beyond the nearest neighbors, the coupling is evaluated by transition charge-transition charge interactions including charge fluxes. We obtained a complete set of model coupling constants, of which some of the largest ones are β12 ) +7.5, β13 ) -4.7, β14 ) -6.0, β15 ) -0.5, β16 ) -0.7, β17 ) -0.8, and β18 ) -0.3 cm-1, where β12 means nearest neighbor, β13 is next to nearest neighbor, and so on. As the spatial separation between amide units gets larger, the transition charge-transition charge interaction scheme reduces to the transition dipole-transition dipole coupling. The form of eq 5 assumes that the frequency fluctuations of transitions from the ground state to one excitons and that from the one excitons to the overtones are strictly correlated. The correlations between different sites can be incorporated numerically to account for correlated energy fluctuations as needed. The simulation of the linear-IR and 2D IR spectra can be performed directly in the frequency domain if a homogeneous width parameter γ is assigned to the amide I′ mode. This amounts to assuming Bloch dynamics in the simulation model.
Isotopomers of an Alanine Rich R-Helix
J. Phys. Chem. B, Vol. 108, No. 29, 2004 10423
The homogeneous spectral line width γ ) 5.5 cm-1, and the inhomogeneous broadening parameter σ ) 4 cm-1, are chosen on the basis of experimental measurements of linear-IR and 2D IR spectra. For the results shown in Figures 2, 4, and 5, the zero-order diagonal anharmonicity was chosen as 15.0 cm-1 for all isotopomers. The 13C natural abundance contribution (1.1%) to the spectra has been taken into account in all cases. The calculated 2D IR spectra are convoluted with a ∼150 cm-1 fwhm Gaussian spectrum centered at 1580 cm-1 to mimic the effect of the laser pulse spectrum. The 2D IR spectra S(-ωτ,ωt,T) consist of rephasing (with -ωτ) and nonrephasing (with +ωτ) parts. Single exciton states, which are the 25 eigenstates of H1, are labeled with a lower case k; and the overtones and combinations labeled by an upper case K (K ) k + k′), are the 25 × 24/2 eigenstates of H2. The overall profile of the 2D IR spectrum is determined by contributions from the orientational pre-factor of the vibrators and the signal strength. Expressions for 2D IR spectra in this Bloch limit can be written as
SR(-ωτ,ωt) )
〈
∑ k,k′ [-i(-ω
z0kz0k′zk0zk′0
4
2
k0
- ωτ) - γ][i(ωk′0 - ωt) + γ]
∑ k,k′,K [-i(-ω
SNR(+ωτ,ωt) )
〈
z0kz0k′zk′KzKk k0
- ωτ) - γ][i(ωKk - ωt) + γ]
∑ k,k′ [-i(ω
2
k0
k0
∑K zKk′z0kzkKzk′0
+
-
z0k′z0kzkKzKk′ k0
(6a)
- ωτ) - γ][i(ωk0 - ωt) + γ]
- ωτ) - γ][i(ωk′0 - ωt) + γ]
∑ k,k′,K [-i(ω
〉
z0k′z0kzk′0zk0
zk0z0kz0k′zk′0 +
∑ k,k′ [-i(ω
-
- ωτ) - γ][i(ωKk′ - ωt) + γ]
〉
(6b)
where the numerators are the projections onto the z-axis of the laboratory of the transition dipoles, between states indicated by the subscripts, calculated from the eigenvectors of H1 and H2. The formula applies specifically to the case, as in the experiment, where the three incident electric field polarizations are parallel. A more general form of these equations is given in a recent report.48 The overall ensemble average (angle brackets) was carried out over 5000 equally weighted 25-unit helices to realize the inhomogeneous broadening and accompanying localization of the excitations. The total 2D IR rephasing signal in eq 6a can be viewed in two parts: one part includes only the |0〉 f |k〉 transitions and one includes both |0〉 f |k〉 and |k〉 f |K〉 transitions. The first term contributes to the positive peaks on diagonal and off diagonal, and the second term contributes to all the negative peaks. The total 2D IR nonrephasing signal in eq 6b can be viewed in three parts: the first part includes only the |0〉 f |k〉 transitions, which contribute to the positive diagonal peaks; the second term contributes to the positive peaks on diagonal (when k′ ) k) and off diagonal (when k′ * k); and the third term contributes to all the negative peaks. Finally, as in 2D NMR,16 the absorptive and dispersive 2D IR spectra can be constructed by taking linear combinations of rephasing and
nonrephasing terms, as described in detail for 2D IR68 and recently used in some examples.15,73 The simulations require a number of input parameters. Although we did measure the anharmonicities of model amides having isotopic substitution, there can be no guarantee that these values are transferable to the alanines of the helix. Therefore, some variations in these parameters are to be expected. The same situation occurs for the zero-order isotope shifts, which are not expected to be transferable with high accuracy from those of model compounds having different force fields. Furthermore the alanines in the helix surely interact over longer ranges than considered here. Our approach is to lump together the remaining interactions, beyond 1-4, and estimate them from the transition charge or dipole interactions. This will also not be exact, even though the distances are large, because of polarization effects that are well-known in the theory of excitons in the optical regime. However, we will show below by simulation that 2D IR spectra of a particular isotopomer is dominated by the coupling of the two isotopically substituted amide transitions, lending support to this approach. The simulated linear-IR spectra, 2D IR absolute magnitude rephasing spectra and 2D IR absorptive spectra are shown in Figures 2, 4, and 5 along with the experimental results. The simulation was not intended to be a least-squares fit to experiment; rather it uses parameters that were independently optimized for helices of this type. There are too many unknowns to justify least squares variations of all the couplings, diagonal energies, isotope shifts, and anharmonicities. Therefore, the general features of the spectra are being compared rather than the precise numerical values of the results. The simulation of linear-IR spectra successfully reproduces the energy separations between the transitions of bulk and isotopes and that between two isotopes implying that the coupling constants of the calculation are in the correct range of magnitude and sign. For example, the 13Cd16O peak position difference between [12,13] and [11,13] is +2.1 (+2.2) cm-1, where the simulation result is in parentheses. This difference between [12,13] and [11,14] is +2.6 (+1.0) cm-1. The value for the 13Cd18O peak is -1.2 (-0.8) cm-1 between [12,13] and [11,13], and -3.0 (-1.7) cm-1 between [12,13] and [11,14], respectively. In addition, the peak separations of 13Cd16O and 13Cd18O in [12,13], [11,13], and [11,14] are 26.9, 23.9, and 24.3 cm-1 in simulation, whereas in the experiment they are 28.0, 24.7, and 22.4 cm-1. Besides these fairly close agreements on the isotope transition peak positions, the intensities of the linear-IR spectra from the full simulation are similar to the observed values, confirming that the signs and magnitudes of the computed couplings given above are reasonable: the ratio of the integrated intensity of the 13Cd16O region to that of 13Cd18O is found to be 3.0 (4.6) for [12,13], 0.68 (1.3) for [11,13], and 0.48 (0.9) for [11,14] where again the experimental values are in parentheses. The ambiguities of the baseline subtraction in the FTIR spectra cause the 13Cd18O amide I′ intensity to be underestimated. By incorporating the 13C natural abundance contribution, the observed peak of [0,11] at 1596 cm-1 is well reproduced (Figure 2a). The major difference between the simulation and experiment in the linear-IR spectra lies in the total width of the bulk helix: the simulated curve is narrower than the experiment in every case. The experimental 2D IR rephasing magnitude spectra are compared with simulations in Figure 4. The general features of the simulated 2D IR resemble the measurements quite reasonably. The diagonal peak positions of the bulk helix and the isotopes are reproduced by the simulation. A stick diagram of
10424 J. Phys. Chem. B, Vol. 108, No. 29, 2004 the complete set of calculated states for [11,13] is shown in Figure 7, which is typical for all the substitutions. The simulation was based on the choice of the zero-order frequency of the bulk helix and the isotope shift of 13Cd18O, which were obtained by matching the simulation to the experimental spectra of [0,11]. This figure also shows the significant spread of the two-exciton states toward the region of the isotopomer overtones and combination band, which dramatizes the approximate nature of “separated pair” modeling. The simulated 2D IR absorptive spectra are given in Figure 5 for comparison with experiment. The ratio of the nonrephasing to the rephasing components used in generating the absorptive spectra is 1.8, which is only slightly different from the factor of 2.3 used to scale the measured R and NR spectra. However, the relative intensities of signals from 13Cd18O and 13Cd16O transitions appear to be similar between simulation and measurements. Detailed comparison shows that the simulated off-diagonal anharmonicities are in the correct order but significantly smaller than the experimental values. Due to the coupling scheme in the simulation, the diagonal anharmonicities are predicted to be slightly less for 13Cd16O than 13Cd18O, which is not what was observed experimentally. These results suggest that the unperturbed diagonal anharmonicities are somehow different for the two isotopomers in the helix. To address these issues and test the sensitivity of the simulated off-diagonal anharmonicity to the parameters being used, we systematically varied certain parameters to investigate their effects. When the zero-order diagonal anharmonicity (set to be equally 15.0 cm-1 for all the CdO groups’ amide I′ modes) is increased by up to 2.0 cm-1, the overtones and combination band energy level of two isotopes change significantly: the largest downshift is observed for the 13Cd18O overtone and the smallest downshift is obtained for the combination band. The 13Cd16O zero-order diagonal anharmonicity would require it to be increased by 10% to generate a ∼0.4 cm-1 increase of the resulting off-diagonal anharmonicity and a ∼1.0 cm-1 increase of the 13Cd16O diagonal anharmonicity. The coupling constants can also be varied in sequence to explore their separate effects on the calculated frequencies. We find that the offdiagonal anharmonicity of [12,13], [11,13], and [11,14] is mainly dependent on the coupling constant associated with the pair of isotopomer levels, and only weakly dependent on the remaining coupling constants. This useful result is illustrated in Figure 8, which shows the variation of the off-diagonal anharmonicity for each isotopomer with the changes, δβ, in each of the three coupling constants considered. The position δβ ) 0 corresponds to the choice of coupling constants listed above from the simulation, but the results are not sensitive to which set of coupling constants in this range is used as the zero within reasonable limits. The slopes d∆ij/d(δβkl) at δβ ) 0 are very informative. In every case the slope for kl ) ij is significantly larger than that for other choices of β . For instance, for [12,13] the slope is 3.6 times larger for β12 than for β14, whereas the result is nearly independent of β13. Similarly, the off-diagonal anharmonicity of [11,13] has a major dependence on the variance of β13, but only a minor dependence on the variance of β14 with a slope 3.3 times smaller in magnitude, and a negligible dependence on the variance of β12. The off-diagonal anharmonicity of [11,14] mainly depends on β14, whereas the slopes for its dependences on β12 and β13 are 4.0 and 3.6 times smaller, respectively. Therefore, it is quite general that as the coupling constant directly relating to the isotopomer pair is increased, the associated off-diagonal anharmonicity will be increased. But increasing the other two coupling constants will only slightly decrease that off-diagonal anharmonicity. These
Fang et al.
Figure 8. Simulation results for off-diagonal anharmonicities of different isotopomer pairs varying as a function of β12, β13, and β14 deviating from their original values. The crossing point for the three curves in each graph represents the set of original values used to generate the 2D IR spectra: β12 ) +7.5, β13 ) -4.7, and β14 ) -6.0 cm-1 with the zero-order diagonal anharmonicity 15.0 cm-1 for all the amide I′ modes. Along the δβ axis, coupling constants assume smaller numerical values to the left of the crossing point.
results explain the approximate result of the second-order perturbation theory that the off-diagonal anharmonicity only depends on that specific pair’s coupling constant, as shown in eq 3. These sensitivities allow us to vary the magnitudes of the coupling constants in the range of the original calculated set to obtain a better fit with the observed off-diagonal anharmonicities. Signs of the Coupling Constants. The signs of the coupling constants, which were mentioned a number of times in the foregoing text, appear to be very well determined by the relative intensity data as computed by the simulation. Changing the signs alters the intensity ratios of the pairs of isotopomers dramatically in each of the compounds studied. Presumably, this situation arises because of the sensitivity of that pair’s eigenvectors to the coupling constant between the isotopically substituted pair, which results in even a simple model yielding the same predictions as from the full simulation. This result is very useful and allows us to read off the signs of the coupling constants either from the linear-IR or from the 2D IR spectral intensities, considering just the linear spectra and confining attention to the pair of isotopically substituted residues, which we will label 16 and 18 to represent the amide I′ mode v ) 1 states of 13Cd16O and 13Cd18O, respectively. A useful function of the intensities of the transitions to the 13Cd18O and 13Cd16O substituted amide modes is
I16 - I18 ) µˆ 18‚µˆ 16 sin θ cos φ I16 + I18
(7)
where φ is a phase that must be 0 or π if it is assumed that the coupling is real: β ) |β|eiφ. The angle θ lies in the range 0 eθ < π so sin θ, a function of the energy gaps and coupling squared,74 is positive definite. Therefore, the sign of the coupling is obtained from
sign[β] ) sign[(I16 - I18)µˆ 18‚µˆ 16]
(8)
where µˆ 18‚µˆ 16 is the scalar product of the unit transition dipoles of the amides. If all the transition dipoles were directed along
Isotopomers of an Alanine Rich R-Helix
J. Phys. Chem. B, Vol. 108, No. 29, 2004 10425
Figure 9. Comparison between the experimental and simulated transitions for the two isotopomer levels of the R-helical chain, in the one- and two-exciton regions. The experimental frequencies are in the left column of each example: the width of the line is indicative of its error bar. The simulation results are shown as thin lines on the right side of each example. The dashed lines represent the average frequency of the isotopomer pair in the one-exciton region and the average of the two overtone states in the two-exciton region, respectively.
carbonyl groups that are close to the helix axis, then the scalar product would be positive and greater than 0.9 for all isotopomers. This result is robust and not changed if the transition dipoles are redirected to ca. 20° to the carbonyl group, at which orientation all the cosines are still greater than 0.42 for a wide range of dihedral angles. This latter choice of transition dipole direction is consistent with the charge distribution and with previous work.44 Therefore, with reference to eq 7, the 13Cd18O mode transition is strengthened over 13Cd16O when φ ) π. This is the case for [11,13] and [11,14] but clearly φ ) 0 for [12,13]. These results dramatize how structural information can be obtained by inspection from the transition intensities. A similar argument can be made in the 2D IR intensities and of course both of these qualitative yet robust approaches are verified by the full simulation. The 1-3 and 1-4 interactions are negative whereas 1-4 is stronger than 1-3 even though the intermode distance has increased. These features arise from the angular part of the intermode potential. For example this would be the case for a dipole-dipole interaction. A summary of the comparison between simulation and experiment is given in Figure 9, which shows only the energy level locations for the isotopomer fundamentals, their overtones, and the combination band. By increasing the three major coupling constants β12, β13, and β14 by about 12% each from the original calculated set, the frequencies are brought close to the values estimated by fitting the experiment to a separated pair model. Figure 9 demonstrates a very reasonable agreement between experiment and simulation in both the one-exciton and two-exciton regions. The resulting off-diagonal anharmonicities are in experimental order and within better than a factor of 2 of experiment: ∆12,13 ) 4.8, ∆11,13 ) 1.8, and ∆11,14 ) 2.1 cm-1. The coupling constants used in this comparison of theory and experiment are β12 ) +8.5 cm-1, β13 ) -5.4 cm-1, and β14 ) -6.6 cm-1. We have not attempted to least-squares fit the experiment and simulation to variations of these coupling constants because the result depends also on all the other coupling constants, the diagonal anharmonicities, and the isotope shifts, all of which have some uncertainties. However, we do infer that the calculated coupling constants are within a few cm-1 in magnitude and have the same sign as those determining the spectra. Figure 9 also shows the simulated average frequency of the isotopomer pair in the one-exciton region and the average of the two overtones in the two-exciton cases. The one-exciton
spectra are in good agreement. Those overtone averages are extremely well determined in the 2D IR experiment because they represent the points halfway between the two main isotopomer peaks representing the v ) 0 f v ) 1 and v ) 1 f v ) 2 peaks of the 2D IR spectra. To a good approximation they are not dependent on the spectral shapes or on small changes in the intensities of these overlapping transitions. The experimental averages agree reasonably with the simulation. However, the combination band from the simulation is always found at slightly higher frequency than the experimental value, resulting in consistently smaller off-diagonal anharmonicities being calculated than those that are observed. The three major coupling constants β12, β13, and β14 have been determined by various model approaches, and their magnitudes are consistent within the few cm-1 range. Leastsquares fitting of the full matrix diagonalization simulation to the experimental 2D IR spectra might provide a slightly improved set of couplings but that fitting is also sensitive to all the couplings in B2 and to the other zero-order parameters. The evaluation of the sensitivities of the results to various input parameters leads to what we will call a preferred set of absolute values and their associated error bars, which include the uncertainty of the measurement: |β12| ) 8.5 (1.8, |β13| ) 5.4 ( 1.0, and |β14| ) 6.6 ( 0.8 cm-1. The signs are independently determined as given above. These results are in reasonable agreement with the computed coupling constants given earlier for a particular pair of dihedral angles. However, in real systems there is a range of dihedral angles that should be included in the simulation, so exact agreement between the calculation and the observation is not expected. For example, if dipeptides with dihedral angles in the range of {-70° e φ e -50°, -60° e ψ e -40°} are considered, the standard deviation of the distribution of calculated β12 values is estimated to be 2.3 cm-1, on the basis of a set of DFT results.48 The standard deviations of β13 and β14 calculated via the transition charge-transition charge interactions are estimated to be 2.1 and 2.0 cm-1 for this same range of structures. Further Comparison between Experiment and Simulation. It can also be seen from Figure 4 that the simulated bulk diagonal peaks are not sufficiently elongated along the diagonal of the 2D IR spectra and the peak intensities of the isotopomer transitions are stronger than those simulated. These differences arise directly from the modeling of the bulk helix states, which excluded a number of potentially important effects such as the
10426 J. Phys. Chem. B, Vol. 108, No. 29, 2004 imperfections in the helix structures, especially at the C-terminal end, and spatial correlations in the couplings and energy fluctuations. The nonhelical segment at the C-terminus in the R-helix, which is known to be more or less random-like and solvated differently from the main helix portion, is very likely to be the source of the high frequency, nonhelical component in the linear spectra. Its isotopomer transitions are presumably buried under the main helical band and not identifiable in the experiments. A real R-helix contains a distribution of dihedral angles and interamide distances, which is not likely to be properly modeled by Gaussian fluctuations of the diagonal frequencies. We did explore the effects of a distribution of dihedral angles, which introduces disorder into the coupling matrix elements and has the effect of broadening the helix transitions for reasonable choices of distributions obtained from the protein data bank. However, we do not yet have a reliable relationship between the dihedral angle and the diagonal energy. Furthermore, correlations among the transition frequency distributions of different amide units are not considered explicitly in the simulation. The shapes of the diagonal peaks in the 2D IR absorptive spectra shown in Figure 5 suggest different correlation factors for the 13Cd18O and 13Cd16O transition frequency fluctuations. Even with all these omissions from the simulation, the general features of the experiments are reproduced. There seems little doubt that the skeleton of the mode structure is determined by a set of coupling constants between the units that must be quite similar to those computed. Furthermore the dynamics must rather closely follow those for a fixed inhomogeneous distribution of vibrational frequencies. We also used the simulation of the 2D IR spectra to compute the peak shifts of the bulk helix and isotopes at particular values of ωt. This amounts to computing the partial spectra S(τ,ωt,T) corresponding to the Fourier transform of the signal S(ωτ,ωt,T) along ωτ. The signals peak at values of τ given by 120, 120, and 350 fs for 12Cd16O, 13Cd16O, and 13Cd18O, respectively, and these results are similar for [12,13], [11,13], and [11,14]. In recent experiments, not reported here, we have measured peak shifts in a similar range, confirming that the magnitudes of the dynamical parameters and inhomogeneous broadening used in the model are reasonable. However, we have not yet been able to evaluate the validity of the assumption of Bloch dynamics incorporated into the simulation method. The shapes of S(τ,ωt,T) at one waiting time, T ) 0, are obtained in the present work and the peak shifts are in the few hundred femtoseconds regime, qualitatively similar to those from the simulation. But to establish these subtle line shape changes requires measurements of the 2D IR spectra as a function of the waiting time T, which is now being addressed in detail in this laboratory. Through these types of experiments the dynamics of different isotopomer levels can be measured accurately. We expect that their local environments may be influenced by hydrogen bonding with the solvent and by the shielding effect of the neighboring lysine side chains.75 Reference to Figure 1 shows that the nearest neighbor interactions do not involve any intervening polarizable medium to modify the through-space interactions. The couplings deduced from 2D IR agree well with those from VCD66 and linear-IR measurements76 and they verify the simple transition charge interaction models.48 The power of 2D IR is that it also provides information on the dynamics of helical and other systems in solutions. The relationship between structural constraints and 2D IR spectroscopy is strengthened by these results. If the coupling β is approximately a dipole interaction, the interaction distance becomes quite well defined inasmuch 3 as it is proportional to x1/|β|. The isotopic substitution
Fang et al. strategy introduced here provides a platform from which to determine the helix backbone conformation. A next step would be to modify the labeling positions, but with fixed label separations, to obtain structural information over the complete chain at the residue level. The simulations also show clearly the couplings between the isotopomers and helix excitons. The spread of these coupling regions along the ωτ axis directly manifests the distribution of exciton states along the diagonal of the helix region and again proves that the helix states are delocalized excitons. According to the simulation these cross-peaks should be spread over a width of ca. 40 cm-1, which is in the range of what is observed. A more detailed study of this feature should prove interesting. Conclusions We succeeded in measuring the 2D IR (rephasing, nonrephasing, absorptive and pump-probe) spectra of a series of doubly isotopically substituted 25-residue alanine rich helical structures. The presence of 13Cd16O and 13Cd18O labels at known residues in the helix chain enabled the vibrational couplings between different amide I′ modes to be measured at the residue level. The couplings between the two isotopomers and between them and the helix exciton band states were clearly identified as significant cross-peaks in different spectral regions. The experimental results were compared with several model analyses including a matrix diagonalization simulation based on averaged Hamiltonian matrices representing the amide I′ vibrator one- and two-exciton manifold of states. The essential experimental spectral features in both the linear-IR and 2D IR spectra agreed reasonably with a set of coupling constants that were derived from transition charge-transition charge interactions for all but the nearest neighbors, for which the coupling was modified by through-bond interaction between the adjacent amide groups. The magnitudes of the couplings incorporated in this numerical calculation were very similar to those deduced from the separated pair modeling of the experimental data, and their signs were consistent with expectations for the isotopomers’ transition intensities in the spectra. The relative intensities of the vibrational transitions in the linear-IR and 2D IR spectra depended upon the spatial separations of the amide modes, the angles between the transition dipole moments and the signs of the associated coupling constants. Second-order perturbation theory estimates showed reasonable agreement with experiment; the dependence of the off-diagonal anharmonicity of an isotopomer pair on principally that specific pair’s coupling constant was verified and analyzed. Though we did not least-squares fit the linear-IR and 2D IR data, the sensitivities of the output off-diagonal anharmonicities on various input parameters such as coupling constants and zeroorder diagonal anharmonicities were investigated explicitly. On the basis of a variety of theoretical approaches and detailed comparisons for the series of isotopomers under study, we obtained the magnitudes and signs of the three major coupling constants β12, β13, and β14. The results clearly showed that the 2D IR signal of one isotopically substituted residue out of 25, representing an optical density of ca. 0.01, could be experimentally observed and studied. Our analysis of the signals with the signal-to-noise level of the data in Figure 4 indicated that one isotopic impurity site in a 35-residue system could have been detected and studied. These conditions meet the requirements for the application of this 2D IR method to single residue substituted small proteins as a function of time following a perturbation such as a temperature jump.
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