Dynamics of Amide-I Modes of the Alanine Dipeptide in D2O - The

Publication Date (Web): March 1, 2005 .... The Journal of Physical Chemistry B 0 (proofing), .... Accounts of Chemical Research 2007 40 (1), 75-83 ...
1 downloads 0 Views 315KB Size
Download PDF
6884

J. Phys. Chem. B 2005, 109, 6884-6891

Dynamics of Amide-I Modes of the Alanine Dipeptide in D2O† Yung Sam Kim and Robin M. Hochstrasser* Department of Chemistry, UniVersity of PennsylVania, Philadelphia, PennsylVania 19104-6323 ReceiVed: NoVember 4, 2004; In Final Form: December 23, 2004

The equilibrium dynamics of the acetyl and amino amide-I groups of the alanine dipeptide were examined separately using 13C isotopic selection and 2D IR. The population relaxation times of the amide transitions were measured to be in the range 500 fs by means of heterodyne transient grating methods. The vibrational frequency correlation functions consisted in all cases of a motionally narrowed part, a component near 800 fs, and a constant part representing a distribution of structures that is static on the few ps time scale. The intermediate time scale is attributed to fluctuations in the stretching and bending of hydrogen bonds between the carbonyl and water.

1. Introduction In protein secondary structures the amide carbonyl group is very often involved in hydrogen bonding either to water or to N-H groups or to both. Although a vast amount of information exists on the average structures of peptides, too little is known from experiments about the dynamics of these hydrogen bonds for a wide range of CdO environments. The frequency of the CdO group is known to be sensitive to both H-bonding and to the local secondary structure. Therefore, at equilibrium, the fluctuations in CdO frequency must be sensitive to fluctuations of these properties. Indeed any of the solvent or internal motions that couple to the mode should contribute to the vibrational frequency correlation function. Stimulated IR echo decays of a small amide, N-methylacetamide,1-3 and various peptides and proteins4 have shown that fast dephasing mechanisms are prominent in the smaller amides and that large inhomogeneous distributions dominate the larger polypeptides. One thing that has become clear is that the traditional Bloch dynamics notion of vibrational dynamics being characterized by two parameters T1 and T2 is not generally applicable to peptides.5 The faster decays of the photon echoes of aqueous carbonyl systems have been attributed to the water dynamics by theoretical modeling.6 There have been many experimental and theoretical studies aimed at understanding the factors that determine the conformations adopted by dipeptides in various solvents,7-13 especially in water. The alanine dipeptide is one of these model systems that often have been considered. In general, it is believed that the solvent polarity is a key factor in determining the hydrogen bonding character and hence the structure of the peptide. In a polar solvent, such as water, the existence of external hydrogen bonds is possible but not in a nonpolar solvent, such as CCl4. Combined CD and NMR,7 13C NMR,10 density functional theory,9 NMR,13 and 2D IR14 experiments agree that the polyproline-like (PII) structure of the alanine dipeptide is present to varying degrees in aqueous solutions. The equilibrium dynamics of peptide structure in association with water are closely linked with their structural propensities so further investigations of these dynamics are essential. †

Part of the special issue “David Chandler Festschrift”. * Corresponding author. Tel: 215-898-8203. Fax: 215-898-0590. E-mail: [email protected]

Recent experimental15-17 and theoretical18 developments in multidimensional infrared spectroscopies and 2D IR, which use pump/probe techniques and time and spectral domain interferometry, have provided ways to explore the equilibrium dynamics of peptides and proteins. The extension to 3D IR records these echo signals along a waiting time axis (T) and provides information on the decay of the frequency correlations of the vibrators in analogy with stimulated photon echo experiments on electronic transitions.19 Such measurements on peptide amide-I IR spectra relate directly to the interaction of the carbonyl groups with water. In this paper we present a study of the alanine dipeptide in D2O using multidimensional infrared spectroscopies to examine two different 13C labelings of the dipeptide to spectrally separate the otherwise overlapping amide modes. By substituting 12Cd O with 13CdO in an amide group we lower the transition frequency of the amide-I mode by ca. 40 cm-1.20 In the present work the aims are to establish features of the equilibrium dynamics of the alanine dipeptide in water. The dipeptides present a new situation because the various amide group contributions are separated in the spectral domain and the 2D IR spectroscopy can be configured to determine the dynamics at different spatial locations in the molecule. This paper is also concerned with establishing the differences that occur in the echo responses from a distribution of states that are spectrally separated compared with those of single oscillators. 2. Materials and Methods The 2D IR procedures and data processing have been described in more detail elsewhere.14 Briefly, the spectra were obtained using heterodyned spectral interferometry.16,20,21 Three 300 nJ infrared pulses with wave vectors k1, k2, and k3 were incident on the sample. The interval between pulse 1 and 2 is denoted as τ, between 2 and 3 as T, and between 3 and the detected signal as t. The beam k1 arrives earlier than k2 by an amount τ in the rephasing scheme, while k2 arrives earlier than k1 by τ in the nonrephasing scheme. In both cases k3 arrives last in our experiments. Below, the rephasing and nonrephasing configurations are denoted R and NR, respectively. The phasematched signal at wave vector -k1+k2+k3 is detected as a function of the time intervals. The local oscillator pulse always preceded the signal pulse by a fixed interval of ∼1.5 ps. The

10.1021/jp0449511 CCC: $30.25 © 2005 American Chemical Society Published on Web 03/01/2005

Equilibrium Dynamics of the Alanine Dipeptide signal and local oscillator pulses were combined at the focal plane of a monochromator having a 64-element MCT array detector (IR Associates, Inc.). The raw data collected using this method were in the form of a two-dimensional array of time, in 2 fs steps, and wavelength, in ∼6 nm steps. For example in the rephasing case, the columns and rows of the array at each choice of waiting time T are the time delays τ and wavelengths, respectively. The objective of the experiment is to measure the third-order infrared response of the sample to the three incident pulses. This response is the signal, in the direction specified, from incident delta function pulse excitations. It is a real function of three time intervals S(τ,T,t) that can be Fourier transformed on any of the time axes to produce complex spectra. In reality the signal is S(τ,T,t) convoluted with the infrared field envelopes of the pulses. This convolution looks mathematically different for the different Fourier transforms of the response. As a shorthand, the signals that are being measured and manipulated will be symbolized by S(τ,T,t)*E as a reminder that they are convolutions. Later the theoretical delta responses will be convoluted with the incident pulses for comparison with the experimental dynamics. The raw data is in the form of S˜ (τ,T,λk)*E, which was converted into S˜ (τ,T,ωm)*E, where λk is the wavelength of the kth (k ) 1...64) detector element and ωm (written as the continuous variable ωt below) corresponds to 400 evenly spaced frequencies. Using procedures described by Joffre,22,23 the processed signals S˜ R(τ,T,ωt)*E and S˜ NR(τ,T,ωt)*E were obtained by Fourier transformation of the S˜ signals along ωt followed by back Fourier transformation of only the positive frequency component. The complex 2D IR spectra SR(-ωτ,T,ωt)*E and SNR(ωτ,T,ωt)*E were obtained by Fourier transformation along τ. The center frequency of the incident pulses was ∼1620 cm-1, the intensity median of the amide-I absorption region spectrum being examined. All the experiments were carried out in D2O solutions. Neat D2O exhibits a weak, pulsewidth-limited, presumed to be nonresonant contribution of 0.3% when all three pulses overlapped at the sample. When T is increased a background signal from an induced absorption grating of D2O begins to appear. This signal is attributed to a thermally induced grating where k3 is diffracted into the direction -k1+k2+k3. It becomes significant (ca. 2% of the signal) at T ) 500 fs and increases in magnitude throughout the experimental time window of T ) 5 ps. Although this thermal grating signal is much smaller than the photon echo from the alanine dipeptide solution within the time range considered, it is very useful for the accurate determination of the time interval between the k1 and k2 beams. By maximizing the grating signal as a function of τ at a large value of T, where the pulse k3 is well separated from the others, the location of τ ) 0 can be determined to an accuracy of ca. 1 fs, which is 5% of a cycle of IR radiation at 1650 cm-1. Samples. The two isotopically labeled alanine dipeptide samples were purchased from Synpep Corporation and the isotopically unlabeled alanine dipeptide sample was purchased from Bachem. They were used without further purification. The concentration of the samples was ∼70 mM in D2O and the optical density of the samples for the amino-end peak was ∼0.14 OD using a 25 µm path length. The cell material was CaF2. The sample contained some trifluoracetic acid (TFA) from the synthetic procedure. Based on our measurements, the mole fractions of TFA molecules in isotopomer I (acetyl-end labeled) and II (amino-end labeled) were 7% and 5% respectively of those of the alanine dipeptide. The extinction coefficient of TFA is max ∼1800 cm-1 M-1, and that of the alanine dipeptide is

J. Phys. Chem. B, Vol. 109, No. 14, 2005 6885

Figure 1. (a) FTIR spectra of the alanine dipeptide in D2O. A: unlabeled; B: amino-end labeled; C: acetyl-end labeled. (b) Schematic diagram representing center frequencies of the transitions for three isotopomers. (c) A pictorial model of alanine dipeptide at its PII structure.

max ∼800 cm-1 M-1. The heterodyned photon echo signal varies as the product of the extinction coefficient and the optical density. 3. Results and Discussion FTIR Spectra of the Alanine Dipeptide. The linear IR spectra of the alanine dipeptide and its isotopomers each show two transitions in the amide-I region as shown schematically in Figure 1. This figure also shows a pictorial model of the alanine dipeptide at its PII structure. In all the spectra there is an additional weak band at 1674 cm-1 from the TFA that is not represented in the scheme of Figure 1 but which shows up clearly in all the 2D IR spectra. The unsubstituted molecule shows only one asymmetric absorption band that fits well to two overlapped Voigt profiles represented by the lines at 1629 and 1642 cm-1 in Figure 1. The assignment of these transitions to the acetyl-ends (N-terminal) and amino-ends (C-terminal) of the peptide is made obvious from the isotopomer spectra. The 1642 cm-1 band undergoes a 40.5 cm-1 shift, while the 1629 cm-1 band shifts by only 2.5 cm-1 when the amino-end is isotopically substituted with 13Cd16O (isotopomer II). Isotopic substitution of only the acetyl group (isotopomer I) leads to an isotope shift of 40 cm-1 of the 1629 cm-1 band, while the 1642 cm-1 band is shifted by only 1 cm-1. The linear spectra are consistent with two approximately localized modes separated

6886 J. Phys. Chem. B, Vol. 109, No. 14, 2005 by 13 cm-1, each with about the same isotope shift of 41 ( 1 cm-1 so that the unsubstituted form has amide modes that are approximately localized on the ends of the peptide. The peak intensity ratios of amide-I modes of the amino-end to acetylend of the three alanine dipeptides were 1:0.935, 1:0.905, and 1:0.870 for the unsubstituted compound and isotopomers I and II, respectively. The coupling between modes is apparently not responsible for these intensity variations.14 The intensity differences are not solely caused by amide-I mode coupling but arise from other terms in the force constant matrix. 2D IR Spectra Mechanisms. The nonrephasing and rephasing contributions to the third-order nonlinear response that describes the multidimensional spectra of vibrators have been described previously.24-26 Usually the responses are subdivided into the various Liouville pathways that contribute to the signals, but in this paper a rather simple description in terms of the relevant third-order density matrix elements is given. In the ensuing discussion the fundamentals of the acetyl- and aminoend oscillators are labeled as 1 and 2, their overtones as 1+1 and 2+2, and the combination as 1+2. In the rephasing process the first pulse k1 generates sub-ensembles of molecules having coherences F01(1) or F02(1) which are allowed to evolve for time period τ. The second pulse k2 creates from these coherences the populations F00(2), F11(2), and coherence F21(2) in the first ensemble and F00(2), F22(2), and coherence F12(2) in the second, which are examined during the waiting time period, T. Through allowed IR transitions undergone by the six resulting groups of molecules, the third pulse k3 creates the coherences F10(3), F20(3), F1+1,1(3), F1+2,1(3), F2+2,2(3), and F1+2,2(3), which each contribute to the third-order macroscopic polarization and to the field radiated at their characteristic frequencies, indicated by their subscripts, leading to the spectrum distributed along the ωt axis. The signal is detected in the direction -k1+k2+k3. The nonrephasing processes lead to these same six coherences but they arise from the coherences being created in the τ period by the k2 pulse arriving at the sample before k1. The dynamics of the frequencies are described by means of the relaxation functions defined by Mukamel.27 The population relaxation times are introduced into the responses as the values measured by heterodyned transient grating experiments. The 2D IR spectra are composed from a set of peaks on the ωτ-axis, corresponding to those seen in the linear spectrum, that become spread out on the ωt-axis to create peaks at the oscillation frequencies of all the coherences mentioned earlier. Summary of the 2D IR Spectra. This paper is mainly concerned with the dynamical processes but the 2D IR spectra of the alanine dipeptide14 will first be summarized. Figure 2 shows the spectra at T ) 0. The V ) 0 f V ) 1 and V ) 1 f V ) 2 transitions of each of the isotopomer bands are clearly observed in these absorptive spectra after the phase was properly adjusted by comparisons of the R and NR spectra with the pump/ probe spectra obtained by pumping with one pulse and dispersing another transmitted probe pulse onto the array detector. To obtain the spectra in Figure 2 we combined S˜ R(τ,T,ωt)*E and S˜ NR(τ,T,ωt)*E along the τ-axis then Fourier transformed the combined data along the τ-axis. In Figure 2 the dashed lines do not pass through the center of each peak, which is due to the finite pulse width instead of delta pulse. The weak crosspeaks in compound II are more apparent than are any crosspeaks in I. These results, when compared with other reported measurements of peptide 2D IR, indicate that the coupling between the amide groups is relatively weak in the alanine dipeptide. A detailed analysis shows that the coupling peaks are at most 3% of the diagonal peaks.14 The other obvious

Kim and Hochstrasser

Figure 2. Absorptive spectra of isotopomers I and II at T ) 0. Dashed lines represent the transition frequencies in linear FTIR spectra. (a) Isotopomer I (ωamino ) 1643 cm-1, ωacetyl ) 1589 cm-1). (b) Isotopomer II (ωamino ) 1601.5 cm-1, ωacetyl ) 1631.5 cm-1).

feature is that the diagonal anharmonicities of the two amides are about the same. A detailed analysis shows that they are 15.5 ( 0.7 cm-1 and 16.0 ( 0.7 cm-1 for the amino-end and acetylend, respectively.14 The pump/probe data also indicate that the transition dipoles for the 0 f 1 and 1 f 2 transitions are roughly equal as expected for harmonic oscillators. Heterodyne Transient Grating. The T1 relaxations are needed to fit the dynamics. These were obtained from the heterodyned transient grating signals for the isotopomer I where the center frequencies of two transitions are more separated than the isotopomer II. For the isotopomer II the center frequencies of two transitions were too close to be spectrally resolved. Figure 3 shows plots of |S˜ (0,T,ωt)*E| for values of ωt and T. The peaks correspond to the V ) 0 f V ) 1 and V ) 1 f V ) 2 transitions of the amino- and acetyl-ends of the peptide and the decay of |S˜ (0,T,ωt)*E| along the T-axis is determined by the loss of population by the V ) 1 state or any intermediate state generated from it that can couple to the k3 pulse. To fit the decay curves in Figure 3 we required double exponential decays which are quoted in the figure caption. All the fits are dominated by a decay in the range of 0.5 ps, and this component is considered here to be the actual T1 lifetime. The larger lifetime components are assumed to arise from relaxed populations that were not originally excited. The lifetimes measured at the V ) 0 f V ) 1 and V ) 1 f V ) 2 transitions were 576 and 507 fs respectively for the amino-end. The corresponding lifetimes for the acetyl-end were 415 and 394 fs. For both amide modes the lifetimes measured at V ) 1 f V ) 2 transitions were a little shorter than those measured at V ) 0 f V ) 1. For the simulations we used 778 and 514 fs as the lifetimes of V ) 1 state for the amino- and acetyl-end, respectively, where at those times the transient grating signals of V ) 0 f V ) 1 decay to 1/e of those at T ) 0. In most cases there was a very small residual absorption amounting to ca. 1-3% of the maximum

Equilibrium Dynamics of the Alanine Dipeptide

J. Phys. Chem. B, Vol. 109, No. 14, 2005 6887

Figure 5. Plots of Σωt|S˜ (τ,T,ωt)*E| vs τ at several representative values of T for isotopomer II. From the top at T ) 0, T ) 200 fs, T ) 500 fs, T ) 900 fs, T ) 1400 fs, and T ) 2000 fs, respectively.

Figure 3. Heterodyned transient grating signals |S˜ (0,T,ωt)*E| at: (a) ωt ) 1651 cm-1; (b) ωt ) 1619 cm-1; (c) at ωt ) 1588 cm-1; and (d) at ωt ) 1567 cm-1. The solid lines are fits to a biexponential decay function A1 exp(-T/τ1) + A2 exp(-T/τ2) with parameters (a) A1 ) 464, τ1 ) 576 fs, A2 ) 167, τ2 ) 1860 fs; (b) A1 ) 444, τ1 ) 507 fs, A2 ) 157, τ2 ) 2513 fs; (c) A1 ) 327, τ1 ) 415 fs, A2 ) 54, τ2 ) 5700 fs; and (d) A1 ) 177, τ1 ) 394 fs, A2 ) 23, τ2 ) 4977 fs.

Figure 4. Contour plots of absolute magnitude spectra |S˜ (τ,T,ωt)| for the isotopomer I and II at T ) 0 and T ) 2 ps. (b) and (d) were multiplied by 10. (a) Isotopomer I at T ) 0. (b) Isotopomer I at T ) 2 ps. (c) Isotopomer II at T ) 0. (d) Isotopomer II at T ) 2 ps.

signal which could be caused by heating of the sample. This dynamics is important in our consideration of the T dependence of the signals. Long after these T1 decays are complete there are populations remaining in states that were not originally excited by the pulses, and these will exhibit an amide-I 2D IR spectrum if they have an anharmonicity that differs from that of the ground state. Waiting Time, T, Dependence. The effect of the time interval T is made very clear by the data in Figure 4 which are in the form of contour plots of the absolute magnitude spectra |S˜ (τ,T,ωt)*E| for the isotopomers I and II at T ) 0 and T ) 2 ps. At T ) 0, the peak signals are shifted out to ca. 300 fs along the τ-axis. At later time T the spectra are becoming more symmetric about the axis τ ) 0. The evolution of this peak position as a function of T is the peak shift. This evolution is also seen as a gradual change in the shape of the 2D IR absorptive spectra.14 The figure also shows the effects of interference between the oscillators: the spectra are evidently not simply the sum of those for independent vibrators. The

0 f 1 and 1 f 2 parts of the 2D IR spectrum are not resolved in these absolute magnitude spectra, so each sample shows only one peak for the acetyl- and one for the amino-ends of the peptide. The interferences evident from Figure 4 are more dramatic for isotopomer II where the resonances are closer. For example, the echo peak shifts and interference-free peak shifts of amino-ends were 303 fs (220 fs) and 266 fs (153 fs) for isotopomer I and isotopomer II, respectively, where the times in the parentheses represent those of interference-free echo peak shifts calculated from simulations that omit from consideration the acetyl-end response. Also clear from Figure 4 is that the apparent peak shifts are much larger for isotopomer I (see Figure 4a). The sharper TFA contribution is also evident at ca. 1670 cm-1. In the spectrum of isotopomer II after 2 ps, when the amino- and acetyl-end amide-I peaks have diminished significantly, a new peak at 1555 cm-1 becomes evident in the 2D IR spectra. The origin of this resonance is not certain, but it appears to be present also at small T values where it is swamped by the main signal. It is seen in the 2D IR of the unlabeled compound but not in any of the linear spectra. Since the new resonance is present in the unlabeled and the amino-end labeled isotopomer, we suggest it corresponds to a mode on the acetyl-end. The mode is displaced in frequency by 13CdO isotope replacement in the NH(CO)CH3 group, which is why it is not seen in isotopomer I. Signals as a function of τ for some representative values of T are shown in Figure 5 to illustrate the occurrence and evolution of the shift of the peak of this plot along the τ-axis. The curves in Figure 5 exhibit a weak shoulder at negative τ, which causes a decrease of the magnitude of the signal at small positive values of τ (∼100 fs). This has the effect of causing an apparent shoulder to appear at negative τ. This shoulder is a manifestation of interference of the fields generated by the two amide oscillators, and it appears in both rephasing and nonrephasing quadrants. The deepest dip happens to appear at around τ ) 100 fs for the alanine dipeptide. The oscillation frequency is the frequency difference of the two oscillators as confirmed by simulations as discussed below. In the case of the two alanine dipeptide isotopomers, the periods of the beat frequencies calculated from the measured separations are 0.62 ps for I and 1.1 ps for II. This interference is not unique for the alanine dipeptide sample, and analogous oscillations are readily observed from a dilute solution consisting of a 1:1 mixture of 12CdO: 13CdO acetone in CCl , where the coupling is known to be 4 exactly zero. The experiments in Figure 5 show plots of Σωt|S˜ (τ,T,ωt)*E| versus T. The usual stimulated photon echo signal consists of the |S(τ,T,t)*E|2 integrated over times t by a slow detector. However, the quantity |S˜ (τ,T,ωt)|2 integrated over all frequencies ωt is identical to |S(τ,T,t)|2 integrated over all times t. Figure 6

6888 J. Phys. Chem. B, Vol. 109, No. 14, 2005

Kim and Hochstrasser simple model of homogeneously broadened transitions. The 2D IR response for two oscillators having the same relaxation time has the form borrowed from NMR:29

SR(τ,0,t) ) e-iω1τ-γτeiω1t-γt + e-iω2τ-γτeiω2t-γt

(1)

It is well known that the echo arises because of the different sign of the phase development on the τ- and the t-axes in each of the terms. A stimulated echo signal E(τ) from such a response using a pulse whose spectrum brackets the spectrum of both resonances is the magnitude squared of S(τ,0,t) integrated over the detector time t: Figure 6. Experimental and simulated echo peak shifts vs T delay for isotopomer I. Experimental peak shifts, *. Simulated peak shifts, solid lines.

E(τ) ) e-2γτ(1/γ + 2cos(δω12τ - φ12)/(δω122 + 4γ2)1/2) (2) which oscillates at the frequency δω12 ) ω1 - ω2 with phase φ12 ) arctan(δω12/2γ). The response of a set of vibrators consists of many terms of this type, but the oscillatory behavior is demonstrated by the simple example. In general the phases depend on the relaxation times and the frequency separations. These modulations also occur in heterodyned 2D IR. For example, the real part of the complex spectrum at the resonance ω1 of this simple response is

Re{S˜ R(τ,0, - ω1)} ) (e - γτ/γ)(cosω1τ + γcos(ω2τ + φ)/(γ2 + ω212)1/2) (3) Figure 7. Experimental and simulated echo peak shifts vs T delay for isotopomer II. Experimental peak shifts, *. Simulated peak shifts, solid lines.

and Figure 7 show the plots of the experimental peak shifts versus T for the acetyl- and amino-ends of both peptides and also show simulated peak shifts. The difference between experimental peak shifts and simulated peak shifts is less than 10% for both ends of both isotopomers. What is plotted versus τ in Figure 6 and Figure 7 is the uniform average of |S˜ (τ,T,ωt)*E| over 30 cm-1 around the amino-end and acetylend peak positions for each of isotopomer I and isotopomer II. Figure 6 and Figure 7 show that after a fast reduction over the first 200 fs, the peak shift changes very little for T less than ca. 1.5 ps (the figures show peak shifts only up to 0.5 ps), as can be verified by substitution of the Fourier transforms defining the different signals. The shifts along τ of the peak of the frequency integrated function |S˜ (τ,T,ωt)| are the same as the stimulated echo peak shifts.28 However, because of the interference, the magnitudes of these peak shifts are greatly increased when there are multiple oscillators excited. In the present case about half the observed peak shift arises from the interference effect as will be seen below from the fits. The different curves in Figure 6 and Figure 7 make it clear that the dynamical responses as seen through the peak shifts are different for the amino- and acetyl-ends of each peptide. Relationship of the Experiments to Theory. The theory of the response functions and the signal treatment for 2D IR spectra has been discussed in numerous sources.25-27 However, some review is needed to place the measurements in the proper context. Furthermore, the variation of the photon echo signal with the waiting time T is expected to be quite different for a set of oscillators than for a single oscillator. This is because of the modulation in the sum of the fields generated by the oscillators, and the effects of interference in the magnitude signals when there is overlap between the broadened transitions. That some modulation expected is obvious from the following

which is a carrier wave at close to ω1 modulated at the difference frequency with phase φ ) arctan(ω21/γ). The modulations, like the interferences in the absolute magnitude, grow larger as the overlap of the two resonances increases and becomes small when |ω21|.γ. It is easy to see that the absolute magnitude spectrum |S˜ (τ,T,ωt)| also oscillates at the difference frequency. In this instance the oscillations are due to the interference arising from the cross terms between the two oscillators in the magnitude squared. When there is an inhomogeneous broadening these principles are not changed, but now the peak of the envelope of the signal along τ is no longer at τ ) 0. More importantly for our experiments, the interference in the magnitude signal influences significantly the shift of the peak of the echo envelope along the τ-axis so that the simple relationships between the peak shift and the vibrational frequency correlation functions4,6,30-32 are no longer valid. In recent work we have shown that the cross-peaks in the 2D IR spectra of the alanine dipeptide are extremely weak, accounting for no more than 3% of the total 2D IR signal.14 This is also apparent from the spectra in Figure 2. Therefore, the dynamics of the response can be examined to sufficient accuracy without considering the coupling between the modes. The dynamics for only the diagonal part of the 2D IR spectrum of two oscillators can be more generally written as

SR(τ,T,t) ) 〈

∑ Ake-iΩ (0,τ)-γ τ/2e-γ T(eiΩ (τ+T,τ+T+t)-γ t/2 k0

kk

kk

k0

kk

k)1,2

mkeiΩk+k,k(τ+T,τ+T+t)-(γk+k,k+k+γkk)t/2)〉 (4) where Ωij(t1,t2) ) ∫tt12ωij(t′) dt′ is the phase accumulated in the interval t2-t1 due to the vibrational frequency being time dependent, the population decay rates of the state k are γkk, the overtone states are labeled as |k + k〉, and mk is the one-half the ratio of the probability of the V ) 1 f V ) 2 transition to that of the V ) 0 f V ) 1 transition of the kth oscillator. For

Equilibrium Dynamics of the Alanine Dipeptide

J. Phys. Chem. B, Vol. 109, No. 14, 2005 6889

a harmonic oscillator, mk ) 1. The average is over the frequency distribution at each instant. In general the frequencies are time dependent, thereby allowing for spectral diffusion. The positive weighting factor of the linear spectral components for each resonance is Ak that has the values given above for the alanine dipeptide. It is a common approach to assume that the rate of decay of the coherence Fk+k,k due to population relaxation is (γk+k,k+k + γkk)/2. If the relaxation of the overtone generates mainly states with one quantum of the amide-I mode, and if harmonic oscillator rules are obeyed, then it is expected that γk+k,k+k/2 ≈ γkk because the modes involved in the vibrational relaxation might be approximately the same for γk+k,k+k and γkk. However, we do not make this assumption and instead set this exponent equal to γk+k,k+k/2 ) Rγkk, where R is a parameter that is allowed to vary from 1/2 to 1. It must be noted that these ad hoc assumptions can be crucial in the interpretation of vibrational dephasing dynamics where 1/T1 is comparable with the decay of the frequency correlation function or pure dephasing part of the relaxation. Finally, if the time dependence of the frequencies of the transitions 0 f 1, having mean $k0 and 1 f 2 having mean $k0 - ∆k are considered to be equal, the signal simplifies to the form

SR(τ,T,t) )

∑ k)1,2

species I (amino-end labeled) a12 (ps-2) a22 (ps-2) a32 (ps-2) τ1 (ps) τ2 (ps)

〈e-ixk(0,τ)+ixk(τ+T,τ+T+t)〉 ) e-γk0(τ+t)-σk(t-τ) /2 2

(6)

One signature of a fixed inhomogeneous distribution of homogeneously broadened transitions as found in Bloch dynamics (eq 6) is a peak shift that is independent of T. When the accumulated phase fluctuations are Gaussian, the average gives the result33

〈e-ixk(0,τ)+ixk(τ+T,τ+T+t)〉 ) -gk(t)+gk(T)-gk(τ)-gk(τ+T)-gk(t+T)+gk(τ+T+t)

e

(7)

in terms of the dephasing relaxation function, gk(t):

∫0t dt1∫0t

1

dt2〈xk(t1)xk(t2)〉

(8)

The transient grating signal for pulses whose spectra bracket the width of the acetyl and amino resonances is the half Fourier transform (HFT) of SR(τ,T,t) along t with τ ) 0.

Ake-γ THFT(ei$ t-g (t)-γ t/2 ∑ k)1,2 kk

k0

k

amino-end

acetyl-end

amino-end

acetyl-end

5.1 ( 1.2 2.3 ( 0.5 2.3 ( 0.3 0.05 ( 0.02 0.8 ( 0.15

7.0 ( 1.5 4.1 ( 0.7 2.3 ( 0.5 0.05 ( 0.02 0.8 ( 0.2

10.3 ( 2.0 1.1 ( 0.3 1.4 ( 0.3 0.05 ( 0.02 0.8 ( 0.2

12.6 ( 2.5 1.7 ( 0.4 1.6 ( 0.3 0.05 ( 0.02 0.8 ( 0.25

〈x(t)x(0)〉 ) a12e-t/τ1 + a22e-t/τ2 + a32

where ∆k is the diagonal anharmonicity of the oscillator k that is thereby assumed to be time independent, $k0 ) ωk0(t) - xk(t) is the mean frequency, and xk(t1,t2) is the accumulated phase fluctuation through the indicated time interval. Equation 5 is the response to which the data are fit. The average in eq 5 can be cast into forms useful for comparison with experimental data if the fluctuations in frequency are assumed to be Gaussian; in that case, the average of the exponential can be written exactly in terms of the exponential of an average that contains correlation functions of the variables. For the Bloch type dynamics where each frequency component in a distribution of width σk is homogeneously broadened with dephasing rate γk0 the average is well known:

gk(t) )

species II (acetyl-end labeled)

which by inspection exhibits four peaks along ωt at ωt ) $1, ωt ) $1 - ∆1, ωt ) $2, and ωt ) $2 - ∆2. The decay of the signal |S˜ R(0,T,ωt)| is dominated by γkk at ωt ) $k - ∆k and ωt ) $k - ∆k, which describes the conditions under which the experiment was performed. The real part of the kth term in eq 9, before taking the absolute magnitude, is the difference in the linear-IR absorption spectra of the 0 f 1 and 1 f 2 transitions. The approximation of eq 7 is used in the interpretation of the echo peak shift data in Figure 6 and Figure 7. To obtain g(t) we employ a sum of exponentials to mimic the correlation function:

Ake-γkk(τ/2+T+t/2)e-i$k0τ+i$k0t ×

〈e-ixk(0,τ)+ixk(τ+T,τ+T+t)〉 (1 - mke-i∆kte-Rγkkt) (5)

|S˜ R(0,T,ωt)| ) |

TABLE 1. Sets of Parameters that Fit Both Linear FTIR Spectra and Apparent Peak Shifts

kk

mkei($k0-∆k)t-gk(t)-(1+2R)γkkt/2)| (9)

(10)

where the inhomogeneous width parameters a1, a2, and a3 with the correlation times τ1 and τ2 are the parameters that were varied to fit the experiments. The best sets of parameters that fit the experimental peak shifts are represented in Table 1. All the sets of parameters in the table provide an adequate fit of the linear FTIR spectra. The population relaxation times of V ) 1 states of amino- and acetyl-ends were assumed to be 778 and 514 fs, respectively, for both isotopomers. The population relaxation rates of overtone states were assumed to be 20% larger than those of their first excited states. In absorptive 2D IR spectra the 1-2 transition was ∼15% wider than the 0-1 transition for both ends. A slightly faster relaxation of overtone states accounted for this difference. In fact, the peak shift measurement was not too sensitive to the population relaxation rate of the overtone states in a range up to twice that of the V ) 1 state. The peak shift data were fitted to |S˜ R(τ,T,ωt)*E|, which is the HFT of eq 5 incorporating eq 7 and a convolution with the three incident fields:

|S˜ R(τ,T,ωt)*E| ) |

Ake-γ (τ/2+T)e-i$ τeg (T)-g (τ)-g (τ+T)HFT ∑ k)1,2 kk

k0

k

k

k

(e-γkkt/2ei$k0te-gk(t)-gk(t+T)+gk(τ+T+t) (1 - mke-i∆kte-Rγkkt ))| (11) The T1 relaxations of the two ends of the molecule are in the range 500 fs as described above. Thus the attribution of the peak shift to the dynamics of the frequency distributions that are excited by the incident pulses is probably only justified up to ca. 500 fs. Beyond that time the gratings created by relaxation of the amide-I mode may contribute to the signals.34-36 It is beyond the scope of this paper to include all the possible relaxation channels of the amide-I modes that would be required to account for the echo signals over a larger time range. Correlation Functions. The simple correlation function of eq 10 and the defined response functions fail to completely capture the detailed shape of the echo peak shift for the aminoend labeled isotopomer but do well with the acetyl-end labeled isotopomer. The best fits are shown in Figures 6 and 7. The values used for the fits were deduced from detailed analysis of

6890 J. Phys. Chem. B, Vol. 109, No. 14, 2005 the 2D IR and linear FTIR spectra.14 For both isotopomers, k ) 1 and 2 represent the amino- and acetyl-end, respectively. For the isotopomer I: A1 ) 1, A2 ) 0.82, ω1 ) 1643 cm-1, ω2 ) 1589 cm-1, γ11 ) 1.29 ps-1, γ22 ) 1.95 ps-1, ∆1 ) 15.5 cm-1, and ∆2 ) 16.0 cm-1. For the isotopomer II: A1 ) 1, A2 ) 0.77, ω1 ) 1632 cm-1, ω2 ) 1602 cm-1, γ11 ) 1.29 ps-1, γ22 ) 1.95 ps-1, ∆1 ) 15.5 cm-1 and ∆2 ) 16.0 cm-1. For all cases, Rk ) 0.6 and mk ) 1. The signal for the amino-end labeled species is much more influenced by the interference because the transitions are closer in frequency. Therefore there is a substantially greater sensitivity to the line shape parameters that define the wings of the bands. Presumably any deficiencies in the response functions are also exposed by the proximity of the interfering transitions. Each of the correlation functions shown in Table 1 exhibits three components. To fit the experimental echo peak shifts, each correlation function required a term that has correlation time ∼0.8 ps to explain the slow decay of echo peak shifts, a constant term that determines the magnitude of echo peak shift at large value of T, and a term that has correlation time ∼50 fs to fit linear FTIR spectra. The fast decaying term, which does not contribute much to the echo peak shifts, is essentially homogeneous broadening. The product a1τ1 is much less than unity in all cases, and so the inhomogeneous distribution with width a1 is motionally narrowed to a dephasing rate, 1/T2 ) a12τ1, which is in the range 0.25-0.6 ps-1. There is an additional uncertainty in the short time response due to residual chirping of the pulses. The amino-end labeled isotopomer appears to dephase somewhat faster than the more widely separated amide-I transitions of the acetylend labeled isotopomer. These fast correlation times are not as well determined as the products a1τ1. Clearly there is a slower evolution of the inhomogeneous distribution on the τ2 ≈ 800 fs time scale that is definitely not motionally narrowed. In addition, there is a distribution having a width a3 that is static on the time scale of the measurement. The widths of these slowly varying distributions are significantly larger for the acetyl-end labeled isotopomer. The correlation functions indicate that for both isotopomers the acetyl-ends are more inhomogeneously broadened than the amino-ends, where the inhomogeneous broadening is mainly determined by the second and third terms of the correlation function. We compared the correlation functions with that reported for N-methylacetamide-D [NMAD; CH3(CO)ND(CH3)].3 For NMAD a correlation function of the form 〈x(t)x(0)〉 ) ∆12e-t/τ1 + ∆02 was used, where ∆1 ) 12.2 ps-1, τ1 ) 0.006 ps-1, and ∆0 ) 1.1 ps-1. The pure dephasing rates for NMAD (∆1τ1 ∼0.07) and the alanine dipeptide (a1τ1 ∼0.13) are similar. However, their static inhomogeneous distribution are quite different. The alanine dipeptide shows a significantly larger static inhomogeneous distribution than NMAD, while the 0.8 ps correlation time is presumed to arise from frequency shifts accompanying hydrogen bonding of the amide carbonyl groups to solvent D2O molecules. However, these experiments need to be repeated with much shorter pulses in order to obtain more accurate information on the correlation functions. The inhomogeneous width is suggested to represent a distribution of dipeptide structures, each with its characteristic frequency. We also compared our experimental results with those reported for the alanine dipeptide in H2O obtained from MD simulations using CHARMM force fields.11 Both experimental and MD simulation results show significant inhomogeneous frequency distributions. Both show that in water the acetyl-end is more inhomogeneously broadened than the amino-end. The difference can be due to the fact that the amino-

Kim and Hochstrasser end is more readily hydrogen bonded to the solvent than is the acetyl-end. The a3 value in each correlation function lies in the range 1.3-1.5 ps-1. The σ values predicted by MD simulations are 1.0 ps-1 and 1.3 ps-1 for the amino- and acetyl-end, respectively, where σ represents the standard deviation of frequencies. However, in contrast to the experimental results, the simulation predicted the amino-end to have a higher transition frequency than the acetyl-end. The existence of a slowly varying set of structures, as represented by the static term in the correlation function, shows that even for a small peptide the energy surface in water is rugged. We know from previous work14 that the alanine dipeptide structure is most likely PII, which arises when the two CdO groups are hydrogen bonded to solvent. The total width associated with the correlation function of the fastest varying part is 1/πc(ln4(a12 + a22))1/2 and ranges from 30 to 40 cm-1. This width is large enough to encompass the hydrogen bond shift of the amide-I transition. From FTIR studies it is wellknown that the amide-I transition shifts to lower frequency by ca. 35 cm-1 when it becomes hydrogen bonded.37 Therefore, if a dynamic distribution of H-bond distances and angles were present in the ensemble we would expect to see an initial width in the range seen in the experimental correlation functions. Therefore, we suggest that the 800 fs time scale dynamics make a significant contribution to the stretching and bending motions of the CdO‚‚‚H-O hydrogen bond. Conclusion Frequency correlation functions were obtained for both amide-I groups of two differently isotopically labeled alanine dipeptides by comparing the measured and simulated echo peak shifts. For both isotopomers reasonable agreement with the simulated echo peak shifts was found. The agreement was best for the two ends of isotopomer I where the effect of interference between two vibrators is less pronounced than in II. Each correlation function comprises two exponentially decaying terms with correlation times of ∼50 fs and ∼0.8 ps and a constant term. The decay of echo peak shift was mainly influenced by the decay term with ∼0.8 ps time constant, which is attributed to the fluctuations in the stretching and bending coordinates of the CdO‚‚‚H-O hydrogen bond. The presence of a significant constant term indicates that there is a significant static inhomogeneous distribution of the center frequencies of these amide transitions. This result signals a distribution of structures that do not rearrange on the few-ps time scale. The correlation functions tell us that for the alanine dipeptide the static inhomogeneous distributions are significant for both amide-I modes and larger than those reported for NMAD in D2O.3 The correlation functions also indicate that for both isotopomers the acetyl-ends are more inhomogeneously broadened than the amino-ends, which was also predicted from MD simulations.11 The population relaxation times obtained from heterodyne transient grating experiments are 576 and 416 fs for the aminoand acetyl-end, respectively. Acknowledgment. The research was supported by a grant from NIH (GM12592) and NSF using instrumentation developed under NIH RR 01348. References and Notes (1) Hamm, P.; Lim, M.; DeGrado, W. F.; Hochstrasser, R. M. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 2036.

Equilibrium Dynamics of the Alanine Dipeptide (2) Hamm, P.; Lim, M.; DeGrado, W. F.; Hochstrasser, R. M. J. Phys. Chem. A 1999, 103, 10049. (3) Zanni, M. T.; Asplund, M. C.; Hochstrasser, R. M. J. Chem. Phys. 2001, 114, 4579. (4) Lim, M.; Hamm, P.; Hochstrasser, R. M. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 15315. (5) Rubtsov, I. G.; Kim, Y. S.; Hochstrasser, R. M. Isr. J. Chem. 2004, 44, 271. (6) Schmidt, J. R.; Corcelli, S. A.; Skinner, J. L. J. Chem. Phys. 2004, 121, 8887. (7) Madison, V.; Kopple, K. D. J. Am. Chem. Soc. 1980, 102, 4855. (8) Drozdov, A. N.; Grossfield, A.; Pappu, R. V. J. Am. Chem. Soc. 2004, 126, 2574. (9) Han, W.-G.; Jalkanen, K. J.; Elstner, M.; Suhai, S. J. Phys. Chem. B 1998, 102, 2587. (10) Mehta, M. A.; Fry, E. A.; Eddy, M. T.; Dedeo, M. T.; Anagnost, A. E.; Long, J. R. J. Phys. Chem. B 2004, 108, 2777. (11) Gnanakaran, S.; Hochstrasser, R. M. J. Am. Chem. Soc. 2001, 123, 12886. (12) Takekiyo, T.; Imai, T.; Kato, M.; Taniguchi, Y. Biopolymers 2004, 73, 283. (13) Poon, C.-D.; Samulski, E. T.; Weise, C. F.; Weisshaar, J. C. J. Am. Chem. Soc. 2000, 122, 5642. (14) Kim, Y. S.; Wang, J.; Hochstrasser, R. M. J. Phys. Chem. B, in press. (15) Hamm, P.; Lim, M.; Hochstrasser, R. M. J. Phys. Chem. B 1998, 102, 6123. (16) Asplund, M. C.; Zanni, M. T.; Hochstrasser, R. M. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 8219. (17) Golonzka, O.; Khalil, M.; Demirdoven, N.; Tokmakoff, A. Phys. ReV. Lett. 2001, 86, 2154. (18) Zhang, W. M.; Chernyak, V.; Mukamel, S. J. Chem. Phys. 1999, 110, 5011. (19) Fleming, G. R.; Cho, M. Ann. ReV. Phys. Chem. 1996, 47, 109.

J. Phys. Chem. B, Vol. 109, No. 14, 2005 6891 (20) Fang, C.; Wang, J.; Kim, Y. S.; Charnley, A. K.; Barber-Armstrong, W.; Smith, A. B., III.; Decatur, S. M.; Hochstrasser, R. M. J. Phys. Chem. B 2004, 108, 10415. (21) Zanni, M. T.; Gnanakaran, S.; Stenger, J.; Hochstrasser, R. M. J. Phys. Chem. B 2001, 105, 6520. (22) Dorrer, C.; Belabas, N.; Likforman, J.-P.; Joffre, M. J. Opt. Soc. Am. B: Opt. Phys. 2000, 17, 1795. (23) Lepetit, L.; Cheriaux, G.; Joffre, M. J. Opt. Soc. Am. B: Opt. Phys. 1995, 12, 2467. (24) Rubtsov, I., V.; Wang, J.; Hochstrasser Robin, M. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 5601. (25) Ge, N.-H.; Zanni, M. T.; Hochstrasser, R. M. J. Phys. Chem. A 2002, 106, 962. (26) Wang, J.; Hochstrasser, R. M. Chem. Phys. 2004, 297, 195. (27) Mukamel, S. Principles of Nonlinear Optical Spectrscopy; Oxford University Press: Oxford, 1995. (28) Hochstrasser, R. M. AdV. Chem. Phys., in press. (29) Ernst, R. R. Principles of Nuclear Magnetic Resonances in One and Two Dimensions; Oxford Univerity Press: Oxford, 1987. (30) Piryatinski, A.; Skinner, J. L. J. Phys. Chem. B 2002, 106, 8055. (31) Cho, M.; Yu, J.-Y.; Joo, T.; Nagasawa, Y.; Passino, S.; Fleming, G. R. J. Phys. Chem. 1996, 100, 11944. (32) de Boeij, W. P.; Pshenichnikov, M. S.; Wiersma, D. A. Chem. Phys. Lett. 1996, 253, 53. (33) Hamm, P.; Lim, M.; Asplund, M.; Hochstrasser, R. M. Chem. Phys. Lett. 1999, 301, 167. (34) Rubtsov, I. V.; Hochstrasser, R. M. J. Phys. Chem. B 2002, 106, 9165. (35) Stenger, J.; Madsen, D.; Dreyer, J.; Hamm, P.; Nibbering, E. T. J.; Elsaesser, T. Chem. Phys. Lett. 2002, 354, 256. (36) Park, J.; Ha, J.-H.; Hochstrasser, R. M. J. Chem. Phys. 2004, 121, 7281. (37) Krimm, S.; Bandekar, J. AdV. Protein Chem. 1986, 38, 181.