Anharmonic Decay of Vibrational States in Proteins

CONTENTS 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 16.2 Computation of Vibrational Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 16.3 Vibrational Energy Transfer in Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

16.3.1 Cytochrome c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 16.3.2 Photoactive Yellow Protein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

16.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Appendix: Force Field for Chromophore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

Normal modes of proteins provide a useful starting point for describing dynamics [1-5], thermodynamics [6-8], and for interpreting vibrational spectra [9-12]. Going beyond the harmonic approximation and exploring the limits of its validity begins with examining anharmonicity. In the limit of infinitesimally tiny atomic displacements the normal modes describe the vibrations exactly. As displacements exceed this infinitesimal limit, anharmonic corrections become increasingly important and can appreciably affect, for instance, the vibrational thermodynamic properties of a molecule [13], as well as the vibrational spectrum [9-12]. Anharmonicity also gives rise to vibrational energy transfer. The transfer and storage of vibrational energy mediate kinetics of chemical reactions [14-22], including photochemical reactions in proteins [23-27], allosteric transitions [28], and charge transfer reactions [14, 29].Aproperdescriptionof vibrational energyflow inmolecules generally requires a quantum mechanical treatment. We address herein the

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quantum mechanical transfer of energy among normal modes of hydrated cytochrome c and photoactive yellow protein (PYP) by anharmonic decay. The calculations provide insight into the vibrational lifetimes themselves, their temperature dependence in different spectral regions, the influence of hydration water on energy transfer, and, for the case of PYP, the influence of vibrational energy transfer on photoisomerization kinetics. To illustrate principles of vibrational energy transfer in a protein, we exam-

ine anharmonic decay of vibrational states in cytochrome c. One particular focus of this part of the chapter is the influence of hydration water on energy transfer in a protein. The analysis here extends our recent work on myoglobin [30], as well as earlier work on bovine pancreatic trypsin inhibitor [9, 10] and other studies on myoglobin [31, 32], for which anharmonic matrix elements coupling someof thenormalmodeswere carriedout. Important insights into the role of anharmonicity in vibrational spectra and energy flow were gained from these early studies. Roitberg et al., demonstrated the importance of anharmonic corrections to vibrational spectra of low temperature proteins and their interpretation [9, 10]. Computational work by Kidera et al. on myoglobin has highlighted the important role played by Fermi resonances that spatially overlap in classical vibrational energy transfer [31, 32]. Prior to studying myoglobin [30], we calculated the anharmonic decay and dephasing rates of the vibrational states of helical and coil segments of myoglobin consisting of 10 to 24 amino acids [33-35]. This work identified a propensity for certain vibrational modes to overlap in space, depending on the range of frequency and their spatial extent (vide infra), a property that restricts the flow of vibrational energy in proteins. This work on vibrational energy transfer complements simulations on vibrational energy flow inmyoglobin and cytochrome c, which have also provided connections between protein structure and the typically anisotropic flow of energy [36-40]. Proteins are of course aperiodic systems, a property that influences pro-

tein vibrations and energy transfer. Indeed, most normal modes of proteins are localized to a relatively small number of atoms of the protein, meaning that the vibrational amplitude for most atoms in most vibrational modes is exponentially small. It turns out that because most vibrational modes of proteins are spatially localized, the anharmonic decay rate is typically only weakly temperature dependent, as discussed below. We have observed this to be the case for the vibrationalmodes ofmyoglobin above 500 cm−1, or even lower [30], andweshall see that the situation isquite similar forhydratedcytochrome c. This trend is consistent with the nearly temperature-independent anharmonic decay rates of high-frequency modes in both myoglobin [41-43] and myoglobin-CO [44] found in pump-probe studies over temperatures from 10 to 310K. We also examine vibrations and energy transfer in PYP. The relevance of

vibrations and energy transfer to protein function is illustrated by the primary events of many photoactive proteins. In bacteriorhodopsin, for example, ultrafast experiments and simulations reveal that reorganization dynamics of the protein following the sizable charge redistribution in the chromophore

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upon photoexcitation occurs by numerous local, small-amplitude motions of charged groups and dipoles throughout the protein [23-25]. The protein does not have time for significant conformational change during the brief period between photoexcitation and isomerization of the chromophore; protein motions responding to charge redistribution of the chromophore are thus largely vibrational. Some of these collective oscillations are dynamically coupled to, and in effect become part of, the reaction coordinate. The dynamic coupling may appear as oscillations in fluorescence decay as the wave packet recrosses the transition state [14] during the course of conformational isomerization. Ultrafast studies [26, 45, 46] on PYP reveal a similar picture. Mataga

et al. [26, 45] have observed coherent oscillations with at least two characteristic frequencies. PYP is a small water-soluble protein of a halophilic photosynthetic bacterium, Ectothiorhodospira halophila, which functions as a photoreceptor for negative phototaxis, specifically avoidance of blue light [47-49]. PYP belongs to a family of blue-light receptor proteins, Xanthopsins [47-49], which contain as their light-sensitive chromophore trans-p-coumaric acid in PYP, a deprotonated coumaric acid thioester (Figure 16.1). The chromophore is positioned in PYP by hydrogen bonding at the head part, O−-phenyl-, and by covalent bonding at the tail part,

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–CO-S-,whichundergoesultrafast twistingbyflipping the thioesterbond following photoexcitation [47-49]. This twistingmotion is coupled to vibrations of the protein matrix. Coherent oscillations appearing in fluorescence decay curves provide information about the frequencies and the nature of these coupled vibrations. Mataga et al. [26, 45] identified coherent oscillations in the fluorescence decay of PYP of roughly 50 and 140 cm−1. Site-directed mutations of the protein can shift the frequency and amplitude of the oscillation as well as the decay rate itself, and correspondingly the rate of conformational change of the chromophore. The amplitude of the lower frequency oscillation appears to be more sensitive to changes in protein environment, though both disappearwhen PYP is denatured [26, 45]. Mataga et al., suggest that the 140 cm−1 is more localized to the chromophore [26], which our analysis corroborates. Recent ab initio calculationson the isolated chromophorebyMataga et al. [21] provide an assignment of the 140 cm−1 mode. We shall see below that the twisting motion of the chromophore is in fact enhanced by coupling to the protein matrix, and that the energy transfer time from this vibration is comparable to the time for decoherence and on the same timescale as fluorescence decay. The rate of conformational change thus appears to be mediated by the rate of transfer of excess vibrational energy from modes closely associated with conformational change. We discuss how energy transfer appears to us to influence the reaction kinetics at the conclusion of this chapter. A number of simulations [50-53] and ab initio studies [54, 55] on PYP have

described PYP dynamics when the chromophore is in its cis and trans conformation, and how the protein aids in stabilizing the transition state [56, 57]. In the following section, we identify and characterize by analysis of normal modes and their lifetimes, the specific vibrations of PYP that appear as oscillations in fluorescence decay measurements [26]. We have recently characterized these vibrations for PYP in the absence of hydration water [27]; we consider here the effect of hydration water on the coupled vibrations of the chromophore and protein matrix. In the following section we summarize the computational methods. In

Section 16.3.1, we present results of our calculations of anharmonic decay rates of vibrational states of cytochrome c, where we discuss the influence of temperature and hydration. In Section 16.3.2, we present a normal mode analysis on hydrated PYP in the S1 state, as well as results for the vibrational lifetimes. Concluding remarks follow in Section 16.4.