ABSTRACT
CONTENTS 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 15.2 Cytochrome c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 15.3 QCF Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
15.3.1 Fermi’s Golden Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 15.3.2 Quantum Correction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 15.3.3 NM Calculations for Cyt c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 15.3.4 Application to VER of the CD Bond in Cyt c . . . . . . . . . . . . . . . . . . 307 15.3.5 Fluctuation of the CD Bond Frequency. . . . . . . . . . . . . . . . . . . . . . . . . 308
15.4 Reduced Model Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 15.4.1 Reduced Model for a Protein. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 15.4.2 Maradudin-Fein Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 15.4.3 Third-Order Coupling Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 15.4.4 Width Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 15.4.5 Temperature Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
15.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 15.5.1 Comparison with Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 15.5.2 Validity of Fermi’s Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 15.5.3 Higher-Order Coupling Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
15.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
The harmonic (or normal mode (NM)) approximation has been a powerful tool for the analysis of few and many-body systems where the essential
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dynamics of the system consists of small oscillations about a well-defined mechanically stable structure. The concept of NMs is appealing in science because it provides a simple view for complex systems such as solids and proteins. Though it had been believed that NMsmay be too simplistic to analyze the dynamics of proteins, it is by nomeans always true; the experimental data of neutron scattering for proteins (B-factor) indicate that the fluctuations for each residue arewell represented by a simplifiedmodel usingNMs [1]. It was also shown that such a large-amplitude motion as the hinge-bending motion in a protein is well described by a NM [2]. Importantly, NMs have been used to refine the x-ray structures of proteins [3]. Recently, large proteins or even protein complexes can be analyzed by using NMs [4-6]. In this chapter, we are concerned with vibrational energy relaxation (VER)
in a protein. This subject is related to our understanding of the functionality of proteins.At themost fundamental level, wemust understand the energy flow (pathway) of an injected energy, that is channeled to do useful work. Due to the advance of laser technology, time-resolved spectroscopy can detect such energy flow phenomena experimentally [7]. To interpret experimental data, and to suggest new experiments, theoretical approaches and simulations are essential as they can provide a detailed view of VER. However, VER in large molecules itself is still a challenging problem in molecular science [8]. This is because VER is a typical many-body problem and estimations of quantum effects are difficult [9]. There is a clear need to test and compare the validity of the existing theoretical methods. Wehere employ twodifferentmethods to estimate theVER rate in a protein,
cytochrome c (see Section 15.2 for details). One is the classical equilibrium simulation method [10] with quantum correction factors (QCFs) [11, 12]. The second is the reducedmodel approach [13], whichhasbeen recently employed by Leitner’s group [14, 15]. The latter approach is based on NM concepts, whichdescribesVERas energy transfers betweenNMsmediatedbynonlinear resonance [16].We concludewith adiscussionof the validity andapplicability of such approaches.
