Non-Markovian theory of vibrational energy relaxation and its applications to biomolecular systems

arXiv:1003.4796v1 [physics.bio-ph] 25 Mar 2010 Non-Markovian theory of vibrational energy relaxation and its applications to biomolecular systems Hiroshi FUJISAKI,1, 2, ∗Yong ZHANG,3, † and John E. STRAUB4, ‡ 1 Department of Physics, Nippon Medical School, 2-297-2 Kosugi-cho, Nakahara, Kawasaki 211-0063, Japan 2 Molecular Scale Team, Integrated Simulation of Living Matter Group, Computational Science Research Program, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan 3 Center for Biophysical Modeling and Simulation and Department of Chemistry, University of Utah, 315 S. 1400 E. Rm. 2020, Salt Lake City, Utah 84112-0850, USA 4 Department of Chemistry, Boston University, Boston, Massachusetts, 02215, USA (Dated: September 29, 2018) ∗Electronic address: fujisaki@nms.ac.jp †Electronic address: zhangy1977@gmail.com ‡Electronic address: straub@bu.edu 1 I. INTRODUCTION Energy transfer (relaxation) phenomena are ubiquitous in nature. At a macroscopic level, the phenomenological theory of heat (Fourier law) successfully describes heat transfer and energy flow. However, its microscopic origin is still under debate. This is because the phe- nomena can contain many-body, multi-scale, nonequilibrium, and even quantum mechanical aspects, which present significant challenges to theories addressing energy transfer phenom- ena in physics, chemistry and biology [1]. For example, heat generation and transfer in nano-devices is a critical problem in the design of nanotechnology. In molecular physics, it is well known that vibrational energy relaxation (VER) is an essential aspect of any quanti- tative description of chemical reactions [2]. In the celebrated RRKM theory of an absolute reaction rate for isolated molecules, it is assumed that the intramolecular vibrational energy relaxation (IVR) is much faster than the reaction itself. Under certain statistical assump- tions, the reaction rate can be derived [3]. For chemical reactions in solutions, the transition state theory and its extension such as Kramer’s theory and the Grote-Hynes theory have been developed [4, 5] and applied to a variety of chemical systems including biomolecular systems [6]. However, one cannot always assume that separation of timescales. It has been shown that a conformational transition (or reaction) rate can be modulated by the IVR rate [7]. As this brief survey demonstrates, a detailed understanding of IVR or VER is essential to study the chemical reaction and conformation change of molecules. A relatively well understood class of VER is a single vibrational mode embedded in (vi- brational) bath modes. If the coupling between the system and bath modes is weak (or assumed to be weak), a Fermi’s-golden-rule style formula derived using 2nd order pertur- bation theory [8–10] may be used to estimate the VER rate. However, the application of such theories to real molecular systems poses several (technical) challenges, including how to choose force fields, how to separate quantum and classical degrees of freedom, or how to treat the separation of timescales between system and bath modes. Multiple solutions have been proposed to meet those challenges leading to a variety of theoretical approaches to the treatment of VER [11–16]. These works using Fermi’s golden rule are based on quantum mechanics and suitable for the description of high frequency modes (more than thermal energy ≃200 cm−1), on which nonlinear spectroscopy has recently focused [17–20]. In this chapter, we summarize our recent work on VER of high frequency modes in 2 biomolecular systems. In our previous work, we have concentrated on the VER rate and mechanisms for proteins [21]. Here we shall focus on the time course of the VER dynamics. We extend our previous Markovian theory of VER to a non-Markovian theory applicable to a broader range of chemical systems [22, 23]. Recent time-resolved spectroscopy can detect the time course of VER dynamics (with femto-second resolution), which may not be accurately described by a single time scale. We derive new formulas for VER dynamics, and apply them to several interesting cases, where comparison to experimental data is available. This chapter is organized as follows: In Sec. II, we briefly summarize the normal mode concepts in protein dynamics simulations, on which we build our non-Markovian VER theory. In Sec. III, we derive VER formulas under several assumptions, and discuss the limitations of our formulas. In Sec. IV, we apply the VER formulas to several situations: the amide I modes in solvated N-methylacetamide and cytochrome c, and two in-plane modes (ν4 and ν7 modes) in a porphyrin ligated to imidazol. We employ a number of approximations in describing the potential energy surface on which the dynamics takes place, including the empirical CHARMM [24] force field and density functional calculations [25] for the small parts of the system (N-methylacetamide and porphyrin). We compare our theoretical results with experiment when available, and find go

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