Renormalization group approach to vibrational energy transfer in protein
aa r X i v : . [ phy s i c s . b i o - ph ] A ug Renormalization group approach to vibrational energytransfer in protein
Shigenori Tanaka
Graduate School of System Informatics, Department of Computational Science,Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, Japan
Abstract
Renormalization group method is applied to the study of vibrational en-ergy transfer in protein molecule. An effective Lagrangian and associatedequations of motion to describe the resonant energy transfer are analyzed interms of the first-order perturbative renormalization group theory that hasbeen developed as a unified tool for global asymptotic analysis. After theelimination of singular terms associated with the Fermi resonance, ampli-tude equations to describe the slow dynamics of vibrational energy transferare derived, which recover the result obtained by a technique developed innonlinear optics [S.J. Lade, Y.S. Kivshar, Phys. Lett. A 372 (2008) 1077].
Email address: [email protected]; Fax:+81-78-803-6621 (Shigenori Tanaka)
Preprint submitted to Elsevier November 6, 2018 . Introduction
The study of vibrational motions in protein system has attracted muchinterest in the context of their relations to biological functions, paying at-tention to the intrinsic flexibility of large biomolecules. A lot of theoreticalinvestigations on the basis of the normal mode or principal component anal-ysis [1–5] have been carried out in combination with molecular mechanics ordynamics simulations for protein systems, thus providing useful informationat molecular level. Vibrational energy transfers between the normal modesare then expected when the nonlinear, anharmonic couplings are taken intoaccout [6, 7]. The dynamical relaxation behavior after the excitation of vi-brational modes is thus central for a deep understanding of the relationshipsamong structure, dynamics and function in protein systems.In this context, Moritsugu et al. [8, 9] studied the intramolecular transferof vibrational energy in myoglobin by means of molecular dynamics simula-tion. They found that the vibrational energy was transferred from a givennormal mode to very few other modes that were selected by the frequencyrelation of the Fermi resonance [10] and the magnitude of the third-ordermode coupling term. It was also observed that the magnitude of the couplingcoefficients was mainly determined by the degree of geometrical overlap ofthe corresponding normal modes. Further, performing a molecular dynamicssimulation with just these few modes, they reproduced the results for all-atom simulation well, thus confirming that only these modes were importantfor the dynamics of vibrational energy transfer.As is well known, the description by naive perturbation theory breaksdown for this kind of resonant energy transfer problem due to the emer-2ence of singular terms associated with the Fermi resonance [10]. Invokedby a theoretical technique for describing the resonant parametric interactionbetween different harmonics in nonlinear optics, Lade and Kivshar [11] pro-posed a simple theory to overcome this difficulty, which provided coupledmode equations for slowly varying envelopes of the vibrational modes. Theythus found that their equations could describe the dynamics of the resonantenergy transfer between the vibrational modes observed in the simulations byMoritsugu et al. [8, 9] with good accuracy, and gave a wealth of suggestionsabout the underlying mechanism of nonlinear dynamics of vibrational energytransfer [12].In this work, the renormalization group (RG) approach which has beendeveloped as a unified tool for global asymptotic analysis [13–15] is employedin order to describe the long-time dynamics of the resonant vibrational en-ergy transfer in protein system. Starting with the same model Lagrangianas studied by Moritsugu et al. [8, 9] and Lade et al. [11], we apply the RGtechnique at the first-order perturbation level and derive the correspondingamplitude equations through the elimination of singular terms associatedwith the Fermi resonance. We then find that the derived equations are es-sentially of the same form as those obtained by Lade and Kivshar [11] atthe lowest-order level. In the following section, the theoretical formulationof the present RG approach is addressed. Some discussions and conclusionare given in Sec. 3.
2. Theory
We start with a model Lagrangian given by Moritsugu et al. [8, 9],3 = X j =1
12 ˙ q j − X j =1 ω j q j − αq q q − βq q , (1)where q j refers to the coordinate of the j th normal mode with the frequency ω j and the dot denotes the time derivative. Two types of third-order couplingamong the modes are considered in the model, which are characterized bysmall coupling parameters α and β . Equations of motion for q j then read¨ q = − ω q − αq q − βq q , ¨ q = − ω q − αq q , ¨ q = − ω q − αq q , ¨ q = − ω q − βq . (2)These equations of motion were found to appropriately reproduce the tempo-ral variations of vibrational energy transfer observed in the all-atom molec-ular dynamics simulation for myoglobin [8, 9].The solution to q j is given by a perturbation theory up to the first orderof α and β as q ( t ) = A e iω t − αA A ω − ( ω + ω ) e i ( ω + ω ) t − αA ∗ A ω − ( ω − ω ) e i ( ω − ω ) t − βA A ω − ( ω + ω ) e i ( ω + ω ) t − βA ∗ A ω − ( ω − ω ) e i ( ω − ω ) t + c . c ., ( t ) = A e iω t − αA A ω − ( ω + ω ) e i ( ω + ω ) t − αA A ∗ ω − ( ω − ω ) e i ( ω − ω ) t + c . c .,q ( t ) = A e iω t − αA A ω − ( ω + ω ) e i ( ω + ω ) t − αA A ∗ ω − ( ω − ω ) e i ( ω − ω ) t + c . c .,q ( t ) = A e iω t − βA ω − ω e iω t − βA A ∗ ω + c . c ., (3)where we have introduced amplitude parameters, A j , and c.c. (or the asterisk)means the complex conjugate.Here, we consider a case of vibrational energy transfer in which the Fermiresonance takes place. According to Moritsugu et al. [8, 9], it is assumedthat two frequency parameters, Ω α = ω − ω − ω and Ω β = 2 ω − ω , satisfysuch a condition as | Ω α | , | Ω β | ≪ ω j . Retaining only the terms of the firstorder of α or β that become large under the Fermi resonance condition onthe right-hand side of Eq. (3), we find q ( t ) ≈ (cid:20) A − αA A Ω α ( ω + ω + ω ) e − i Ω α t − βA ∗ A Ω β ω e − i Ω β t (cid:21) e iω t + c . c .,q ( t ) ≈ (cid:20) A + αA A ∗ Ω α ( ω + ω − ω ) e i Ω α t (cid:21) e iω t + c . c .,q ( t ) ≈ (cid:20) A + αA A ∗ Ω α ( ω − ω + ω ) e i Ω α t (cid:21) e iω t + c . c .,q ( t ) ≈ (cid:20) A + βA Ω β (2 ω + ω ) e i Ω β t (cid:21) e iω t + c . c .. (4)5ccording to the RG technique [13–15], we express the amplitude param-eters A j in Eq. (4) up to the first order of α or β as A = ˜ A ( τ ) [1 + αZ α ( τ ) + βZ β ( τ )] ,A = ˜ A ( τ ) [1 + αZ α ( τ )] ,A = ˜ A ( τ ) [1 + αZ α ( τ )] ,A = ˜ A ( τ ) [1 + βZ β ( τ )] , (5)where we have introduced a time variable τ and renormalization constants Z jα , Z jβ . In order to eliminate the singular terms in Eq. (4) up to the lowestorder, the renormalization constants are required to satisfy the followingequations: ˜ A ( τ ) Z α ( τ ) = ˜ A ( τ ) ˜ A ( τ ) e − i Ω α τ Ω α ( ω + ω + ω ) , ˜ A ( τ ) Z β ( τ ) = 2 ˜ A ∗ ( τ ) ˜ A ( τ ) e − i Ω β τ Ω β ω , ˜ A ( τ ) Z α ( τ ) = − ˜ A ( τ ) ˜ A ∗ ( τ ) e i Ω α τ Ω α ( ω + ω − ω ) , ˜ A ( τ ) Z α ( τ ) = − ˜ A ( τ ) ˜ A ∗ ( τ ) e i Ω α τ Ω α ( ω − ω + ω ) , A ( τ ) Z β ( τ ) = − ˜ A ( τ ) e i Ω β τ Ω β (2 ω + ω ) . (6)Since the amplitude parameters A j should not depend on the introducedtime variable τ , we call for identities as dA j dτ = 0 . (7)We then obtain the RG equations [13–15] up to the first order of α or β as d ˜ A ( τ ) dτ + (cid:20) α dZ α ( τ ) dτ + β dZ β ( τ ) dτ (cid:21) ˜ A ( τ ) = 0 ,d ˜ A ( τ ) dτ + α dZ α ( τ ) dτ ˜ A ( τ ) = 0 ,d ˜ A ( τ ) dτ + α dZ α ( τ ) dτ ˜ A ( τ ) = 0 ,d ˜ A ( τ ) dτ + β dZ β ( τ ) dτ ˜ A ( τ ) = 0 , (8)from Eq. (5), considering d ˜ A j ( τ ) /dτ ∼ O ( α ) or O ( β ). We also note thelowest-order relations as˜ A ( τ ) dZ α ( τ ) dτ = − i ˜ A ( τ ) ˜ A ( τ ) e − i Ω α τ ω + ω + ω , ˜ A ( τ ) dZ β ( τ ) dτ = − i ˜ A ∗ ( τ ) ˜ A ( τ ) e − i Ω β τ ω , ˜ A ( τ ) dZ α ( τ ) dτ = − i ˜ A ( τ ) ˜ A ∗ ( τ ) e i Ω α τ ω + ω − ω , A ( τ ) dZ α ( τ ) dτ = − i ˜ A ( τ ) ˜ A ∗ ( τ ) e i Ω α τ ω − ω + ω , ˜ A ( τ ) dZ β ( τ ) dτ = − i ˜ A ( τ ) e i Ω β τ ω + ω , (9)from Eq. (6). Combining Eqs. (8) and (9), we finally obtain differentialequations, d ˜ A ( τ ) dτ = iα ˜ A ( τ ) ˜ A ( τ ) e − i Ω α τ ω + ω + ω + 2 iβ ˜ A ∗ ( τ ) ˜ A ( τ ) e − i Ω β τ ω ,d ˜ A ( τ ) dτ = iα ˜ A ( τ ) ˜ A ∗ ( τ ) e i Ω α τ ω + ω − ω ,d ˜ A ( τ ) dτ = iα ˜ A ( τ ) ˜ A ∗ ( τ ) e i Ω α τ ω − ω + ω ,d ˜ A ( τ ) dτ = iβ ˜ A ( τ ) e i Ω β τ ω + ω , (10)to determine the renormalized amplitudes ˜ A j ( τ ).We now equate the time variable τ with time t in Eq. (10), and solvingthe differential equations with the initial condition of q j (0) = ˜ A j (0) + ˜ A ∗ j (0),we can obtain ˜ A j ( t ). Thus, we find a solution to q j as q j ( t ) = ˜ A j ( t ) e iω j t + ˜ A ∗ j ( t ) e − iω j t . (11)The energy of each mode can then be evaluated in the harmonic approxima-tion [8, 9, 11]. 8 . Discussion and Conclusion In the preceding section we have derived an amplitude equation, Eq. (10),on the basis of the RG approach. This equation is free from the singularitiesassociated with the Fermi resonance, which have been observed in the naiveperturbation theory, and describes a nonlinear, slow dynamics of vibrationalenergy transfer [12] in terms of the renormalized amplitudes ˜ A j . The presentrenormalization approach has thus succeeded in an effective separation of thetwo time domains associated with the fast dynamics governed by ω j and theslow (“envelope”) dynamics for vibrational energy transfer in protein.Lade and Kivshar [11] analyzed the same model as in the present study,and derived an analogous set of amplitude equations by integrating out thefast vibrational motions in an intuitive way developed in the field of nonlinearoptics. By solving the resultant equations numerically, they found that thelong-time behavior of the vibrational energy transfer in the model Eq. (1) canbe described very well, thus demonstrating the excellence of their method.Taking into account the condition that | Ω α | and | Ω β | ≪ ω j , their derivedequations are essentially identical to the present Eq. (10). That is, theirresult can be recovered in terms of the first-order perturbation theory inthe RG approach, and can be systematically improved in the more generalframework of the present theory.In order to explain the dynamical behavior of vibrational energy trans-fer observed in their all-atom molecular dynamics simulation for myoglobin,Moritsugu et al. [8, 9] considered a model system given by Eq. (1). Perform-ing a molecular dynamics simulation for the model system with the param-eter set of ω = 5 . ω = 2 . ω = 3 . ω = 10 . α = − .
13 and9 = 0 .
12, they reproduced the temporal evolution of the resonant energytransfers among the normal modes quantitatively. Lade and Kivshar [11]then solved their amplitude equations numerically with the same parameterset as that by Moritsugu et al., and found a good agreement between the tworesults for the time course of energy variations of the normal modes. The en-ergy transfer dynamics observed in their studies is essentially a renormalizedone in the sense that a slow dynamics emerges in contrast to the normal-mode dynamics characterized by ω j . The present work reproduces the resultby Lade and Kivshar at the lowest-order level, and provides a more generalframework for describing the resonant vibrational energy transfer in proteinon the basis of the RG theory. This line of theoretical development wouldthus give a key tool for performing a coarse-grained description of proteindynamics in a comprehensive way. 10 eferenceseferences