Nagaoka Laboratory Graduate School of Information Science, Nagoya University

From Molecular Motion to Nonequilibrium System

Fig.1 Solution composed of water molecules and an HF molecule.

Our goal is to elucidate the origin of non-equilibrium nature which can be seen in natural phenomena. We address this issue by employing both microscopic and macroscopic theories. Macroscopically, we analyze the time evolution of spatial distribution of particles using Fokker-Planck equation etc. On the other hand, microscopically, we analyze the time evolution of molecular motion at the atomic level using molecular dynamics (MD) simulation.

Coarse-grained formulation for condensed reaction systems

Condensed systems are treated as statistical ensembles consisting of ~1023 (Avogadro number) molecules. Because an accurate analysis of the system requires parallel computation of Newtonian equation with the ~1023 degrees of freedom, it is practically impossible to execute such a huge number of degrees of freedom system even if we can use the state-of-the-art supercomputer. Thus it is indispensable to employ coarse graining to reduce the number of degrees of freedom.

Kramers proposed stochastic equations of motion (e.g., Langevin formalism and Fokker-Planck formalism), where the dynamics is described along reaction coordinates of phenomenon. These formalisms assume that solvent dynamics is sufficiently faster than solute dynamics. In other words, this means that we can neglect the correlations between solute dynamics and the faster solvent dynamics. However, such an assumption does not hold in real systems. To improve the traditional methodology of coarse-graining, Mori et al developed formalisms to separate multiple dynamics across different time scales. The earlier Mori's study draw our attention to chemical reaction system which led us to indicate the importance of mode-coupling between reactant molecules.

Fig.2 Induction of rotational modes by mode coupling
[Reference] M. Nagaoka, T. Okamoto and Y. Maruyama, J. Chem. Phys., 117, 5594 (2002).

Statistical analysis of relaxation process by multiple MD simulations

Fig.3 Spatial distribution of the number of solvent oxygen atoms

It is obvious that, since many degrees of freedom in solvent molecules around a solute molecule correlates with those in the solute molecule, chemical reactions cannot be described only by the internal degrees of freedom of solute. To simulate relaxation processes including all the degrees of freedom, MD simulation is a useful methodology. As a typical example, a hetero-nuclear solute diatomic molecule, HF, in an equilibrium aqueous solvent was vibrationally excited. By conducting excitation MD simulation for various times with ensemble averaging, the spatial distribution of the solvent oxygen atoms was obtained with high precision (Fig.3).

As a result of space-time-resolved analyses of kinetic energy and power spectra, it was revealed that energy relaxation occurs anisotropically because of the antisymmetricity of hetero-nuclear solute structure. It was also found that vibrational relaxation process of the solute molecule and structural fluctuations of solvent molecules play important roles in intermolecular vibrational couplings.

Fig.4 Power spectra of HF in aqueous solvent. (red: equilibrium state, black: vibrationally excited state)

[Reference] T. Okamoto and M. Nagaoka, Chem. Phys. Lett., 407, 444 (2005).

Statistical analysis of relaxation process by perturbation ensemble MD method

As shown in the above study, it was revealed that ensemble averages over many MD trajectories of vibrationally excited systems provide us with information on the relaxation process of solute molecules. By utilizing perturbation ensemble MD (PEMD) method, in which perturbation procedure is added to the ensemble averaging, it is possible to analyze relaxation processes with higher precisions. In the PEMD method, many pairs of perturbed MD (PMD) with the perturbation and unperturbed MD (UMD) simulations without the perturbation are executed for ensemble averaging. The number of trajectory pairs should be determined to obtain target precision which is necessary to obtain experimental precision.

Fig.5 An example of the PEMD method to analyze a relaxation process of photolyzed carbonmonoxy myoglobin (MbCO). Structures ensemble averaged over 2,000 PMD (red) and UMD (blue) trajectories at 20 ps after the perturbation (ligand photolysis).

As an example of PEMD method, a relaxation process of photolyzed carbonmonoxy myoglobin (MbCO) was analyzed (Figure 5). Each pair of PMD and UMD was executed starting from the same initial structure with and without the perturbation (photolysis of heme-bound CO). By ensemble averaging over 2,000 pairs of PMD and UMD trajectories at 20 ps after the perturbation, the structural relaxation process after the perturbation was obtained with high precision. The result is consistent with the time-resolved X-ray crystallography. Moreover, it was also possible to analyze vibrational energy relaxation processes from the heme to the myoglobin and solvent.

M. Takayanagi, H. Okumura, M. Nagaoka, J. Phys. Chem. B, 111, 864 (2007).
M. Takayanagi, C. Iwahashi, M. Nagaoka, J. Phys. Chem. B, 114, 12340 (2010).
M. Takayanagi and M. Nagaoka, Theoret. Chem. Acc., 130, 1115 (2011).
M. Nagaoka, I. Yu, M. Takayanagi, in "Proteins: Energy, Heat and Signal Flow" (CRC Press) (Eds., D.M. Leitner, J.E. Straub)
(Okamoto, Koyano, Takayanagi, Takenaka)


質量数12の炭素原子(12C)12g中に含まれている原子の数は、6.02×1023個で、これをアボガドロ数という。原子や分子やイオンの量はアボガドロ数個の集団を単位とすると便利なので、6.02×1023個の粒子の集合を1 molと言う。1 molあたりの粒子の数をアボガドロ定数とよび、6.02×1023 mol-1である。






ブラウン運動する粒子に関する分布関数 f(x, t) に関する運動方程式の一つであり、分布関数 f(x, t) に関する2階の偏微分方程式



或るポテンシャル中のブラウン運動を記述する確率微分方程式。簡単なLangevin方程式は、質量 m のブラウン粒子の加速度 a が、ポテンシャルによる力 F(x) と、その速度に比例する粘性力 -βv と、ノイズ項 ηの和として表現される。












温度や圧力などの熱力学的な変数で指定される状態を巨視的状態と言うのに対して、個々の原子 i の位置 q i や運動量 p i などの力学的な変数で指定される状態を微視的状態と言う。熱平衡状態にある孤立系においては、どの微視的状態も等しい確率で実現されるというのが等重率の原理である。ミクロカノニカル集団(あるいはアンサンブル)は、{q i, p i} からなる相空間において、当エネルギー面上の微視的状態が等確率で実現されることから、定義される。ミクロカノニカル集団は、孤立系が十分放置されて熱平衡に達した場合を表現している。






時間変化のない平衡状態とは異なり、時間の原点をずらしても変化しない状態。例えば、流速が一定の系の状態。定常な確率過程 x(t) について、その相関関数 R(t1, t2) = < x(t1) x(t2) >は、τ = t1 - t2 の関数となり、R(t1, t2) = R(τ) = R(-τ) を満たす。