C-1-09
Random Matrix Theory Analysis of Molecular Dynamics
Simulation
Masanori Yamanaka 1
yamanaka@phys.cst.nihon-u.ac.jp

1

Department of Physics, College of Science and Technology, Nihon University, 1-8
Kanada-Surugai, Chiyoda-ku, Tokyo 101-8308, Japan

Keywords: Molecular dynamics, All atom MD with solvent, Random matrix theory, Principal
component analysis, Protein-ligand interaction, Protein-protein interaction, Domain decomposition,Correlation matrix, Variance-covariance matrix
We apply the random matrix theory [1,2,3] to analyze the time series data of motion of atoms ofproteins which is produced by the molecular dynamics simulation [4]. We perform all-atom
molecular dynamics simulation with solvent with the maximum duration of 1000ns. We study adata set with the different sampling-time intervals, 0.01fs, 0.1fs, 1fs, 10fs, 100fs, 1ps, and 10ps toobserve the characteristic motion at each time scale. The variance-covariance matrices areconstructed from the time series data. We calculate the probability density of thevariance-covariance matrix, eigenvalue distribution, unfolded eigenvalue distribution of thenearest-neighbor and the next nearest-neighbor level spacings, inverse participation ratio, etc.
They are the fundamental quantities which charact erize the universality class in the random matrix
theory. The results of the nearest-neighbor and the next nearest-neighbor level spacings agree wellwith the Gaussian orthogonal and Gaussian symplectic ensembles, respectively. On the other hand,
the raw eigenvalue distribution has a crossover behavior between the universal and non -universal
classes as a function of the sampling-time interval. Following the random matrix theory, weclassify the dynamically-correlated sectors of the protein domain by analyzing the eigenvalues
outside of the bulk and the inverse participation ratio, and decompose them into subsectors. Ourmethod is an attempt to improve and refine the principal compo nent analysis of protein domain [5].
As an example, we demonstrate our method in the bovine eye lens protein gamma-B(gamma-II)-crystallin, PDBID:4GCR to study the domain decomposition, and ubiquitin carboxy
terminal hydrolase L1 bound to ubiquitin vinylmethylester, PDBID:3KW5 to study theprotein-ligand and protein-protein dynamical interactions.
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Math. Phys. 4, 713, 1963.
[2] L. Laloux, P. Cizeau, J.-P. Bouchaud, and M. Potters, Phys. Rev. Lett. 83, 1467, 1999; V. Plerou,P. Gopikrishnan, B. Rosenow, L.A.N. Amaral, and H.E. Stanley, Phys. Rev. Lett. 83 , 1471, 1999;
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[3] The first attempt to the protein is reported by R. Potestio, F. Caccioli, and P. Vivo, Phy. Rev.
Lett. 103, 268101, 2009 by the solvable elastic network models. Here, we perform all-atom
molecular dynamics simulation with water.
[4] M. Yamanaka, Journal of the Physical Society of Japan 82, 083801, 2013.
[5] For a review, see Akio Kitao, Normal Mode Analysis: Theory and Applications to Biologicaland Chemical Systems. Chapter 12: edited by Qiang Cui & Ivet Bahar, , Chapman & Hall/CRC,Boca Raton, FL, USA, 233-251, 2006, and references therein.