Parallel solution of large-scale eigenvalue problem for master equation in protein folding dynamics

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Abstract

It is known that a master equation characterizes time evolution of trajectories and transition of states in protein folding dynamics. Solution of the master equation may require calculating eigenvalues for the corresponding eigenvalue problem. In this paper, we numerically study the folding rate for a dynamic problem of protein folding by solving a large-scale eigenvalue problem. Three methods, the implicitly restarted Arnoldi, Jacobi–Davidson, and QR methods are employed in solving the corresponding large-scale eigenvalue problem for the transition matrix of master equation. Comparison shows that the QR method demands tremendous computing resource when the length of sequence L>10 due to extremely large size of matrix and CPU time limitation. The Jacobi–Davidson method may encounter convergence issue, for cases of L>9. The implicitly restarted Arnoldi method is suitable for solving problems among them. Parallelization of the implicitly restarted Arnoldi method is successfully implemented on a PC-based Linux cluster. The parallelization scheme mainly partitions the operation of matrix. For the Arnoldi factorization, we replicate the upper Hessenberg matrix Hm for each processor, and distribute the set of Arnoldi vectors Vm among processors. Each processor performs its own operation. The algorithm is implemented on a PC-based Linux cluster with message passing interface (MPI) libraries. Numerical experiment performing on our 32-nodes PC-based Linux cluster shows that the maximum difference among processors is within 10%. A 23-times speedup and 72% parallel efficiency are achieved when the matrix size is greater than 2×106 on the 32-nodes PC-based Linux cluster. This parallel approach enables us to explore large-scale dynamics of protein folding.

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Yiming Li received his B.S. degrees in applied mathematics and electronics engineering, his M.S. degree in applied mathematics, and his Ph.D. degree in electronics from the National Chiao Tung University (NCTU), Hsinchu, Taiwan, in 1996, 1998, and 2001, respectively. He is currently an Associate Professor with the Department of Communication Engineering, NCTU. He is a Deputy Director of the Modeling and Simulation Center and conducts the Parallel and Scientific Computing Laboratory at NCTU. His

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    Yiming Li received his B.S. degrees in applied mathematics and electronics engineering, his M.S. degree in applied mathematics, and his Ph.D. degree in electronics from the National Chiao Tung University (NCTU), Hsinchu, Taiwan, in 1996, 1998, and 2001, respectively. He is currently an Associate Professor with the Department of Communication Engineering, NCTU. He is a Deputy Director of the Modeling and Simulation Center and conducts the Parallel and Scientific Computing Laboratory at NCTU. His research areas include computational science and engineering, in particular, for biology, electronics and physics. He has authored over 120 research papers appearing in international book chapters, journals, and conferences. He has organized and served on various international conferences and has served as a reviewer and editor for many international journals. Dr. Li is a member of Phi Tau Phi, Sigma Xi, ACM and IEEE, and is included in Who's Who in the World. He was the recipient of the 2002 Research Fellowship Award presented by the Pan Wen-Yuan Foundation, Taiwan and the 2006 Outstanding Young Electrical Engineer Award from Chinese Institute of Electrical Engineering, Taiwan.

    Shao-Ming Yu received his B.S. and M.S. degrees in Computer and Information Science from NCTU in 2002 and 2004. Currently he is pursuing his Ph.D. degree at the Department of Computer Science of NCTU. His research interests focus on modeling and simulation of semiconductor nanodevices, parallel and scientific computation, evolutionary algorithms, and design optimization. He is a student member of IEEE.

    Yih-Lang Li received his B.S. degree in nuclear engineering and his M.S. and his Ph.D. degrees in computer science from the National Tsing Hua University, Hsinchu, Taiwan, in 1987, 1990, and 1996, respectively. In February 2003, he joined the faculty of the Department of Computer Science, NCTU, where he is currently an Assistant Professor. Prior to joining the faculty of NCTU, from 1995 to 1996 and from 1998 to 2003, he was a Software Engineer and an Associate Manager at Springsoft Corporation, Hsinchu, where he was heavily involved in the development of verification and synthesis tools for custom-based layout. His research interests include physical synthesis, parallel architecture, and VLSI testing.

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