Numerical Solutions of Master Equation for Protein Folding Kinetics

Numerical solution of a master equation involves the calculation of eigenpairs for the corresponding transition matrix. In this paper, we computationally study the folding rate for a kinetics problem of protein folding by solving a large-scale eigenvalue problem. Three numerical methods, the implicitly restarted Arnoldi, the Jacobi-Davidson, and the QR methods are applied to solve the corresponding large scale eigenvalue problem of the transition matrix of master equation. Comparison among three methods is performed in terms of the computational efficiency. It is found that the QR method demands tremendous computing resource when the length of sequence L>10 due to the extremely large size of matrix and CPU time limitation. The Jacobi-Davidson method may encounter convergence issue, for some testing cases with L>9. The implicitly restarted Arnoldi method is suitable for solving the problem among three solution methods. Numerical examples with various residues are investigated