Abstract
One method of computing the electrostatic energy of a biomolecule in a solution uses a continuum representation of the solution via the Poisson–Boltzmann equation. This can be solved in many ways, and we consider a Monte Carlo method of our design that combines the Walk-on-Spheres and Walk-on-Subdomains algorithms. In the course of examining the Monte Carlo implementation of this method, an issue was discovered in the Walk-on-Subdomains portion of the algorithm which caused the algorithm to sometimes take an abnormally long time to complete. As the problem occurs when a walker repeatedly oscillates between two subdomains, it is something that could cause a large increase in runtime for any method that used a similar algorithm. This issue is described in detail and a potential solution is examined.
References
[1] H. M. Berman, J. Westbrook, Z. Feng, G. Gilliland, T. N. Bhat, H. Weissig, I. N. Shindyalov and P. E. Bourne, The protein data bank, Nucleic Acids Res. 28 (2000), 235–242. 10.1093/nar/28.1.235Search in Google Scholar PubMed PubMed Central
[2] R. N. Cardinal and M. R. F. Aitken, ANOVA for the Behavioral Sciences Researcher, Psychology Press, London, 2013. 10.4324/9780203763933Search in Google Scholar
[3] M. O. Fenley, M. Mascagni, J. McClain, A. R. J. Silalahi and N. A. Simonov, Using correlated Monte Carlo sampling for efficiently solving the linearized Poisson–Boltzmann equation over a broad range of salt concentration, J. Chem. Theory Comput. 6 (2009), 300–314. 10.1021/ct9003806Search in Google Scholar PubMed PubMed Central
[4] J. A. Given, J. B. Hubbard and J. F. Douglas, A first-passage algorithm for the hydrodynamic friction and diffusion-limited reaction rate of macromolecules, J. Chem. Phys. 106 (1997), 3761–3771. 10.1063/1.473428Search in Google Scholar
[5] C.-O. Hwang, M. Mascagni and N. A. Simonov, Monte Carlo methods for the linearized Poisson–Boltzmann equation, Advances in Numerical Analysis, Nova Science, Hauppauge (2004). Search in Google Scholar
[6] T. Mackoy, R. C. Harris, J. Johnson, M. Mascagni and M. O. Fenley, Numerical optimization of a walk-on-spheres solver for the linear Poisson–Boltzmann equation, Commun. Comput. Phys. 13 (2013), 195–206. 10.4208/cicp.220711.041011sSearch in Google Scholar
[7] M. Mascagni and N. A. Simonov, Monte Carlo method for calculating the electrostatic energy of a molecule, Computational Science—ICCS 2003. Part I, Lecture Notes in Comput. Sci. 2657, Springer, Berlin (2003), 63–72. 10.1007/3-540-44860-8_7Search in Google Scholar
[8] M. Mascagni and N. A. Simonov, Monte Carlo methods for calculating some physical properties of large molecules, SIAM J. Sci. Comput. 26 (2004), no. 1, 339–357. 10.1137/S1064827503422221Search in Google Scholar
[9] M. E. Muller, Some continuous Monte Carlo methods for the Dirichlet problem, Ann. Math. Statist. 27 (1956), 569–589. 10.1214/aoms/1177728169Search in Google Scholar
[10] A. Pal and S. Reuveni, First passage under restart, Phys. Rev. Lett. 118 (2017), Article ID 030603. 10.1103/PhysRevLett.118.030603Search in Google Scholar PubMed
[11] K. K. Sabelfeld, Monte Carlo Methods in Boundary Value Problems, Springer, Berlin, 1991. 10.1007/978-3-642-75977-2Search in Google Scholar
[12] K. K. Sabelfeld and N. A. Simonov, Stochastic Methods for Boundary Value Problems. Numerics for High-dimensional PDEs and Applications, De Gruyter, Berlin, 2016. 10.1515/9783110479454Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
