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A Pseudospectral solution of a bistable Fokker–Planck equation that models protein folding

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  • Philipp, Lucas
  • Shizgal, Bernie D.

Abstract

We consider the one-dimensional bistable Fokker–Planck equation proposed by Polotto et al. (2018), with specific drift and diffusion coefficients so as to model protein folding. In this paper, a pseudospectral method is used to solve the Fokker–Planck equation in terms of the eigenvalues (λn) and eigenfunctions (ψn) of the Fokker–Planck operator. Nonclassical polynomials, constructed orthogonal with respect to the equilibrium distribution of the Fokker–Planck equation, are used as basis functions. The eigenvalues determined with the pseudospectral method are compared with the Wentzel–Kramers–Brillouin (WKB) and the SUperSYmmetric (SUSY) Wentzel–Kramers–Brillouin (SWKB) approximations. The eigenvalues calculated differ significantly from those reported by Polotto et al. A detailed study of the role of the lowest non-zero eigenvalue, λ1, to model the rate coefficient for the transition between the bistable states is provided.

Suggested Citation

  • Philipp, Lucas & Shizgal, Bernie D., 2019. "A Pseudospectral solution of a bistable Fokker–Planck equation that models protein folding," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 522(C), pages 158-166.
    • Handle: RePEc:eee:phsmap:v:522:y:2019:i:c:p:158-166
    DOI: 10.1016/j.physa.2019.01.146
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    References listed on IDEAS

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    1. van den Brink, Alec Maassen & Dekker, H., 1997. "Reaction rate theory: weak- to strong-friction turnover in Kramers' Fokker-Planck model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 237(3), pages 515-553.
    2. Polotto, Franciele & Drigo Filho, Elso & Chahine, Jorge & Oliveira, Ronaldo Junio de, 2018. "Supersymmetric quantum mechanics method for the Fokker–Planck equation with applications to protein folding dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 493(C), pages 286-300.
    3. Felderhof, B.U., 2008. "Diffusion in a bistable potential," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(21), pages 5017-5023.
    4. Borges, G.R.P. & Filho, Elso Drigo & Ricotta, R.M., 2010. "Variational supersymmetric approach to evaluate Fokker–Planck probability," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(18), pages 3892-3899.
    5. Caldas, Denise & Chahine, Jorge & Filho, Elso Drigo, 2014. "The Fokker–Planck equation for a bistable potential," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 412(C), pages 92-100.
    6. Zheng, W.M., 1983. "On the extra factor 12√π in the Kramers approximation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 117(1), pages 171-178.
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