Skip to content
Licensed Unlicensed Requires Authentication Published online by De Gruyter May 17, 2022

Reduced basis method for the nonlinear Poisson–Boltzmann equation regularized by the range-separated canonical tensor format

Cleophas Kweyu, Lihong Feng, Matthias Stein and Peter Benner

Abstract

The Poisson–Boltzmann equation (PBE) is a fundamental implicit solvent continuum model for calculating the electrostatic potential of large ionic solvated biomolecules. However, its numerical solution encounters severe challenges arising from its strong singularity and nonlinearity. In (P. Benner, V. Khoromskaia, B. Khoromskij, C. Kweyu, and M. Stein, “Regularization of Poisson-Boltzmann type equations with singular source terms using the range-separated tensor format,” SIAM J. Sci. Comput., vol. 43, no. 1, pp. A415–A445, 2021; C. Kweyu, V. Khoromskaia, B. Khoromskij, M. Stein, and P. Benner, “Solution decomposition for the nonlinear Poisson-Boltzmann equation using the range-separated tensor format,” arXiv:2109.14073, 2021), the effect of strong singularities was eliminated by applying the range-separated (RS) canonical tensor format (P. Benner, V. Khoromskaia, and B. N. Khoromskij, “Range-separated tensor format for many-particle modeling,” SIAM J. Sci. Comput., vol. 40, no. 2, pp. A1034–A1062, 2018; B. N. Khoromskij, “Range-separated tensor representation of the discretized multidimensional Dirac delta and elliptic operator inverse,” J. Comput. Phys., vol. 401, p. 108998, 2020) to construct a solution decomposition scheme for the PBE. The RS tensor format allows deriving a smooth approximation to the Dirac delta distribution in order to obtain a regularized PBE (RPBE) model. However, solving the RPBE is still computationally demanding due to its high dimension N , where N is always in the millions. In this study, we propose to apply the reduced basis method (RBM) and the (discrete) empirical interpolation method ((D)EIM) to the RPBE in order to construct a reduced order model (ROM) of low dimension N N , whose solution accurately approximates the nonlinear RPBE. The long-range potential can be obtained by lifting the ROM solution back to the N -space while the short-range potential is directly precomputed analytically, thanks to the RS tensor format. The sum of both provides the total electrostatic potential. The main computational benefit is the avoidance of computing the numerical approximation of the singular electrostatic potential. We demonstrate in the numerical experiments, the accuracy and efficacy of the reduced basis (RB) approximation to the nonlinear RPBE (NRPBE) solution and the corresponding computational savings over the classical nonlinear PBE (NPBE) as well as over the RBM being applied to the classical NPBE.

AMS Subject Classification: 65F30; 65F50; 65N35; 65F10

Corresponding author: Cleophas Kweyu, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, D-39106 Magdeburg, Germany; and Department of Mathematics and Physics, Moi University, P.O. Box 3900-30100, Eldoret, Kenya, E-mail:

Acknowledgement

The authors thank the following organizations for financial and material support on this project: International Max Planck Research School (IMPRS) for Advanced Methods in Process and Systems Engineering and Max Planck Society for the Advancement of Science (MPG).

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] N. A. Baker, “Poisson-Boltzmann methods for biomolecular electrostatics,” Methods Enzymol., vol. 383, pp. 94–118, 2004. https://doi.org/10.1016/s0076-6879(04)83005-2.Search in Google Scholar

[2] N. A. Baker, “Biomolecular applications of Poisson–Boltzmann equation,” in Reviews in Computational Chemistry, vol. 21, Hoboken, NJ, USA, John Wiley & Sons, 2005.10.1002/0471720895.ch5Search in Google Scholar

[3] N. A. Baker and J. Wagoner, “Solvation forces on biomolecular structures: a comparison of explicit solvent and Poisson-Boltzmann models,” J. Comput. Chem., vol. 25, pp. 1623–1629, 2004. https://doi.org/10.1002/jcc.20089.Search in Google Scholar

[4] B. Z. Lu, Y. C. Zhou, M. J. Holst, and J. A. McCammon, “Recent progress in numerical methods for Poisson-Boltzmann equation in biophysical applications,” Commun. Comput. Phys., vol. 3, no. 5, pp. 973–1009, 2008.Search in Google Scholar

[5] F. Fogolari, A. Brigo, and H. Molinari, “The Poisson-Boltzmann equation for biomolecular electrostatics: a tool for structural biology,” J. Mol. Recogn., vol. 15, no. 6, pp. 377–392, 2002. https://doi.org/10.1002/jmr.577.Search in Google Scholar

[6] M. T. Neves-Petersen and S. Petersen, “Protein electrostatics: a review of the equations and methods used to model electrostatic equations in biomolecules - applications in biotechnology,” Biotechnol. Annu. Rev., vol. 9, pp. 315–395, 2003. https://doi.org/10.1016/S1387-2656(03)09010-0.Search in Google Scholar

[7] M. Stein, R. R. Gabdoulline, and R. C. Wade, “Cross-species analysis of the glycoliticmpathway by comparison of molecular interaction fields,” Mol. Biosyst., vol. 6, pp. 162–174, 2010. https://doi.org/10.1039/b912398a.Search in Google Scholar

[8] M. J. Holst, “Multilevel methods for the Poisson-Boltzmann equation,” Ph.D. thesis, Urbana-Champaign, IL, USA, Numerical Computing Group, University of Illinois, 1994.Search in Google Scholar

[9] F. Dong, B. Oslen, and N. A. Baker, “Computational methods for biomolecular electrostatics,” Methods Cell Biol., vol. 84, no. 1, pp. 843–870, 2008. https://doi.org/10.1016/S0091-679X(07)84026-X.Search in Google Scholar

[10] N. A. Baker, M. J. Holst, and F. Wang, “The adaptive multilevel finite element solution of the Poisson-Boltzmann equation on massively parallel computers,” IBM J. Res. Dev., vol. 45, pp. 427–438, 2001. https://doi.org/10.1147/rd.453.0427.Search in Google Scholar

[11] J. Wang and R. Luo, “Assessment of linear finite difference Poisson-Boltzmann solvers,” J. Comput. Chem., vol. 31, pp. 1689–1698, 2010. https://doi.org/10.1016/j.cpc.2015.08.029.Search in Google Scholar

[12] M. Holst, N. Baker, and F. Wang, “Adaptive multilevel finite element solution of the Poisson-Boltzmann equation: algorithms and examples,” J. Comput. Chem., vol. 21, pp. 1319–1342, 2000. https://doi.org/10.1002/1096-987X(20001130)21:15<1319::AID-JCC1>3.0.10.1002/1096-987X(20001130)21:15<1319::AID-JCC1>3.0.CO;2-8Search in Google Scholar

[13] A. H. Boschitsch and M. O. Fenley, “Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzman equation,” J. Comput. Chem., vol. 25, no. 7, pp. 935–955, 2004. https://doi.org/10.1002/jcc.20000.Search in Google Scholar

[14] H. X. Zhou, “Boundary element solution of macromolecular electrostatics: inteaction energy between two proteins,” Biophys. J., vol. 65, no. 2, pp. 955–963, 1993. https://doi.org/10.1016/S0006-3495(93)81094-4.Search in Google Scholar

[15] D. Xie, “New solution decomposition and minimization scheme for Poisson-Boltzmann equation in calculation of biomolecular electrostatics,” J. Comput. Phys., vol. 275, pp. 294–309, 2014. https://doi.org/10.1016/j.jcp.2014.07.012.Search in Google Scholar

[16] L. Chen, M. J. Holst, and J. Xu, “The finite element approximation of the nonlinear Poisson-Boltzmann equation,” SIAM J. Numer. Anal., vol. 45, no. 6, pp. 2298–2320, 2007. https://doi.org/10.1137/060675514.Search in Google Scholar

[17] M. Mirzadeh, M. Theillard, A. Helgadottir, D. Boy, and F. Gibou, “An adaptive, finite difference solver for the nonlinear Poisson-Boltzmann equation with applications to biomolecular computations,” Commun. Comput. Phys., vol. 13, no. 1, pp. 150–173, 2013. https://doi.org/10.4208/cicp.290711.181011s.Search in Google Scholar

[18] P. Benner, V. Khoromskaia, B. Khoromskij, C. Kweyu, and M. Stein, “Regularization of Poisson-Boltzmann type equations with singular source terms using the range-separated tensor format,” SIAM J. Sci. Comput., vol. 43, no. 1, pp. A415–A445, 2021. https://doi.org/10.1137/19M1281435.Search in Google Scholar

[19] C. Kweyu, V. Khoromskaia, B. Khoromskij, M. Stein, and P. Benner, “Solution decomposition for the nonlinear Poisson-Boltzmann equation using the range-separated tensor format,” arXiv:2109.14073, 2021.Search in Google Scholar

[20] P. Benner, V. Khoromskaia, and B. N. Khoromskij, “Range-separated tensor format for many-particle modeling,” SIAM J. Sci. Comput., vol. 40, no. 2, pp. A1034–A1062, 2018. https://doi.org/10.1137/16m1098930.Search in Google Scholar

[21] L. Ji, Y. Chen, and Z. Xu, “A reduced basis method for the nonlinear Poisson-Boltzmann equation,” Adv. Appl. Math. Mech., vol. 11, pp. 1200–1218, 2019. https://doi.org/10.4208/aamm.OA-2018-0188.Search in Google Scholar

[22] S. Chaturantabut and D. C. Sorensen, “Nonlinear model reduction via discrete empirical interpolation,” SIAM J. Sci. Comput., vol. 32, no. 5, pp. 2737–2764, 2010. https://doi.org/10.1137/090766498.Search in Google Scholar

[23] M. A. Grepl, Y. Maday, N. C. Nguyen, and A. T. Patera, “Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations,” ESAIM Math. Model. Numer. Anal., vol. 41, no. 3, pp. 575–605, 2007. https://doi.org/10.1051/m2an:2007031.10.1051/m2an:2007031Search in Google Scholar

[24] M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera, “An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations,” C. R. Math. Acad. Sci. Paris, vol. 339, no. 9, pp. 667–672, 2004. https://doi.org/10.1016/j.crma.2004.08.006.Search in Google Scholar

[25] B. N. Khoromskij, “Range-separated tensor representation of the discretized multidimensional Dirac delta and elliptic operator inverse,” J. Comput. Phys., vol. 401, p. 108998, 2020. https://doi.org/10.1016/j.jcp.2019.108998.Search in Google Scholar

[26] K. A. Sharp and B. Honig, “Electrostatic interactions in macromolecules: theory and applications,” Annu. Rev. Biophys. Chem., vol. 19, pp. 301–332, 1990. https://doi.org/10.1146/annurev.bb.19.060190.001505.Search in Google Scholar

[27] C. Kweyu, L. Feng, M. Stein, and P. Benner, “Fast solution of the Poisson-Boltzmann equation with nonaffine parametrized boundary conditions using the reduced basis method,” Comput. Visual Sci., vol. 23, p. 15, 2020. https://doi.org/10.1007/s00791-020-00336-z.Search in Google Scholar

[28] F. Fogolari, P. Zuccato, G. Esposito, and P. Viglino, “Biomolecular electrostatics with the linearized Poisson-Boltzmann equation,” Biophys. J., vol. 76, no. 1, pp. 1–16, 1999. https://doi.org/10.1016/S0006-3495(99)77173-0.Search in Google Scholar

[29] C. Qin, H. Meng-Juei, W. Jun, and L. Ray, “Performance of nonlinear finite-difference Poisson-Boltzmann solvers,” J. Chem. Theor. Comput., vol. 6, no. 1, pp. 203–211, 2010. https://doi.org/10.1021/ct900381r.Search in Google Scholar PubMed PubMed Central

[30] I. Chern, J. Liu, and W. Wang, “Accurate evaluation of electrostatics for macromolecules in solution,” Methods Appl. Anal., vol. 10, no. 2, pp. 309–328, 2003. https://doi.org/10.4310/maa.2003.v10.n2.a9.Search in Google Scholar

[31] Encyclopedia of Mathematics, Newton potential, 2018. Available at: http://www.encyclopediaofmath.org/index.php?title=Newton_potential&oldid=33114 [accessed: 12 03, 2018].Search in Google Scholar

[32] V. Khoromskaia and B. N. Khoromskij, “Grid-based lattice summation of electrostatic potentials by assembled rank-structured tensor approximation,” Comput. Phys. Commun., vol. 185, no. 12, pp. 3162–3174, 2014. https://doi.org/10.1016/j.cpc.2014.08.015.Search in Google Scholar

[33] B. N. Khoromskij and V. Khoromskaia, “Multigrid accelerated tensor approximation of function related multidimensional arrays,” SIAM J. Sci. Comput., vol. 31, no. 4, pp. 3002–3026, 2009. https://doi.org/10.1137/080730408.Search in Google Scholar

[34] W. Rocchia, E. Alexov, and B. Honig, “Extending the applicability of the nonlinear Poisson-Boltzmann equation: multiple dielectric constants and multivalent ions,” J. Phys. Chem., vol. 105, no. 28, pp. 6507–6514, 2001. https://doi.org/10.1021/jp010454y.Search in Google Scholar

[35] B. A. Luty, M. E. Davis, and J. A. McCammon, “Solving the finite-difference nonlinear Poisson-Boltzmann equation,” J. Comput. Chem., vol. 13, no. 9, pp. 1114–1118, 1992. https://doi.org/10.1002/jcc.540130911.Search in Google Scholar

[36] H. Oberoi and N. M. Allewell, “Multigrid solution of the nonlinear Poisson-Boltzmann equation and calculation of titration curves,” Biophys. J., vol. 65, no. 1, pp. 48–55, 1993. https://doi.org/10.1016/S0006-3495(93)81032-4.Search in Google Scholar

[37] M. Holst and F. Saied, “Numerical solution of the nonlinear Poisson-Boltzmann equation: developing more robust and efficient methods,” J. Comput. Chem., vol. 16, pp. 337–364, 1995. https://doi.org/10.1002/jcc.540160308.Search in Google Scholar

[38] A. I. Shestakov, J. L. Milovich, and A. Noy, “Solution of the nonlinear Poisson-Boltzmann equation using pseudo-transient continuation and the finite element method,” Commun. Comput. Phys., vol. 247, pp. 62–79, 2002. https://doi.org/10.1006/jcis.2001.8033.Search in Google Scholar PubMed

[39] Y. Notay, “An aggregation-based algebraic multigrid method,” Electron. Trans. Numer. Anal., vol. 37, pp. 123–146, 2010.Search in Google Scholar

[40] J. S. Hesthaven, G. Rozza, and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Switzerland, Springer International Publishing, 2016.10.1007/978-3-319-22470-1Search in Google Scholar

[41] P. Benner, S. Gugercin, and K. Willcox, “A survey of model reduction methods for parametric systems,” SIAM Rev., vol. 57, no. 4, pp. 483–531, 2015. https://doi.org/10.1137/130932715.Search in Google Scholar

[42] J. L. Eftang, “Reduced basis methods for parametrized partial differential equations,” Ph.D. thesis, Trondheim, Norway, Norwegian University of Science and Technology, 2011.10.1007/978-3-642-15337-2_15Search in Google Scholar

[43] G. Rozza, D. B. P. Huynh, and A. T. Patera, “Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations,” Arch. Comput. Methods Eng., vol. 15, no. 3, pp. 229–275, 2008. https://doi.org/10.1007/s11831-008-9019-9.Search in Google Scholar

[44] S. Volkwein, Model Reduction Using Proper Orthogonal Decomposition. Lecture Notes, Konstanz, University of Konstanz, 2013.Search in Google Scholar

[45] L. Feng and P. Benner, “A robust algorithm for parametric model order reduction based on implicit moment matching,” in Reduced Order Methods for Modeling and Computational Reduction, MS&A Series, vol. 9, Berlin, Heidelberg, New York, Springer-Verlag, 2014, pp. 159–186. chapter 6.10.1007/978-3-319-02090-7_6Search in Google Scholar

[46] L. Feng, M. Mangold, and P. Benner, “Adaptive POD-DEIM basis construction and its application to a nonlinear population balance system,” AIChE J., pp. 3832–3844, 2017, https://doi.org/10.1002/aic.15749.Search in Google Scholar

[47] M. Holst and F. Saied, “Multigrid solution of the Poisson-Boltzmann equation,” J. Comput. Chem., vol. 14, pp. 105–113, 1993. https://doi.org/10.1002/jcc.540140114.Search in Google Scholar

[48] S. Vergara-Perez and M. Marucho, “MPBEC, a Matlab program for biomolecular electrostatic calculations,” Comput. Phys. Commun., vol. 198, pp. 179–194, 2016. https://doi.org/10.1016/j.cpc.2015.08.029.Search in Google Scholar

[49] M. H. le Du, P. Marchot, P. E. Bougis, and J. C. Fontecilla-Camps, “1.9 Angstrom resolution structure of fasciculine 1, an anti-acetylcholinesterase toxin from green mamba snake venom,” J. Biol. Chem., vol. 267, pp. 22122–22130, 1992. https://doi.org/10.1016/s0021-9258(18)41644-4.Search in Google Scholar

[50] M. G. Hinds, T. Maurer, J. Zhang, and N. A. Nicola, “Solution structure of Leukemia inhibitory factor,” Biol. Chem., vol. 273, pp. 13738–13745, 1998. https://doi.org/10.1074/jbc.273.22.13738.Search in Google Scholar PubMed

[51] C. Kweyu, M. Hess, L. Feng, M. Stein, and P. Benner, “Reduced basis method for Poisson-Boltzmann Equation,” in ECCOMAS Congress 2016 - Proc. of the VII European Congress on Computational Methods in Applied Sciences and Engineering, vol. 2, M. Papadrakakis, V. Papadopoulos, G. Stefanou, and V. Plevris, Eds., Athens, National Technical University of Athens, 2016, pp. 4187–4195.10.7712/100016.2103.5891Search in Google Scholar

[52] J. O. Bockris and A. K. N. Reddy, Modern Electrochemistry: Ionics, New York, Plenum Press, 1998.Search in Google Scholar

[53] N. A. Baker, D. Sept, S. Joseph, M. J. Holst, and J. A. McCammon, “Electrostatics of nanosystems: application to microtubules and the ribosome,” Proc. Natl. Acad. Sci. U.S.A., vol. 98, no. 18, pp. 10037–10041, 2001. https://doi.org/10.1073/pnas.181342398.Search in Google Scholar PubMed PubMed Central

Received: 2021-03-11
Revised: 2022-04-01
Accepted: 2022-04-26
Published Online: 2022-05-17

© 2022 Walter de Gruyter GmbH, Berlin/Boston