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A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore

Published online by Cambridge University Press:  03 June 2015

Jehanzeb Hameed Chaudhry,
Jeffrey Comer,
Aleksei Aksimentiev and
Luke N. Olson
Jehanzeb Hameed Chaudhry*
Affiliation:
Department for Mathematics, Colorado State University, Fort Collins, CO 80523, USA
Jeffrey Comer*
Affiliation:
Department for Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Aleksei Aksimentiev*
Affiliation:
Department for Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Luke N. Olson*
Affiliation:
Department for Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
*
Corresponding author.Email:jehanzeb@colostate.edu
Email:jeffcomer@gmail.com
Email:aksiment@illinois.edu
Email:lukeo@illinois.edu
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Abstract

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore. We describe a novel, robust finite element solver that combines the applications of the Newton’s method to the nonlinear Galerkin form of the equations, augmented with stabilization terms to appropriately handle the drift-diffusion processes.

To make direct comparison with particle-based simulations possible, our method is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain. We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates, calculations of the ionic current though a nanopore subject to an external electric field, and modeling the effect of a DNA molecule on the ion concentration and nanopore current.

Keywords

82.20.Wt87.15.A-83.10.Rs87.10.Kn87.10.EdSteric effectsnucleic acidsionic currentcontinuum transport theoryion channelsPNPSUPGfinite element method

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Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 1 , January 2014, pp. 93 - 125
Copyright
Copyright © Global Science Press Limited 2014

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References

[1] Neher, E., Sakmann, B., Steinbach, J., The extracellular patch clamp: A method for resolving currents through individual open channels in biological membranes, Pflug. Arch. Eur. J. Physiol. 375 (1978) 219–228.Google Scholar
[2] Sakmann, B., Neher, E., Patch clamp techniques for studying ionic channels in excitable membranes, Ann. Rev. Physiol. 46 (1984) 455–472.Google Scholar
[3] Mathé, J., Visram, H., Viasnoff, V., Rabin, Y., Meller, A., Nanopore unzipping of individual DNA hairpin molecules, Biophys. J. 87 (2004) 3205–3212.Google Scholar
[4] Zhao, Q., Sigalov, G., Dimitrov, V., Dorvel, B., Mirsaidov, U., Sligar, S., Aksimentiev, A., Timp, G., Detecting SNPs using a synthetic nanopore, Nano Lett. 7 (2007) 1680–1685. CrossRefGoogle ScholarPubMed
[5] Zhao, Q., Comer, J., Dimitrov, V., Aksimentiev, A., Timp, G., Stretching and unzipping nucleic acid hairpins using a synthetic nanopore, Nucl. Acids Res. 36 (2008) 1532–1541.Google Scholar
[6] Kang, X. F., Cheley, S., Guan, X. Y., Bayley, H., Stochastic detection of enantiomers, J. Am. Chem. Soc. 128 (2006) 10684–10685.CrossRefGoogle ScholarPubMed
[7] Kasianowicz, J. J., Brandin, E., Branton, D., Deamer, D. W., Characterization of individual polynucleotide molecules using a membrane channel, Proc. Natl. Acad. Sci. USA 93 (1996) 13770–13773.Google Scholar
[8] Akeson, M., Branton, D., Kasianowicz, J. J., Brandin, E., Deamer, D. W., Microsecond time-scale discrimination among polycytidylic acid, polyadenylic acid, and polyuridylic acid as homopolymers or as segments within singe RNA molecules, Biophys. J. 77 (1999) 3227–3233.Google Scholar
[9] Branton, D., Deamer, D., Marziali, A., Bayley, H., Benner, S., Butler, T., Di, M. Ventra, Garaj, S., Hibbs, A., Huang, X., et al., The potential and challenges of nanopore sequencing, Nature Biotech. 26 (10) (2008) 1146–1153.Google Scholar
[10] Clarke, J., Wu, H., Jayasinghe, L., Patel, A., Reid, S., Bayley, H., Continuous base identification for single-molecule nanopore DNA sequencing, Nature Nanotech. 4 (2009) 265–270.Google Scholar
[11] Derrington, I., Butler, T., Collins, M., Manrao, E., Pavlenok, M., Niederweis, M., Gundlach, J., Nanopore DNA sequencing with MspA, Proc. Natl. Acad. Sci. USA 107 (2010) 16060.Google Scholar
[12] Im, W., Roux, B., Ions and counterions in a biological channel: a molecular dynamics study of OmpF porin from Escherichia coli in an explicit membrane with 1 M KCl aqueous salt solution, J. Mol. Biol. 319 (2002) 1177–1197.Google Scholar
[13] Im, W., Roux, B., Ion permeation and selectivity of OmpF porin: A theoretical study based on molecular dynamics, brownian dynamics, and continuum electrodiffusion theory, J. Mole. Bio. 322 (2002) 851 –869.Google Scholar
[14] Noskov, S. Y., Im, W., Roux, B., Ion permeation through the a-hemolysin channel: Theoretical studies based on Brownian Dynamics and Poisson-Nernst-Plank electrodiffusion theory, Biophys. J. 87 (2004) 2299–2309.CrossRefGoogle Scholar
[15] Aksimentiev, A., Heng, J. B., Timp, G., Schulten, K., Microscopic kinetics of DNA translocation through synthetic nanopores, Biophys. J. 87 (2004) 2086–2097.Google Scholar
[16] Aksimentiev, A., Schulten, K., Imaging alpha-hemolysin with molecular dynamics: Ionic conductance, osmotic permeability and the electrostatic potential map, Biophys. J. 88 (2005) 3745–3761.Google Scholar
[17] Comer, J., Dimitrov, V., Zhao, Q., Timp, G., Aksimentiev, A., Microscopic mechanics of hairpin DNA translocation through synthetic nanopores, Biophys. J. 96 (2009) 593–608.Google Scholar
[18] Pezeshki, S., Chimerel, C., Bessonov, A. N., Winterhalter, M., Kleinekathofer, U., Understanding ion conductance on a molecular level: An all-atom modeling of the bacterial porin OmpF, Biophys. J. 97 (2009) 1898–1906.Google Scholar
[19] Luo, Y., Egwolf, B., Walters, D., Roux, B., Ion selectivity of α-hemolysin with a β-cyclodextrin adapter.I. Single ion potential of mean force and diffusion coefficient, J. Phys. Chem. B (2009) 2035–2042.Google Scholar
[20] Aksimentiev, A., Deciphering ionic current signatures of DNA transport through a nanopore, Nanoscale 2 (2010) 468–483.Google Scholar
[21] Maffeo, C., Bhattacharya, S., Yoo, J., Wells, D., Aksimentiev, A., Modeling and simulation of ion channels, Chem. Rev. 112 (2012) 6250–6284.Google Scholar
[22] Allen, M. P., Tildesley, D. J., Computer Simulation of Liquids, Oxford University Press, New York, 1987.Google Scholar
[23] Carr, R., Comer, J., Ginsberg, M., Aksimentiev, A., Atoms-to-microns model for small solute transport through sticky nanochannels, Lab Chip 11 (2011) 3766–3773.Google Scholar
[24] Comer, J., Aksimentiev, A., Predicting the DNA sequence dependence of nanopore ion current using atomic-resolution Brownian dynamics, J. Phys. Chem. C 116 (2012) 3376–3393.Google Scholar
[25] Davis, M. E., McCammon, J. A., Electrostatics in biomolecular structure and dynamics, Chem. Rev. 90 (1990) 509–521.Google Scholar
[26] Barcilon, V., Chen, D.-P. and Eisenberg, R. S., Ion flow through narrow membrane channels. Part II, SIAM J. Appl. Math. 52 (1992) 1405–1425.Google Scholar
[27] Kurnikova, M. G., Coalson, R. D., Graf, P., Nitzan, A., A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transport through the gramicidin A channel, Biophys J. 76 (1999) 642–656.Google Scholar
[28] Lu, B., Zhou, Y., Huber, G. A., Bond, S. D., Holst, M. J., McCammon, J. A., Electrodiffusion: A continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution, J. Chem. Phys. 127 (2007) 135102 (17 pages).Google Scholar
[29] Kilic, M. S., Bazant, M. Z., Ajdari, A., Steric effects in the dynamics of electrolytes at large applied voltages. II. Modified Poisson-Nernst-Planck equations, Phys. Rev. E 75 (2007) 021503.Google Scholar
[30] Bolintineanu, D. S.,Sayyed-Ahmad, A., Davis, H. T., Kaznessis, Y. N., Poisson-Nernst-Planck models of nonequilibrium ion electrodiffusion through a protegrin transmembrane pore, PLoS Comput. Biol. 5 (1) (2009) e1000277.Google Scholar
[31] Cardenas, A. E., Coalson, R. D., Kurnikova, M. G., Three-dimensional Poisson-Nernst-Planck theory studies: Influence of membrane electrostatics on gramicidin A channel conductance, Biophys. J. 79 (1) (2000) 80–93.Google Scholar
[32] Cohen, H., Cooley, J., The numerical solution of the time-dependent Nernst-Planck equations, Biophys. J. 5 (1965) 145–162.Google Scholar
[33] Lu, B., Zhou, Y. C., Huber, G. A., Bond, S. D., Holst, M. J., McCammon, J. A., Electrodiffusion: A continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution, J. Chem. Phys. 127 (2007) 135102.Google Scholar
[34] Lu, B., Holst, M. J., Andrew, J. McCammon, Zhou, Y. C., Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: Finite element solutions, J. Comput. Phys. 229 (2010) 6979–6994.CrossRefGoogle ScholarPubMed
[35] Lu, B., Zhou, Y., Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes II: Size effects on ionic distributions and diffusion-reaction rates, Biophys. J. 100 (2011) 2475–2485.Google Scholar
[36] Zheng, Q., Chen, D., Wei, G.-W., Second-order Poisson-Nernst-Planck solver for ion transport, J. Comput. Phys. 230 (2011) 5239–5262.Google Scholar
[37] Logg, A., Wells, G. N., DOLFIN: Automated finite element computing, ACM Trans. Math. Softw. 37 (2010) 20:1–20:28.CrossRefGoogle Scholar
[38] Braess, D., Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory, 3rd Edition, Cambridge University Press, 2007.CrossRefGoogle Scholar
[39] Bochev, P. B., Gunzburger, M. D., Shadid, J. N., Stability of the SUPG finite element method for transient advection-diffusion problems, Comp. Meth. Appl. Mech. Engr. 193 (23-26) (2004) 2301–2323.Google Scholar
[40] Hughes, T. J. R., Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations, Int. J. Numer. Meth. Fluids 7 (1987) 1261–1275.Google Scholar
[41] Hughes, T. J. R., Franca, L. P., Hulbert, G. M., A new finite element formulation for computational fluid dynamics: VIII. The galerkin/least-squares method for advective-diffusive equations, Comp. Meth. Appl. Mech. Engr. 73 (1989) 173–189.Google Scholar
[42] Franca, L. P., Frey, S. L., Hughes, T. J. R., Stabilized finite element methods: I. Application to the advective-diffusive model, Comp. Meth. Appl. Mech. Engr. 95 (1992) 253–276.Google Scholar
[43] T, T.E., Stabilized finite element formulations for incompressible flow computations, Adv. Appl. Mech. 28 (1991) 1–44.Google Scholar
[44] Hughes, T. J., Mallet, M., Akira, M., A new finite element formulation for computational fluid dynamics: II. Beyond SUPG, Comp. Meth. Appl. Mech. Engr. 54 (1986) 341–355.Google Scholar
[45] Gilson, M. K., Davis, M. E., Luty, B. A., McCammon, J. A., Computation of electrostatic forces on solvated molecules using the poisson-boltzmann equation, J. Phys. Chem. 97 (1993) 3591–3600.Google Scholar
[46] Zhou, Z., Payne, P., Vasquez, M., Kuhn, N., Levitt, M., Finite-difference solution of the Poisson-Boltzmann equation: Complete elimination of self-energy, J. Comput. Chem. 17 (1996) 1344–1351.Google Scholar
[47] Chaudhry, J., Bond, S., Olson, L., Finite element approximation to a finite-size modified Poisson-Boltzmann equation, J. Sci. Comput. 47 (2011) 347–364, 10.1007/s10915-010-9441-7.Google Scholar
[48] Bond, S. D., Chaudhry, J. H., Cyr, E. C., Olson, L. N., A first-order system least-squares finite element method for the Poisson-Boltzmann equation, J. Comput. Chem. 31 (8) (2010) 1625–1635.Google Scholar
[49] Chern, I.‐L., Liu, J.‐G., Wang, W.‐C., Accurate evaluation of electrostatics for macromolecules in solution, Meth. Appl. Anal. 10 (2003) 309–328.Google Scholar
[50] Aksoylu, B., Bond, S. D., Cyr, E. C., Holst, M. J., Goal-oriented adaptivity and multilevel pre-conditioning for the Poisson-Boltzmann equation, J. Sci. Comput. 52 (2012) 202–225.Google Scholar
[51] Bazant, M. Z., Kilic, M. S., Storey, B. D., Ajdari, A., Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions, Adv. Colloid Interface Sci. 152 (1-2) (2009) 48–88.Google Scholar
[52] Heng, J., Aksimentiev, A., Ho, C., Marks, P., Grinkova, Y., Sligar, S., Schulten, K., Timp, G., The electromechanics of dna in a synthetic nanopore, Biophys. J. 90 (2006) 1098–1106.Google Scholar
[53] Drew, H. R., Wing, R. M., Takano, T., Broka, C., Tanaka, S., Itakura, K., Dickerson, R. E., Structure of a b-dna dodecamer: Conformation and dynamics, Proc. Natl. Acad. Sci. USA 78 (1981) 2179–2183.Google Scholar
[54] URL http://www.rcsb.org/pdb/ Google Scholar
[55] Dolinsky, T., Czodrowski, P., Li, H., Nielsen, J., Jensen, J., Baker, G. K. N., Pdb2pqr: Expanding and upgrading automated preparationof biomolecular structures for molecular simulations, Nucleic Acids Res. 35 (2007) W522–W525.Google Scholar
[56] Dolinsky, T., Nielsen, J., McCammon, J., Baker, N., Pdb2pqr: An automated pipeline for the setup, execution, and analysis of Poisson-Boltzmann electrostatics calculations, Nucleic Acids Res. 35 (2004) W665–W667.Google Scholar
[57] Yu, Z., Holst, M. J., Cheng, Y., McCammon, J. A., Feature-preserving adaptive mesh generation for molecular shape modeling and simulation, J. Mol. Graph. Model. 26 (2008) 1370 –1380.Google Scholar
[58] Geuzaine, C., Remacle, J.-F., Gmsh: A 3-d finite element mesh generator with built-in preand post-processing facilities, Int. J. Numer. Meth. Engr. 79 (2009) 1309–1331.Google Scholar
[59] Smeets, R. M. M., Keyser, U. F., Krapf, D., Wu, M.-Y., Dekker, N. H., Dekker, C., Salt dependence of ion transport and dna translocation through solid-state nanopores, Nano Letters 6 (2006) 89–95.Google Scholar
[60] Young, M., Jayaram, B., Beveridge, D., Local dielectric environment of B-DNA in solution: Results from a 14 ns molecular dynamics trajectory, J. Phys. Chem. B 102 (1998) 7666–7669.Google Scholar
[61] Roux, B., The calculation of the potential of mean force using computer simulations, Comp. Phys. Commun. 91 (1995) 275–282.Google Scholar
[62] Phillips, J. C., Braun, R., Wang, W., Gumbart, J., Tajkhorshid, E., Villa, E., Chipot, C., Skeel, R. D., Kale, L., Schulten, K., Scalable molecular dynamics with NAMD, J. Comp. Chem. 26 (2005) 1781–1802.Google Scholar
[63] MacKerell, A. D., Jr., , Bashford, D., Bellott, M., Dunbrack, R. L., Jr., , Evanseck, J., Field, M. J., Fischer, S., Gao, J., Guo, H., Ha, S., Joseph, D., Kuchnir, L., Kuczera, K., Lau, F. T. K., Mattos, C., Michnick, S., Ngo, T., Nguyen, D. T., Prodhom, B., Reiher, I. W. E., Roux, B., Schlenkrich, M., Smith, J., Stote, R., Straub, J., Watanabe, M., Wiorkiewicz-Kuczera, J., Yin, D., Karplus, M., Allatom empirical potential for molecular modeling and dynamics studies of proteins, J. Phys. Chem. B 102 (1998) 3586–3616.Google Scholar
[64] Beglov, D., Roux, B., Finite representation of an infinite bulk system: Solvent boundary potential for computer simulations, J. Chem. Phys. 100 (1994) 9050–9063.Google Scholar
[65] Im, W., Roux, B., Brownian dynamics simulations of ions channels: A general treatment of electrostatic reaction fields for molecular pores of arbitrary geometry, J. Chem. Phys. 115 (2001) 4580.Google Scholar
[66] Wells, D. B., Abramkina, V., Aksimentiev, A., Exploring transmembrane transport through a-hemolysin with grid-steered molecular dynamics, J. Chem. Phys. 127 (2007) 125101.Google Scholar