Stabilized finite element methods to simulate the conductances of ion channels

Abstract We have previously developed a finite element simulator, ichannel, to simulate ion transport through three-dimensional ion channel systems via solving the Poisson–Nernst–Planck equations (PNP) and Size-modified Poisson–Nernst–Planck equations (SMPNP), and succeeded in simulating some ion channel systems. However, the iterative solution between the coupled Poisson equation and the Nernst–Planck equations has difficulty converging for some large systems. One reason we found is that the NP equations are advection-dominated diffusion equations, which causes troubles in the usual FE solution. The stabilized schemes have been applied to compute fluids flow in various research fields. However, they have not been studied in the simulation of ion transport through three-dimensional models based on experimentally determined ion channel structures. In this paper, two stabilized techniques, the SUPG and the Pseudo Residual-Free Bubble function (PRFB) are introduced to enhance the numerical robustness and convergence performance of the finite element algorithm in ichannel. The conductances of the voltage dependent anion channel (VDAC) and the anthrax toxin protective antigen pore (PA) are simulated to validate the stabilization techniques. Those two stabilized schemes give reasonable results for the two proteins, with decent agreement with both experimental data and Brownian dynamics (BD) simulations. For a variety of numerical tests, it is found that the simulator effectively avoids previous numerical instability after introducing the stabilization methods. Comparison based on our test data set between the two stabilized schemes indicates both SUPG and PRFB have similar performance (the latter is slightly more accurate and stable), while SUPG is relatively more convenient to implement.

[1]  J. Douglas,et al.  Stabilized mixed methods for the Stokes problem , 1988 .

[2]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[3]  Junichi Matsumoto,et al.  A Relationship between Stabilized FEM and Bubble Function Element Stabilization Method with Orthogonal Basis for Incompressible Flows , 2005 .

[4]  Aleksei Aksimentiev,et al.  A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore. , 2014, Communications in computational physics.

[5]  Nathan A. Baker,et al.  PDB2PQR: an automated pipeline for the setup of Poisson-Boltzmann electrostatics calculations , 2004, Nucleic Acids Res..

[6]  Lin Li,et al.  On the Dielectric “Constant” of Proteins: Smooth Dielectric Function for Macromolecular Modeling and Its Implementation in DelPhi , 2013, Journal of chemical theory and computation.

[7]  J S Wall,et al.  Anthrax protective antigen forms oligomers during intoxication of mammalian cells. , 1994, The Journal of biological chemistry.

[8]  D. Gillespie.  Energetics of divalent selectivity in a calcium channel: the ryanodine receptor case study. , 2008, Biophysical journal.

[9]  Minxin Chen,et al.  A parallel finite element simulator for ion transport through three‐dimensional ion channel systems , 2013, J. Comput. Chem..

[10]  Sergey M Bezrukov,et al.  Blocking anthrax lethal toxin at the protective antigen channel by using structure-inspired drug design. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[11]  Andrzej Lewenstam,et al.  Application of Nernst–Planck and Poisson equations for interpretation of liquid-junction and membrane potentials in real-time and space domains , 2001 .

[12]  Peter A. Markowich,et al.  The Stationary Semiconductor Device Equations. , 1987 .

[13]  Benzhuo Lu,et al.  VCMM: a visual tool for continuum molecular modeling. , 2014, Journal of molecular graphics & modelling.

[14]  S. Chung,et al.  Brownian dynamics study of ion transport in the vestibule of membrane channels. , 1998, Biophysical journal.

[15]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems , 1987 .

[16]  Nir Gavish,et al.  Poisson–Nernst–Planck equations with steric effects — non-convexity and multiple stationary solutions , 2017, 1703.07164.

[17]  Alvaro L. G. A. Coutinho,et al.  Numerical multiscale methods , 2012 .

[18]  Alessandro Russo,et al.  Approximation of the Stokes problem by residual-free macro bubbles , 1996 .

[19]  Alessandro Russo,et al.  On the choice of a stabilizing subgrid for convection?diffusion problems , 2005 .

[20]  Marek Behr,et al.  Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows , 1993 .

[21]  Bjrn Engquist.  Encyclopedia of Applied and Computational Mathematics , 2016 .

[22]  T L Croxton,et al.  Liquid junction potentials calculated from numerical solutions of the Nernst-Planck and Poisson equations. , 1989, Journal of theoretical biology.

[23]  Benzhuo Lu,et al.  Parallel Adaptive Finite Element Algorithms for Solving the Coupled Electro-diffusion Equations , 2013 .

[24]  G. Hummer,et al.  Ion transport through membrane-spanning nanopores studied by molecular dynamics simulations and continuum electrostatics calculations. , 2005, Biophysical journal.

[25]  Sunhwan Jo,et al.  Web interface for brownian dynamics simulation of ion transport and its applications to beta‐barrel pores , 2012, J. Comput. Chem..

[26]  Benzhuo Lu,et al.  Molecular surface-free continuum model for electrodiffusion processes. , 2008, Chemical physics letters.

[27]  A. Nitzan,et al.  A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transport through the gramicidin A channel. , 1999, Biophysical journal.

[28]  Minxin Chen,et al.  riangulated manifold meshing method preserving molecular surface topology , 2012 .

[29]  R. Codina.  Comparison of some finite element methods for solving the diffusion-convection-reaction equation , 1998 .

[30]  Y. C. Zhou,et al.  Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes II: size effects on ionic distributions and diffusion-reaction rates. , 2011, Biophysical journal.

[31]  Michael J. Holst,et al.  Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: Finite element solutions , 2010, J. Comput. Phys..

[32]  Bob Eisenberg.  Ionic Channels in Biological Membranes: Natural Nanotubes , 1998 .

[33]  S. Chung,et al.  Tests of continuum theories as models of ion channels. I. Poisson-Boltzmann theory versus Brownian dynamics. , 2000, Biophysical journal.

[34]  Thomas J. R. Hughes,et al.  A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems , 1986 .

[35]  F Friedberg,et al.  Effects of Metal Binding on Protein Structure , 1974, Quarterly Reviews of Biophysics.

[36]  T. Hughes,et al.  Stabilized finite element methods. I: Application to the advective-diffusive model , 1992 .

[37]  J. Milne,et al.  pH‐dependent permeabilization of the plasma membrane of mammalian cells by anthrax protective antigen , 1993, Molecular microbiology.

[38]  Endre Süli,et al.  Residual-free bubbles for advection-diffusion problems: the general error analysis , 2000, Numerische Mathematik.

[39]  Isaac Harari,et al.  A bubble‐stabilized finite element method for Dirichlet constraints on embedded interfaces , 2007 .

[40]  Marek Danielewski,et al.  EIS simulations for ion-selective site-based membranes by a numerical solution of the coupled Nernst–Planck–Poisson equations , 2006 .

[41]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[42]  Johannes Grotendorst,et al.  Modern methods and algorithms of quantum chemistry , 2000 .

[43]  Sanjay R. Mathur,et al.  A multigrid method for the Poisson–Nernst–Planck equations , 2009 .

[44]  J. Grotendorst,et al.  Modern methods and algorithms of quantum chemistry : winterschool 21. - 25. February 2000 Forschungszentrum Jülich : proceedings / org. by John von Neumann Institute for Computing , 2000 .

[45]  Shin-Ho Chung,et al.  Tests of continuum theories as models of ion channels. II. Poisson-Nernst-Planck theory versus brownian dynamics. , 2000, Biophysical journal.

[46]  J. Lear,et al.  Permeation through an open channel: Poisson-Nernst-Planck theory of a synthetic ionic channel. , 1997, Biophysical journal.

[47]  Claes Johnson,et al.  Finite element methods for linear hyperbolic problems , 1984 .

[48]  Serdar Kuyucak,et al.  Models of permeation in ion channels , 2001 .

[49]  Alessandro Russo,et al.  Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion problems , 1998 .

[50]  J. Ruppersberg.  Ion Channels in Excitable Membranes , 1996 .

[51]  Guillermo Hauke,et al.  a Unified Approach to Compressible and Incompressible Flows and a New Entropy-Consistent Formulation of the K - Model. , 1994 .

[52]  Benzhuo Lu,et al.  Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution. , 2007, The Journal of chemical physics.

[53]  L. Franca,et al.  Stabilized Finite Element Methods , 1993 .

[54]  Jens Schröter,et al.  A diagnostic stabilized finite-element ocean circulation model , 2003 .

[55]  Minxin Chen,et al.  TMSmesh: A Robust Method for Molecular Surface Mesh Generation Using a Trace Technique. , 2011, Journal of chemical theory and computation.

[56]  T. Hughes,et al.  The Galerkin/least-squares method for advective-diffusive equations , 1988 .

[57]  Thomas J. R. Hughes,et al.  A comparative study of different sets of variables for solving compressible and incompressible flows , 1998 .

[58]  Qiong Zheng,et al.  Second-order Poisson-Nernst-Planck solver for ion transport , 2011, J. Comput. Phys..

[59]  Aleksei Aksimentiev,et al.  Predicting the DNA sequence dependence of nanopore ion current using atomic-resolution Brownian dynamics. , 2012, The journal of physical chemistry. C, Nanomaterials and interfaces.

[60]  Robert S. Eisenberg,et al.  Two- and Three-Dimensional Poisson–Nernst–Planck Simulations of Current Flow Through Gramicidin A , 2002, J. Sci. Comput..

[61]  Damien Farrell,et al.  Protein dielectric constants determined from NMR chemical shift perturbations. , 2013, Journal of the American Chemical Society.

[62]  Thomas J. R. Hughes,et al.  Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations , 1984 .

[63]  Benzhuo Lu,et al.  An ionic concentration and size dependent dielectric permittivity Poisson-Boltzmann model for biomolecular solvation studies. , 2014, The Journal of chemical physics.

[64]  Michael Habeck,et al.  Structure of the human voltage-dependent anion channel , 2008, Proceedings of the National Academy of Sciences.

[65]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: II. Beyond SUPG , 1986 .

[66]  W. Im,et al.  Theoretical and computational models of biological ion channels , 2004, Quarterly Reviews of Biophysics.

[67]  T. Nguyen,et al.  Three-dimensional Model of the Pore Form of Anthrax Protective Antigen. Structure and Biological Implications , 2004, Journal of biomolecular structure & dynamics.