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Guozhen Zhang

Poisson-Nernst-Planck Theory Approach to the calculation of ion transport through protein channels. Guozhen Zhang. Ion transport through protein channels. “We human beings consist to about 70% of salt water. This year's Nobel Prize in Chemistry rewards two

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Guozhen Zhang

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  1. Poisson-Nernst-Planck Theory Approach to the calculation of ion transport through protein channels Guozhen Zhang

  2. Ion transport through protein channels “We human beings consist to about 70% of salt water. This year's Nobel Prize in Chemistry rewards two scientists whose discoveries have clarified how salts (ions) and water are transported out of and into the cells of the body… This is of great importance for our understanding of many diseases of e.g. the kidneys, heart, muscles and nervous system. “ (Press release of the Nobel Prize in Chemistry 2003; Doyle, et al., 1998)

  3. Theoretical study of ion channels • Kinetic models • Electrodiffusion models • Stochastic models • Molecular Dynamics • Brownian Dynamics (Kurnikova, et al., 1999; Coalson and Kurnikova, 2005)

  4. Poisson-Nernst-Planck theory • Basic idea • Numerical solution • Validity • Application to Gramicidin A channel • Improvement • Summary

  5. Preconditions of PNP theory • Coarse grained approximation mobile ions —> continuous charge distribution surroundings —> 3D grid with different dielectric constant • High-friction assumption Brownian motion —> Smoluchowski equation • Steady-state assumption the particle flux is time-independent (Kurnikova, et al., 1999)

  6. standard PNP theory • Nernst-Planck equation • Poisson equation • Total Potential Energy (Kurnikova, et al., 1999)

  7. Solving 3D Poisson equation on a cubic grid • 1D case a. division of grid where the lattice cell extends from (j-1/2)×h to (j-1/2)×h b. discretization of Poisson equation on the grid • 3D case where ∠ijis the 3D generalization of the matrix defined in the above equation, bi(D) are the effective source terms associated with the Dirichlet boundary condition. (Graf, et al., 2000)

  8. i,j+1,k i,j,k+1 i-1,j,k i, j, k i+1,j,k i,j,k-1 i,j-1,k ; Solving 3D NP Eq. by successive over-relaxation i. Flux ii. Steady-state flux condition iii. Concentration for central point Where , is the number of nearest-neighbor lattice points (Cárdenas, et al., 2000; Kurnikova, et al., 1999) iv. SOR iteration equation

  9. Calibration of the accuracy of the 3D code (Kurnikova, et al., 1999)

  10. Application to Gramicidin A channel (Kurnikova, et al., 1999)

  11. (Kurnikova, et al., 1999)

  12. (Kurnikova, et al., 1999)

  13. Comparison with experiments (Kurnikova, et al., 1999)

  14. standard PNP theory • Nernst-Planck equation • Poisson equation • Total Potential Energy (Kurnikova, et al., 1999)

  15. Dielectric-Energy PNP theory Nernst-Planck equation Poisson equation The free energy of ions of species i in solution (Graf, et al., 2004; Coalson and Kurnikova, 2005)

  16. Performance of DSEPNP (Coalson and Kurnikova, 2005)

  17. Potential of Mean Force PNP theory • The protein structure used in both BD and DSEPNP simulations is taken to be rigid, while in reality the protein structure responds dynamically to an ion’s presence. Such a defect usually exhibits very small superlinear currents for voltages up to 200mV for narrow channels. • This issue can in principle be solved by a full atomistic simulation which requires complete sampling of the system configuration space. But it’s formidable for current computing capability. • Limited sampling of the environment configurational space has been introduced to deal with the problem. A combined MD/continuum electrostatics approach is then proposed to obtain ΔGSIP at an average solvent effect level, which is then used in PNP formalism. Such a procedure is termed PMFPNP. (Coalson and Kurnikova, 2005)

  18. Results of the PMFPNP calculations • The overall structure of peptide doesn’t change much over the course of MD trajectory, so the ΔGDSE contribution to the overall ΔGSIP doesn’t vary much. • Small local distortions of pore-lining parts of the peptide (especially carbonyl groups) significantly stabilize cations as they move through it. • PMFPNP theory is able to account for effects that are beyond the reach of primitive PNP theory, namely, saturation of ion current through the channel as the concentration of bathing solutions increases to a sufficiently high value. (Coalson and Kurnikova, 2005)

  19. The saturation mechanism (Coalson and Kurnikova, 2005)

  20. Summary • 3D PNP theory is of conceptual simplicity. It relies on a caricature of the microscopic world in which background media are treated as dielectric slabs and the mobile ions of interest are “smeared out” into a continuous charge distribution. • The inherent restriction of the theory is mainly due to its simplicity. It may be unrealistic for treating certain properties of certain ion channels. Also, the mean-field continuum solvent/ion theory of this type is inadequate to accurately describe the underlying dynamics. • Despite of these restrictions, PNP theory will continue to play a useful role in computing and understanding the kinetics of ion permeation through (wider) biological channels. (Coalson and Kurnikova, 2005)

  21. References • Cárdenas, A. E., R. D. Coalson, and M. G. Kurnikova. 2000. Three-Dimensional Poisson-Nernst-Planck Theory Studies: Influence of Membrane Electrostatics on Gramicidin A Channel Conductance. Biophys. J. 79 : 80-93. • Coalson, R. D., and M. G. Kurnikova. 2005. Poisson–Nernst–Planck Theory Approach to the Calculation of Current Through Biological Ion Channels. IEEE T Nanobiosci. 4 : 81-93. • Doyle, D.A., J. M. Cabral, R. A. Pfuetzner, A. Kuo, J. M. Gulbis, S. L. Cohen, B.T. Chait, and R. MacKinnon. 1998. The structure of the potassium channel: Molecular basis of K+ conduction and selectivity. Science. 280 : 69-77. • Graf, P., A. Nitzan, M. G. Kurnikova, and R. D. Coalson. 2000. A dynamic lattice Monte Carlo model of ion transport in inhomogeneous dielectric environments :Method and implementation. J. Phys. Chem. B. 104 : 12324-12338. • Graf, P., M. Kurnikova, R. Coalson, and A. Nitzan. 2004. Comparison of dynamic lattice Monte Carlo simulations and the dielectric self-energy Poisson-Nernst-Planck continuum theory for model ion channels. J. Phys. Chem. B. 108 : 2006-2015. • Kurnikova, M. G., R. D. Coalson, P. Graf, and A. Nitzan. 1999. A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transport through the gramicidinA channel. Biophys. J. 76 : 642–656.

  22. Thank you!

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