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1、Poisson-Nernst-Planck Theory Approach to the calculation of ion transport through protein channelsGuozhen ZhangIon transport through protein channels“We human beings consist to about 70% of salt water. This years NobelPrize in Chemistry rewards twoscientists whose discoveries haveclarified how salts
2、 (ions) and waterare transported out of and into thecells of the body This is of great importance for our understanding ofmany diseases of e.g. the kidneys, heart, muscles and nervous system. “(Press release of the Nobel Prize in Chemistry 2003; Doyle, et al., 1998)Theoretical study of ion channels
3、Kinetic models Electrodiffusion models Stochastic models Molecular Dynamics Brownian Dynamics(Kurnikova, et al., 1999; Coalson and Kurnikova, 2005)Poisson-Nernst-Planck theory Basic idea Numerical solution Validity Application to Gramicidin A channel Improvement SummaryPreconditions of PNP theory Co
4、arse grained approximationmobile ions continuous charge distributionsurroundings 3D grid with different dielectric constant High-friction assumptionBrownian motion Smoluchowski equation Steady-state assumptionthe particle flux is time-independent(Kurnikova, et al., 1999)standard PNP theory Nernst-Pl
5、anck equation Poisson equation Total Potential Energy(Kurnikova, et al., 1999)Solving 3D Poisson equation on a cubic grid 1D casea. division of gridwhere the lattice cell extends from (j-1/2)h to (j-1/2)h b. discretization of Poisson equation on the grid 3D casewhere ij is the 3D generalization of t
6、he matrix defined in the above equation, bi (D)are the effective source terms associated with the Dirichlet boundary condition. (Graf, et al., 2000)i,j+1,ki,j-1,ki-1,j,ki+1,j,ki,j,k+1i,j,k-1i, j, kSolving 3D NP Eq. by successive over-relaxationi. Flux ii. Steady-state flux conditioniii. Concentratio
7、n for central pointWhere , is the number of nearest-neighbor lattice pointsiv. SOR iteration equation;(Crdenas, et al., 2000; Kurnikova, et al., 1999)Calibration of the accuracy of the 3D code(Kurnikova, et al., 1999)Application to Gramicidin A channel(Kurnikova, et al., 1999)(Kurnikova, et al., 199
8、9)(Kurnikova, et al., 1999)Comparison with experiments(Kurnikova, et al., 1999)standard PNP theory Nernst-Planck equation Poisson equation Total Potential Energy(Kurnikova, et al., 1999)Dielectric-Energy PNP theoryNernst-Planck equationPoisson equationThe free energy of ions of species i in solution
9、 (Graf, et al., 2004; Coalson and Kurnikova, 2005)Performance of DSEPNP(Coalson and Kurnikova, 2005)Potential of Mean Force PNP theoryThe protein structure used in both BD and DSEPNP simulations is taken to be rigid, while in reality the protein structure responds dynamically to an ions presence. Su
10、ch 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 its formidable for current computing capability.Limit
11、ed 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 GSIP at an average solvent effect level, which is then used in PNP formalism. Such a procedure is termed PMFPNP.(Coalson and K
12、urnikova, 2005)Results of the PMFPNP calculationsThe overall structure of peptide doesnt change much over the course of MD trajectory, so the GDSE contribution to the overall GSIP doesnt vary much. Small local distortions of pore-lining parts of the peptide (especially carbonyl groups) significantly
13、 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, 20
14、05)The saturation mechanism(Coalson and Kurnikova, 2005)Summary3D 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 distributi
15、on. 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 rest
16、rictions, 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)ReferencesCrdenas, A. E., R. D. Coalson, and M. G. Kurnikova. 2000. Three-Dimensional Poisson-Nernst- Planck Theory Studies: Influence of Membrane Electrostatics on Gramic
