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 about70%of salt water.This years Nobel Prize in Chemistry rewards two scientists whose discoveries have clarified how salts
2、(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 2019;Doyle,et al.,2019)Theoretical study of ion channels Kineti
3、c models Electrodiffusion models Stochastic models Molecular Dynamics Brownian Dynamics(Kurnikova,et al.,2019;Coalson and Kurnikova,2019)Poisson-Nernst-Planck theory Basic idea Numerical solution Validity Application to Gramicidin A channel Improvement SummaryPreconditions of PNP theory Coarse grain
4、ed 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.,2019)standard PNP theory Nernst-Planck equ
5、ation Poisson equation Total Potential Energy iii0()()();1,.,c Rc RV RiN fii1()()4()()NiRRRqc R ii()()()V RU RqR(Kurnikova,et al.,2019)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
6、 the grid 3D case where ij is 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)i,j+1,ki,j-1,ki-1,j,ki+1,j,ki,j,k+1i,j,k-1i,j,kx1ijx1ijy1jjy1jjz1kjz1kjSolving 3D NP Eq.by successive
7、 over-relaxation2/)(2/)(,1,1,1,1x1jijijijijijijijiiccVVccaDDji.Flux x1ijii.Steady-state flux condition001111x1x1zkzkyjyjiijjjjjjjiii.Concentration for central pointeffNiiifeNiiiiDVVDcVVcf10100)(2/(1)(2/(1 Where ,is the number of nearest-neighbor lattice points2/)(0DDDiieffNiv.SOR iteration equation0
8、)1(wccwcoldii1w;(Crdenas,et al.,2000;Kurnikova,et al.,2019)Calibration of the accuracy of the 3D code(Kurnikova,et al.,2019)Application to Gramicidin A channel(Kurnikova,et al.,2019)(Kurnikova,et al.,2019)(Kurnikova,et al.,2019)Comparison with experiments(Kurnikova,et al.,2019)standard PNP theory Ne
9、rnst-Planck equation Poisson equation Total Potential Energy iii0()()();1,.,c Rc RV RiN fii1()()4()()NiRRRqc R ii()()()V RU RqR(Kurnikova,et al.,2019)Dielectric-Energy PNP theory)()()()(0iiiirrcrcrD)()()(iSIPmobileiirGrqriiimobile)(4)()(rcqrr)()()(iDSEproteiniiSIPrGrqrGNernst-Planck equationPoisson
10、equationThe free energy of ions of species i in solution)()(4)()(fiiirrcqrr)()()(iDSEiirGrqri21DSEi2()(,)Grq g r r(Graf,et al.,2019;Coalson and Kurnikova,2019)Performance of DSEPNP(Coalson and Kurnikova,2019)Potential of Mean Force PNP theoryThe protein structure used in both BD and DSEPNP simulatio
11、ns is taken to be rigid,while in reality the protein structure responds dynamically to an ions 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 comp
12、lete sampling of the system configuration space.But its 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 GSIP at an average sol
13、vent effect level,which is then used in PNP formalism.Such a procedure is termed PMFPNP.(Coalson and Kurnikova,2019)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.Sma
14、ll 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 conce
15、ntration of bathing solutions increases to a sufficiently high value.(Coalson and Kurnikova,2019)The saturation mechanism(Coalson and Kurnikova,2019)Summary3D PNP theory is of conceptual simplicity.It relies on a caricature of the microscopic world in which background media are treated as dielectric
16、 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 t
17、ype 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,2019)ReferencesCrdenas,A.E.,R.D.Coalson
18、,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.2019.PoissonNernstPlanck Theory Approach to the Calculation of Current Through Biological Ion Chan
19、nels.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.2019.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
20、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.2019.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.108:2019-2019.Kurnikova,M.G.,R.D.Coalson,P.Graf,and A.Nitzan.2019.A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transport through the gramicidinA channel.Biophys.J.76:642656.Thank you!
