Uniform Convergence of the Generalized Scharfetter-Gummel ...

Uniform Convergence of the Generalized Scharfetter-Gummel ...

TIMS National Taiwan University A Poisson-Fermi Theory for Biological Ion Channels: Models and Methods Jinn-Liang Liu National Hsinchu University of Education In collaboration with Bob Eisenberg Rush University Medical Center, Chicago, USA

Dec. 25, 2013 1 1 Ion Outlin e

Channel Poisson-Fermi & Nernst-Planck Numerical Methods Results Outlook 2 Ion Channel Biological ion channels seem to be a

precondition for all living matter. A. L. Hodgkin & A. Huxley (Nobel Prize in Physiology or Medicine 1963) for their discoveries concerning " the ionic mechanisms in the nerve cell membrane ". (Action Potential) E. Neher & B. Sakmann (Nobel Prize in Physiology or Medicine 1991) for their discoveries concerning "the function

of single ion channels in cells". (Current Measurement) P. Agre & R. MacKinnon (Nobel Prize in Chemistry 2003) for their discoveries concerning "channels in cell membranes". (Crystal Structures) Hypothesized Ion Channel Confirmed

Ion Channel Saw Ion Channel Poisson-Fermi*-Nernst*Planck* Steric & Correlation in Poisson(Steric & Correlation) Boltzmann:

100-Year Old Problems since Gouy (1910) & Developments: Chapman (1913) Historical Bjerrum (1918), Debye*-Huckel (1923), Stern* (1924), Onsager * (1936), Kirkwood (1939), DuttaBagchi (1950), Grahame (1950), Eigen*-Wicke (1954), Borukhov-AndelmanOrland (1997),

Santangelo (2006), Eisenberg-Hyon-Liu (2010), Bazant-Storey-Kornyshev (2011), * Nobel Laureates Wei-Zheng-Chen-Xia (2012) 5 Steric Effects

Steric effects arise from the fact that each atom within a molecule occupies a certain amount of space. (Wiki) 6 Steric Effects Boltzmann => Infinity => Unphysical Fermi => Finite =>

Saturation 7 Correlation Effects Bazant, Storey, Kornyshev (PRL 2011) No Cor. Yes, Cor.

8 Fermi Distribution Borukhov-Andelman-Orland (Steric) (PRL 1997) Blow Up? Fermi Like Dist. oblems: 1. Single Size (Same a) for all Ions

2. Energy functional blows up as a goes to 0 (phenomenological or analytical?) Bazant, Storey, Kornyshev (PRL 2011) 9 Free Energy of Electrolyte Solutions

differe nt sizes 10 Global Electrochemical Potential Local Electrochemical Potential

Fermi Distribution Global Probabilit ies Local Probabiliti es

Water New Probabilit y New 11 Fermi Distribution

differen t valence s Steric Functional Saturation New Condition

Liu-Eisenberg (2013 JPC) improves Borukhov- Andelman-Orland (1997 PRL) differe nt sizes 12

4th-Order Poisson Eq. Santangelo (PRE (Correlation) 2006) 13 4th-Order Poisson Eq. (Correlation)

14 2nd-Order Poisson(JCoP Fermi Liu Eqs 2013) New Charge

Neutrality Boundary Conditions New 15 Poisson-Fermi-Nernst-Planck

Model New 16 Nernst-Planck implies Fermi in Equilibrium 17

Numerical Methods Gramicidin A (GA) Channel 18 Simulation Domain Zheng, Chen, Wei (JCoP 2011)

19 Molecular Surface Molecular Surface (MS) generated in 3D uniform grid by a probe ball Rolling Ball Algorithm

Shrake-Rupley (1973)1-3 Error: Ball (Water) Radius: 1.4 20 Numerical Rolling BallMethods

Method for MS 7-Point Finite Difference Method Weis Matched Interface and Boundary Method (MIB) Linear Solver: SOR Chern-Liu-Wangs Method for Singular Charges Newtons Method Continuity Method on Steric Functional

and Correlation Length 21 Simplified MIB 22 Results (SMIB vs MIB: Zheng,MIB) Chen, Wei (JCoP

2011) MIB > 27-pt FDM SMIB = 7-pt FDM h= 0.25 Matrix Size = 4,096,000 MS Error: 1-3

23 SMIB vs MIB (Ex) SMIB is simpler, efficient, accurate but mid-point Ca channel pore interface.

radius is only ~1. May not have sufficient pts for high order MIB. 24 Order and Time of SMIB (Ex)

25 Born Ion Model (Ex) Classic Example of Solvation MIBPB: Geng, Yu, Wei (JChP 26

Charged Wall Models 27 Charged Wall Models (MC) Solution: NaCl Solution: CaCl

Overscreening Impossible by 28 L-Type Ca Channel Model 1 Boda, Valisk, Henderson, Eisenberg, Gillespie, Nonner (JGenPhysio 2009)

29 PF vs PB Oxygen Ion Concentration Profile 30

Dielectric Function 31 Ca Binding Curves (MC) # Na+ # Ca 32

L-Type Ca Channel Model 2 (All Spheres-PF Model) Experiments: Almers, McCleskey, Palade (JPhysio 1984) Anomalous Mole Fraction Effect 10-8 M Variation in [Ca2+] Femto Ampere (fA) Current Half

Current Block 33 All Spheres-PF Model Molecular Dynamics: Lipkind, Fozzard (Biochem. 2001) Barreiro, Guimaraes, Alencastro (Protein Eng. 2002)

Lipkind-Fozzard Binding Site All Spheres 34 All Spheres-PF (MolecularContinuum) All Spheres

PF 35 PF Results (w. MD) Experimental Data (Halved Current) PF Shows Flexibility of Side Chains: Pore Radius Variation = 2.3 (2 by MD) Steric is Critical:

36 PF Results (w. NCX Exp) J. Liao, ..., Y. Jiang (Science Crystal 2012) Structure of Sodium /Calcium Exchanger (NCX) PF = 2.3

Exp = 2.4 37 Mol-PF Potential Profile 38 Mol-PF Concentration Profile 39

Mol-PF Dielectric Profile 40 Mol-PF Binding Curve 41 Outlook

Equilibrium and Nonequilibrium Systems PFNP vs Experiments Ca, Na, KcsA Channels Sodium Calcium Exchanger (NCX) Mathematical and Numerical Analysis Biophysical Analysis

42 Than k You 43

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