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Poisson-Fermi Theory for Ion Channels: Models and Methods

Explore the Poisson-Fermi theory for biological ion channels, its historical context, steric effects, correlations, and numerical methods. Discover insights into molecular interactions and the future outlook for the field.

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Poisson-Fermi Theory for Ion Channels: Models and Methods

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  1. Jinn-Liang Liu National Hsinchu University of EducationIn collaboration withBob EisenbergRush University Medical Center, Chicago, USA Dec. 25, 2013 TIMS National Taiwan University A Poisson-Fermi Theory for Biological Ion Channels:Models and Methods 1

  2. Outline • Ion Channel • Poisson-Fermi & Nernst-Planck • Numerical Methods • Results • Outlook

  3. Ion Channel Biological ion channels seem to be a precondition for all living matter.

  4. A. L. Hodgkin & A. Huxley (Nobel Prize in Physiology or Medicine 1963) for their discoveries concerning " the ionicmechanisms in the nervecell membrane ". (Action Potential) HypothesizedIon Channel E. Neher & B. Sakmann (Nobel Prize in Physiology or Medicine 1991) for their discoveries concerning "the function of singleion channels in cells". (Current Measurement) ConfirmedIon Channel P. Agre & R. MacKinnon (Nobel Prize in Chemistry 2003) for their discoveries concerning "channels in cell membranes". (Crystal Structures) SawIon Channel

  5. Poisson-Fermi*-Nernst*-Planck*(Steric & Correlation) Steric & Correlation in Poisson-Boltzmann: 100-Year Old Problems since Gouy (1910) & Chapman (1913) Historical Developments: Bjerrum (1918), Debye*-Huckel (1923), Stern* (1924), Onsager * (1936), Kirkwood (1939), Dutta-Bagchi (1950), Grahame (1950), Eigen*-Wicke (1954), Borukhov-Andelman-Orland (1997), Santangelo (2006), Eisenberg-Hyon-Liu (2010), Bazant-Storey-Kornyshev (2011), Wei-Zheng-Chen-Xia (2012) * Nobel Laureates 5

  6. Steric Effects Steric effects arise from the fact that each atom within a molecule occupies a certain amount of space. (Wiki) 6

  7. Steric Effects Boltzmann => Infinity => Unphysical Fermi => Finite => Saturation 7

  8. Correlation Effects Bazant, Storey, Kornyshev (PRL 2011) No Cor. Yes, Cor. 8

  9. Fermi Distribution (Steric) Borukhov-Andelman-Orland (PRL 1997) Blow Up? Fermi Like Dist. Problems: 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

  10. Free Energy of Electrolyte Solutions different sizes 10

  11. Global Electrochemical Potential Global Probabilities Local Electrochemical Potential Local Probabilities New Water Probability Fermi Distribution New 11

  12. Fermi Distribution different valences Steric Functional different sizes Saturation Condition New Liu-Eisenberg (2013 JPC) improves Borukhov- Andelman-Orland (1997 PRL) 12

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

  14. 4th-Order Poisson Eq. (Correlation) 14

  15. 2nd-Order Poisson-Fermi Eqs Liu (JCoP 2013) New Charge Neutrality Boundary Conditions New 15

  16. Poisson-Fermi-Nernst-Planck Model New 16

  17. Nernst-Planck implies Fermi in Equilibrium 17

  18. Numerical Methods Gramicidin A (GA) Channel 18

  19. Simulation Domain Zheng, Chen, Wei (JCoP 2011) 19

  20. Molecular Surface • Molecular Surface (MS) generated in 3D uniform grid by a probe ball Rolling Ball Algorithm Shrake-Rupley (1973) Ball (Water) Radius: 1.4Å Error: 1-3 Ų 20

  21. Numerical Methods • Rolling Ball Method for MS • 7-Point Finite Difference Method • Wei’s Matched Interface and Boundary Method (MIB) • Linear Solver: SOR • Chern-Liu-Wang’s Method for Singular Charges • Newton’s Method • Continuity Method on Steric Functionaland Correlation Length 21

  22. Simplified MIB 22

  23. Results (SMIB vs MIB) MIB: Zheng, Chen, Wei (JCoP 2011) MIB > 27-pt FDM SMIB = 7-pt FDM h= 0.25ŠMatrix Size = 4,096,000 MS Error: 1-3 Ų 23

  24. SMIB vs MIB (Ex) SMIB is simpler, efficient, accurate but mid-point interface. Ca channel pore radius is only ~1Å. May not have sufficient pts for high order MIB. 24

  25. Order and Time of SMIB (Ex) 25

  26. Born Ion Model (Ex)Classic Example of Solvation MIBPB: Geng, Yu, Wei (JChP 2007) 26

  27. Charged Wall Models 27

  28. Charged Wall Models (MC) Solution: NaCl Solution: CaCl₂ OverscreeningImpossible by PB 28

  29. L-Type Ca Channel Model 1 Boda, Valiskó, Henderson, Eisenberg, Gillespie, Nonner (JGenPhysio 2009) 29

  30. PF vs PB Oxygen Ion Concentration Profile 30

  31. Dielectric Function 31

  32. Ca Binding Curves (MC) # Na+ # Ca²⁺ 32

  33. 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

  34. All Spheres-PF Model Molecular Dynamics: Lipkind, Fozzard (Biochem. 2001) Barreiro, Guimaraes, Alencastro (Protein Eng. 2002) All Spheres Lipkind-Fozzard Binding Site 34

  35. All Spheres-PF (Molecular-Continuum) All Spheres PF 35

  36. PF Results (w. MD) Experimental Data (Halved Current) • PF Shows Flexibility of Side Chains: Pore Radius Variation = 2.3 Å (2 Å by MD) • Steric is Critical: 36

  37. PF Results (w. NCX Exp) J. Liao, ..., Y. Jiang (Science 2012) Crystal Structure of Sodium /Calcium Exchanger (NCX) PF = 2.3 Å Exp = 2.4 Å 37

  38. Mol-PF Potential Profile 38

  39. Mol-PF Concentration Profile 39

  40. Mol-PF Dielectric Profile 40

  41. Mol-PF Binding Curve 41

  42. Outlook • Equilibrium and Nonequilibrium Systems • PFNP vs Experiments • Ca, Na, KcsA Channels • Sodium Calcium Exchanger (NCX) • Mathematical and Numerical Analysis • Biophysical Analysis 42

  43. Thank You

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