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Jinn-Liang Liu National Hsinchu University of Education In collaboration with Bob Eisenberg Rush University Medical Center, Chicago, USA Dec. 25, 2013.

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Presentation on theme: "Jinn-Liang Liu National Hsinchu University of Education In collaboration with Bob Eisenberg Rush University Medical Center, Chicago, USA Dec. 25, 2013."— Presentation transcript:

1 Jinn-Liang Liu National Hsinchu University of Education In collaboration with Bob Eisenberg Rush University Medical Center, Chicago, USA Dec. 25, 2013 1 A Poisson-Fermi Theory for Biological Ion Channels: Models and Methods 1 TIMS National Taiwan University

2 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 P. Agre & R. MacKinnon (Nobel Prize in Chemistry 2003) for their discoveries concerning "channels in cell membranes". (Crystal Structures) 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) 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) Hypothesized Ion Channel Confirmed Ion Channel Saw Ion Channel

5 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

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

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

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

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

10 10 different sizes Free Energy of Electrolyte Solutions

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

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

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

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

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

16 16 Poisson-Fermi-Nernst-Planck Model New

17 17 Nernst-Planck implies Fermi in Equilibrium

18 18 Numerical Methods Gramicidin A (GA) Channel

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

20 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

21 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 Functional and Correlation Length

22 22 Simplified MIB

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

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

25 25 Order and Time of SMIB (Ex)

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

27 27 Charged Wall Models

28 28 Charged Wall Models (MC) Overscreening Impossible by PB Solution: NaClSolution: CaCl ₂

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

30 30 PF vs PB Oxygen Ion Concentration Profile

31 31 Dielectric Function

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

33 33 L-Type Ca Channel Model 2 (All Spheres-PF Model) Experiments: Almers, McCleskey, Palade (JPhysio 1984) Anomalous Mole Fraction Effect Femto Ampere (fA) Current 10 -8 M Variation in [Ca 2+ ] Block Half Current

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

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

36 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:

37 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

38 38 Mol-PF Potential Profile

39 39 Mol-PF Concentration Profile

40 40 Mol-PF Dielectric Profile

41 41 Mol-PF Binding Curve

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

43 43 Thank You

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