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Gating Modeling of Ion Channels Chu-Pin Lo ( 羅主斌 ) Department of Applied Mathematics Providence University 2010/01/12 (Workshop on Dynamics for Coupled Systems, CMMSC)
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Outline Cardiac Electrophysiology Modeling Techniques (electrical part) Full Current Flux Form: PNP model Gating Modeling (1). Experiment Measurements for Gating Issues (2). Classical Kinetics (3). Hodgkin-Huxley Theory (cell scale) (4). Markovian Process Method (channel scale) (5). Smoluchowski model (channel scale) Pharmacological Applications
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Cardiac Electrophysiology
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Electrophysiology of the cardiac muscle cell
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ECG & Action Potentials Single Cell Action Potential (Microscopic) ECG (Macroscopic)
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Macroscopic property Mesoscopic property
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Computing of ECG ( 心電圖 )
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Isotropic, space homogeneous of conductive tensor, and infinite media ECG= Where and
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Computing of ECG ( 心電圖 ), Cont. Bounded media, piecewise constant and isotropic conductive tensor ECG= boundary element method
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Computing of ECG ( 心電圖 ), Cont. Real case (finite media, anisotropic and space heterogeneity conductive tensor) finite difference, finite element, finite volume methods
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Cellular Basis of ECG
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Modeling Techniques (electrical part)
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Modeling Approaches (cell and channel scale) Poisson-Nernst Planck+Density functional Theory (for full open flux) (channel scale) Barrier model (for full open flux) (channel scale) Hodgkin-Huxley Theory (for gating issue)(cell scale) Markovian Process Method (for gating issue)(channel scale) Smoluchowski model (for gating issue)(channel scale)
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(sub)channel scale
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Current Form: single channel and single cell (1)Single channel current: I_s=(gating factor/open probability) ‧ (full open flux) (2) Single cell current: I_t=(total channels number) ‧ I_s
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Tissue scale
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Macroscopic property Mesoscopic property
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Organ scale
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Rat Left Ventricle
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Fiber-Sheet Structure
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Incorporation of fiber-sheet structure into bidomain Model
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Full Current Flux Form: Poisson-Nernst-Planck Model (PNP) & Density Functional Theory (DFT)
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PNP model (continuum model) Nernst- Planck equation (derived from molecular Langevin equation) continuity equation
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Poisson equation for electrostatic potential
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Density Functional Theory (DFT): excess chemical potential description (finite size charged particle)
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Simulation Results: flux form
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Simulation Result: Permeation Selectivity for Ca2+
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Two famous flux form: (1). Goldman-Hodgkin-Katz (GHK) current form Conditions: short channel Or low ionic concentrations of either side of the membrane Or constant field PNP with only ideal electrochemical potential (point particle)
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Two famous flux form: (2). Linear I-V relation (Ohm’s law) Conditions: long channel high ionic concentrations of either side of the membrane PNP with only ideal electrochemical potential (point particle)
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Gating Modeling Experiment Measurements for Gating Issues Classical Kinetics Hodgkin-Huxley Theory (cell scale) Markovian Process Method (channel scale) Smoluchowski model (channel scale)
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Ion Channel Structure
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Experiment Measurements for Gating Issues
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Fluctuation analysis Single-channel recording Gating current
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Fluctuation Analysis
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Single Channel Recording
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Single channel recording Mean open (shut) time The time to first opening of a channel (first-latency distribution) Number of times that a channel opens before inactivation Conditional probability that an open period of a certain length is followed immediately by a closed period of a certain length Hidden Markov analysis
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Complement to classical kinetics (single channel recording) macro current single channel current
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Hidden Markov Analysis
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Gating Current
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Gating Mechanism: gating current (two states transition) Conformational change of channel protein Gating current (charge): energy supply one-step conformational change probability ratio of open to closed states by Boltzmann equation open probability of channel
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Bertil Hille, 2001
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Gating Mechanism: gating current (multiple states transition)
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Gating Mechanism: gating current (multiple states transition):conti Bertil Hille, 2001
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Classical Kinetics
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Gating Mechanism: Classical kinetics
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Gating Issue: Hodgkin-Huxley Model (single cell model)
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stimulus current capacitance current Ionic currents
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Model Formalism and Experimental Protocol Design
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Activation (steady state) protocol: tail current analysis
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Inactivation (steady state) protocol
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Recovery protocol (1)
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Recovery protocol (2) Modeling formula for recovery kinetics
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Time course determination: time constant activation deactivation inactivation recovery
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Deactivation experimental protocol (used for time constant determination of deactivation phase)
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Gating Issue: Markov Model (single channel and cell model, discrete protein state)
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Example 1 (Fitzhugh, 1965) (Markovian version of HH model) INa channel IK channel
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Example 2 (Vandenberg, Bezanilla, Perozo, 1990,1991)(match the single channel recording and gating current measure) INa channel IK channel
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Example 3 INa IK transition rate
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Comparison (INa)
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Comparison (action potential)
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Differences between Examples Activation and inactivation are kinetically independent in example 1 and dependent in example 2,3 Fast activation and slow inactivation in examples 1,2; slow activation and fast inactivation in example 3
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Relation between HH & Markov Models
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Relation between HH & Markov Models, Conti.
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transition rate determination
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Gating issue: Smoluchowski Model (Fokker-Planck type model in energy landscape, continuuum protein state)
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Probability Flux Calculation (Fokker-Planck Equation) Smoluchowski Model :
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Example1
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Example 2
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Potential of mean field (PMF)
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Langevin Equation
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Computation of rate constant rate constant = 1/T mfp mean first passage time (mfp)
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Computation of Gating Current master equation gating current
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Example 3
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Potential Calculation Linearized Poisson- Boltzman with transmembrane potential effect
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Movie
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Pharmacological Applications
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Thanks for your Attention !
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