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)

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

Cardiac Electrophysiology

Electrophysiology of the cardiac muscle cell

ECG & Action Potentials Single Cell Action Potential (Microscopic) ECG (Macroscopic)

Macroscopic property Mesoscopic property

Computing of ECG ( 心電圖 )

Isotropic, space homogeneous of conductive tensor, and infinite media ECG= Where and

Computing of ECG ( 心電圖 ), Cont. Bounded media, piecewise constant and isotropic conductive tensor ECG= boundary element method

Computing of ECG ( 心電圖 ), Cont. Real case (finite media, anisotropic and space heterogeneity conductive tensor) finite difference, finite element, finite volume methods

Cellular Basis of ECG

Modeling Techniques (electrical part)

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)

(sub)channel scale

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

Tissue scale

Macroscopic property Mesoscopic property

Organ scale

Rat Left Ventricle

Fiber-Sheet Structure

Incorporation of fiber-sheet structure into bidomain Model

Full Current Flux Form: Poisson-Nernst-Planck Model (PNP) & Density Functional Theory (DFT)

PNP model (continuum model) Nernst- Planck equation (derived from molecular Langevin equation) continuity equation

Poisson equation for electrostatic potential

Density Functional Theory (DFT): excess chemical potential description (finite size charged particle)

Simulation Results: flux form

Simulation Result: Permeation Selectivity for Ca2+

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)

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)

Gating Modeling Experiment Measurements for Gating Issues Classical Kinetics Hodgkin-Huxley Theory (cell scale) Markovian Process Method (channel scale) Smoluchowski model (channel scale)

Ion Channel Structure

Experiment Measurements for Gating Issues

Fluctuation analysis Single-channel recording Gating current

Fluctuation Analysis

Single Channel Recording

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

Complement to classical kinetics (single channel recording) macro current single channel current

Hidden Markov Analysis

Gating Current

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

Bertil Hille, 2001

Gating Mechanism: gating current (multiple states transition)

Gating Mechanism: gating current (multiple states transition):conti Bertil Hille, 2001

Classical Kinetics

Gating Mechanism: Classical kinetics

Gating Issue: Hodgkin-Huxley Model (single cell model)

stimulus current capacitance current Ionic currents

Model Formalism and Experimental Protocol Design

Activation (steady state) protocol: tail current analysis

Inactivation (steady state) protocol

Recovery protocol (1)

Recovery protocol (2) Modeling formula for recovery kinetics

Time course determination: time constant activation deactivation inactivation recovery

Deactivation experimental protocol (used for time constant determination of deactivation phase)

Gating Issue: Markov Model (single channel and cell model, discrete protein state)

Example 1 (Fitzhugh, 1965) (Markovian version of HH model) INa channel IK channel

Example 2 (Vandenberg, Bezanilla, Perozo, 1990,1991)(match the single channel recording and gating current measure) INa channel IK channel

Example 3 INa IK transition rate

Comparison (INa)

Comparison (action potential)

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

Relation between HH & Markov Models

Relation between HH & Markov Models, Conti.

transition rate determination

Gating issue: Smoluchowski Model (Fokker-Planck type model in energy landscape, continuuum protein state)

Probability Flux Calculation (Fokker-Planck Equation) Smoluchowski Model :

Example1

Example 2

Potential of mean field (PMF)

Langevin Equation

Computation of rate constant rate constant = 1/T mfp mean first passage time (mfp)

Computation of Gating Current master equation gating current

Example 3

Potential Calculation Linearized Poisson- Boltzman with transmembrane potential effect

Movie

Pharmacological Applications

Thanks for your Attention !