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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)
OutlineCardiac 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 )
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
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 supplyone-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
activationdeactivation
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.
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/Tmfp
mean first passage time (mfp)
Computation of Gating Currentmaster
equation
gating current
Example 3
Potential CalculationLinearized Poisson-
Boltzman with transmembrane potential
effect
Movie
Pharmacological Applications
Thanks for your
Attention !