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
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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)
Mesoscopic
property
Macroscopic
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
Mesoscopic
property
Macroscopic
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
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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
inactivation
deactivation
activation
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
mean first passage
time (mfp)
rate constant = 1/Tmfp
Computation of Gating Current
master
equation
gating
current
Example 3
Potential Calculation
Linearized PoissonBoltzman with
transmembrane potential
effect
Movie
Pharmacological Applications
Thanks for your
Attention !