Jinn-Liang Liu

1
Quantum Poisson-Nernst-Planck Modelfor
Biological Ion Channels

• Jinn-Liang Liu
• National Hsinchu University of Education
• with
• Hsin-Hung Wu and Ren-Chuen Chen

IOP, NCTU, May 5, 2011
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Outline

• Ion Channel

• Classical PNP Model

• Quantum PNP Model

• Numerical Methods

• Numerical Results

• Conclusion

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Ion Channel
Biological ion channels seem to be a
precondition for all living matter.
Nervous system (050)
Action Potentials (324)
Potassium Channel (142)
K radius 0.133 nm, Na radius 0.095
nm (0.133-0.095)/0.133 28.6
Ion Channel
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A. L. Hodgkin A. Huxley (Nobel Prize in
Physiology or Medicine 1963) for their
discoveries concerning " the ionic mechanisms in
the nerve cell membrane ".
E. Neher B. Sakmann (Nobel Prize in Physiology
or Medicine 1991) for their discoveries
concerning "the function of single ion channels
in cells".
HodgkinHuxley Model on Ion Channel
P. Agre R. MacKinnon (Nobel Prize in Chemistry
2003) for their discoveries concerning "channels
in cell membranes".
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Ion Channel
Ion channels regulate the flow of ions across the
membrane in all cells. (Ions in water are the
liquid of Life.)
Figures from Bob Eisenberg
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KcsA channel
membrane
-
protein
Selectivity filter
-
-
-
_
interior
exterior
-
-
-
-
_
gate
membrane
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Ion Channel
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Classical PNP (Drift-Diffusion) Model
number current density
electric field
number density for the mobile ions
diffusion coefficient
mobility coefficient
protein charge
ionic charge
electrostatic potential
number density for the permanent fixed charge
protein
dielectric coefficient
total electric current density
PNP Model
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Change Variable
Let
then
0
(Slotboom)
to obtain
thermal voltage
PNP Model
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Self-Adjoint PNP Model
PNP Model
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3D?1D
cross sectional area A(z)
cylindrical symmetry (r, ?, z)
r
neglect r direction
z
Gausss (Divergence ) Theorem (u vector field )
outward normal to that surface
volume of an arbitrary shaped region in R3 that
includes the point x
surface of V
For
then
flux u across the surface S
PNP Model
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1D PNP Model
PNP Model
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Bohms Quantum Potential
Schrodinger Eq.
Quantum Potential
QPNP Model
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QP Equation
Let
Rewrite
as
QPNP Model
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Self-Adjoint QPNP Model
where
QPNP Model
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1D QPNP Model
QPNP Model
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Numerical Methods
DD Eqn.
J
by the Scharfetter-Gummel method
and
where
Bernoulli function
Potential Eqn.

• Finite different method

Quantum Potential Eqn.
Numerical Methods
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Generalized Gummel Algorithm
Initial Guess
Update Potential
Solve Potential Eqn.
Solve DD Eqn.
Update ?
Update Density
Update
Solve Quantum Potential Eqn.
or
Stop
Numerical Methods
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Monotone Iteration

• Global Convergence

• Optimal Convergence

Numerical Methods
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Results and Features

• Global and Optimal Convergence of Monotone
  Iterative Method

• Well-Posedness of the Generalized Gummel Iteration

• Single Finite Element Subspace

• Highly Parallel for Linear Solver and I-V
  Calculation (by Self-Adjoint and Monotone)

Numerical Methods
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Simulation of the K Channel
5 nm
5 nm
11 nm
1 nm
3.5 nm
V 0mV
V -10mV -100mV
Numerical Results
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Numerical Results
V -100 millivolts I picoamperes

Reference 4.3 1/15 22.2 0.3 22.5 154
DD model 4.58 1/11 24.24 0.39 24.63 155
QCDD model 4.48 1/13 22.67 0.31 22.98 152
Experimental 4.5 0.05I 20-30 200
Numerical Results
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Numerical Results
Electrostatic potential
PNP
QPNP
Numerical Results
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Numerical Results
Density
PNP
QPNP
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Numerical results for the K channel
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Numerical Results
Quantum potentials
Cl
K
Numerical Results
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Conclusion
PNP
QPNP
It is shown that the I-V curve of this channel is
corrected by the quantum potential equations with
current drive reduced by about 6.6 comparing
with that of the classical model along.
Conclusion
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Thank You