Jinn-Liang Liu

National Hsinchu University of Educationwith

Hsin-Hung Wu and Ren-Chuen Chen

1

Quantum Poisson-Nernst-Planck Modelfor Biological Ion Channels

1

April 22-23, 2011 WCM, NCTU

Conclusion

2

Outline

Numerical Results

Classical PNP Model Quantum PNP Model

Numerical Methods

Ion Channel

3Ion Channel

Ion Channel

Nervous system (0:50)

Action Potentials (3:24)

Potassium Channel (1:42)

K+ radius: 0.133 nm, Na+ radius: 0.095 nm(0.133-0.095)/0.133 = 28.6%

Biological ion channels seem to be a precondition for all living matter.

P. Agre & R. MacKinnon(Nobel Prize in Chemistry 2003)for their discoveries concerning "channels in cell membranes".

Hodgkin–Huxley Model on Ion Channel

E. Neher & B. Sakmann(Nobel Prize in Physiology or Medicine 1991)for their discoveries concerning "the function of single ion channels in cells".

A. L. Hodgkin & A. Huxley (Nobel Prize in Physiology or Medicine 1963)for their discoveries concerning " the ionic mechanisms in the nerve cell membrane ".

Ion ChannelIon channels regulate the flow of ions across the membrane in all cells.

(Ions in water are the liquid of Life.)

Figures from Bob Eisenberg

6

選擇性濾嘴中的氧原子形成的環境和濾嘴外水分子中的氧原子形成的環境相同。以能量的觀點來解釋，鈉離子留在水中會有較佳的水合能。

membrane

鉀離子進入通道後閘門必須開啟，離子才能進入細胞內。感應器構形的改變造成了閘道開和關的動作。不同的感應器會回應不同的訊號來開啟或關閉閘道。

Selectivity filter

membrane

interiorexterior

+

-+

-

+

-

+

-

+

-

+

-

+

-

+

-

_

_

protein

KcsA channel

gate

Ion Channel

7

Classical PNP (Drift-Diffusion) Model

PNP Model

number current density electric field

mobility coefficient

number density for the mobile ions

diffusion coefficient

electrostatic potential

protein charge

number density for the permanent fixed charge protein

ionic charge

dielectric coefficient

total electric current density

8

Change Variable

Let

then

to obtain

(Slotboom)

thermal voltage

= 0

PNP Model

9

Self-Adjoint PNP Model

PNP Model

10

Gauss’s (Divergence ) Theorem (u: vector field )

For

3D1D

volume of an arbitrary shaped region in R3 that includes the point x

surface of V

outward normal to that surface

cross sectional area A(z)

cylindrical symmetry (r, θ, z)neglect r direction

r

z

then

flux u across the surface S

PNP Model

11

1D PNP Model

PNP Model

12

Bohm’s Quantum Potential

Schrodinger Eq.

QPNP Model

Quantum Potential

13

QP Equation

Let

Rewrite

QPNP Model

as

14

Self-Adjoint QPNP Model

where

QPNP Model

15

1D QPNP Model

QPNP Model

Finite different method

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Numerical Methods

Numerical Methods

Potential Eqn.

by the Scharfetter-Gummel methodJ

where and

DD Eqn.:

Quantum Potential Eqn.

Bernoulli function

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Generalized Gummel Algorithm

Solve Potential Eqn.

Solve Quantum Potential Eqn.

Solve DD Eqn.

Update Potential

Update DensityUpdate ζ

orStop

Initial Guess

Update

Numerical Methods

18

Monotone Iteration

Global Convergence

Numerical Methods

Optimal Convergence

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Results and Features

Well-Posedness of the Generalized Gummel Iteration

Global and Optimal Convergence of Monotone Iterative Method

Single Finite Element Subspace Highly Parallel for Linear Solver

and I-V Calculation (by Self-Adjoint and Monotone)

Numerical Methods

20Numerical Results

11 nm

3.5 nm

1 nm

5 nm

Simulation of the K Channel

5 nm

V = 0mV V = -10mV ~ -100mV

Numerical Results

21

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

V = -100 millivolts I : picoamperes

Numerical Results

Numerical Results

Electrostatic potential

PNP QPNP

22Numerical Results

Numerical Results

Density

PNP QPNP

Numerical results for the K channel 23

Numerical Results

Quantum potentials

Cl K

Numerical Results

25

Conclusion

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

PNP QPNP

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~Thank You~