Title: Poisson-Nernst-Planck Theory Approach to the calculation of ion transport through protein channels
1Poisson-Nernst-Planck Theory Approach to the
calculation of ion transport through protein
channels
2Ion transport through protein channels
- We human beings consist to about
- 70 of salt water. This year's Nobel
- Prize in Chemistry rewards two
- scientists whose discoveries have
- clarified how salts (ions) and water
- are transported out of and into the
- cells of the body This is of great
- importance for our understanding of
- many diseases of e.g. the kidneys,
- heart, muscles and nervous system.
(Press release of the Nobel Prize in Chemistry
2003 Doyle, et al., 1998)
3Theoretical study of ion channels
- Kinetic models
- Electrodiffusion models
- Stochastic models
- Molecular Dynamics
- Brownian Dynamics
(Kurnikova, et al., 1999 Coalson and Kurnikova,
2005)
4Poisson-Nernst-Planck theory
- Basic idea
- Numerical solution
- Validity
- Application to Gramicidin A channel
- Improvement
- Summary
5Preconditions of PNP theory
- Coarse grained approximation
- mobile ions gt continuous charge distribution
- surroundings gt 3D grid with different
dielectric constant - High-friction assumption
- Brownian motion gt Smoluchowski equation
- Steady-state assumption
- the particle flux is time-independent
(Kurnikova, et al., 1999)
6standard PNP theory
- Nernst-Planck equation
-
- Poisson equation
- Total Potential Energy
-
(Kurnikova, et al., 1999)
7Solving 3D Poisson equation on a cubic grid
- 1D case
- a. division of grid
-
- where the lattice cell extends from
(j-1/2)h to (j-1/2)h - b. discretization of Poisson equation on the
grid -
- 3D case
-
-
- where ?ij is the 3D generalization of the matrix
defined in the above equation, bi (D) - are the effective source terms associated with
the Dirichlet boundary condition.
(Graf, et al., 2000)
8Solving 3D NP Eq. by successive over-relaxation
i. Flux
ii. Steady-state flux condition
iii. Concentration for central point
Where ,
is the number of nearest-neighbor lattice points
(Cárdenas, et al., 2000 Kurnikova, et al., 1999)
iv. SOR iteration equation
9Calibration of the accuracy of the 3D code
(Kurnikova, et al., 1999)
10Application to Gramicidin A channel
(Kurnikova, et al., 1999)
11(Kurnikova, et al., 1999)
12(Kurnikova, et al., 1999)
13Comparison with experiments
(Kurnikova, et al., 1999)
14standard PNP theory
- Nernst-Planck equation
-
- Poisson equation
- Total Potential Energy
-
(Kurnikova, et al., 1999)
15Dielectric-Energy PNP theory
Nernst-Planck equation
Poisson equation
The free energy of ions of species i in solution
(Graf, et al., 2004 Coalson and Kurnikova, 2005)
16Performance of DSEPNP
(Coalson and Kurnikova, 2005)
17Potential of Mean Force PNP theory
- The protein structure used in both BD and DSEPNP
simulations is taken to be rigid, while in
reality the protein structure responds
dynamically to an ions presence. Such a defect
usually exhibits very small superlinear currents
for voltages up to 200mV for narrow channels. - This issue can in principle be solved by a full
atomistic simulation which requires complete
sampling of the system configuration space. But
its formidable for current computing capability. - Limited sampling of the environment
configurational space has been introduced to deal
with the problem. A combined MD/continuum
electrostatics approach is then proposed to
obtain ?GSIP at an average solvent effect level,
which is then used in PNP formalism. Such a
procedure is termed PMFPNP.
(Coalson and Kurnikova, 2005)
18Results of the PMFPNP calculations
- The overall structure of peptide doesnt change
much over the course of MD trajectory, so the
?GDSE contribution to the overall ?GSIP doesnt
vary much. - Small local distortions of pore-lining parts of
the peptide (especially carbonyl groups)
significantly stabilize cations as they move
through it. - PMFPNP theory is able to account for effects that
are beyond the reach of primitive PNP theory,
namely, saturation of ion current through the
channel as the concentration of bathing solutions
increases to a sufficiently high value.
(Coalson and Kurnikova, 2005)
19The saturation mechanism
(Coalson and Kurnikova, 2005)
20Summary
- 3D PNP theory is of conceptual simplicity. It
relies on a caricature of the microscopic world
in which background media are treated as
dielectric slabs and the mobile ions of interest
are smeared out into a continuous charge
distribution. - The inherent restriction of the theory is mainly
due to its simplicity. It may be unrealistic for
treating certain properties of certain ion
channels. Also, the mean-field continuum
solvent/ion theory of this type is inadequate to
accurately describe the underlying dynamics. - Despite of these restrictions, PNP theory will
continue to play a useful role in computing and
understanding the kinetics of ion permeation
through (wider) biological channels.
(Coalson and Kurnikova, 2005)
21References
- Cárdenas, A. E., R. D. Coalson, and M. G.
Kurnikova. 2000. Three-Dimensional
Poisson-Nernst-Planck Theory Studies Influence
of Membrane Electrostatics on Gramicidin A
Channel Conductance. Biophys. J. 79 80-93. - Coalson, R. D., and M. G. Kurnikova. 2005.
PoissonNernstPlanck Theory Approach to the
Calculation of Current Through Biological Ion
Channels. IEEE T Nanobiosci. 4 81-93. - Doyle, D.A., J. M. Cabral, R. A. Pfuetzner, A.
Kuo, J. M. Gulbis, S. L. Cohen, B.T. Chait, and
R. MacKinnon. 1998. The structure of the
potassium channel Molecular basis of
K conduction and selectivity. Science. 280
69-77. - Graf, P., A. Nitzan, M. G. Kurnikova, and R. D.
Coalson. 2000. A dynamic lattice Monte Carlo
model of ion transport in inhomogeneous
dielectric environments Method and
implementation. J. Phys. Chem. B. 104
12324-12338. - Graf, P., M. Kurnikova, R. Coalson, and A.
Nitzan. 2004. Comparison of dynamic lattice Monte
Carlo simulations and the dielectric self-energy
Poisson-Nernst-Planck continuum theory for model
ion channels. J. Phys. Chem. B. 108 2006-2015. - Kurnikova, M. G., R. D. Coalson, P. Graf, and A.
Nitzan. 1999. A lattice relaxation algorithm for
three-dimensional Poisson-Nernst-Planck theory
with application to ion transport through the
gramicidinA channel. Biophys. J. 76 642656.
22Thank you!