Poisson-Nernst-Planck Theory Approach to the calculation of ion transport through protein channels

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Poisson-Nernst-Planck Theory Approach to the
calculation of ion transport through protein
channels

• Guozhen Zhang

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Ion 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)
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Theoretical study of ion channels

• Kinetic models
• Electrodiffusion models
• Stochastic models
• Molecular Dynamics
• Brownian Dynamics

(Kurnikova, et al., 1999 Coalson and Kurnikova,
2005)
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Poisson-Nernst-Planck theory

• Basic idea
• Numerical solution
• Validity
• Application to Gramicidin A channel
• Improvement
• Summary

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Preconditions 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)
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standard PNP theory

• Nernst-Planck equation
• Poisson equation
• Total Potential Energy

(Kurnikova, et al., 1999)
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Solving 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)
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Solving 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
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Calibration of the accuracy of the 3D code
(Kurnikova, et al., 1999)
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Application to Gramicidin A channel
(Kurnikova, et al., 1999)
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(Kurnikova, et al., 1999)
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(Kurnikova, et al., 1999)
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Comparison with experiments
(Kurnikova, et al., 1999)
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standard PNP theory

• Nernst-Planck equation
• Poisson equation
• Total Potential Energy

(Kurnikova, et al., 1999)
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Dielectric-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)
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Performance of DSEPNP
(Coalson and Kurnikova, 2005)
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Potential 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)
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Results 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)
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The saturation mechanism
(Coalson and Kurnikova, 2005)
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Summary

• 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)
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References

• 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.

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Thank you!