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Framework Model of Ion Permeation

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Outside a steady state diffusion process subject to potential - U ... J., 2006) compares 17O shift due to metal ion binding with that due to proton binding. ... – PowerPoint PPT presentation

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Title: Framework Model of Ion Permeation


1
Framework Model of Ion Permeation
  • Mark F. Schumaker
  • Department of Mathematics
  • Washington State University

2
Organization of Talk
  • Single-Particle Model
  • The Stationary Arrival Property of Diffusers from
    a Continuum to an Absorbing Boundary
  • Grotthuss Model (Proton Conduction)
  • Framework Model of Potassium Conduction In KcsA

3
Single-Particle Model
  • Pete McGill, Ph.D. Washington State University,
    1995
  • MFS

4
Prior Results
  • Levitt (1986) Single-particle boundary
    conditions
  • Jakobsson and Chiu (1987) and (1989) Model of
    Na conduction through gramicidin

5
Molecular Model of Gramicidin (Pomès and Roux)
6
Potential of Mean Force for Na in Gramicidin
(Roux and Karplus, 1993)
7
State Diagram of the Single Particle Model
8
State Diagram and Random Walk
9
Agmon-Hopfield Transition Rates
is the diffusion coefficient associated with site
i.
is the potential of mean force associated with
site i.

10
Smoluchowski Equation with Nonlocal Boundary
Conditions

11
Smoluchowski Equation with Nonlocal Boundary
Conditions

12
Smoluchowski Equation with Nonlocal Boundary
Conditions

Nonlocal boundary conditions. Pointed out by
Hong-Ming Yin. WSU.
13
Trajectories Underlying the Single-Particle Model
14
Time-Independent Single Particle Model
15
Trajectories With Nonlocal Boundary Conditions
Schumaker, Journal of Chemical Physics, 2002
16
Smoluchowski Equation with Mean Field Boundary
Conditions

17
Trajectories with Mean Field Boundary Conditions

Schumaker, Journal of Chemical Physics, 2002
18
Comparing Nonlocal and Mean Field Currents
Mapes and Schumaker, Bulletin of Mathematical
Biology, 2006
19
The Stationary Arrival Process of Diffusers from
a Continuum to an Absorbing Boundary
  • Boaz Nadler, Tamara Naeh, Zeev Schuss

20
The Arrival Process of New Diffusers To An
Absorbing Boundary is Poissonian
  • Smooth boundary with absorbing and reflecting
    components
  • Outside a steady state diffusion process subject
    to potential -ÑU
  • Poisson rate equal to the total flux on the
    absorbing boundary as calculated by continuum
    theory

Nadler, Naeh and Schuss, SIAM Journal Applied
Math, 2001
21
Recirculating Diffusers
  • In addition to new particles arriving at a
    boundary, recirculating particles must be
    considered.
  • The figure illustrates a simulation scheme of
    Nadler, Naeh and Schuss

Figure from Nadler, Naeh and Schuss, SIAM Journal
of Applied Math, 2003
22
In a steady 1D flow, recirculating diffusers make
no contribution to net particle flux
  • At any spatial point x, the unidirectional flux
    of recirculating particles to the right equals
    the unidirectional flux of recirculating
    particles to the left.
  • Integrate with conc. fixed at , recirculating
    particles make no contribution to concentration.

Schumaker, Bulletin of Mathematical Biology, in
press
23
Implications for Framework Models
  • Poisson distributed entrances may be regarded as
    a 1D approximation of diffusion from an infinite
    bath.
  • Within the 1D approximation, appropriate
    particle fluxes and concentrations may be
    calculated (assuming other features of the model
    are correct).

24
Grotthuss Model
  • Regis Pomès (Toronto) and Benoit Roux (then at
    the University of Montreal)
  • Joseph Gowen, Jeffrey Markham, Sara Morrison and
    David Busath at BYU and Tim Cross at FSU
  • Eric Mapes (WSU), MFS

25
Gramicidin With Excess Proton (Pomès and Roux)
26
Gramicidin Without Excess Proton (Pomès and Roux)
27
Proton Occupation Water Reorientation Potentials
(PM6 water model) and Grotthuss Model State
Diagram
Schumaker, Pomès and Roux, Biophys. J. 2000.
28
The Lumped State Approximation
29
Two Grotthuss Models
30
Comparison with Proton Conductance Data

Using the lumped-state approximation
31
Sensitivity Analysis
  • Required over 106 current evaluations
  • With analytical formula, an evaluation requires
    only 0.5 ms on a 1 GHz PC

Gowen et al., Biophysical Journal, 2002
32
Conclusion of Sensitivity Analysis
33
PMF for water reorientation PM6 vs TIP3P
PM6
TIP3P
Pomès and Roux, (Biophys. J., 2002)
34
A recent experimental result consistent with the
framework model for proton conduction
  • NMR study of Checkmenev et al. (Biophys. J.,
    2006) compares 17O shift due to metal ion binding
    with that due to proton binding. They conclude
    these processes are different.

35
Potassium Conduction in KcsA
  • Simon Bernèche and
  • Benoit Roux

36
Structure of the KcsA K Channel
Roux. Ann. Rev. Biophy. Biomol. Structure, 2005
(based on Doyle et al. Science, 1998)
37
Projections of 3D Potential of Mean Force
Bernèche and Roux (PNAS, 2003)
38
Comparison with Experiment
Bernèche and Roux, PNAS 2003 (Experimental
results of LeMasurier, Heginbotham and Miller, J.
Gen. Physiol., 2001)
39
Conclusions
  • Framework models of ion permeation in narrow ion
    channels provide a useful comparison between MD
    and experiment.
  • They seem limited by a requirement for a
    configuration space with a small number of
    important variables.
  • But they can also be viewed as a generalization
    of the methods of enzyme kinetics.
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