Multidimensional Smoluchowski diffusion equation:

1
10
Multi-dimensional Smoluchowski diffusion equation
where is the probability
distribution of the coordinates at time t,
is the diffusion coefficient of the
tagged particle, is the Boltzmanns
constant, T is the temperature, is the
friction coefficient, is the effective
potential.
2
29
The nonlinear form of the Poisson-Boltzmann
equation where is the
position-dependent dielectric constant at point
r, is the local electrostatic
potential, is the charge density of the
solute, is the charge of the ions, is
the number density of the ions in the bulk
solute, is a core-repulsive potential
excluding the ions from the interior of the
solute.
Solute ion solvent, the second term of the
right hand side of the eq is the
total space-dependent charge density from the
mobile ions.
3
31
Linearization of Eq. (29) with respect to the
potential yields the Debye-Huckel
approximation where is the
space-dependent screening factor which varies
from zero, in the solvent-excluded regions, to
, in the bulk solvent.
4
48
The linearized Poisson-Boltzmann equation for the
total average electrostatic potential in the
presence of a membrane potential
where is a Heaviside step-function
equal to 0 on side I and 1 on side II, and
is the coupling parameter varying between 0
and 1 to scale the protein charges.
is the total average electrostatic potential
at point r, with potential charges scaled by
, and imposed membrane potential , which
governs the movement of charged species across
the cell membrane. bulk water protein region
membrane (fig 7)
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82
Poisson-Boltzmann voltage equation The step
function is
The pore region is the region from which all
ions other than the permeating species are
excluded, and the bulk region contains the
electrolytic solutions. Ion channel lipid
membrane (fig 8)
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123
According to the algorithm of Ermark, the
Brownian Dynamics trajectory of the n ions is
where is the Brownian dynamics time step,
is the diffusion coefficient of the
ion, is a random Gaussian vector with zero
average and ,
is the
effective force.
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124
The total multi-ion potential of mean force is
written as where is the spherically
symmetric potential of mean force between ions i
and j in an isotropic bulk solution, is
the non- electrostatic repulsive core overlap
potential (excluding the ions from the interior
of the protein or the membrane), is the
static field arising from the channel charges and
the transmembrane potential, and is the
electrostatic reaction-field free energy of the
ions in the system.
8
130
Given that the system contains ions of type
. The destruction probability of an ion of
type is where is the expectancy
for the number of ions of type from the
bulk density and the volume of the buffer
region, is the chemical potential, is the
charge in the multi-ion potential of mean force
of the system due to particle removal.
9
132
Poisson-Nernst-Plank continuum electrodiffusion
theory represents the average ion fluxes in terms
of density and potential gradients where
is the diffusion coefficient, is
the density, is an effective
potential acting on the ions.
10
135
The Poisson equation (Jackson, 1962) where
is the charge density of the channel,
is the position-dependent dielectric
constant at point r, is the average
electrostatic potential arising from all the
interactions in the system, is the charge
of the ions, is the charge Density of
ion .
11
The linearized Poisson-Boltzmann equation for the
total average electrostatic potential in the
presence of a membrane potential where
is the position-dependent dielectric constant at
point r, is the total
average electrostatic potential at point r, with
potential charges scaled by , and imposed
membrane potential , which governs the
movement of charged species across the cell
membrane.
is a Heaviside step-function equal to 0
on side I and 1 on side II, and
is the coupling parameter varying between 0
and 1 to scale the protein charges.
is the charge density of the solute.
12
Poisson-Boltzmann voltage equation of the ion
channel membrace system with asymmetrical
solutions on sides I and II The step
function is
The pore region is the region from which all
ions other than the permeating species are
excluded and the bulk region contains the
electrolytic solutions.
13
The Grand Canonical Monte Carlo Brownian Dynamics
Algorithm
The trajectory of the ions in the inner and
buffer regions
The total multi-ion potential of mean force is
where is the Brownian dynamics time step,
is the diffusion coefficient of the
ion, is a random Gaussian vector with zero
average and ,
is the
effective force.
is the spherically symmetric potential
of mean force between ions i and j in an
isotropic bulk solution, is the
non-electrostatic repulsive core overlap
potential (excluding the ions from the interior
of the protein or the membrane), is the
static field arising from the channel charges and
the transmembrane potential, and is the
electrostatic reaction-field free energy of the
ions in the system.
14
Poisson-Nernst-Plank equations
where is the diffusion coefficient,
is the density, is an
effective potential acting on the ions,
is the charge density of the channel, is
the position-dependent dielectric constant at
point r, is the average electrostatic
potential arising from all the interactions in
the system, is the charge of the ions.