Biomedical Engineering Reference
In-Depth Information
where
m
i
,
v
i
are the mass, charge and velocity of ion i. Water molecules are not
included explicitly, but are present implicitly in the form of a friction coefficient
m
i
g
i
=
q
i
,
D
i
and a stochastic force
F
R
arising from random collisions of water
molecules with ions, obeying the fluctuation-dissipation theorem. The term
q
i
E
i
is
the force on a particle with charge
q
i
due to the electric field
E
i
at the position of
particle i. In a first approximation, this field is due to the partial charges plus an
applied external field arising from the transmembrane potential. However, this term
should also include the effect of multiple ions and reaction field terms (image charge
effects) due to moving ions near regions with a low dielectric constant. Finally,
F
S
is a short-range repulsive force between ions and possibly between ions and protein.
This short-range force could be modelled as a hard sphere potential or as the repul-
sive part of a Lennard-Jones potential. When the friction is large and the motions are
overdamped, the inertial term
m
i
d
v
kT
/
/
dt
may be neglected, leading to the simplified
form
D
i
kT
(
v
i
=
q
i
E
i
+
F
s
)+
F
R
(2.8)
This is the approximation made in Brownian dynamics. This form has been used
in several ion channel simulations. When the free energy profile changes rapidly
on the scale of the mean free path of an ion, the full Langevin equation including
inertial effects must be used. BD simulations require only a few input parameters:
in its simplest form the diffusion coefficients of the different species of ions and
the charge on the ions. However, the model can be refined. The ion channel is
present as a set of partial charges, and some form of interaction potential between
the mobile ions and the protein must be specified (see above). The result of BD
simulations is a large set of trajectories for ions, from which macroscopic properties
such as conductance and ion selectivity can be calculated by counting ions crossing
the channel. In addition, the simulations yield molecular details of the permeation
paths for different types of ions. Such simulations have been performed of a series
of different systems, including simplified ion channel models [11, 31, 34, 75, 82],
gramicidin A [42], KcsA [23, 30] and OmpF porin and mutants [61, 86].
Although Brownian dynamics simulations are conceptually simple, in practice
they can give rise to a number of problems for which there may not be an obvious so-
lution. Representing all solvent effects by a random force and a diffusion coefficient
is a drastic simplification, particularly for narrow regions where ions interact very
strongly with one or two highly oriented water molecules. A recent study on grami-
cidin suggested continuum electrostatics calculations cannot provide a potential for
Brownian dynamics that is accurate enough to reproduce the experimental data on
gramicidin A [42]. Such calculations require several parameters such as dielectric
constants whose values cannot be derived from basic principles [23]. In the con-
text of pKa calculations for titratable amino acids by continuum calculations this has
been addressed in great detail, and it has been shown that the dielectric constant for a
protein depends on the approximations made; it might best be chosen differently for
different types of interactions [101]. Brownian dynamics simulations so far do not
take protein flexibility into account, but there is evidence from MD simulations that
this can be quite critical for e.g., potassium channels and gramicidin A. Finally, even
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