Chemistry Reference
In-Depth Information
We discussed above solution of the PNP equations, which couple Poisson and
diffusion problems to solve for the steady-state transport.
4,20
The PNP theory
is a mean-field theory where, like in the PB equation, the ions are assumed to
be pointlike and uncorrelated. In addition, the surrounding solvent is treated
as a dielectric continuum. These methods are thus adequate for studying
transport through pores that are much larger than the size of the ions, but
unsatisfactory for some of the most important ion channels in biology where
the pore size is comparable to the ion size. Nevertheless, by incorporating
physically reasonable diffusion constants and dielectric profiles, decent
results can be obtained in some cases, and if the pores are larger, accurate
results are possible.
In the field of continuum-modeled ion transport through channels,
pioneering work has been done by the Eisenberg
4
and Coalson
20
groups.
Kurnikova et al.
245,246
first developed a numerical real-space three-
dimensional solver for the PNP equations and applied that method to
examining cation transport through the gramicidin channel. In their work,
a successive overrelaxation method (on a single grid) was employed to obtain
convergence. MGmethods can significantly accelerate the rate of convergence
to the solution, as discussed above.
More recently, Cheng et al.
153
developed an FE numerical technique to
model diffusion in large biomolecular systems. Their simulations do not
assume steady-state diffusion, but rather, solve the time-dependent Smolu-
chowski equation directly; the steady-state limit may emerge as the solution
progresses. Rates for inhibitor binding to acetylcholinesterase were computed
for several ionic strengths. Electrostatic steering was found to be important.
If we want to go beyond the rather severe physical limitations of the PNP
theory, a first step is to model the transport via Brownian dynamics.
11,132,247-249
Here the ions are modeled discretely, but the solvent and surrounding macromo-
lecules are treated as dielectric continua. Additionally, when considering ion
transport through channels, the protein is typically maintained in a fixed config-
uration. Ion correlations are accounted for with this approach, however, at least
approximately. Even though it is a challenging goal to model ion transport at the
all-atom level, efforts are progressing on this front as well.
250,251
The necessary
system size for ion channels as an example is often over 100,000 atoms, and one
ion transit event can be expected to occur on the tens of nanoseconds time scale.
To compute currents accurately, simulations approaching microseconds are
thus required.
EXISTING REAL-SPACE AND MULTIGRID CODES
Compiled here are existing electronic structure, electrostatics, and trans-
port codes that utilize real-space and MG or multiscale methods. The vitality
of this field of computational science is self-evident.
