Atomistic Simulations of Ion Channels - Computational Neuroscience: A Comprehensive Approach - page 45

Poisson's equation for the ion channel system:

4p r

Â
i

. [ e ( r ) f ( r )= − ( r )+ z i en i ( r ) (2.5)

= 1

(where the first term on the right-hand side is the charge density of the fixed charges
in the channel and the membrane, and the second term is the average charge density
of the mobile charges) combined with the steady-state equation for drift-diffusion to
accommodate the fluxes of the mobile ions:

;

n i ( )

kT

r

0 = . J i = . − D i ( n i ( n i ( r )+ m i ( r )) i = 1 , ··· , N (2.6)

Here m i ( ,

which assumes the chemical potential can be approximated by the electrostatic po-
tential and only depend on z, the depth inside the channel and membrane. In this
case, ions interact through the average potential

r ) is the chemical potential. In its simplest form, this could just be z i e f ( z )

)
can also include other interactions, providing a way to improve the theory. In these
equations, c i ,

f ( z ) . However, like before m i ( r

z i ,and D i are respectively the concentrations, fluxes, valences, and
diffusion constants of the ion species i. These two equations are coupled, because
the flux changes the potential due to the mobile charges, and the potential changes
the flux. In practice, they are solved simultaneously to self-consistency using numer-
ical methods. When all the fluxes J i are zero, and n i ∼

J i ,

, again with
f the average potential, these equations reduce to the normal Poisson-Boltzmann
equation. Thus, the PNP equations are an extension of the PB equation, and the
same assumptions as in PB theory underly PNP. To single out one assumption: ions
interact with each other only through the average charge density. This may be prob-
lematic, as in ion channels these interactions are discrete: a binding site with an
average occupancy of 0.25 will enter the average charge density as a charge of 0.25,
but this does not reflect the real situation. It may be argued that this may not be
too serious a problem in that the ion channel walls have such a high charge density
that ion-ion interactions are less important, and the average charge density is good
enough. However, this remains to be determined in specific cases.

exp ( − z i e f / ( kT ))

2.2.3 Brownian dynamics

A different approach, that maintains some of the benefits of atomistic simulation,
is provided by Brownian dynamics (BD). In BD, typically ions and the ion channel
are represented explicitly whereas solvent and lipids are represented implicitly (see
also Figure 2.3) . In these simulations, ions move stochastically in a potential that
is a combination of ion-ion interactions, ion-protein interactions, and a mean field.
These three components can be treated at different levels of complexity, analogous
to the calculation of the electrostatic potential for use in the Nernst-Planck equation.
In Brownian dynamics simulations the trajectories of individual ions are calculated
using the Langevin equation:
m i d v i

dt = −

g i v i + F R + q i E i + F s (2.7)