Abstract
We present a discrete Markovian model to describe the
transport of potassium ions in the selectivity filter
of the $KcsA$ $K^+$ channel 1.
Recent work by MacKinnon et al. 2 has documented that an array of two
$K^+$ ions alternating in position with two water molecules partake in
single-file transport through the filter. To capture this geometry,
we construct a d=1 dimensional lattice model with alternating positive
ions and neutral particles of equal mass (the slight mass difference
between $K^+$ ions and water molecules is neglected). Elaboration of the
model allows one to calculate the mean exit time of one member of this quartet
starting from a given initial position inside the channel and
averaging over all sterically-allowed joint configurations. We then
proceed in stages to incorporate the relevant interactions in the
model. First, we take into account only the hard-core (excluded volume)
interactions between the ions and the water molecules and find the
unexpected result that, as a consequence of considering 'gaps'
(vacant sites) in the channel, fluctuations in the positions of the
particles may actually decrease the exit time relative to single-particle
transport. The role of electrostatic interactions is then
investigated: ion-ion repulsions, interactions between the $K^+$ ions
and carbonyl oxygens along the channel (modeled as dipoles), and
interactions experienced by an ion owing to the presence of a
transmembrane potential. For specific choices of atomic/molecular
parameters, we find exit times which approximate closely the case
where electrostatic interactions are absent,
thus providing evidence for the kind of delicate 'charge balance'
proposed previously by McKinnon in 2. In quantifying the
time dependence of the flow (by determining the eigenvalue spectrum of
the associated stochastic master equation), we identify regimes where
more than one time scale may govern the transport of ions through the
channel. Finally, we discuss the strengths and limitations of the Markovian model developed here, and the generalizations that would need to be undertaken to bring the model in closer correspondence with experimental studies of
multi-ion permeation in biological channels.\\
1) E. Abad and J.J. Kozak, Physica A (in press), doi:10.1016/j.physa.2007.02.092.\\
2) J. H. Morais-Cabral, Y. Zhou and R.M. MacKinnon, Nature 414 (2001) 37
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