Fluctuation assisted diffusion through ion channels

We present a discrete Markovian model to describe the transport of potassium ions in the selectivity filter of the KcsAK+ channel. Recent work by MacKinnon et al. has documented that an array of two K+ ions alternating in position with water molecules partake in single-file transport through the filter. To capture this geometry, we construct a d=1 dimensional lattice model with two 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 et al. Finally, 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. The final section is given over to a discussion of 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.