  A discrete rate theory for general multi-ion channels is presented, in which
the continuous dynamics of ion diffusion is reduced to transitions between
Markovian discrete states. In an open channel, the ion permeation process
involves three types of events: an ion entering the channel, an ion escaping
from the channel, or an ion hopping between different energy minima in the
channel. The continuous dynamics leads to a hierarchy of Fokker-Planck
equations, indexed by channel occupancy. From these the mean escape times and
splitting probabilities (denoting from which side an ion has escaped) can be
calculated. By equating these with the corresponding expressions from the
Markov model the Markovian transition rates can be determined. The theory is
illustrated with a two-ion one-well channel. The stationary probability of
states is compared with that from both Brownian dynamics simulation and the
hierarchical Fokker-Planck equations. The conductivity of the channel is also
studied, and the optimal geometry maximizing ion flux is computed.
