Discrete stochastic modeling of ion channels

This dissertation explores stochastic models for the study of ion transport in biological cells. Analysis of the models is given to explain certain features of ion transport observed by biophysicists. For example, results in this dissertation explain why in single ion channels, the average time it takes an ion to cross the channel is the same in either direction, even if there is an electric potential difference across the channel. The result given here is much stronger. It states that the distribution of these paths is equivalent when they are time-reversed. Therefore, not only is the mean duration of these paths equal, but any statistical function of them, such as the variance of the passage time, is equal under time-reversal.; Furthermore, it is found that if a system of interacting ions is in reversible equilibrium (net flux is zero) then the equivalence of the left-to-right trans paths with the time-reversed right-to-left trans paths still holds. However, if the system is in equilibrium, but not reversible equilibrium, then such equivalence need not hold.; This dissertation discusses requirements for multidimensional processes, in discrete and continuous space, to be reversible. The reversibility of these models is exploited in order to show that the distribution of left-to-right trans paths is equal to that of the time-reversed right-to-left trans paths. A discretization procedure which preserves the reversibility of diffusion models is suggested, and simulation of these discrete models is simple.; Although most of this dissertation considers one-dimensional models and assumes that baths are large and rapidly mixing, an idealized three-dimensional model is analyzed in equilibrium. An exact series expansion for the concentration allows us to explore the behavior of the concentration near the boundaries between the channel and the baths.; Kernel representations are obtained for the concentration and flux away from equilibrium, where the kernel functions have a probabilistic interpretation. This is done in a discrete space model and also in its diffusion limit. In particular, the kernel representation is given for the flux at a boundary for the diffusion model. This requires special attention since one of the kernel functions exhibits a singularity.