  Mathematically modeling complex transport phenomena at the molecular level
can be a powerful tool for identifying transport mechanisms and predicting
macroscopic properties. We use two different stochastic time series models,
parameterized from long molecular dynamics (MD) simulation trajectories of a
cross-linked HII phase lyotropic liquid crystal (LLC) membrane, in order to
predict solute mean squared displacements (MSDs) and solute flux, and thus
solute selectivity, in macroscopic length pores. First, using anomalous
diffusion theory, we show how solute dynamics can be modeled as a fractional
diffusion process subordinate to a continuous time random walk. From the MD
simulations, we parameterize the distribution of dwell times, hop lengths
between dwells and correlation between hops. We explore two variations of the
anomalous diffusion modeling approach. The first applies a single set of
parameters to the solute displacements and the second applies two sets of
parameters based on the solute's radial distance from the closest pore center.
Next, we generalize Markov state models, treating the configurational states of
the system as a Markov process where each state has distinct transport
properties. For each state and transition between states, we parameterize the
distribution and temporal correlation structure of positional fluctuations as a
means of characterization and to allow us to predict solute MSDs. Qualitative
differences between MD and Markov state dependent model-generated trajectories
may limit its usefulness. Finally, we demonstrate how one can use these models
to estimate flux of a solute across a macroscopic-length pore and, based on
those quantities, the membrane's selectivity towards each solute. This work
helps to connect microscopic chemically-dependent solute motions that do not
follow simple diffusive behavior with macroscopic membrane performance.
