  We have developed a molecular mean-field theory -- fourth-order
Poisson-Nernst-Planck-Bikerman theory -- for modeling ionic and water flows in
biological ion channels by treating ions and water molecules of any volume and
shape with interstitial voids, polarization of water, and ion-ion and ion-water
correlations. The theory can also be used to study thermodynamic and
electrokinetic properties of electrolyte solutions in batteries, fuel cells,
nanopores, porous media including cement, geothermal brines, the oceanic
system, etc. The theory can compute electric and steric energies from all atoms
in a protein and all ions and water molecules in a channel pore while keeping
electrolyte solutions in the extra- and intracellular baths as a continuum
dielectric medium with complex properties that mimic experimental data. The
theory has been verified with experiments and molecular dynamics data from the
gramicidin A channel, L-type calcium channel, potassium channel, and
sodium/calcium exchanger with real structures from the Protein Data Bank. It
was also verified with the experimental or Monte Carlo data of electric
double-layer differential capacitance and ion activities in aqueous electrolyte
solutions. We give an in-depth review of the literature about the most novel
properties of the theory, namely, Fermi distributions of water and ions as
classical particles with excluded volumes and dynamic correlations that depend
on salt concentration, composition, temperature, pressure, far-field boundary
conditions etc. in a complex and complicated way as reported in a wide range of
experiments. The dynamic correlations are self-consistent output functions from
a fourth-order differential operator that describes ion-ion and ion-water
correlations, the dielectric response (permittivity) of ionic solutions, and
the polarization of water molecules with a single correlation length parameter.
