  We develop a mesoscopic approach to model the non-equilibrium behavior of
membranes at the cellular scale. Relying on lattice Boltzmann methods, we
develop a solution procedure to recover the Nernst-Planck equations and Gauss's
law. A general closure rule is developed to describe mass transport across the
membrane, which is able to account for protein-mediated diffusion based on a
coarse-grained representation. We demonstrate that our model is able to recover
the Goldman equation from first principles and show that hyper-polarization
occurs when membrane charging dynamics are controlled by multiple relaxation
timescales. The approach provides a promising way to characterize
non-equilibrium behaviors that arise due to the role of membranes in mediating
transport based on realistic three-dimensional cell geometries.
