Molecular Simulation/Membrane permeability

The flux of a solute across a membrane is calculated from its permeability coefficient, P, and the concentration gradient across the bilayer (ΔC)
 * $$ J = -P \cdot \Delta C $$

P can be calculated using molecular simulation using the solubility-diffusion model,
 * $$ \frac{1}{P} = \int^L_{-L} \frac{e^{w(z)/{k_BT}}}{D(z)} dz $$

w(z) is the potential of mean force of the solute along the transmembrane axes, z. D(z) is the diffusion coefficient profile. The interval [L,-L] spans the membrane.

Derivation
The solubility diffusion model can be derived from the Nernst-Planck equation,
 * $$J(z) = -D(z) \frac{\textrm{d}C(z)}{\textrm{d}z} - C(z) D(z) \frac{\textrm{d} \left( w(z)/k_BT \right) }{\textrm{d}z} $$

In this equation, J(z) is the flux of the solute through the membrane at the depth z. C(z) is the concentration of the solute. w(z) and D(z) are the potential of mean force and diffusivity, respectively.