The flux of a solute across a membrane is calculated from its permeability coefficient, P, and the concentration gradient across the bilayer (ΔC)

${\displaystyle J=-P\cdot \Delta C}$

P can be calculated using molecular simulation using the solubility-diffusion model,^[1][2]

${\displaystyle {\frac {1}{P}}=\int _{-L}^{L}{\frac {e^{w(z)/{k_{B}T}}}{D(z)}}dz}$

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,

${\displaystyle J(z)=-D(z){\frac {{\textrm {d}}C(z)}{{\textrm {d}}z}}-C(z)D(z){\frac {{\textrm {d}}\left(w(z)/k_{B}T\right)}{{\textrm {d}}z}}}$

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.

References