Molecular Simulation/Membrane permeability

File:Molecular Dynamics Simulation of DPPC Lipid Bilayer.webm The flux of a solute across a membrane is calculated from its permeability coefficient, P, and the concentration gradient across the bilayer (ΔC)

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

P can be calculated using molecular simulation using the solubility-diffusion model,

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

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,

$J(z)=-D(z)\frac{\text{d}C(z)}{\text{d}z}-C(z)D(z)\frac{\text{d}(w(z)/k_{B}T)}{\text{d}z}$

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.