In the previous formula, a is the lattice parameter, l is the total length of a lipid within the bilayer, and h is the size of the lipid head (see
A similar approach was used to account for heterogeneity in molecular transport. To this aim, we introduced a space-dependent diffusion coefficient (
In the previous formula, and correspond to the diffusion coefficients of the guest molecule when considered in pure water and in the lipid bilayer, respectively, while w is the thickness of the water layer in which the continuous change between and takes place; therefore, w accounts for the reduced mobility of water molecules in the vicinity of the lipid heads [44 (link)].
Typical values of for nanoscopic objects are found in the range m2/s. For instance, at 25 °C, one has m2/s for amino acids [49 ,50 ,51 (link),52 ,53 ], and m2/s for ibuprofen [54 (link)], aspirin [55 (link)], and paracetamol [56 (link)], respectively. The value of is expected to be dependent on temperature, T. When small temperature differences are considered (such as estimation of at physiological temperature starting from room-temperature measurements), a simple yet effective approach to estimate the effect of T is to assume a Stokes–Einstein relation , where is Boltzmann’s constant, R is the size of the particle, and is the temperature-dependent viscosity of water. This approach has enabled accurate predictions of transport of glucose molecules in monolinolein-based cubic phases [57 (link)]. Unless stated otherwise, in our simulations, we consider m2/s.
As for the diffusion coefficient in the lipid phase, , one expects its value to be significantly smaller than due to the lower fluidity of the lipid membrane as compared to water. For instance, the three-dimensional self-diffusion of lipids for various monoacylglycerols with cubic symmetry has been reported to be m2/s [26 (link)], which gives values in the range m2/s for the lateral diffusion coefficient when accounting for the geometric constraint imposed by the minimal surface at the mid-plane of the lipid bilayer [58 (link)]. Amino acids and drugs such as the ones mentioned above are smaller than lipid molecules, so that is expected to be somewhat larger for them. Here, we fix , based on molecular dynamics simulations of paracetamol in DPPC [47 ].
Finally, the parameter w was set in accordance with experimental evidence and molecular dynamics simulations, which point to the existence of 3–4 layers of water with reduced mobility in proximity of the lipid heads [8 (link),40 ,44 (link)]. The specific value of this thickness was selected to be nm (