The hydropathy of the guest molecule is described by considering a periodic potential of mean force U(z) such as the one depicted in Figure 1b (we show only one periodicity), which corresponds to the equation
U(z)=ΔUifzlhΔU+8ΔUbΔUh2lhz2if lh<zl34hΔUb8ΔUbΔUh2l12hz2if l34h<zl12hΔUb8ΔUbh2l12hz2if l12h<zl14h8ΔUbh2lz2if l14h<zl0if l<za2.
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 Figure 1). In order to set reasonable values for these parameters, we considered a=6.65 nm, which corresponds to fully-hydrated Lα lamellar mesophases obtained by water/dipalmitoylphosphatidylcholine mixtures at 43 °C [46 ]. Moreover, we also set l=2.365 nm and h=1 nm based on the electron-density profile computed in Ref. [47 ] for the same mixture. As for the energy parameters, ΔU is the free-energy difference between the plateaus corresponding to the lipid tails and the water region. Therefore, ΔU>0 for hydrophilic molecules (such as in Figure 1b), while ΔU<0 in the case of hydrophobicity. The parameter ΔUb introduces a barrier in correspondence with the lipid heads, which can mimic the kinetic barriers associated with the permeability of the membrane; moreover, ΔUb can be employed to introduce depletion of molecules from the lipid heads ( ΔUb>0 ) or the tendency to sit at the water/lipid interface typical of amphiphilic molecules, which is the case for many proteins ( ΔUb<0 ) [48 ]. The use of parabolic fragments in Equation (1) allows for tuning the potential between 0, ΔU and ΔUb , while ensuring that both U(z) and its derivative are continuous throughout space (see Figure 1b), which avoids undesirable numerical instabilities in the simulations.
A similar approach was used to account for heterogeneity in molecular transport. To this aim, we introduced a space-dependent diffusion coefficient D(z) (Figure 1c): D(z)=DlipifzlDlip+2DwatDlipw2lz2if l<zl+12wDwat2DwatDlipw2l+wz2if l+12w<zl+wDwatif l+w<z12a.
In the previous formula, Dwat and Dlip 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 Dlip and Dwat takes place; therefore, w accounts for the reduced mobility of water molecules in the vicinity of the lipid heads [44 (link)].
Typical values of Dwat for nanoscopic objects are found in the range 1010109 m2/s. For instance, at 25 °C, one has Dwat=0.70.9×109 m2/s for amino acids [49 ,50 ,51 (link),52 ,53 ], and Dwat=0.5,0.7,0.8×109 m2/s for ibuprofen [54 (link)], aspirin [55 (link)], and paracetamol [56 (link)], respectively. The value of Dwat is expected to be dependent on temperature, T. When small temperature differences are considered (such as estimation of Dwat 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 Dwat=kBT/(6πη(T)R) , where kB is Boltzmann’s constant, R is the size of the particle, and η(T) 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 Dwat=0.7×109 m2/s.
As for the diffusion coefficient in the lipid phase, Dlip , one expects its value to be significantly smaller than Dwat 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 1.11.3×1011 m2/s [26 (link)], which gives values in the range 1.72×1011 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 Dlip is expected to be somewhat larger for them. Here, we fix Dlip=0.09Dwat , 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 w=0.96 nm (Figure 1c), in order to ensure that Dwat is reached exactly at z=a/2 , thus avoiding a discontinuity in the derivative of D(z) , which would have occurred for larger values of w.
Free full text: Click here