Modeling Guest Molecule Hydropathy in Lipid Bilayers

The hydropathy of the guest molecule is described by considering a periodic potential of mean force $U\left(z\right)$ such as the one depicted in Figure 1b (we show only one periodicity), which corresponds to the equation
$$U\left(z\right)=\left\{\begin{array}{cc}\Delta U\hfill & \mathrm{if}\left|z\right|\le l-h\hfill \\ \Delta U+8\frac{\Delta U_b-\Delta U}{h^2}{\left(l-h-\left|z\right|\right)}^2\hfill & \mathrm{if} l-h<\left|z\right|\le l-\frac{3}{4}h\hfill \\ \Delta U_b-8\frac{\Delta U_b-\Delta U}{h^2}{\left(l-\frac{1}{2}h-\left|z\right|\right)}^2\hfill & \mathrm{if} l-\frac{3}{4}h<\left|z\right|\le l-\frac{1}{2}h\hfill \\ \Delta U_b-8\frac{\Delta U_b}{h^2}{\left(l-\frac{1}{2}h-\left|z\right|\right)}^2\hfill & \mathrm{if} l-\frac{1}{2}h<\left|z\right|\le l-\frac{1}{4}h\hfill \\ 8\frac{\Delta U_b}{h^2}{\left(l-\left|z\right|\right)}^2\hfill & \mathrm{if} l-\frac{1}{4}h<\left|z\right|\le l\hfill \\ 0\hfill & \mathrm{if} l<\left|z\right|\le \frac{a}{2}\hfill \end{array}\right..$$
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 $L_{\alpha }$ 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, $\Delta U$ is the free-energy difference between the plateaus corresponding to the lipid tails and the water region. Therefore, $\Delta U>0$ for hydrophilic molecules (such as in Figure 1b), while $\Delta U<0$ in the case of hydrophobicity. The parameter $\Delta U_b$ introduces a barrier in correspondence with the lipid heads, which can mimic the kinetic barriers associated with the permeability of the membrane; moreover, $\Delta U_b$ can be employed to introduce depletion of molecules from the lipid heads ( $\Delta U_b>0$ ) or the tendency to sit at the water/lipid interface typical of amphiphilic molecules, which is the case for many proteins ( $\Delta U_b<0$ ) [48 ]. The use of parabolic fragments in Equation (1) allows for tuning the potential between $0, \Delta U$ and $\Delta U_b$ , while ensuring that both $U\left(z\right)$ 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 $D\left(z\right)$ (Figure 1c): $$D\left(z\right)=\left\{\begin{array}{cc}{D}_{lip}\hfill & \mathrm{if}\left|z\right|\le l\hfill \\ D_{lip}+2\frac{D_{wat}-D_{lip}}{w^2}{\left(l-\left|z\right|\right)}^2\hfill & \mathrm{if} l<\left|z\right|\le l+\frac{1}{2}w\hfill \\ D_{wat}-2\frac{D_{wat}-D_{lip}}{w^2}{\left(l+w-\left|z\right|\right)}^2\hfill & \mathrm{if} l+\frac{1}{2}w<\left|z\right|\le l+w\hfill \\ D_{wat}\hfill & \mathrm{if} l+w<\left|z\right|\le \frac{1}{2}a\hfill \end{array}\right..$$
In the previous formula, $D_{wat}$ and $D_{lip}$ 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 $D_{lip}$ and $D_{wat}$ takes place; therefore, w accounts for the reduced mobility of water molecules in the vicinity of the lipid heads [44 (link)].
Typical values of $D_{wat}$ for nanoscopic objects are found in the range ${10}^{-10}–{10}^{-9}$ m²/s. For instance, at 25 °C, one has $D_{wat}=0.7-0.9\times {10}^{-9}$ m²/s for amino acids [49 ,50 ,51 (link),52 ,53 ], and $D_{wat}=0.5,0.7,0.8\times {10}^{-9}$ m²/s for ibuprofen [54 (link)], aspirin [55 (link)], and paracetamol [56 (link)], respectively. The value of $D_{wat}$ is expected to be dependent on temperature, T. When small temperature differences are considered (such as estimation of $D_{wat}$ 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 $D_{wat}=k_{B}T/\left(6\pi \eta \left(T\right)R\right)$ , where $k_B$ is Boltzmann’s constant, R is the size of the particle, and $\eta \left(T\right)$ 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 $D_{wat}=0.7\times {10}^{-9}$ m²/s.
As for the diffusion coefficient in the lipid phase, $D_{lip}$ , one expects its value to be significantly smaller than $D_{wat}$ 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 $1.1–1.3\times {10}^{-11}$ m²/s [26 (link)], which gives values in the range $1.7–2\times {10}^{-11}$ 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 $D_{lip}$ is expected to be somewhat larger for them. Here, we fix $D_{lip}=0.09D_{wat}$ , 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 $D_{wat}$ is reached exactly at $z=a/2$ , thus avoiding a discontinuity in the derivative of $D\left(z\right)$ , which would have occurred for larger values of w.