2.1. Lateral Diffusion Coefficient

Lateral diffusion depends on the nearby molecules. It is a fast and spontaneous movement relative to the translational movements between the two layers. Ensemble averaged mean–square displacement (MSD) was evaluated for the bilayer systems to assess lipid diffusion. The average MSD for the DPPC molecule was calculated with the gmx msd subroutine of the GROMACS software. Diffusion coefficients are calculated from time– and ensemble–mean squared displacement of the chosen molecular species by restarting the MSD computation at even time intervals across the trajectory. The linear part of the MSD curves, (i.e., the MSD-t plot) was fitted to determine D_α. The α parameter, which characterizes the diffusion mechanism [24], was obtained from a non-linear fit to the general form of the equation for the MSD by assuming that the diffusion of the lipids progresses in two dimensions:

Normal diffusion corresponds to a linear time dependence and α = 1, while anomalous diffusion resembles non-linear dependence of the MSD vs. time. Values of α between 0 and 1 characterize sub-diffusion and above 1, super-diffusion. Typically, the motion of lipids in membranes is considered as sub-diffusive or normal with transient sub-diffusion [25].

The resulting MSD curve exhibits good averaging at short time intervals while becoming gradually worse at longer simulation times. Here, the full trajectories were used by restarting the MSD calculation every 50 ps. For the lipids movement, the MSD, W(t) = <(Δr(t))²> = <(r(t + t₀) − r(t₀))²>, is typically used to calculate the fractional self-diffusion coefficient:

where n_f represents the number of the translational degrees of freedom, the r(t) is the position of the centre of mass (COM) of the simulated molecules at time t, Δr(t) is the deviation of the r(t₀) and r(t + t₀) position vectors of the simulated molecules at time t₀ and t + t₀ respectively, and the brackets denote ensemble average (i.e., mean squared deviation of the position vectors at time t₀). Although biological systems are characterized by time length scales far beyond current atomistic simulations (t tends to infinity in Equation (2)), time remains an elusive parameter in MD simulations due to the finite computational resources. In order to test whether a simulation is long enough, we can test if the molecules explore a sufficiently diverse region of configuration space by calculating the MSD and comparing the outcome to the simulation box length (see Computational Methods). As permeation is a slow process requiring several nanoseconds, we followed standard simulation procedures to obtain statistically significant information such as to either drag the molecules into the membrane or to place them within the membrane to start with. These processes ensured sampling the membrane–penetrant interactions across the whole simulation space during the given simulation time scale.