Computing the lateral diffusion coefficient of lipids on lipid membranes

In biological membranes, lateral diffusion refers to the movement of constituent molecules, such as lipids and proteins, within each leaflet of the bilayer. Understanding this process is crucial for elucidating membrane dynamics, molecular interactions, and the physical basis of cellular transport. The present study aims to compute the lateral diffusion coefficient (D) of lipid molecules subjected to a harmonic potential within a model lipid membrane. The stochastic dynamics of individual lipid molecules arise from both random thermal collisions with the surrounding medium and deterministic forces characterized by a potential energy function U(r). The system was modeled using the overdamped form of the Langevin equation, appropriate for regimes with large frictional coefficients and small diffusion constants. Numerical simulations were performed using the Euler-Maruyama integration scheme, implemented in Python. The mean square displacement (MSD) of lipid molecules was computed from the simulated trajectories, from which the diffusion coefficient Dwas extracted. The calculated value, D = 1.28 x 10^-12 m^2/s, indicates a diffusion process consistent with thermally driven motion in a viscous membrane environment. The linear relationship observed between MSD and lag time confirms that membrane components exhibit unrestricted lateral mobility in the absence of strong intermolecular constraints or structural barriers. These results contribute to a quantitative understanding of lipid dynamics under harmonic confinement and demonstrate the applicability of stochastic numerical methods for modeling molecular transport processes in complex biological membranes. Keywords: Lateral diffusion; Lipid membranes; Mean square displacement; Langevin dynamics