Stochastic process description of lipid flip-flop

Since lipid bilayers are self-assembled macroscopic aggregates, their constituent lipid molecules can spontaneously transition between the two leaflets. This so-called "flip-flop" is almost universally described via first-order chemical kinetics: the net "flux" leaving a given leaflet is proportional to the number of lipids it contains. However, this model ignores interactions, such as those arising from packing or non-ideal mixing, and restricting the analysis to macroscopic rate equations misses fluctuations. Here, we employ tools from the field of stochastic processes to examine the impact of stress and non-ideal mixing on lipid flip-flop, and we discuss several methods for quantifying the associated fluctuations-ranging from stochastic trajectories to evolution equations for probability densities. We show that differential stress strongly enhances the rate at which lipid abundance asymmetry decays, while compositional relaxation in mixed systems can be closer to ideal under suitable conditions. For the case of binary systems in the presence of packing constraints, we employ a linear noise approximation to the system's master equation and show that it leads to an easily manageable Ornstein-Uhlenbeck process for the fluctuations of (and correlations between) compositions. We also show how to include non-ideal mixing, which leads to large and very slow compositional fluctuations as we approach the critical point.