Effects of passive phospholipid flip-flop and asymmetric external fields on bilayer phase equilibria

Compositional asymmetry between the leaflets of bilayer membranes modifies their phase behaviour, and is thought to influence other important features such as mechanical properties and protein activity. We address here how phase behaviour is affected by passive phospholipid \textit{flip-flop}, such that the compositional asymmetry is not fixed. We predict transitions from "pre flip-flop" behaviour to a restricted set of phase equilibria that can persist in the presence of passive flip-flop. Surprisingly, such states are not necessarily symmetric. We further account for external symmetry-breaking, such as a preferential substrate interaction, and show how this can stabilise strongly asymmetric equilibrium states. Our theory explains several experimental observations of flip-flop mediated changes in phase behaviour, and shows how domain formation and compositional asymmetry can be controlled in concert, by manipulating passive flip-flop rates and applying external fields.