“…There is extensive literature on how particles—nearby, attached, or inserted into a membrane—affect membrane shape. Examples include the effects of different types of lipids [21–30] polymers [31–35] nanoparticles [36–40] phospholipid-surfactant mixtures [41], protein pumps [20, 42], adsorbed BAR proteins [43–48] inter-calated curvature-inducing proteins [49–51] and crowds of sterically-repelling proteins [52–56] (see also reviews in [57–59] Generally speaking, the total free energy of such systems contains, besides the Helfrich free energy, the free energy of the particles and a term accounting for the interaction of the membrane with the particles. Expressions for the free energy of the particles often contain terms penalising phase boundaries [22, 41, 43, 49, 60] and terms accounting for interactions between the particles and for entropy, either through a Flory-Huggins theory for a mixture of occupied and unoccupied sites [21, 27, 33, 41, 43], or a Ginzburg-Landau expansion thereof [22, 49, 61].…”