Membrane curvature induced by proximity of anionic phospholipids can initiate endocytosis

The plasma membrane is uniquely enriched in phosphatidylserine (PtdSer). This anionic phospholipid is restricted almost exclusively to the inner leaflet of the plasmalemma. Because of their high density, the headgroups of anionic lipids experience electrostatic repulsion that, being exerted asymmetrically, is predicted to favor membrane curvature. We demonstrate that cholesterol limits this repulsion and tendency to curve. Removal of cholesterol or insertion of excess PtdSer increases the charge density of the inner leaflet, generating foci of enhanced charge and curvature where endophilin and synaptojanin are recruited. From these sites emerge tubules that undergo fragmentation, resulting in marked endocytosis of PtdSer. Shielding or reduction of the surface charge or imposition of outward membrane tension minimized invagination and PtdSer endocytosis. We propose that cholesterol associates with PtdSer to form nanodomains where the headgroups of PtdSer are maintained sufficiently separated to limit spontaneous curvature while sheltering the hydrophobic sterol from the aqueous medium.


Description of the system
We consider a lipid bilayer consisting of two monolayers. Each monolayer has an overall surface charge, which comes from all charged lipids: PtdSer, PtdIns(4,5)P2 etc. The charge densities (charge per unit area) in the outer and inner monolayers will be denoted by !"# and !" , respectively.
Each monolayer resists bending deformation quantified by the total curvature, , (the sum of two principle curvatures) of the monolayer surface.
The bending energy per unit area of the monolayer surface, ! , is determined by Helfrich model as The aqueous solution bathing the bilayer is be characterized by the dielectric constant, , and t the concentration, ! , of 1:1 electrolyte (ionic strength). The characteristic length of charge screening within the solution referred to as Debay length, ! , is given by where ! is the dielectric constant of vacuum, ! is the product of Boltzmann constant and the absolute temperature, and is the elementary charge. For the physiological ionic strength ! = 100mM, the Debay length is ! =1nm.

Spontaneous curvature of monolayer
The electrostatic free energy of each monolayer, related to its charge density, , results in a monolayer tendency to bend such that the monolayer surface adopts a convex shape (Scheme 1).

Scheme 1 Spontaneous monolayer curvature
The curvature of this favorable monolayer shape is referred to as the spontaneous curvature, ! . We define the curvature as illustrated in Scheme 1 to be positive. The previous work suggests that, for moderate surface charge densities, the electrostatically induced spontaneous curvature of a monolayer is given by where ~0.5 is the distance between the level of the glycerol backbone within the monolayer representing the monolayer neutral surface and the effective plane of the headgroup electric charges.
For the following, it is convenient to express the monolayer spontaneous curvature through the fraction, , of charged lipids within the monolayer, where is the area per lipid in the monolayer plane and it is assumed that a charged lipid polar head carries lipids carries one elementary charge, .

Spontaneous curvature of a bilayer
The spontaneous curvature of a bilayer, ! , is determined by the spontaneous curvatures of its inner, ! !" , and outer, ! !"# , monolayer. In the approximation of moderate curvatures it can be presented as Scheme 2 Spontaneous bilayer curvature Using Eqs.(5), (7), and we obtain for the bilayer spontaneous curvature or the radius of a preferable bilayer cylinder

Conclusion
Eqs.8 and 9 represent the major result of the computation, which can be used to estimate the membrane spontaneous curvature depending on the fraction of charged lipids in the inner and outer monolayer. Once we have numbers for !" and !"# before and after the cholesterol extraction, we can estimate the corresponding spontaneous curvatures.
For the moment, let us make a simplified estimation for the fraction of charged lipids in the inner monolayer, !" , needed to generate tubules of 25nm radius (50nm diameter). Assume, for simplicity, that the outer monolayer is absolutely neutral, !"# = 0. We get that the required !" =0.3.