Quadrupolar ordering of phospholipid molecules in narrow necks of phospholipid vesicles

Shapes of phospholipid vesicles that involve narrow neck(s) were studied theoretically. It is taken into account that phospholipid molecules are intrinsically anisotropic with respect to the membrane normal and that they exhibit quadrupolar orientational ordering according to the difference between the local principal membrane curvatures. Direct interactions between oriented molecules were considered within a linear approximation of the energy coupling with the deviatoric field. The equilibrium shapes of axisymmetric closed vesicles were studied by minimization of the free energy of the phospholipid bilayer membrane under relevant geometrical constraints. The variational problem was stated by a system of Euler-Lagrange differential equations that revealed a singularity in the derivative of the meridian curvature at points where the effect of the orientational ordering exactly counterbalances the effect of the isotropic bending. The system of Euler-Lagrange differential equations was solved numerically to yield consistently related equilibrium orientational distribution of the phospholipid molecules and vesicle shape. According to our estimation of the model constants the formation of the neck is promoted if direct interactions between the oriented molecules are taken into account. It was shown that the energy of the equilibrium shapes is considerably affected by the quadrupolar ordering of phospholipid molecules.