A series representation of the nonlinear equation for axisymmetrical fluid membrane shape

Abstract

Whatever the fluid lipid vesicle is modeled as the spontaneous-curvature, bilayer-coupling, or the area-difference elasticity, and no matter whether a pulling axial force applied at the vesicle poles or not, a universal shape equation presents when the shape has both axisymmetry and up-down symmetry. This equation is a second order nonlinear ordinary differential equation about the sine $sinψ(r)$ of the angle $ψ(r)$ between the tangent of the contour and the radial axis $r$. However, analytically there is not a generally applicable method to solve it, while numerically the angle $ψ(0)$ can not be obtained unless by tricky extrapolation for $r=0$ is a singular point of the equation. We report an infinite series representation of the equation, in which the known solutions are some special cases, and a new family of shapes related to the membrane microtubule formation, in which $sinψ(0)$ takes values from 0 to $π/2$, is given.