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

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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.