Resting shape and spontaneous membrane curvature of red blood cells

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A boundary-value problem is formulated describing the biconcave resting shape of normal red blood cells, based on local constitutive equations for the membrane tensions and bending moments. The fundamental physical assumption is that curvature-dependent anisotropic membrane stress resultants accompanied by isotropic bending moments arise from isotropic tensions developing in each leaflet of the lipid bilayer, while the cytoskeleton is unstressed in the resting configuration. Families of equilibrium resting shapes parametrized by the spontaneous bilayer curvature and cell sphericity compare favourably with the average shape of normal red blood cells. The successful comparison supports Helfrich's notion of a non-zero spontaneous curvature whose magnitude is nearly equal to the negative of the equivalent cell radius defined with respect to the membrane surface area. The structure of the solution space suggests a minimum spontaneous curvature below which the cell sphericity is lower than that of the red blood cell, independent of the transmural pressure. The computed cell shapes also compare favourably with the shapes of swollen red blood cells, though for a different value of the spontaneous curvature. The dependence of the spontaneous curvature on the cell volume is attributed to in-plane elastic tensions developing due to the deformation of the cytoskeleton. An alternative formulation based on a non-local model for the monolayer tensions is found to be incapable of predicting non-spherical shapes.