 Solution

A lipid bilayer is fluid within its plane, so a bare equilibrated bilayer has no static in-plane shear modulus. Stretching its area is nevertheless expensive, whereas bending is much softer. The Helfrich membrane energy expresses this distinction throughwith membrane bending rigidity $\kappa$, mean curvature $H$, Gaussian curvature $K$, spontaneous curvature $C_0$ and membrane tension $\sigma$. Area dilation additionally costs approximately $K_A(\Delta A{)}^2/(2A_0)$. For a closed surface of fixed topology, the Gauss-Bonnet theorem makes the integrated Gaussian-curvature term constant.

These energies control vesicle shapes, budding and the ease of forming membrane tubes. Asymmetric leaflet composition or bound proteins can change the preferred curvature. Real cell membranes can also be coupled to a cytoskeleton, which supplies shear elasticity absent from a free lipid bilayer. Thus a red-cell shape cannot generally be inferred from the bare bilayer model alone.

Two useful measurements probe different terms. Optical tracking of thermal membrane flickering measures fluctuation amplitudes; for small height modes, a normalization with $h_q=A^{-1}\int h{e}^{-i\mathbf{q}\cdot \mathbf{x}}{d}^{2}x$ gives $⟨|h_q{|}^2⟩=k_{B}T/[A(\kappa q^4+\sigma q^2)]$. Micropipette aspiration controls pressure while measuring the membrane drawn into a narrow pipette, linking apparent area change to tension and, at larger strain, area dilation. The observables test the elastic coefficients rather than merely showing a membrane image.