Membrane Undulations

Helfrich bending elasticity describes membrane thermal fluctuations. The free energy F = ∫(κ/2)(2H)² dA where H is mean curvature and κ is the bending modulus (~20 kBT for DPPC).

Bending κ (kBT)20

Temperature (K)310

Tension σ5×10⁻³ mN/m

Modes12

Helfrich: F = ∫ [κ/2 · (2H)² + σ] dA | Mode amplitude: ⟨|h_q|²⟩ = kBT / [κq⁴ + σq²] · L²