BPS2026 – Long-wavelength membrane undulation in all-atom detail

The Helfrich Hamiltonian describes the energetics of lipid membrane mechanics, predicting that the long-wavelength height spectrum ⟨|h(q)| 2 ⟩ scales like q −4 , with an amplitude that is inversely proportional to the bending modulus k c . In principle, this offers a direct way to obtain the bending modulus from a molecular dynamics simulation. In practice, very large system sizes (ca. 10,000 lipids) are needed to observe the long-wavelength regime. Several authors have therefore suggested methods that obtain the bending modulus from smaller systems by analysis of fluctuations in molecular tilt or by analysis of spontaneously sampled differences in local curvature. However, we find some disagreement among these methods when obtaining the bending modulus from the same simulation data. We address this with an all-atom molecular dynamics simulation of a single-component DOPC bilayer of size 60 × 60 nm 2 containing 10,330 lipids. This system pushes the accessible wavevector range down to q min ≈ 2π/L ≈ 0.10 nm −1 , enabling direct analysis of the long-wavelength regime. We compute the height spectrum ⟨|h(q)| 2 ⟩ and fit (1) the Helfrich asymptote via the q 4 ⟨|h(q)| 2 ⟩ plateau and (2) the curvature-tilt model of Terzi and Deserno via nonlinear regression over the intermediate-q window where the tilt contributions matter. Fits for the asymptotic regime (flat fit) and the nonlinear curvature-tilt fit were rigorously estimated (using goodness-of-fit criteria, AIC criteria, and χ 2 ). Uncertainties were quantified using Flyvbjerg-Petersen blocking. These two methods yield statistically indistinguishable estimations of the bending modulus k c . To our knowledge, this is the first all-atom simulation that simultaneously (1) reaches the Helfrich long-wavelength asymptote directly and (2) verifies consistency with the curvature-tilt theory.