Interpretation of fluctuation spectra in lipid bilayer simulations

Abstract

Atomic resolution and coarse-grained simulations of dimyristoylphosphatidylcholine lipid bilayers were analyzed for fluctuations perpendicular to the bilayer using a completely Fourier-based method. We find that the fluctuation spectrum of motions perpendicular to the bilayer can be decomposed into just two parts: 1), a pure undulation spectrum proportional to q(-4) that dominates in the small-q regime; and 2), a molecular density structure factor contribution that dominates in the large-q regime. There is no need for a term proportional to q(-2) that has been postulated for protrusion fluctuations and that appeared to have been necessary to fit the spectrum for intermediate q. We suggest that earlier reports of such a term were due to the artifact of binning and smoothing in real space before obtaining the Fourier spectrum. The observability of an intermediate protrusion regime from the fluctuation spectrum is discussed based on measured and calculated material constants.

Figure 1

Schematic drawing illustrating the monolayer surfaces and the definitions of the bilayer undulations, u(x,y), and the thickness, h(x,y).

Figure 2

Static number density structure factor, S_ρ(q), for the UA and CG systems (normalized as described in the text) shown versus q. At small q the structure factor reaches the limiting value, k_BT (〈u²〉 + 〈h²〉)/aK_A, given by the compressibility equation (Eq. 16). This value is indicated (dotted line) for K_A = 0.234 N/m. For large q, it approaches (〈u₂〉 + 〈h²〉)/2 (derived in Section S2 in the Supporting Material). (Inset) The corresponding radial distribution function g(r) (the inverse two-dimensional Fourier transform of 〈|ρ(q)|²〉). The value of g(r) approaches 1 at large distances in these units, which corresponds to the bulk number density. The function g(r) was calculated by inverse numerical Fourier transform of 〈|ρ(q)|²〉 and also by direct calculation in real space; the results were very similar.

Figure 3

Undulation spectra S_u(q) for the UA and CG systems versus wave vector q. Fits are shown to the small-q data of Eq. 3, with k_c = 7.5 × 10⁻²⁰ J for the UA system and 15 × 10⁻²⁰ J for the CG system. The number density structure factors S_ρ(q) are plotted to illustrate the agreement in the intermediate-q and large-q regimes.

Figure 4

(a) Data for the UA model. The full S_u(q) spectrum (squares) and our suggested undulation spectrum S_u(q) − S_ρ(q) (diamonds); the latter is fitted to the theoretical q⁻⁴ line (Eq. 3). Also shown to illustrate our nonpreferred spectrum is S_u(q) (circles) obtained by binning in real space, fitted to Eq. 4 with γ_p = 0.1 N/m. (b) The undulation spectra of the UA and CG models with density structure factor subtracted. The scale is linear for the spectra and logarithmic for q. (Solid lines) Fits to q⁻⁴ lines. (Dotted line) Effect of adding a protrusion tension.

Figure 5

Fluctuation spectra S_u(q) (squares) and S_u(q) − S_ρ(q) (diamonds) in the UA (top) and CG (bottom) models calculated with the direct Fourier method, compared to S_u(q) (circles) calculated with spline interpolation. The direct Fourier method follows a straight q⁻⁴ line (Eq. 3, dashed) whereas the interpolation method introduces an artificial smoothing that gives rise to a q⁻² broadening (Eq. 4, dotted) of the intermediate spectrum q values.