A combined molecular/continuum-modeling approach to predict the small-angle neutron scattering of curved membranes

Abstract

This paper develops a framework to compute the small-angle neutron scattering (SANS) from highly curved, dynamically fluctuating, and potentially inhomogeneous membranes. This method is needed to compute the scattering from nanometer-scale membrane domains that couple to curvature, as predicted by molecular modeling. The detailed neutron scattering length density of a small planar bilayer patch is readily available via molecular dynamics simulation. A mathematical, mechanical transformation of the planar scattering length density is developed to predict the scattering from curved bilayers. By simulating a fluctuating, curved, surface-continuum model, long time- and length-scales can be reached while, with the aid of the planar-to-curved transformation, the molecular features of the scattering length density can be retained. A test case for the method is developed by constructing a coarse-grained lipid vesicle following a protocol designed to relieve both the osmotic stress inside the vesicle and the lipid-number stress between the leaflets. A question was whether the hybrid model would be able to replicate the scattering from the highly deformed inner and outer leaflets of the small vesicle. Matching the scattering of the full (molecular vesicle) and hybrid (continuum vesicle) models indicated that the inner and outer leaflets of the full vesicle were expanded laterally, consistent with previous simulations of the Martini forcefield that showed thinning in small vesicles. The vesicle structure is inconsistent with a zero-tension leaflet deformed by a single set of elastic parameters, and the results show that this is evident in the scattering. The method can be applied to translate observations of any molecular model's neutron scattering length densities from small patches to large length and timescales.

Keywords: Curvature; Modulated phases; Neutron scattering; Vesicles.

Figure 1:

The POPC (with NC3) and POPE (with NH3) lipids used in the study. The interaction site names (e.g., C2B) are referred to in the text regarding the pivotal plane of bending.

Figure 2:

A rendering of a ca. 18 nm diameter POPC/POPE Martini vesicle the scattering from which that of the hybrid continuum/planar-molecular modeling is compared. To the left of the vertical dotted line the near lipids are hidden to show the vesicle interior. Tail interaction sites are light while surface sites are dark. The inner and outer leaflet lipids are tinted separately.

Figure 3:

An illustration of how a surface continuum model, combined with a scattering length density profile from molecular simulation, β_LA(z), is used to model the scattering centers from a dynamically fluctuating vesicle of a particular size. A frame of a simulation of a Martini vesicle is shown (water is removed from the image for clarity). In the inset, a portion of the lipids are removed and the continuum model is shown. The underlying approximately spherical continuum model is shown in dark grey while the deformed NSLD is shown for both the outer and inner leaflets. The shapes of the inner and outer leaflets are discontinuous because, in this model, the leaflets deform semi-independently (they meet at the dark grey continuum-modeled midplane). The inner leaflets of the Martini model and continuum-modeled density are both tinted red.

Figure 4:

The geometric area-per-lipid (4πR² divided by the number of lipids) of a Martini site in a ca. 18 nm diameter vesicle plotted against the same site’s height above the bilayer midplane in a planar simulation. The form is expected to be linear, as per Eq. 2. The area-per-lipid for the planar system, plotted as a horizontal line, is the projected area of the bilayer. The inner leaflet curve (red) plots both the NH3 sites of POPE as well as the NC3 sites of POPC. The outer leaflet curve (blue) has only POPC.

Figure 5:

The scattering computed directly from that of a ca. 13 nm diameter Martini vesicle (red) and that of a hybrid continuum/planar-molecular simulation of the same system (blue).

Figure 6:

The scattering computed directly from that of ca. 18 nm diameter Martini vesicle (red) and that of a hybrid continuum/planar-molecular simulation of the same system (blue).