Determination of electron density profiles and area from simulations of undulating membranes

Abstract

The traditional method for extracting electron density and other transmembrane profiles from molecular dynamics simulations of lipid bilayers fails for large bilayer systems, because it assumes a flat reference surface that does not take into account long wavelength undulations. We have developed what we believe to be a novel set of methods to characterize these undulations and extract the underlying profiles in the large systems. Our approach first obtains an undulation reference surface for each frame in the simulation and subsequently isolates the long-wavelength undulations by filtering out the intrinsic short wavelength modes. We then describe two methods to obtain the appropriate profiles from the undulating reference surface. Most combinations of methods give similar results for the electron density profiles of our simulations of 1024 DMPC lipids. From simulations of smaller systems, we also characterize the finite size effect related to the boundary conditions of the simulation box. In addition, we have developed a set of methods that use the undulation reference surface to determine the true area per lipid which, due to undulations, is larger than the projected area commonly reported from simulations.

Figure 1

Snapshots of DMPC bilayers with (A) 64 lipids and (B) 1024 lipids. Water is not displayed, but phosphorus atoms are shown (enlarged spheres, emphasizing the top and bottom monolayers). (Dashed lines) Global bilayer reference planes, defined at the z position of the bilayer center of mass. (Solid lines) Local bilayer reference surfaces. Local bilayer normals (solid arrows) and global bilayer normals (dashed arrows) emphasize that the local orientation of the bilayer normals varies for the larger system size in panel B. (C) EDP and (D) form factors, F(q), for 64, 128, 256, and 1024 lipids, all obtained by the artifact-prone z-bin method.

Figure 2

(A) Log-log plot of the one-dimensional unfiltered spectrum S_u(q) = N〈|u(q)|²〉 determined using the DF and RI methods for selected atoms P and TC (terminal carbons) versus the magnitude of the wave vector q, for the last 10 ns from the μs-long simulation of the 1024-lipid DMPC system. (B) L4 filtered spectra. Undulations theoretically follow the dashed q⁻⁴-lines.

Figure 3

Single frame URS for 1024 DMPC lipids, selecting the TC atoms and using q₀ = 1.15 nm⁻¹. (A and B) Using the DF method. (C and D) Using the RI method. (A and C) Using the L4 filter. (B and D) Using the ID filter.

Figure 4

Definitions for surface referencing. (Upper circle) The k^th atom [z(r_k)]. (Lower circle) Reference position on the URS ($\tilde{u}\left(\mathbf{r}\right)$, undulating solid line) with normal vector $\widehat{\mathbf{n}}$ (solid arrow) that deviates from the z axis by angle θ. The OA method assigns the distance from the URS to be z_OA (dashed-dotted arrow).

Figure 5

Comparisons of EDP for 1024 DMPC lipids upon variation of URS methods (DF versus RI) and surface referencing methods (OA versus UC) using q₀ = 1.15 nm⁻¹ and not varying TC:ID. (Black line) From z-bin 1024.

Figure 6

RMSD measuring the difference between EDP determined for a range of q₀ values compared to the EDP obtained at q₀ = 1.15 nm⁻¹ for the methods indicated in the legend.

Figure 7

Comparison of DF:ID:TC:UC results for 32, 64, 128, 256, and 1024-lipid systems with q₀ = 1.15 nm⁻¹. (Inset) Subtle differences near the maximum of the EDP, highlighting finite size effects for the smaller systems.