Determining Structural and Mechanical Properties from Molecular Dynamics Simulations of Lipid Vesicles

Abstract

We have developed an algorithm to determine membrane structure, area per lipid, and bending rigidity from molecular dynamics simulations of lipid vesicles. Current methods to extract structure from vesicle simulations define densities relative to the global center of mass of the vesicle. This approach ignores the long-wavelength fluctuations (undulations) that develop across the sphere and broaden the underlying structure. Our method establishes a local reference frame by defining a radially undulating reference surface (URS) and thereby removes the broadening effect of the undulations. Using an arc-length low-pass filter, we render the URS by defining the bilayer midplane on an equi-angular θ, ϕ-grid (colatitude, longitude). This surface is then expanded onto a truncated series of spherical harmonics. The spherical harmonic coefficients characterize the long-wavelength fluctuations that define both the local reference frame-used to determine the bilayer's structure-and the area per lipid (A_L) along the undulating surface. Additionally, the resulting power spectrum of spherical harmonic coefficients can be fit to a Helfrich continuum model for membrane bending in spherical geometry to extract bending rigidity (k_c). k_c values determined for both DMPC and DMPC + cholesterol (30 mol %) vesicles are consistent with values from corresponding flat-patch systems determined using an independent, previously published spectral method. These new tools to accurately extract structure, A_L, and k_c should prove invaluable in evaluating the construction and equilibration of lipid vesicle simulations.

Figure 1

A. Snapshot of a 34 nm DMPC-lipid vesicle. B. Cutaway of the vesicle. C. An arc-length filter is used to define a surface for both inner and outer monolayers (blue and red selection beads respectively). D. r_und(θ, ϕ) is determined as the average of the inner and outer monolayers (color-map indicates fluctuations about the average radius, r₀, (units in nm).

Figure 2

SPHA decomposes r_und(θ, ϕ) into a series of standing waves on a sphere with degree l and order m.

Figure 3

Undulation spectra for a range of arc-length cutoff wavenumbers, (q_cut = 0.5 nm^–1: blue; 1.5 nm^–1: green; 2.0 nm^–1: red; and 2.5 nm^–1: cyan) for the DMPC vesicle system. The crossover wavenumber q₀ = 1.5 nm^–1 is denoted by the black dashed line. k_c values for each q_cut are detailed in the legend.

Figure 4

Comparisons of component number densities for DMPC vesicle and flat-patch systems. Vesicle profiles are dashed-lines. Flat-patch profiles are filled-distributions with headgroup (black/gray), carbonyl-glycerol (red/pink), and acyl-chain (blue/cyan). (A), (B) ρ_rbin for the high-water and low-water DMPC system, respectively. (C), (D) ρ_uc for high-water and low-water DMPC system. The same flat-patch profile has been included in all panels.

Figure 5

(A) A_L trajectory for the high-water DMPC vesicle system with total vesicle (blue), inner monolayer (green), outer monolayer (red), and flat-patch system (cyan). (B) A_L trajectory for the low-water DMPC system.

Figure 6

Power spectra for both vesicle systems (A) with corresponding k_c fit for DMPC (B) and DMPC + cholesterol (C). Undulation power spectrum and k_c-fits for flat-patch systems. For all panels, DMPC (black) and DMPC + cholesterol (red).