Algorithmic Improvements for Extracting Lipid Bilayer Elastic Moduli from Height and Director Fluctuations

Many important cellular processes take place at membranes, for example signaling and sensing, energy conversion, and metabolism. In order to understand how the physical properties of cell membranes enable these processes, we focus on bilayer shape changes. Shape changes in biomembranes arise in many circumstances, such as endocytosis, along the fusion pathway, and when accommodating transmembrane proteins. When they occur at a scale not much larger than the membrane thickness, standard curvature elasticity needs to be amended by contributions coming from lipid tilt. The energetics of lipid bilayer deformations across different scales can be described by continuum theories that contain a set of elastic moduli. In order to get information about the associated elastic moduli, we use atomistic model simulations of 13 different lipid membranes and analyze the power spectra of their bilayer shape and lipid director fluctuations. We rely on trajectories by Venable et al. [CPL 192, 60 (2015)], who have recently conducted such an analysis.

The energetics of lipid membrane deformations can be described by continuum theories for idealized lields representing lipid positions and orientations. In computer simulations, these moduli can be determined by fitting power spectra of fluctuating fields to the predictions of those continuum theories. This data analysis involves numerous choices, such as how to define a membrane surface or how to determine the fitting range, which significantly affect the observables of interest. We discuss strategies that may lead to objective choices irrespective of the theory. We in particular discuss systematic effects connected with: (1) interpolation of height and directional fields; (2) normalization and averaging of lipid directors; (3) ignoring extremely tilted lipid molecules. Overall, the systematic shifts in the moduli arising from equally plausible choices are often larger than the statistical uncertainties given that current computation technology gives increasingly precise statistical data. We propose a tentative set of criteria based on which the relative merits of such choices could be evaluated.

The data analysis process involves an expansion in Fourier modes, which is typically done by interpolating the continuum fields on a grid and performing a finite Fourier transform. Unfortunately this discretization creates artifacts, such as spectral damping and aliasing. Here we revisit an alternative, proposed by Kopelevich and Nagle [JCP 143, 154702 (2015)] but not in wide-spread use, which calculates the Fourier amplitudes by least-squares fitting of the modes to the off-lattice molecular positions and orientations. We compare these spectra to those derived via a conventional grid-based method, with and without corrections for spectral damping and aliasing. Our analysis suggests that the least-squares analysis has multiple advantages compared to the traditional method and is therefore a promising alternative for calculating power spectra. Additionally, we discuss other statistical aspects such as correcting for time correlations in the power spectra, determining small-scale cutoffs, getting uncertainties on the parameters, and simultaneously fitting different spectra.

Lastly, we visited the extensive derivation of height-height correlation function of a membrane stack discussed in Ref. [40]. More specifically, the effect of a curvature-tilt coupling term on the correlation function is investigated. The correlation function is key to determine the elastic moduli from X-ray diffuse scattering analysis of a bilayer stack. The preliminary results show that the new term has a negligible effect on the large-scale moduli (bending and bulk), while the tilt modulus is rather sensitive to the coupling.