The intermediate scattering function for lipid bilayer membranes: from nanometers to microns

Curvature fluctuations of fluid vesicles reveal hydrodynamic dissipation within the bilayer

The biological function of membranes is closely related to their softness, which is often studied through the membranes' thermally driven fluctuations. Typically, the analysis assumes that the relaxation rate of a pure bending deformation is determined by the competition between membrane bending rigidity and viscous dissipation in the surrounding medium. Here, we reexamine this assumption and demonstrate that viscous flows within the membrane dominate the dynamics of bending fluctuations of nonplanar membranes with a radius of curvature smaller than the Saffman-Delbrück length. Using flickering spectroscopy of giant vesicles made of dipalmitoylphosphatidylcholine, DPPC:cholesterol mixtures and pure diblock-copolymer membranes, we experimentally detect the signature of membrane dissipation in curvature fluctuations. We show that membrane viscosity can be reliably obtained from the short time behavior of the shape time correlations. The results indicate that the DPPC:cholesterol membranes behave as a Newtonian fluid, while the polymer membranes exhibit more complex rheology. Our study provides physical insights into the time scales of curvature remodeling of biological and synthetic membranes.

Computing the influence of lipids and lipid complexes on membrane mechanics

Methodology for extracting the spontaneous curvature, bending modulus, and neutral surface of a lipid bilayer is described. The "SPEX" method is a robust technique for computing the bilayer bending modulus while allowing for resolution of the spontaneous curvature of specific interacting lipids and complexes, and the dependence of spontaneous curvature on wavelength. The method is described referring to the publicly available MembraneAnalysis.jl software package.

Dynamic structure factor of undulating vesicles: finite-size and spherical geometry effects with application to neutron spin echo experiments

We consider the dynamic structure factor (DSF) of quasi-spherical vesicles and present a generalization of an expression that was originally formulated by Zilman and Granek (ZG) for scattering from isotropically oriented quasi-flat membrane plaquettes. The expression is obtained in the form of a multi-dimensional integral over the undulating membrane surface. The new expression reduces to the original stretched exponential form in the limit of sufficiently large vesicles, i.e., in the micron range or larger. For much smaller unilamellar vesicles, deviations from the asymptotic, stretched exponential equation are noticeable even if one assumes that the Seifert-Langer leaflet density mode is completely relaxed and membrane viscosity is neglected. To avoid the need for an exhaustive numerical integration while fitting to neutron spin echo (NSE) data, we provide a useful approximation for polydisperse systems that tests well against the numerical integration of the complete expression. To validate the new expression, we performed NSE experiments on variable-size vesicles made of a POPC/POPS lipid mixture and demonstrate an advantage over the original stretched exponential form or other manipulations of the original ZG expression that have been deployed over the years to fit the NSE data. In particular, values of the membrane bending rigidity extracted from the NSE data using the new approximations were insensitive to the vesicle radii and scattering wavenumber and compared very well with expected values of the effective bending modulus ([Formula: see text]) calculated from results in the literature. Moreover, the generalized scattering theory presented here for an undulating quasi-spherical shell can be easily extended to other models for the membrane undulation dynamics beyond the Helfrich Hamiltonian and thereby provides the foundation for the study of the nanoscale dynamics in more complex and biologically relevant model membrane systems.

Physical limits to membrane curvature sensing by a single protein

Membrane curvature sensing is essential for a diverse range of biological processes. Recent experiments have revealed that a single nanometer-sized septin protein has different binding rates to membrane-coated glass beads of 1-µm and 3-µm diameters, even though the septin is orders of magnitude smaller than the beads. This sensing ability is especially surprising since curvature-sensing proteins must deal with persistent thermal fluctuations of the membrane, leading to discrepancies between the bead's curvature and the local membrane curvature sensed instantaneously by a protein. Using continuum models of fluctuating membranes, we investigate whether it is feasible for a protein acting as a perfect observer of the membrane to sense micron-scale curvature either by measuring local membrane curvature or by using bilayer lipid densities as a proxy. To do this, we develop algorithms to simulate lipid density and membrane shape fluctuations. We derive physical limits to the sensing efficacy of a protein in terms of protein size, membrane thickness, membrane bending modulus, membrane-substrate adhesion strength, and bead size. To explain the experimental protein-bead association rates, we develop two classes of predictive models: (i) for proteins that maximally associate to a preferred curvature and (ii) for proteins with enhanced association rates above a threshold curvature. We find that the experimentally observed sensing efficacy is close to the theoretical sensing limits imposed on a septin-sized protein. Protein-membrane association rates may depend on the curvature of the bead, but the strength of this dependence is limited by the fluctuations in membrane height and density.

Membrane free-energy landscapes derived from atomistic dynamics explain nonuniversal cholesterol-induced stiffening

All lipid membranes have inherent morphological preferences and resist deformation. Yet adaptations in membrane shape can and do occur at multiple length scales. While this plasticity is crucial for cellular physiology, the factors controlling the morphological energetics of lipid bilayers and the dominant mechanisms of membrane remodeling remain to be fully understood. An ongoing debate regarding the universality of the stiffening effect of cholesterol underscores the challenges facing this field, both experimentally and theoretically, even for simple lipid mixtures. On the computational side, we have argued that enhanced-sampling all-atom molecular dynamics simulations are uniquely suited for the quantification of membrane conformational energetics, as they minimize a priori assumptions and permit analysis of bilayers in deformed states. To showcase this approach, we examine reported inconsistencies between alternative experimental measurements of bending moduli for cholesterol-enriched membranes. Specifically, we analyze lipid bilayers with different chain saturation and compute free-energy landscapes for curvature deformations distributed over areas from ∼5 to ∼60 nm2 . These enhanced simulations, totaling over 100 μs of sampling time, enable us to directly quantify both bending and tilt moduli and to dissect the contributing factors and molecular mechanisms of curvature generation at each length scale. Our results show that the effects of cholesterol on bending rigidity are lipid-specific and suggest that this specificity arises from differences in the torsional dynamics of the acyl chains. In summary, we demonstrate that quantitative relationships can now be established between lipid structure and bending energetics, paving the way for addressing open fundamental questions in cell membrane mechanics.

Neutron scattering studies on dynamics of lipid membranes

Neutron scattering methods are powerful tools for the study of the structure and dynamics of lipid bilayers in length scales from sub Å to tens to hundreds nm and the time scales from sub ps to μs. These techniques also are nondestructive and, perhaps most importantly, require no additives to label samples. Because the neutron scattering intensities are very different for hydrogen- and deuterium-containing molecules, one can replace the hydrogen atoms in a molecule with deuterium to prepare on demand neutron scattering contrast without significantly altering the physical properties of the samples. Moreover, recent advances in neutron scattering techniques, membrane dynamics theories, analysis tools, and sample preparation technologies allow researchers to study various aspects of lipid bilayer dynamics. In this review, we focus on the dynamics of individual lipids and collective membrane dynamics as well as the dynamics of hydration water.

Membrane free-energy landscapes derived from atomistic dynamics explain nonuniversal cholesterol-induced stiffening

All lipid membranes have inherent morphological preferences and resist deformation. Yet adaptations in membrane shape can and do occur at multiple length scales. While this plasticity is crucial for cellular physiology, the factors controlling the morphological energetics of lipid bilayers and the dominant mechanisms of membrane remodeling remain unclear. An ongoing debate regarding the universality of the stiffening effect of cholesterol underscores the challenges facing this field, both experimentally and theoretically, even for simple lipid mixtures. On the computational side, we have argued that enhanced- sampling all-atom molecular dynamics simulations are uniquely suited for quantification of membrane conformational energetics, not only because they minimize a-priori assumptions, but also because they permit analysis of bilayers in deformed states. To showcase this approach, we examine reported inconsistencies between alternative experimental measurements of bending moduli for cholesterol-enriched membranes. Specifically, we analyze lipid bilayers with different chain saturation, and compute free-energy landscapes for curvature deformations distributed over areas from ∼5 to ∼60 nm 2 . These enhanced simulations, totaling over 100 microseconds of sampling time, enable us to directly quantify both bending and tilt moduli, and to dissect the contributing factors and molecular mechanisms of curvature generation at each length scale. Our results show that cholesterol effects are lipid-specific, in agreement with giantvesicle measurements, and explain why experiments probing nanometer scale lipid dynamics diverge. In summary, we demonstrate that quantitative structure-mechanics relationships can now be established for heterogenous membranes, paving the way for addressing open fundamental questions in cell membrane mechanics.

Significance

Elucidating the energetics and mechanisms of membrane remodeling is an essential step towards understanding cell physiology. This problem is challenging, however, because membrane bending involves both large-scale and atomic-level dynamics, which are difficult to measure simultaneously. A recent controversy regarding the stiffening effect of cholesterol, which is ubiquitous in animal cells, illustrates this challenge. We show how enhanced molecular-dynamics simulations can bridge this length-scale gap and reconcile seemingly incongruent observations. This approach facilitates a conceptual connection between lipid chemistry and membrane mechanics, thereby providing a solid basis for future research on remodeling phenomena, such as in membrane trafficking or viral infection.

Nanoscale Bending Dynamics in Mixed-Chain Lipid Membranes

Lipids that have two tails of different lengths are found throughout biomembranes in nature, yet the effects of this asymmetry on the membrane properties are not well understood, especially when it comes to the membrane dynamics. Here we study the nanoscale bending fluctuations in model mixed-chain 14:0–18:0 PC (MSPC) and 18:0–14:0 PC (SMPC) lipid bilayers using neutron spin echo (NSE) spectroscopy. We find that despite the partial interdigitation that is known to persist in the fluid phase of these membranes, the collective fluctuations are enhanced on timescales of tens of nanoseconds, and the chain-asymmetric lipid bilayers are softer than an analogous chain-symmetric lipid bilayer with the same average number of carbons in the acyl tails, di-16:0 PC (DPPC). Quantitative comparison of the NSE results suggests that the enhanced bending fluctuations at the nanosecond timescales are consistent with experimental and computational studies that showed the compressibility moduli of chain-asymmetric lipid membranes are 20% to 40% lower than chain-symmetric lipid membranes. These studies add to growing evidence that the partial interdigitation in mixed-chain lipid membranes is highly dynamic in the fluid phase and impacts membrane dynamic processes from the molecular to mesoscopic length scales without significantly changing the bilayer thickness or area per lipid.

The bending rigidity of the red blood cell cytoplasmic membrane

An important mechanical property of cells is the membrane bending modulus, κ. In the case of red blood cells (RBCs) there is a composite membrane consisting of a cytoplasmic membrane and an underlying spectrin network. Literature values of κ are puzzling, as they are reported over a wide range, from 5 kBT to 230 kBT. To disentangle the contribution of the cytoplasmic membrane from the spectrin network, we investigated the bending of red blood cell cytoplasmic membranes (RBCcm) in the absence of spectrin and adenosine triphosphate (ATP). We used a combination of X-ray diffuse scattering (XDS), neutron spin-echo (NSE) spectrometry and Molecular Dynamics (MD) simulations. Our results indicate values of κ of order 4 kBT to 6 kBT, relatively small compared to literature values for most single component lipid bilayers. We suggest two ways this relative softness might confer biological advantage.

Effect of encapsulated protein on the dynamics of lipid sponge phase: a neutron spin echo and molecular dynamics simulation study

Lipid membranes are highly mobile systems with hierarchical, time and length scale dependent, collective motions including thickness fluctuations, undulations, and topological membrane changes, which play an important role in membrane interactions. In this work we have characterised the effect of encapsulating two industrially important enzymes, β-galactosidase and aspartic protease, in lipid sponge phase nanoparticles on the dynamics of the lipid membrane using neutron spin echo (NSE) spectroscopy and molecular dynamics (MD) simulations. From NSE, reduced membrane dynamics were observed upon enzyme encapsulation, which were dependent on the enzyme concentration and type. By fitting the intermediate scattering functions (ISFs) with a modified Zilman and Granek model including nanoparticle diffusion, an increase in membrane bending rigidity was observed, with a larger effect for β-galactosidase than aspartic protease at the same concentration. MD simulations for the system with and without aspartic protease showed that the lipids relax more slowly in the system with protein due to the replacement of the lipid carbonyl-water hydrogen bonds with lipid-protein hydrogen bonds. This indicates that the most likely cause of the increase in membrane rigidity observed in the NSE measurements was dehydration of the lipid head groups. The dynamics of the protein itself were also studied, which showed a stable secondary structure of protein over the simulation, indicating no unfolding events occurred.

Measuring the bending modulus of lipid bilayers with cholesterol

The effect of cholesterol on the bending modulus of DOPC lipid bilayers has become a controversial topic that has implications for methods of measuring the bending modulus. Recent results using neutron spin echo and nuclear magnetic resonance relaxation methods that involve linear transport properties have conflicted with earlier results from purely equilibrium experiments that do not involve linear transport properties. A general discussion indicates how one can be misled by data obtained by methods that involve linear transport properties. It is then shown specifically how the recent neutron spin echo results can be interpreted to agree with the earlier purely equilibrium experimental results, thereby resolving that conflict. Regarding the nuclear magnetic resonance relaxation method, it is noted that current interpretation of the data is unclear regarding the identity of the modulus that is involved, and an alternative interpretation is explored that does not disagree with the results of the equilibrium experiments.

Phonon-mediated lipid raft formation in biological membranes

Short-wavelength collective molecular motions, also known as phonons, have recently attracted much interest in revealing dynamic properties of biological membranes through the use of neutron and X-ray scattering, infrared and Raman spectroscopies, and molecular dynamics simulations. Experimentally detecting unique vibrational patterns such as, shear phonon excitations, viscoelastic crossovers, transverse acoustic phonon gaps, and continuous and truncated optical phonon modes in cellular membranes, to name a few, has proven non-trivial. Here, we review recent advances in liquid thermodynamics that have resulted in the development of the phonon theory of liquids. The theory has important predictions regarding the shear vibrational spectra of fluids, namely the emergence of viscoelastic crossovers and transverse acoustic phonon gaps. Furthermore, we show that these vibrational patterns are common in soft (non-crystalline) materials, including, but not limited to liquids, colloids, liquid crystals (mesogens), block copolymers, and biological membranes. The existence of viscoelastic crossovers and acoustic phonon gaps define the self-diffusion properties of cellular membranes and provide a molecular picture of the transient nature of lipid rafts (Bolmatov et al., 2020). Importantly, the timescales (picoseconds) for the formation and dissolution of transient lipid rafts match the lifetime of the formation and breakdown of interfacial water hydrogen bonds. Apart from acoustic propagating phonon modes, biological membranes can also support more energetic non-propagating optical phonon excitations, also known as standing waves or breathing modes. Importantly, optical phonons can be truncated due to the existence of finite size nanodomains made up of strongly correlated lipid-cholesterol molecular pairs. These strongly coupled molecular pairs can serve as nucleation centers for the formation of stable rafts at larger length scales, due to correlations of spontaneous fluctuations (Onsager's regression hypothesis). Finally and importantly, molecular level viscoelastic crossovers, acoustic phonon gaps, and continuous and truncated optical phonon modes may offer insights as to how lipid-lipid and lipid-protein interactions enable biological function.

On the calculation of the potential of mean force between atomistic nanoparticles

We study the potential of mean force (PMF) between atomistic silica and gold nanoparticles in the vacuum by using molecular dynamics simulations. Such an investigation is devised in order to fully characterize the effective interactions between atomistic nanoparticles, a crucial step to describe the PMF in high-density coarse-grained polymer nanocomposites. In our study, we first investigate the behavior of silica nanoparticles, considering cases corresponding to different particle sizes and assessing results against an analytic theory developed by Hamaker for a system of Lennard-Jones interacting particles (H.C. Hamaker, Physica A 4, 1058 (1937)). Once validated the procedure, we calculate effective interactions between gold nanoparticles, which are considered both bare and coated with polyethylene chains, in order to investigate the effects of the grafting density [Formula: see text] on the PMF. Upon performing atomistic molecular dynamics simulations, it turns out that silica nanoparticles experience similar interactions regardless of the particle size, the most remarkable difference being a peak in the PMF due to surface interactions, clearly apparent for the larger size. As for bare gold nanoparticles, they are slightly interacting, the strength of the effective force increasing for the coated cases. The profile of the resulting PMF resembles a Lennard-Jones potential for intermediate [Formula: see text], becoming progressively more repulsive for high [Formula: see text] and low interparticle separations.

Bending elastic properties of a block copolymer-rich lamellar phase doped by a surfactant: a neutron spin-echo study

The influence of the short alkyl-chain ionic surfactant OTAB on the dynamic behavior of an inverse block copolymer-rich lamellar phase was investigated by neutron spin-echo spectroscopy (NSE). The observed intermediate scattering function can be described by a sum of two contributions. For high scattering vectors the model of Zilman-Granek plus a slow diffusional mode can be used to describe the experimental data and the bending elastic modulus κ for a polymer-rich membrane is calculated. At low scattering vectors the relaxation curves are strongly influenced by de Gennes narrowing arising from the structure factor of the Lα phase. Hence, the computed relaxation rates in this q-range are inversely proportional to the static structure factor. The present study demonstrates the necessity to directly investigate the dynamic behavior of lamellar phases and that an analysis of the width of the Bragg peaks can be insufficient to derive information about the single membrane elasticity, especially when both κ and B[combining macron] depend on the composition of the membrane.

Bending stiffness of biological membranes: what can be measured by neutron spin echo?

Large vesicles obtained by the extrusion method represent adequate membrane models to probe membrane dynamics with neutron radiation. Particularly, the shape fluctuations around the spherical average topology can be recorded by neutron spin echo (NSE). In this paper we report on the applicable theories describing the scattering contributions from bending-dominated shape fluctuations in diluted vesicle dispersions, with a focus on the relative relevance of the master translational mode with respect to the internal fluctuations. Different vesicle systems, including bilayer and non-bilayer membranes, have been scrutinized. We describe the practical ranges where the exact theory of bending fluctuations is applicable to obtain the values of the bending modulus from experiments, and we discuss about the possible internal modes that could be alternatively contributing to shape fluctuations.