Quantifying uncertainty in trans-membrane stresses and moments in simulation.

The lateral stress profile of a lipid bilayer constitutes a valuable link between molecular simulation and mesoscopic elastic theory. Even though it is frequently calculated in simulations, its statistical precision (or that of observables derived from it) is often left unspecified. This omission can be problematic, as uncertainties are prerequisite to assessing statistical significance. In this chapter, we provide a comprehensive yet accessible overview of the statistical error analysis for the lateral stress profile. We detail two relatively simple but powerful techniques for generating error bars: block-averaging and bootstrapping. Combining these methods allows us to reliably estimate uncertainties, even in the presence of both temporal and spatial correlations, which are ubiquitous in simulation data. We illustrate these techniques with simple examples like stress moments, but also more complex observables such as the location of stress profile extrema and the monolayer neutral surface.

Biomembranes balance many types of leaflet asymmetries.

Many biological membranes host different lipid species in their two leaflets. Since their spontaneous curvatures are typically not the same, this compositional asymmetry generally entails bending torques, which can be counteracted by differential stress-the difference between the two leaflet tensions. This stress, in turn, can affect elastic parameters or phase behavior of the membrane or each individual leaflet, or push easily flippable species, especially cholesterol, from the compressed leaflet into the tense leaflet. In short, breaking the symmetry of a single observable (to wit: composition), essentially breaks all other symmetries as well, with many potentially interesting consequences. This brief report examines the elastic aspects of this interplay, focusing on some elementary conditions of mechanical and thermodynamic equilibrium, but also shows how this poses novel questions that we are only beginning to appreciate.

Cell-Surface Binding of DNA Nanostructures for Enhanced Intracellular and Intranuclear Delivery.

DNA nanostructures (DNs) have found increasing use in biosensing, drug delivery, and therapeutics because of their customizable assembly, size and shape control, and facile functionalization. However, their limited cellular uptake and nuclear delivery have hindered their effectiveness in these applications. Here, we demonstrate the potential of applying cell-surface binding as a general strategy to enable rapid enhancement of intracellular and intranuclear delivery of DNs. By targeting the plasma membrane via cholesterol anchors or the cell-surface glycocalyx using click chemistry, we observe a significant 2 to 8-fold increase in the cellular uptake of three distinct types of DNs that include nanospheres, nanorods, and nanotiles, within a short time frame of half an hour. Several factors are found to play a critical role in modulating the uptake of DNs, including their geometries, the valency, positioning and spacing of binding moieties. Briefly, nanospheres are universally preferable for cell surface attachment and internalization. However, edge-decorated nanotiles compensate for their geometry deficiency and outperform nanospheres in both categories. In addition, we confirm the short-term structural stability of DNs by incubating them with cell medium and cell lysate. Further, we investigate the endocytic pathway of cell-surface bound DNs and reveal that it is an interdependent process involving multiple pathways, similar to those of unmodified DNs. Finally, we demonstrate that cell-surface attached DNs exhibit a substantial enhancement in the intranuclear delivery. Our findings present an application that leverages cell-surface binding to potentially overcome the limitations of low cellular uptake, which may strengthen and expand the toolbox for effective cellular and nuclear delivery of DNA nanostructure systems.

Asymmetric membrane "sticky tape" enables simultaneous relaxation of area and curvature in simulation.

Biological lipid membranes are generally asymmetric, not only with respect to the composition of the two membrane leaflets but also with respect to the state of mechanical stress on the two sides. Computer simulations of such asymmetric membranes pose unique challenges with respect to the choice of boundary conditions and ensemble in which such simulations are to be carried out. Here, we demonstrate an alternative to the usual choice of fully periodic boundary conditions: The membrane is only periodic in one direction, with free edges running parallel to the single direction of periodicity. In order to maintain bilayer asymmetry under these conditions, nanoscale "sticky tapes" are adhered to the membrane edges in order to prevent lipid flip-flop across the otherwise open edge. In such semi-periodic simulations, the bilayer is free to choose both its area and mean curvature, allowing for minimization of the bilayer elastic free energy. We implement these principles in a highly coarse-grained model and show how even the simplest examples of such simulations can reveal useful membrane elastic properties, such as the location of the monolayer neutral surface.

Elastic and thermodynamic consequences of lipid membrane asymmetry.

Many cellular lipid bilayers consist of leaflets that differ in their lipid composition - a non-equilibrium state actively maintained by cellular sorting processes that counter passive lipid flip-flop. While this lipidomic aspect of membrane asymmetry has been known for half a century, its elastic and thermodynamic ramifications have garnered attention only fairly recently. Notably, the torque arising when lipids of different spontaneous curvature reside in the two leaflets can be counterbalanced by a difference in lateral mechanical stress between them. Such membranes can be essentially flat in their relaxed state, despite being compositionally strongly asymmetric, but they harbor a surprisingly large but macroscopically invisible differential stress. This hidden stress can affect a wide range of other membrane properties, such as the resistance to bending, the nature of phase transitions in its leaflets, and the distribution of flippable species, most notably sterols. In this short note we offer a concise overview of our recently proposed basic framework for capturing the interplay between curvature, lateral stress, leaflet phase behavior, and cholesterol distribution in generally asymmetric membranes, and how its implied signatures might be used to learn more about the hidden but physically consequential differential stress.

Distribution of cholesterol in asymmetric membranes driven by composition and differential stress.

Many lipid membranes of eukaryotic cells are asymmetric, which means the two leaflets differ in at least one physical property, such as lipid composition or lateral stress. Maintaining this asymmetry is helped by the fact that ordinary phospholipids rarely transition between leaflets, but cholesterol is an exception: its flip-flop times are in the microsecond range, so that its distribution between leaflets is determined by a chemical equilibrium. In particular, preferential partitioning can draw cholesterol into a more saturated leaflet, and phospholipid number asymmetry can force it out of a compressed leaflet. Combining highly coarse-grained membrane simulations with theoretical modeling, we investigate how these two driving forces play against each other until cholesterol's chemical potential is equilibrated. The theory includes two coupled elastic sheets and a Flory-Huggins mixing free energy with a χ parameter. We obtain a relationship between χ and the interaction strength between cholesterol and lipids in either of the two leaflets, and we find that it depends, albeit weakly, on lipid number asymmetry. The differential stress measurements under various asymmetry conditions agree with our theoretical predictions. Using the two kinds of asymmetries in combination, we find that it is possible to counteract the phospholipid number bias, and the resultant stress in the membrane, via the control of cholesterol mixing in the leaflets.

Fluid-gel coexistence in lipid membranes under differential stress.

A widely conserved property of many biological lipid bilayers is their asymmetry. In addition to having distinct compositions on its two sides, a membrane can also exhibit different tensions in its two leaflets, a state known as differential stress. Here, we examine how this stress can influence the phase behavior of the constituent lipid monolayers of a single-component membrane. For temperatures sufficiently close to, but still above, the main transition, molecular dynamics simulations show the emergence of finite gel domains within the compressed leaflet. We describe the thermodynamics of this phenomenon by adding two empirical single-leaflet free energies for the fluid-gel transition, each evaluated at its respective asymmetry-dependent lipid density. Finite size effects arising in simulation are included in the theory through a geometry-dependent interfacial term. Our model reproduces the phase coexistence observed in simulation. It could therefore be used to connect the "hidden variable" of differential stress to experimentally observable properties of the main phase transition. These ideas could be generalized to any first-order bilayer phase transition in the presence of asymmetry, including liquid-ordered/liquid-disordered phase separation.

Identifying systematic errors in a power spectral analysis of simulated lipid membranes.

The elastic properties of lipid membranes can be measured by monitoring their thermal fluctuations. For instance, comparing the power spectra of membrane shape or lipid director fluctuations with predictions based on suitable continuum theories gives access to bending-, tilt-, and twist-moduli. However, to do so in a computer simulation, we must first define a continuum surface shape and lipid director field from the discrete configurations of lipid molecules in a typically fairly small box. Here, we show that the required mapping choices, as well as the details of the subsequent data analysis, can shift the measured values of these moduli by far more than their statistical uncertainties. We investigate the resulting systematic errors on the basis of atomistic simulation trajectories for 13 different lipids, previously published by Venable et al. [Chem. Phys. Lipids 192, 60-74 (2015)]. Specifically, we examine mapping choices for surface- and tilt-field definitions, normalizing and averaging lipid directors, accounting for wave vector dependent time autocorrelations, error propagation, and finding the right fitting range. We propose a set of criteria that may help to decide upon a particular combination of choices underlying the fluctuation analysis, and we make several recommendations based on these. While systematic shifts in observables that are extracted from large-wavelength limits vanish, in principle, for sufficiently large system size, no such exact limit exists for intrinsically local parameters, such as the twist modulus or the splay-tilt coupling, and so not all potential choices can be trivially verified.

Stiffening transition in asymmetric lipid bilayers: The role of highly ordered domains and the effect of temperature and size.

Cellular membranes consist of a large variety of lipids and proteins, with a composition that generally differs between the two leaflets of the same bilayer. One consequence of this asymmetry is thought to be the emergence of differential stress, i.e., a mismatch in the lateral tension of the two leaflets. This can affect a membrane's mechanical properties; for instance, it can increase the bending rigidity once the differential stress exceeds a critical threshold. Using coarse-grained molecular dynamics simulations based on the MARTINI model, we show that this effect arises due to the formation of more highly ordered domains in the compressed leaflet. The threshold asymmetry increases with temperature, indicating that the transition to a stiffened regime might be restricted to a limited temperature range above the gel transition. We also show that stiffening occurs more readily for larger membranes with smaller typical curvatures, suggesting that the stiffening transition is easier to observe experimentally than in the small-scale systems accessible to simulation.

Emerging applications at the interface of DNA nanotechnology and cellular membranes: Perspectives from biology, engineering, and physics.

DNA nanotechnology has proven exceptionally apt at probing and manipulating biological environments as it can create nanostructures of almost arbitrary shape that permit countless types of modifications, all while being inherently biocompatible. Emergent areas of particular interest are applications involving cellular membranes, but to fully explore the range of possibilities requires interdisciplinary knowledge of DNA nanotechnology, cell and membrane biology, and biophysics. In this review, we aim for a concise introduction to the intersection of these three fields. After briefly revisiting DNA nanotechnology, as well as the biological and mechanical properties of lipid bilayers and cellular membranes, we summarize strategies to mediate interactions between membranes and DNA nanostructures, with a focus on programmed delivery onto, into, and through lipid membranes. We also highlight emerging applications, including membrane sculpting, multicell self-assembly, spatial arrangement and organization of ligands and proteins, biomechanical sensing, synthetic DNA nanopores, biological imaging, and biomelecular sensing. Many critical but exciting challenges lie ahead, and we outline what strikes us as promising directions when translating DNA nanostructures for future in vitro and in vivo membrane applications.

Stabilizing Leaflet Asymmetry under Differential Stress in a Highly Coarse-Grained Lipid Membrane Model.

We present a version of the coarse-grained Cooke lipid model, modified to simulate asymmetric lipid membranes. It is inspired by a method employed by Wang et al. [ Commun. Comput. Phys. 2013, 13, 1093-1106] for artificially penalizing lipid flip-flop but copes more robustly with differential stress, at the cost of one additional bead per lipid and the concomitant increase in computational overhead. Bilayer asymmetry ultimately breaks down beyond a system size dependent critical differential stress, which can be predicted from a simple analytical model. We remeasure many important material parameters for the new model and find it to be consistent with typical fluid lipid membranes. Maintaining a stable stress asymmetry has many applications, and we give two examples: (i) connecting monolayer stress to lipid number asymmetry in order to directly measure the monolayer area modulus and (ii) finding its strain-dependent higher-order correction by monitoring the equilibrium bilayer area.

Dynamics of active nematic defects on the surface of a sphere.

A nematic liquid crystal confined to the surface of a sphere exhibits topological defects of total charge +2 due to the topological constraint. In equilibrium, the nematic field forms four +1/2 defects, located at the corners of a regular tetrahedron inscribed within the sphere, since this minimizes the Frank elastic energy. If additionally the individual nematogens exhibit self-driven directional motion, the resulting active system creates large-scale flow that drives it out of equilibrium. In particular, the defects now follow complex dynamic trajectories which, depending on the strength of the active forcing, can be periodic (for weak forcing) or chaotic (for strong forcing). In this paper we derive an effective particle theory for this system, in which the topological defects are the degrees of freedom, whose exact equations of motion we subsequently determine. Numerical solutions of these equations confirm previously observed characteristics of their dynamics and clarify the role played by the time dependence of their global rotation. We also show that Onsager's variational principle offers an exceptionally transparent way to derive these dynamical equations, and we explain the defect mobility at the hydrodynamics level.

Probing Nanoparticle/Membrane Interactions by Combining Amphiphilic Diblock Copolymer Assembly and Plasmonics.

Understanding the interactions between nanoparticles (NPs) and boundaries of cells is crucial both for their toxicity and therapeutic applications. Besides specific receptor-mediated endocytosis of surface-functionalized NPs, passive internalization is prompted by relatively unspecific parameters, such as particle size and charge. Based on theoretical treatments, adhesion to and bending of the cell membrane can induce NP wrapping. Experimentally, powerful tools are needed to selectively probe possible membrane-NP motifs at very dilute conditions and avoid dye labeling. In this work, we employ surface resonance-enhanced dynamic light scattering, surface plasmon resonance, electron microscopy, and simulations for sensing interactions between plasmonic AuNPs and polymersomes. We distinguish three different interaction scenarios at nanomolar concentrations by tuning the surface charge of AuNPs and rationalize these events by balancing vesicle bending and electrostatic/van der Waals AuNP and vesicle adhesion. The clarification of the physical conditions under which nanoparticles passively translocate across membranes can aid in the rational design of drugs that cannot exploit specific modes of cellular uptake and also elucidates physical properties that render nanoparticles in the environment particularly toxic.

Spontaneous Curvature, Differential Stress, and Bending Modulus of Asymmetric Lipid Membranes.

Lipid bilayers can exhibit asymmetric states, in which the physical characteristics of one leaflet differ from those of the other. This most visibly manifests in a different lipid composition, but it can also involve opposing lateral stresses in each leaflet that combine to an overall vanishing membrane tension. Here, we use theoretical modeling and coarse-grained simulation to explore the interplay between a compositional asymmetry and a nonvanishing differential stress. Minimizing the total elastic energy leads to a preferred spontaneous curvature that balances torques due to both bending moments and differential stress, with sometimes unexpected consequences. For instance, asymmetric flat bilayers, whose specific areas in each leaflet are matched to those of corresponding tensionless symmetric flat membranes, still exhibit a residual differential stress because the conditions of vanishing area strain and vanishing bending moment differ. We also measure the curvature rigidity of asymmetric bilayers and find that a sufficiently strong differential stress, but not compositional asymmetry alone, can increase the bending modulus. The likely cause is a stiffening of the compressed leaflet, which appears to be related to its gel transition but not identical with it. We finally show that the impact of cholesterol on differential stress depends on the relative strength of elastic and thermodynamic driving forces: if cholesterol solvates equally well in both leaflets, it will redistribute to cancel both leaflet tensions almost completely, but if its partitioning free energy prefers one leaflet over the other, the resulting distribution bias may even create differential stress. Because cells keep most of their lipid bilayers in an asymmetric nonequilibrium steady state, our findings suggest that biomembranes are elastically more complex than previously thought: besides a spontaneous curvature, they might also exhibit significant differential stress, which could strongly affect their curvature energetics.

A consistent quadratic curvature-tilt theory for fluid lipid membranes.

The tilt of a lipid molecule describes the deviation of its orientation away from the local normal of its embedding membrane. Tilt is the subleading degree of freedom after a membrane's geometry, and it becomes relevant at scales comparable to lipid bilayer thickness. Building on earlier work by Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)], who envisioned lipid membranes as thin prestressed fluid elastic films, and Terzi and Deserno [J. Chem. Phys. 147, 084702 (2017)], who discovered a new coupling term between splay and tilt divergence, we construct a theory of membrane elasticity that is quadratic in geometry and tilt and complete at order 1/length2. We show that a general and consistent treatment of both lateral and transverse depth-dependent shear stresses creates several contributions to the elastic energy density, of which only a subset had previously been identified. Apart from the well-known penalty of lipid twist (the curl of tilt), these terms generate no qualitatively new phenomenology, but they quantitatively revise the connections between the moduli of a tilt-curvature theory and its underlying microscopic foundation. In particular, we argue that the monolayer Gaussian curvature modulus κ¯m, widely believed to be equal to the second moment of the transmonolayer stress profile, acquires a second contribution from lipid twist, which is always negative. This could resolve the long-standing conundrum that many measured values of κ¯m appeared to have a sign that violates basic stability considerations. We also show that the previously discovered novel coupling between splay and tilt divergence is not simply proportional to κ¯m but acquires its own splay-tilt coupling modulus, κst,m. We explore the predictions of our theory for various elastic moduli and their mutual interrelations and use an extensive set of existing atomistic molecular dynamics simulations for 12 different lipid types to collectively reason about such predictions. We find that bending rigidities are captured fairly well by existing theories, while reliable predictions for local moduli, especially the splay-tilt coupling modulus, remain challenging.

Mechanical properties of lipid bilayers: a note on the Poisson ratio.

We investigate the Poisson ratio ν of fluid lipid bilayers, i.e., the question how area strains compare to the changes in membrane thickness (or, equivalently, volume) that accompany them. We first examine existing experimental results on the area- and volume compressibility of lipid membranes. Analyzing them within the framework of linear elasticity theory for homogeneous thin fluid sheets leads us to conclude that lipid membrane deformations are to a very good approximation volume-preserving, with a Poisson ratio that is likely about 3% smaller than the common soft matter limit . These results are fully consistent with atomistic simulations of a DOPC membrane at varying amount of applied lateral stress, for which we instead deduce ν by directly comparing area- and volume strains. To assess the problematic assumption of transverse homogeneity, we also define a depth-resolved Poisson ratio ν(z) and determine it through a refined analysis of the same set of simulations. We find that throughout the membrane's thickness, ν(z) is close to the value derived assuming homogeneity, with only minor variations of borderline statistical significance.

Optimal Coarse-Grained Site Selection in Elastic Network Models of Biomolecules.

Elastic network models, simple structure-based representations of biomolecules where atoms interact via short-range harmonic potentials, provide great insight into a molecule's internal dynamics and mechanical properties at extremely low computational cost. Their efficiency and effectiveness have made them a pivotal instrument in the computer-aided study of proteins and, since a few years, also of nucleic acids. In general, the coarse-grained sites, i.e. those effective force centers onto which the all-atom structure is mapped, are constructed based on intuitive rules: a typical choice for proteins is to retain only the C α atoms of each amino acid. However, a mapping strategy relying only on the atom type and not the local properties of its embedding can be suboptimal compared to a more careful selection. Here, we present a strategy in which the subset of atoms, each of which is mapped onto a unique coarse-grained site of the model, is selected in a stochastic search aimed at optimizing a cost function. The latter is taken to be a simple measure of the consistency between the harmonic approximation of an elastic network model and the harmonic model obtained through exact integration of the discarded degrees of freedom. The method is applied to two representatives of structurally very different types of biomolecules: the protein adenylate kinase and the RNA molecule adenine riboswitch. Our analysis quantifies the substantial impact that an algorithm-driven selection of coarse-grained sites can have on a model's properties.

The role of scaffold reshaping and disassembly in dynamin driven membrane fission.

The large GTPase dynamin catalyzes membrane fission in eukaryotic cells, but despite three decades of experimental work, competing and partially conflicting models persist regarding some of its most basic actions. Here we investigate the mechanical and functional consequences of dynamin scaffold shape changes and disassembly with the help of a geometrically and elastically realistic simulation model of helical dynamin-membrane complexes. Beyond changes of radius and pitch, we emphasize the crucial role of a third functional motion: an effective rotation of the filament around its longitudinal axis, which reflects alternate tilting of dynamin's PH binding domains and creates a membrane torque. We also show that helix elongation impedes fission, hemifission is reached via a small transient pore, and coat disassembly assists fission. Our results have several testable structural consequences and help to reconcile mutual conflicting aspects between the two main present models of dynamin fission-the two-stage and the constrictase model.

Responsive behavior of a branched-chain polymer network: a molecular dynamics study.

Smart polymer hydrogels, which can undergo structural and volume phase transitions in response to external stimuli, have gained much attention for their widespread technological applications. Compared to linear polymers, branched chains offer more extensive opportunities to rationally design functional materials, since they permit more extensive structural tunability-for instance by adjusting the balance between hydrophobic and hydrophilic units, the grafting fraction of backbone monomers, or the side chain length, topology, and solubility. Here we conduct coarse-grained molecular dynamics simulations to assess how well generic physical principles capture this complex interplay of tuning parameters, specifically when building networks from complex branched chains with a hydrophobic backbone. Swollen chains collapse upon reducing side chain solubility, length, and grafting density, but neither the sharpness of this transition nor its dynamic range, if measured via chain extension, depends monotonically on these parameters. Networks comprising such chains are more swollen and exhibit even sharper transitions, but their higher responsiveness goes along with a swelling ratio that falls behind that of single chains.