Preferred Curvature Research Articles

Emulsions stabilized by phospholipids

HypothesisPhospholipids are amphiphilic molecules able to adsorb at oil/water interfaces and thus used to stabilize parenteral emulsions. Yet, their low preferred curvature, which sensitively depends on molecular structures and interactions, favors the formation of lamellar phases and sets constraints on the system formulation. Combining phase studies, structural interfacial characterizations, and stability monitoring for different water/phospholipid/oil systems should shine a light on the mechanisms at play and thus tools to optimize formulations. ExperimentsFour phase diagrams were established for ternary aqueous systems containing either DOPC or POPC as the phospholipid and hexadecane or miglyol 812 as the oil. Droplet interfaces were probed using small-angle neutron scattering and the amount of adsorbed lipid was determined using separation and Raman spectroscopy. The metastability of both nano and macro emulsions was systematically assessed over weeks using light scattering. FindingsWe show that nanoemulsion droplets are stabilized by a lipid monolayer and display excellent metastability if the preferred curvature is positive and large enough, even without any added charges or at high ionic strengths. In contrast, macroemulsion droplets are stabilized with a lipid multilayer, which should possess a positive preferred curvature but also a good enough interfacial anchorage, which is lost upon increasing the preferred curvature. Overall, we provide a rationale for understanding the impact of molecular changes in the formulation on emulsion metastability, through the analysis of the lipid film preferred curvature, layering, and interfacial anchorage.

Effect of cardiolipin on the lamellarity and elongation of liposomes hydrated in PBS

Lamellarity and shape are important factors in the formation of vesicles and determine their role in biological systems and pharmaceutical applications. Cardiolipin (CL) is a major lipid in many biological membranes and exerts a great influence on their structural organization due to its particular structure and physico-chemical properties. Here, we used small-angle X-ray and neutron scattering to study the effects of CL with different acyl chain lengths and saturations (CL14:0, CL18:1, CL18:2) on vesicle morphology and lamellarity in membrane models containing mixtures of phosphatidylcholine and phosphatidylethanolamine with different acyl chain lengths and saturations (C14:0 and C 18:1). Measurements were performed in the presence of Phosphate Buffer Saline (PBS), at 37°C, to better reflect physiological conditions, which resulted in strong effects on vesicle morphology, depending on the type and amount of CL used. The presence of small quantities of CL (from 2.5%) reduced inter-membrane correlations and increased perturbation of the membrane, an effect which is enhanced in the presence of matched shorter saturated acyl chains, and mainly unilamellar vesicles (ULV) are formed. In extruded vesicles, employed for SANS experiments, flattened vesicles are observed partly due to the hypertonic effect of PBS, but also influenced by the type of CL added.Our experimental data from SAXS and SANS revealed a strong dependence on CL content in shaping the membrane microstructure, with an apparent optimum in the PC:CL mixture in terms of promoting reduced correlations, preferred curvature and elongation. However, the use of PBS caused distinct differences from previously published studies in water in terms of vesicle shape, and highlights the need to investigate vesicle formation under physiological conditions in order to be able to draw conclusions about membrane formation in biological 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.

Mechanics of Spacetime with Surface Tension and Preferred Curvature

In previous publications, it was argued that because energy and matter are aligned in the thin increment of present time for an observer, spacetime should and must have surface tension. In this paper, we review how to apply 4-dimensional continuum mechanics with imaginary time coordinates to derive a mechanical model of spacetime with surface tension. Then, to continue model development, we discuss the concept of preferred curvature from the physical chemistry of surfaces and attempt to apply those concepts to spacetime geometry. We show how the model exhibits quantum fluctuations at the Plank scale and components resembling dark matter and dark energy at the cosmic scale.

Curvature-driven instabilities in thin active shells.

Spontaneous material shape changes, such as swelling, growth or thermal expansion, can be used to trigger dramatic elastic instabilities in thin shells. These instabilities originate in geometric incompatibility between the preferred extrinsic and intrinsic curvature of the shell, which may be modified by active deformations through the thickness and in plane, respectively. Here, we solve the simplest possible model of such instabilities, which assumes the shells are shallow, thin enough to bend but not stretch, and subject to homogeneous preferred curvatures. We consider separately the cases of zero, positive and negative Gauss curvature. We identify two types of supercritical symmetry-breaking instability, in which the shell’s principal curvature spontaneously breaks discrete up/down symmetry and continuous planar isotropy. These are then augmented by inversion instabilities, in which the shell jumps subcritically between up/down broken symmetry states and rotation instabilities, in which the curvatures rotate by 90° between states of broken isotropy without release of energy. Each instability has a thickness-independent threshold value for the preferred extrinsic curvature proportional to the square root of Gauss curvature. Finally, we show that the threshold for the isotropy-breaking instability is the same for deep spherical caps, in good agreement with recently published data.

Stability of cubic single network phases in diblock copolymer melts

AbstractWe used self‐consistent field theory (SCFT) to examine the stability of three cubic single network structures, single gyroid, single diamond, and single primitive, in neat diblock copolymer melts. Arguments related to packing frustration, as measured by the standard deviation in the mean curvature of the interface relative to its mean, that explain the relative stability of double gyroid, double diamond and double primitive also extend to the relative stability of their single networks. However, this packing frustration measure fails to account for the relative stabilities of single and double networks, for example, single gyroid versus double gyroid, and arguments for a preferred curvature fail to explain the selection of double gyroid as the segregation strength increases. Rather, the larger interfacial areas per unit volume of single networks, arising from decreased domain sizes, are found to be a generic factor that leads to metastability of the cubic single networks relative to double gyroid in the network‐forming region of the morphology diagram. These results clarify the origins of the metastability of single cubic network phases in neat diblock copolymer melts.

Stress accumulation versus shape flattening in frustrated, warped-jigsaw particle assemblies

Geometrically frustrated assembly has emerged as an attractive paradigm for understanding and engineering assemblies with self-limiting, finite equilibrium dimensions. We propose and study a novel 2D particle based on a so-called ‘warped jigsaw’ (WJ) shape design: directional bonds in a tapered particle favor curvature along multi-particle rows that frustrate 2D lattice order. We investigate how large-scale intra-assembly stress gradients emerge from the microscopic properties of the particles using a combination of numerical simulation and continuum elasticity. WJ particles can favor anisotropic ribbon assemblies, whose lateral width may be self-limiting depending on the relative strength of cohesive to elastic forces in the assembly, which we show to be controlled by the range of interactions and degree of shape misfit. The upper limits of self-limited size are controlled by the crossover between two elastic modes in assembly: the accumulation of shear with increasing width at small widths giving way to unbending of preferred row curvature, permitting assembly to grow to unlimited sizes. We show that the stiffness controlling distinct elastic modes is governed by combination and placement of repulsive and attractive binding regions, providing a means to extend the range of accumulating stress to sizes that are far in excess of the single particle size, which we corroborate via numerical studies of discrete particles of variable interactions. Lastly, we relate the ground-state energetics of the model to lower and upper limits on equilibrium assembly size control set by the fluctuations of width along the ribbon boundary.

Curvature strains as a global orchestrator of morphogenesis

Successful morphogenesis on the scale of organs or organisms requires strict coordination between the constituent cells whose action on the local scale must be orchestrated accurately to achieve a functional shape on the global scale. We present a theoretical model in which morphogenetic information is encoded only through a locally preferred curvature, but with cell dynamics which simultaneously ensures that these interactions globally achieve morphogenesis and correct cell-neighbor exchanges to avoid cell stretches. This is achieved by a cell-cell interaction potential that drives correct cell intercalation to reorganize the cell sheet dynamically during the deformation processes. We demonstrate morphogenesis of simple three-dimensional shapes and study the effects of fixed cell neighbor connectivity and noisy cell division.

A general computational framework for the dynamics of single- and multi-phase vesicles and membranes

The dynamics of thin, membrane-like structures are ubiquitous in nature. They play especially important roles in cell biology. Cell membranes separate the inside of a cell from the outside, and vesicles compartmentalize proteins into functional microregions, such as the lysosome. Proteins and/or lipid molecules also aggregate and deform membranes to carry out cellular functions. For example, some viral particles can induce the membrane to invaginate and form an endocytic vesicle that pulls the virus into the cell. While the physics of membranes has been extensively studied since the pioneering work of Helfrich in the 1970's, simulating the dynamics of large scale deformations remains challenging, especially for cases where the membrane composition is spatially heterogeneous. Here, we develop a general computational framework to simulate the overdamped dynamics of membranes and vesicles. We start by considering a membrane with an energy that is a generalized functional of the shape invariants and also includes line discontinuities that arise due to phase boundaries. Using this energy, we derive the internal restoring forces and construct a level set-based algorithm that can stably simulate the large-scale dynamics of these generalized membranes, including scenarios that lead to membrane fission. This method is applied to solve for shapes of single-phase vesicles using a range of reduced volumes, reduced area differences, and preferred curvatures. Our results match well the experimentally measured shapes of corresponding vesicles. The method is then applied to explore the dynamics of multiphase vesicles, predicting equilibrium shapes and conditions that lead to fission near phase boundaries.

Flagellar Cooperativity and Collective Motion in Sperm

Sperm have thin structures known as flagella whose motion must be regulated in order to reach the egg for fertilization. Large numbers of sperm are typically needed in this process and some species have sperm that exhibit collective or aggregate motion when swimming in groups. The purpose of this study is to model planar motion of flagella in groups to explore how collective motion may arise in three-dimensional fluid environments. We use the method of regularized Stokeslets and a three-dimensional preferred curvature model to simulate groups of undulating flagella, where flagellar waveforms are modulated via hydrodynamic coupling with other flagella and surfaces. We find that collective motion of free-swimming flagella is an unstable phenomenon in long-term simulations unless there is an external mechanism to keep flagella near each other. However, there is evidence that collective swimming can result in significant gains in velocity and efficiency. With the addition of an ability for sperm to attach and swim together as a group, velocities and efficiencies can be increased even further, which may indicate why some species have evolved mechanisms that enable collective swimming and cooperative behavior in sperm.

Vesicle budding induced by binding of curvature-inducing proteins.

Vesicle budding induced by protein binding that generates an isotropic spontaneous curvature is studied using a mean-field theory. Many spherical buds are formed via protein binding. As the binding chemical potential increases, the proteins first bind to the buds and then to the remainder of the vesicle. For a high spontaneous curvature and/or high bending rigidity of the bound membrane, it is found that a first-order transition occurs between a small number of large buds and a large number of small buds. These two states coexist around the transition point. The proposed scheme is simple and easily applicable to many interaction types, so we investigate the effects of interprotein interactions, the protein-insertion-induced changes in area, the variation of the saddle-splay modulus, and the area-difference-elasticity energy. The differences in the preferred curvatures for curvature sensing and generation are also clarified.

Sequence-Dependent Analysis of Collagen Mechanics

Extracellular matrix mechanics influence diverse cellular functions, yet surprisingly little is known about the mechanical properties of their constituent collagens. While collagen type IV is an integral component of basement membranes, it has received far less attention than the more abundant fibrillar collagens. In this work, we used atomic force microscopy to image different collagen types and analyze their sequence-dependent mechanics. By analyzing their flexibility in a sequence-dependent manner, we learned that discontinuities in the triple-helix-defining sequence (Gly-X-Y) in collagen IV lead to a generally more flexible polymer with notable flexible “hinges” that correlate with non-helical regions. We contrast these findings by studying collagen III - a continuously triple-helical collagen - in which we found that it also displays variable flexibility along its contour, most notably possessing a high flexibility region near the matrix metalloprotease (MMP) binding site. This result represents the first demonstration of a unique mechanical signature of the MMP site along collagen and offers the opportunity to examine the interplay between sequence, thermal stability, and mechanical properties. Surprisingly, we found that proline content did not correlate with local flexibility in either collagen type. We also found that physiologically relevant changes in pH and chloride concentration did not alter the flexibility of collagen IV, indicating such environmental changes are not used to control its compaction during secretion. Furthermore, we found that collagen IV exhibits oscillatory local curvature, while collagen III displays no strong preferred curvature along its contour. Our sequence-dependent curvature results show that the collagenous domain of collagen IV is structurally different than collagen III, beyond just enhanced flexibility, and we discuss possibilities for this structural profile - it could be conveying information related to collagen IV's mechanism of assembly.

Swimming Through Parameter Subspaces of a Simple Anguilliform Swimmer

Computational scientists have investigated swimming performance across a multitude of different systems for decades. Most models depend on numerous model input parameters and performance is sensitive to those parameters. In this article, parameter subspaces are qualitatively identified in which there exists enhanced swimming performance for an idealized, simple swimming model that resembles a Caenorhabditis elegans, an organism that exhibits an anguilliform mode of locomotion. The computational model uses the immersed boundary method to solve the fluid-interaction system. The 1D swimmer propagates itself forward by dynamically changing its preferred body curvature. Observations indicate that the swimmer's performance appears more sensitive to fluid scale and stroke frequency, rather than variations in the velocity and acceleration of either its upstroke or downstroke as a whole. Pareto-like optimal fronts were also identified within the data for the cost of transport and swimming speed. While this methodology allows one to locate robust parameter subspaces for desired performance in a straight-forward manner, it comes at the cost of simulating orders of magnitude more simulations than traditional fluid-structure interaction studies.

Mechanisms of negative membrane curvature sensing and generation by ESCRT III subunit Snf7.

Certain proteins have the propensity to bind to negatively curved membranes and generate negative membrane curvature. The mechanism of action of these proteins is much less studied and understood than those that sense and generate positive curvature. In this work, we use implicit membrane modeling to explore the mechanism of an important negative curvature sensing and generating protein: the main ESCRT III subunit Snf7. We find that Snf7 monomers alone can sense negative curvature and that curvature sensitivity increases for dimers and trimers. We have observed spontaneous bending of Snf7 oligomers into circular structures with preferred radius of ~20 nm. The preferred curvature of Snf7 filaments is further confirmed by the simulations of preformed spirals on a cylindrical membrane surface. Snf7 filaments cannot bind with the same interface to flat and curved membranes. We find that even when a filament has the preferred radius, it is always less stable on the flat membrane surface than on the interior cylindrical membrane surface. This provides an additional energy for membrane bending which has not been considered in the spiral spring model. Furthermore, the rings on the cylindrical spirals are bridged together by helix 4 and hence are extra stabilized compared to the spirals on the flat membrane surface.

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

On time scales shorter than lipid flip-flop time, lipid bilayers in cells assume asymmetric non-equilibrium steady states in which the physical characteristics of the two leaflets are different. The most notable difference is in lipid composition, but it can also involve different lateral stresses in each leaflet that add up to an overall (usually vanishing) tension. We explore the interplay between these two sources of asymmetry using theoretical modeling and coarse-grained simulation. By minimizing total elastic energy, we find a preferred spontaneous curvature that balances torques between bending moments and differential stress, which can at times lead to unexpected consequences. For example, compositionally asymmetric flat bilayers whose area per lipid in each leaflet is matched with corresponding symmetric tensionless flat bilayers, still exhibit residual differential stress, demonstrating that the conditions of zero area strain and zero bending moment are not identical. Furthermore, we observe that the bending modulus can significantly increase if differential stress exceeds a certain minimum threshold, but that compositional asymmetry alone cannot cause this. We attribute this effect to a stiffening in the compressed leaflet, which seems to be related to its gel transition, but not exactly coinciding with it. Finally, we show that the presence of rapidly flip-flopping lipid species (such as cholesterol) will not necessarily lead to total elimination of differential stress, but may in fact even induce it, depending on whether the partitioning energy prefers one leaflet over the other.

Adsorption of a Helical Filament Subject to Thermal Fluctuations.

We consider semiflexible chains governed by preferred curvature and twist and their flexural and twist moduli. These filaments possess a helical rather than straight three-dimensional (3D) ground state and we call them helical filaments (H-filament). Depending on the moduli, the helical shape may be smeared by thermal fluctuations. Secondary superhelical structures are expected to form on top of the specific local structure of biofilaments, as is documented for vimentin. We study confinement and adsorption of helical filaments utilizing both a combination of numerical simulations and analytical theory. We investigate overall chain shapes, transverse chain fluctuations, loop and tail distributions, and energy distributions along the chain together with the mean square average height of the monomers . The number fraction of adsorbed monomers serves as an order parameter for adsorption. Signatures of adsorbed helical polymers are the occurrence of 3D helical loops/tails and spiral or wavy quasi-flat shapes. None of these arise for the Worm-Like-Chain, whose straight ground state can be embedded in a plane.

Competing pathways for the invagination of clathrin-coated membranes.

Clathrin-mediated endocytosis is the major pathway by which eukaryotic cells take up extracellular material, but it is still elusive which physical pathways are being taken during membrane invagination. From a continuum point of view, it can be driven by increases in coat stiffness, preferred curvature or line tension. Here we develop a comprehensive theoretical framework that can be solved analytically and that predicts the consequences of these different scenarios. We find that for the case of increasing stiffness or preferred curvature, curvature will be acquired gradually with growth, while for increasing line tension, the lattice must have grown to a certain size before a flat-to-curved transition can occur. At low membrane tension, the critical value for coat stiffness is 30 kBT, for preferred curvature it is 200 nm, and for line tension it is 6 pN. For high membrane tension, critical coat stiffness is 150 kBT and critical preferred curvature is 70 nm. In the mixed case when a coat with finite rigidity but increasing line tension is considered, a cup-to-sphere transition can occur for a line tension of 6 pN. The flat-to-curved and the cup-to-sphere transitions driven by line tension are both suppressed by high membrane tension.

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.

Differential or Curvature Stress? Modus Vivendi

Differential or Curvature Stress? Modus Vivendi

Curvature induction and sensing of the F-BAR protein Pacsin1 on lipid membranes via molecular dynamics simulations

F-Bin/Amphiphysin/Rvs (F-BAR) domain proteins play essential roles in biological processes that involve membrane remodelling, such as endocytosis and exocytosis. It has been shown that such proteins transform the lipid membrane into tubes. Notably, Pacsin1 from the Pacsin/Syndapin subfamily has the ability to transform the membrane into various morphologies: striated tubes, featureless wide and thin tubes, and pearling vesicles. The molecular mechanism of this interesting ability remains elusive. In this study, we performed all-atom (AA) and coarse-grained (CG) molecular dynamics simulations to investigate the curvature induction and sensing mechanisms of Pacsin1 on a membrane. From AA simulations, we show that Pacsin1 has internal structural flexibility. In CG simulations with parameters tuned from the AA simulations, spontaneous assembly of two Pacsin1 dimers through lateral interaction is observed. Based on the complex structure, we show that the regularly assembled Pacsin1 dimers bend a tensionless membrane. We also show that a single Pacsin1 dimer senses the membrane curvature, binding to a buckled membrane with a preferred curvature. These results provide molecular insights into polymorphic membrane remodelling.