Vesicle shape, molecular tilt, and the suppression of necks

Stationary shapes of axisymmetric vesicles beyond lowest-energy configurations

We conduct a systematic exploration of the energy landscape of vesicle morphologies within the framework of the Helfrich model. Vesicle shapes are determined by minimizing the elastic energy subject to constraints of constant area and volume. The results show that pressurized vesicles can adopt higher-energy spindle-like configurations that require the action of point forces at the poles. If the internal pressure is lower than the external one, multilobed shapes are predicted. We utilize our results to rationalize experimentally observed spindle shapes of giant vesicles in a uniform AC electric field.

Active morphogenesis of patterned epithelial shells

Shape transformations of epithelial tissues in three dimensions, which are crucial for embryonic development or in vitro organoid growth, can result from active forces generated within the cytoskeleton of the epithelial cells. How the interplay of local differential tensions with tissue geometry and with external forces results in tissue-scale morphogenesis remains an open question. Here, we describe epithelial sheets as active viscoelastic surfaces and study their deformation under patterned internal tensions and bending moments. In addition to isotropic effects, we take into account nematic alignment in the plane of the tissue, which gives rise to shape-dependent, anisotropic active tensions and bending moments. We present phase diagrams of the mechanical equilibrium shapes of pre-patterned closed shells and explore their dynamical deformations. Our results show that a combination of nematic alignment and gradients in internal tensions and bending moments is sufficient to reproduce basic building blocks of epithelial morphogenesis, including fold formation, budding, neck formation, flattening, and tubulation.

Geometric Induction in Chiral Superfluids

We explore the properties of chiral superfluid thin films coating a curved surface. Because of the vector nature of the order parameter, a geometric gauge field emerges and leads to a number of observable effects such as anomalous vortex-geometric interaction and curvature-induced mass and spin supercurrents. We apply our theory to several well-known phases of chiral superfluid ^{3}He and derive experimentally observable signatures. We further discuss the cases of flexible geometries where a soft surface can adapt itself to compensate for the strain from the chiral superfluid. The proposed interplay between geometry and chiral superfluid order provides a fascinating avenue to control and manipulate quantum states with strain.

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.

Additional contributions to elastic energy of lipid membranes: Tilt-curvature coupling and curvature gradient

Lipid bilayer membranes under biologically relevant conditions are flexible thin laterally fluid films consisting of two unimolecular layers (monolayers) each about 2 nm thick. On spatial scales much larger than the bilayer thickness, the membrane elasticity is well determined by its shape. The classical Helfrich theory considers the membrane as an elastic two-dimensional (2D) film, which has no particular internal structure. However, various local membrane heterogeneities can result in a lipids tilt relative to the membrane surface normal. On the basis of the classical elasticity theory of 3D bodies, Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)10.1007/s101890070003] derived the most general energy functional, taking into account the tilt and lipid monolayer curvature. Recently, Terzi and Deserno [J. Chem. Phys. 147, 084702 (2017)10.1063/1.4990404] showed that Hamm and Kozlov's derivation was incomplete because the tilt-curvature coupling term had been missed. However, the energy functional derived by Terzi and Deserno appeared to be unstable, thereby being invalid for applications that require minimizations of the overall energy of deformations. Here, we derive a stable elastic energy functional, showing that the squared gradient of the curvature was missed in both of these works. This change in the energy functional arises from a more accurate consideration of the transverse shear deformation terms and their influence on the membrane stability. We also consider the influence of the prestress terms on the stability of the energy functional, and we show that it should be considered small and the effective Gaussian curvature should be neglected because of the stability requirements. We further generalize the theory, including the stretching-compressing deformation modes, and we provide the geometrical interpretation of the terms that were previously missed by Hamm and Kozlov. The physical consequences of the new terms are analyzed in the case of a membrane-mediated interaction of two amphipathic peptides located in the same monolayer. We also provide the expression for director fluctuations, comparing it with that obtained by Terzi and Deserno.

Membrane stress and torque induced by Frank's nematic textures: A geometric perspective using surface-based constraints

An elastic membrane with embedded nematic molecules is considered as a model of anisotropic fluid membrane with internal ordering. By considering the geometric coupling between director field and membrane curvature, the nematic texture is shown to induce anisotropic stresses additional to Canham-Helfrich elasticity. Building upon differential geometry, analytical expressions are found for the membrane stress and torque induced by splaying, twisting, and bending of the nematic director as described by the Frank energy of liquid crystals. The forces induced by prototypical nematic textures are visualized on the sphere and on cylindrical surfaces.

Novel tilt-curvature coupling in lipid membranes

On mesoscopic scales, lipid membranes are well described by continuum theories whose main ingredients are the curvature of a membrane's reference surface and the tilt of its lipid constituents. In particular, Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)] have shown how to systematically derive such a tilt-curvature Hamiltonian based on the elementary assumption of a thin fluid elastic sheet experiencing internal lateral pre-stress. Performing a dimensional reduction, they not only derive the basic form of the effective surface Hamiltonian but also express its emergent elastic couplings as trans-membrane moments of lower-level material parameters. In the present paper, we argue, though, that their derivation unfortunately missed a coupling term between curvature and tilt. This term arises because, as one moves along the membrane, the curvature-induced change of transverse distances contributes to the area strain-an effect that was believed to be small but nevertheless ends up contributing at the same (quadratic) order as all other terms in their Hamiltonian. We illustrate the consequences of this amendment by deriving the monolayer and bilayer Euler-Lagrange equations for the tilt, as well as the power spectra of shape, tilt, and director fluctuations. A particularly curious aspect of our new term is that its associated coupling constant is the second moment of the lipid monolayer's lateral stress profile-which within this framework is equal to the monolayer Gaussian curvature modulus, κ¯_m. On the one hand, this implies that many theoretical predictions now contain a parameter that is poorly known (because the Gauss-Bonnet theorem limits access to the integrated Gaussian curvature); on the other hand, the appearance of κ¯_m outside of its Gaussian curvature provenance opens opportunities for measuring it by more conventional means, for instance by monitoring a membrane's undulation spectrum at short scales.

Stresses in curved nematic membranes

Ordering configurations of a director field on a curved membrane induces stress. In this work, we present a theoretical framework to calculate the stress tensor and the torque as a consequence of the nematic ordering; we use the variational principle and invariance of the energy under Euclidean motions. Euler-Lagrange equations of the membrane as well as the corresponding boundary conditions also appear as natural results. The stress tensor found includes attraction-repulsion forces between defects; likewise, defects are attracted to patches with the same sign in Gaussian curvature. These forces are mediated by the Green function of the Laplace-Beltrami operator of the surface. In addition, we find nonisotropic forces that involve derivatives of the Green function and the Gaussian curvature, even in the normal direction to the membrane. We examine the case of axial membranes to analyze the spherical one. For spherical vesicles we find the modified Young-Laplace law as a consequence of the nematic texture. In the case of spherical cap with defect at the north pole, we find that the force is repulsive with respect to the north pole, indicating that it is an unstable equilibrium point.

Bistable curvature potential at hyperbolic points of nematic shells

Nematic shells are colloidal particles coated with nematic liquid crystal molecules which may freely glide and rotate on the colloid's surface while keeping their long axis on the local tangent plane. We describe the nematic order on a shell by a unit director field on an orientable surface. Equilibrium fields can then be found by minimising the elastic energy, which in general is a function of the surface gradient of the director field. We learn how to extract systematically out of this energy a fossil component, related only to the surface and its curvatures, which expresses a curvature potential for the molecular torque. At hyperbolic points on the colloid's surface, and only there, the alignment preferred by the curvature potential may fail to be a direction of principal curvature. There the fossil energy becomes bistable.

Tuning interfacial patterns of molecular bonds via surface morphology

Many studies have demonstrated that the mechanical properties of the extracellular matrix can significantly influence the morphology, strength and lifetime of focal adhesions. However, how the morphology of the contact surface affects the pattern formation of the molecular bonds still remains largely unknown. Here, by simplifying the cell and extracellular matrix to two opposing elastic bodies and considering the lateral diffusion as well as the bonding/debonding of molecular bonds, we study the clustering behavior of receptor-ligand bonds between curved surfaces and the phase diagrams of cluster patterns. We reveal the important role of surface morphology and bond kinetics in regulating the patterns of bond clusters. We further investigate the segregation dynamics of the interfacial bonds under various loading speeds, and we show that the average interfacial stress is rate-dependent while the rupture stress is rate-independent. Finally, we demonstrate that programmable patterning of bond clusters can be achieved through the designed surface morphology.

Shape and Dynamics of Adhesive Cells: Mechanical Response of Open Systems

Cell adhesion is an essential biological process. However, previous theoretical and experimental studies ignore a key variable, the changes of cellular volume and pressure, during the dynamic adhesion process. Here, we treat cells as open systems and propose a theoretical framework to investigate how the exchange of water and ions with the environment affects the shape and dynamics of cells adhered between two adhesive surfaces. We show that adherent cells can be either stable (convex or concave) or unstable (spontaneous rupture or collapse) depending on the adhesion energy density, the cell size, the separation of two adhesive surfaces, and the stiffness of the flexible surface. Strikingly, we find that the unstable states vanish when cellular volume and pressure are constant. We further show that the detachments of convex and concave cells are very different. The mechanical response of adherent cells is mainly determined by the competition between the loading rate and the regulation of the cellular volume and pressure. Finally, we show that as an open system the detachment of adherent cells is also significantly influenced by the loading history. Thus, our findings reveal a major difference between living cells and nonliving materials.

Incorporation of Soft Particles into Lipid Vesicles: Effects of Particle Size and Elasticity

The interaction between particles and lipid biomembranes plays an essential role in many fields such as endocytosis, drug delivery, and intracellular traffic. Here we conduct a theoretical study on the incorporation of elastic particles of different sizes and rigidities into a lipid vesicle through adhesive wrapping. It is shown that while the incorporation of relatively small particles involves smooth shape evolution, the vesicle wrapping of large particles exhibits a discontinuous shape transition, followed by a protrusion of the vesicle membrane at infinitesimal cost of elastic deformation energy. Moreover, softer particles require stronger adhesion energy to achieve successful internalization and delay the onset of discontinuous shape transition to a higher wrapping degree. Depending on the adhesion energy, particle-vesicle size, and rigidity ratios, and the spontaneous curvature of the vesicle, a rich variety of wrapping phase diagrams consisting of stable and metastable states of no-wrapping, partial-wrapping, and full-wrapping are established. The underlying mechanism of the discontinuous shape transformation of the vesicle and the relation between the uptake proneness and uptake efficiency are discussed. These results shed further light on the elasticity effects in cellular uptake of elastic particles and may provide rational design guidelines for controlled endocytosis and diagnostics delivery.

Terasaki spiral ramps in the rough endoplasmic reticulum

We present a model describing the morphology as well as the assembly of "Terasaki ramps," the recently discovered helicoidal connections linking adjacent sheets of the rough endoplasmic reticulum (ER). The fundamental unit is a localized symmetric double-ramped "parking garage" formed by two separated gently pitched, approximately helicoidal, ramps of opposite chiralities. This geometry is stabilized by a short-range repulsive interaction between ramps associated with bending energy which opposes the long-range attraction associated with tension. The ramp inner boundaries are themselves stabilized by the condensation of membrane-shaping proteins along their length. A mechanism for parking garage self-assembly is proposed involving the nucleation of dipoles at the center of tubular three-way junctions within the smooth ER. Our predictions are compared with the experimental data.

Recent theoretical advances in elasticity of membranes following Helfrich's spontaneous curvature model

Recent theoretical advances in elasticity of membranes following Helfrich's famous spontaneous curvature model are summarized in this review. The governing equations describing equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are presented. Several analytic solutions to these equations and their corresponding configurations are demonstrated.

Morphology transition in lipid vesicles due to in-plane order and topological defects

Complex morphologies in lipid membranes typically arise due to chemical heterogeneity, but in the tilted gel phase, complex shapes can form spontaneously even in a membrane containing only a single lipid component. We explore this phenomenon via experiments and coarse-grained simulations on giant unilamellar vesicles of 1,2-dipalmitoyl-sn-glycero-3-phosphocholine. When cooled from the untilted L(α) liquid-crystalline phase into the tilted gel phase, vesicles deform from smooth spheres to disordered, highly crumpled shapes. We propose that this shape evolution is driven by nucleation of complex membrane microstructure with topological defects in the tilt orientation that induce nonuniform membrane curvature. Coarse-grained simulations demonstrate this mechanism and show that kinetic competition between curvature change and defect motion can trap vesicles in deeply metastable, defect-rich structures.

Exocytotic fusion pore stability and topological defects in the membrane with orientational degree of ordering

Regulated exocytosis is a process that strongly depends on the formation and stability of the fusion pore. It was indicated experimentally and theoretically that narrow and highly curved fusion pore may be stabilized by accumulation of anisotropic membrane components possessing orientational ordering. On the other hand, narrow fusion pore may also undergo repetitive opening and closing, disruption in the so called kiss and run process or become completely opened in the process of full fusion of the vesicle with the membrane. In this paper we attempt to elucidate the subtle interplay between the stabilizing and destabilizing processes in the fusion neck. A possible physical mechanism which may lead to disruption of the stable fusion pore or complete fusion of the vesicle with the membrane is discussed. It is indicated that topologically driven defects of the in-plane orientational membrane ordering in the region of the fusion pore may disrupt the fusion. Alternatively, it may facilitate repetitive opening and closing of the fusion pore or induce full fusion of the vesicle with the target membrane.

Frustrated order on extrinsic geometries

We study, numerically and theoretically, defects in an anisotropic liquid that couple to the extrinsic geometry of a surface. Though the intrinsic geometry tends to confine topological defects to regions of large Gaussian curvature, extrinsic couplings tend to orient the order along the local direction of maximum or minimum bending. This additional frustration is generically unavoidable, and leads to complex ground-state thermodynamics. Using the catenoid as a prototype, we show, in contradistinction to the well-known effects of intrinsic geometry, that extrinsic curvature expels disclinations from the region of maximum curvature above a critical coupling threshold. On catenoids lacking an "inside-outside" symmetry, defects are expelled altogether above a critical neck size.

Monte Carlo studies of the XY model on two-dimensional curved surfaces

To explore the interaction between topological defects and curvature in materials with orientational order, we perform Monte Carlo studies of the two-dimensional XY model on the surface of curved substrates. Each curved surface is patterned with a random lattice constructed via random sequential absorption, and an XY spin is positioned at each lattice site. Spins lie in the plane locally tangent to the surface and interact with neighbors defined via a distance cutoff. We demonstrate that the relative phase associated with vortices is significant in curved geometries and plays a role in microstructural evolution. We also observe that any nonuniform curvature, e.g., on the surface of a torus, induces spontaneous segregation of positive and negative vortices and promotes the formation of deeply metastable defect microstructures. Though qualitative in nature, these observations provide novel insights into the patterning of topological defects in curved geometries and suggest that the Kosterlitz-Thouless transition may be altered in geometries with nonuniform curvature.

Monte Carlo simulations of fluid vesicles with in-plane orientational ordering

We present a method for simulating fluid vesicles with in-plane orientational ordering. The method involves computation of local curvature tensor and parallel transport of the orientational field on a randomly triangulated surface. It is shown that the model reproduces the known equilibrium conformation of fluid membranes and work well for a large range of bending rigidities. Introduction of nematic ordering leads to stiffening of the membrane. Nematic ordering can also result in anisotropic rigidity on the surface leading to formation of membrane tubes.

The tethered infinitesimal tori and spheres algorithm: a versatile calculator for axisymmetric problems in equilibrium membrane mechanics

Constrained minimization of energy functionals is a central part, and usually the difficult part, of solving problems in the equilibrium mechanics of biological and biomimetic membranes. The inherent difficulties of the conventional variational-calculus approach prevents the numerical calculation involved from being made routine in the analyses of experimental results. We have developed a simulated annealing-based computational technique for routinizing the task of constrained minimization of energy functionals governing whole, or small patches of whole, fluid membranes with axisymmetry, spherical topology, and no domains of inhomogeneity. In this article, we describe the essential principles of the technique and apply it to five examples to demonstrate its versatility. It gives membrane shapes that are automatically stable to axisymmetric perturbations. Presently, it can account for constraints on 1), the membrane area or the effective membrane tension; 2), the enclosed volume or the effective pressure difference across the membrane thickness; and 3), the axial end-to-end distance or the applied axial point force.

Curvature-driven lipid sorting in a membrane tubule

Motivated by recent experiments that implicate the mechanical properties of membranes in lipid sorting, we examine the interplay of lipid composition and curvature in membrane tubules. We study how the coupling between membrane composition and membrane bending stiffness and tension affects tubule formation. Drawing a tubule from a vesicle leads to a rearrangement of composition in which the phase of higher flexibility segregates into the highly curved tubule. For point forcing, the force vs extension curve can have a sharp drop just as the tubule begins to form.