Helfrich Theory Research Articles

Inferring biophysical properties of membranes during endocytosis using machine learning.

Endocytosis is a fundamental cellular process in eukaryotic cells that facilitates the transport of molecules into the cell. With the help of fluorescence microscopy and electron tomography, researchers have accumulated extensive geometric data of membrane shapes during endocytosis. These data contain rich information about the mechanical properties of membranes, which are hard to access via experiments due to the small dimensions of the endocytic patch. In this study, we propose an approach that combines machine learning with the Helfrich theory of membranes to infer the mechanical properties of membranes during endocytosis from a dataset of membrane shapes extracted from electron tomography. Our results demonstrate that machine learning can output solutions that both match the experimental profile and satisfy the membrane shape equations derived from Helfrich theory. The learning results show that during the early stage of endocytosis, the inferred membrane tension is negative, indicating the presence of strong compressive forces at the boundary of the endocytic invagination. Our method presents a generic framework for extracting membrane information from super-resolution imaging.

Enhancement in electrical response of liquid crystals with carbon nanotubes doping

ABSTRACT The Freedericksz transition threshold is investigated in a homogeneously aligned carbon nanotube (CNT) doped liquid crystal (LC) planar cell. By neglecting the aggregate and bundle formation of CNTs in a low concentration of CNT doped LC cell, the reduction in the Freedericksz transition threshold is calculated theoretically. Flexoelectric polarisation tensorial expansion in terms of the order parameter and the generalised Helfrich theory is used to estimate the flexoelectric coefficients in the mixture. Dipolar and quadrupolar polarisations are studied in the flexoelectric coefficients of the mixture. It is shown that doping CNTs changes the flexoelectric coefficients through the order parameter of LCs. According to these calculations, for the LC molecules with a dipole moment parallel to their molecular axis, up to 4-fold increase in flexoelectric coefficient is expected, which is in good accordance with experimental reports. Also, it is shown that in the absence of dipolar flexoelectricity, the quadrupolar polarisation is responsible for the changes of the flexoelectric coefficients.

Membrane buckling and the determination of Gaussian curvature modulus.

Biological membranes can exhibit various morphology due to the fluidity of the lipid molecules within the monolayers. The shape transformation of membranes has been well described by the classical Helfrich theory, which consists only a few phenomenological parameters, including the mean and the Gaussian curvature modulus. Though various methods have been proposed to measure the mean curvature modulus, determining the Gaussian curvature modulus remains difficult both in experiments and in simulations. In this paper we study the buckling process of a rectangular membrane and a circular membrane subject to compressive stresses and under different boundary conditions. We find that the buckling of a rectangular membrane takes place continuously, while the buckling of a circular membrane can be discontinuous depending on the boundary conditions. Furthermore, our results show that the stress-strain relationship of a buckled circular membrane can be used to determine the Gaussian curvature modulus effectively.

Indentation of pore-spanning lipid membranes: Spring-stiffening or -softening responses and apparent stiffness prediction

Atomic force microscopy (AFM)-based nanoindentation has emerged as a key nanomechanical mapping method to probe mechanical properties of two-dimensional engineering and biological materials. In comparison with extensive analytical studies on the indentation of freestanding solid thin films, surprisingly little attention has been drawn on analytical predictions of mechanical responses of freestanding lipid membranes to indentation in the framework of Helfrich membrane theory. Here we perform systematic theoretical studies with formulated analysis on indentation of nanopore-spanning lipid membranes. Two types of lipid membranes are considered. One is the membrane subject to uniform area stretching and the membrane is found to exhibit a spring-stiffening behavior during indentation, while the other is the membrane of constant tension and a spring-softening behavior at intermediate indentation is exhibited. The smaller the membrane tension is, the clearer the spring-softening feature is. For both types of membranes, analytical solutions of load–depth relationship at small indentation and apparent membrane stiffnesses are obtained at small and large tension. For the stretched lipid membranes, a cubic nonlinearity arising from membrane stretching is specified. Roles of membrane bending rigidity, (pre-)tension, or area compression modulus, and indenter size in membrane indentation are elucidated. Moreover, pressure of contact between the membrane and indenter and a line load of contact acting along the contact edge are obtained based on the shell theory. Our results provide fundamental insights on nonlinear mechanical behaviors of the lipid membrane indentation.

Wrapping of a vesicle nanoparticle with variable bending stiffness by membrane

Cellular uptake of nanoparticle (NP) is an important biological process involving mechanically and structurally heterogeneous environments, such as biophysical heterogeneity of single extracellular vesicles and biomechanical heterogeneity of small viral capsids. Despite of these heterogeneous environments, poor understanding of membrane interacting with vesicle NP of non-uniform mechanical properties still exists. To address this issue, combining the continuum Canham–Helfrich theory of membrane and the stochastic Markov method on state transition, we establish a theoretical model of the wrapping of a NP with any variable bending stiffness. As NPs are hypothesized with two typical types of non-uniform bending stiffness distributions (Gaussian-like and piecewise), through both minimum energy and stochastic dynamic approaches, it is identified that the membrane tends to encapsulate the NP with a sealing near soft region of NP. Because complete wrapping of NP with a sealing near its stiff region requires large elastic deformation of membrane. In addition, during cellular uptake, rotation phenomenon of the NP with variable bending stiffness is found, which is reminiscent of previous observations on homogeneous NP as reported in literature. These predictions are verified by relevant Monte Carlo simulations. Our findings are of interest to not only the fundamental understanding of cellular uptake but also the applications in the design of nanocarriers for drug delivery.

Application of the Helfrich elasticity theory to the morphology of red blood cells

In this work, we present in detail, in an accessible manner for undergraduate and graduate physics students, the model of spontaneous curvature, due to Helfrich, that quantitatively explains why the red blood cells in their natural state adopt a biconcave shape. The main hypothesis is that the equilibrium cell shape satisfies the principle of minimum free energy. Therefore, in the model, an expression for the membrane free energy is postulated based on the Helfrich theory. In that approximation, the membrane is modelled as a two-dimensional surface and the energy is written as a function of the surface principal curvatures and three parameters, including the spontaneous curvature, c0, which is associated with the chemical composition of the membrane. The negative values for c0 induce invaginations in the cell membrane. The model predicts the discocyte-spherocyte transition for the red blood cell. In the article, the concepts involved in the theory are developed in detail, and an algorithm that allows obtaining the contour of the cell is presented in detail as supplementary material.

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.

Flexoelectric coefficients enhancement via doping carbon nanotubes in nematic liquid crystal host.

Flexoelectric coefficients of carbon nanotube (CNT) doped nematic liquid crystals (NLCs) are studied based on the Helfrich theory. Weak and hard anchoring conditions between the NLC molecules and CNTs are considered. The volume fraction of the CNTs in nematic host is assumed to be low, which makes nanotubes aggregation phenomena negligible. Also, the length of doped CNTs is assumed to be lower than 10μm, so these rigid rods with low concentration cannot possess any flexoelectric polarization by themselves, only their presence modifies the flexoelectric coefficients of the NLC system. The Landau-de Gennes theory is used to calculate the order parameter changes in the medium. Also, the numerical density definition is renewed in the presence of nanotubes. It is shown that in the nematic phase the flexoelectric coefficients increase along with the increase of the coupling strength and temperature. The enhancement in flexoelectric coefficients is more significant in hard anchoring conditions than in the weak anchoring case. The flexoelectric coefficients increase up to 5-fold is calculated near the phase transition temperature, which is in good accordance with the experimental reported data.

Energetics and Stability of Neck Formation in Yeast and Mammalian Endocytosis

The formation and constriction of membrane necks is a fundamental step that immediately precedes fission in membrane remodeling processes like endocytosis, viral egress, and cytokinesis. Even in endocytosis, the formations of invaginations and buds are different in yeast vs mammalian cells. In mammalian cells, structural and biophysical studies have identified that constriction of the neck by dynamin helices triggers membrane fission. However, yeast cells do not have dynamin. A detailed physical understanding of the force and energetic requirements for neck constriction in yeast vs mammalian cells is difficult to ascertain using experimental techniques due to the small length scales nearing self contact. In this work, we apply a modified Helfrich theory of lipid membranes to investigate membrane necking in yeast and mammalian endocytosis. Specifically, we seek to understand how applied forces and protein-induced spontaneous curvature contribute to induce necking and energy distribution on the membrane. Application of a radial constricting force, mimicking the action of dynamin, leads to a snapthrough instability in the shape of the membrane during budding. We observe a large energy barrier that can be overcome by force to achieve a narrow neck. This instability is eliminated by applying a region of spontaneous curvature, contracting the radius or widening the force. Application of a pulling force, essential to yeast endocytosis, also leads to snapthrough instability for a rigid protein coat. Here, the instability is eliminated by ensuring uniform rigidity of coat and membrane. We extend the analysis to images of clathrin coated buds from mammalian and yeast experiments and track the process of neck formation. To do this, we implement an image processing framework that was recently implemented by the authors. Our model sheds insight into the defining characteristics of membrane necking and related snapthrough instabilities.

Elastic and dynamic properties of membrane phase-field models

.Phase-field models have been extensively used to study interfacial phenomena, from solidification to vesicle dynamics. In this article, we analyze a phase-field model that captures the relevant physical features that characterize biological membranes. We show that the Helfrich theory of elasticity of membranes can be applied to phase-field models, allowing to derive the expressions of the stress tensor, lateral stress profile and elastic moduli. We discuss the relevance and interpretations of these magnitudes from a phase-field perspective. Taking the sharp-interface limit we show that the membrane macroscopic equilibrium equation can be derived from the equilibrium condition of the phase-field interface. We also study two dynamic models that describe the behaviour of a membrane. From the study of the relaxational behaviour of the membrane we characterize the relevant dynamics of each model, and discuss their applications.Graphical abstract

Calculating Membrane Area Compressibility from MD Simulations: a Novel Computational Framework

Like thin elastic sheets, lipid bilayers can bend and stretch, and are characterized by mechanical constants such as bending rigidity (Kc) and area compressibility (Ka), which define the free energy of bilayer deformation and thus have direct implications in quantifying the energetics of protein-membrane interactions. Limited experimental data for Kc and Ka of different lipid mixtures makes such calculations challenging. Existing computational methods for calculating Ka from MD simulations exhibit strong dependence on bilayer size, or require multiple simulations with constraints on the area, or tension. We present a novel computational framework that is independent of system size and requires analysis only of a single sufficiently converged simulation trajectory. The method is based on Helfrich's theory of elasticity by relating relative changes in area to relative changes in thickness. The distribution of local fluctuations in the thickness of each bilayer leaflet is used to construct a potential of mean force and recover the bilayer Ka from a quadratic fit. The rapid calculation utilizes freely available software packages. We present results for a variety of single and multicomponent bilayers of phosphatidylcholine, phosphatidylethanolamine and sphingomyelin, utilizing simulation results at different temperatures and bilayer sizes. The calculated compressibility moduli range from 170 to 320 mN/m and agree for the majority of tested cases with values reported from experiments and other computational methods. We further validate our method in a comparison with an existing polymer brush model and confirm the linear relationship between thickness and (Kc/Ka) ˆ (1/2), all calculated from the simulation trajectories.

Energetics of Membrane Necking: Role of Forces and Protein Induced Curvature

The formation and constriction of membrane necks is a fundamental step that immediately precedes fission in membrane remodeling processes like endocytosis, viral egress, and cytokinesis. Extensive experimental studies have identified key molecular players in these processes, such as the GTPase dynamin in the case of several endocytic pathways. Structural and biophysical studies have identified that constriction of the neck by dynamin helices triggers membrane fission. However, a detailed physical understanding of the force and energetic requirement for neck constriction is difficult to ascertain using experimental techniques due to the small length scales nearing self contact. In this work, we apply a modified Helfrich theory of lipid membranes to investigate membrane necking in cylindrical tubes. Specifically, we seek to understand how applied forces and protein-induced spontaneous curvature contribute to induce necking and energy distribution on the membrane. Application of a radial constricting force, mimicking the action of dynamin, leads to a snapthrough instability in the shape of the membrane. We observe a large energy barrier that can be overcome by force to achieve a narrow neck. This instability is eliminated by applying a region of spontaneous curvature. Additionally, we achieve a closed, tear drop shaped bud when we apply a squeezing tangential force to the membrane. Finally, an out of plane twisting force is also considered in a cylindrical co-ordinate system. These results will help to elucidate the mechanical requirements for dynamin-catalyzed membrane necking and other techniques of constriction and scission.

Modelling A Biological Membrane as a Two Phase Viscous Fluid with Curvature Elasticity

We develop a general model of a multicomponent membrane where we treat the membrane as a two phase viscous fluid flowing on a time dependent surface. We present the basic theory of differential geometry, and use it to describe the shape evolution of the membrane. Using Flory--Huggins--de Gennes theory combined with Cahn--Hilliard theory to describe the free energy of a mixture and Helfrich theory to describe the bending energy of a membrane, we use a minimum energy dissipation argument to derive equations of motion for the two phase fluid. By examining two specific parametrizations of the surface, we explore situations under which the membrane undergoes phase separation, and demonstrate the possibility of curvature induced instability.

The Effect of Variable Spontaneous Curvature on Dynamic Evolution of Two-Phase Vesicle

This article aims to study the effect of non-uniform distribution of spontaneous curvature on shape transformation of two-phase vesicles via an evolutionary method. Their dynamic evolution is developed based on conventional Helfrich theory, considering bending of the membrane and friction in the surrounding fluid in each phase with variable spontaneous curvature. The variation of spontaneous curvature is assumed to be a function of arc length in each domain considering the effects of inducing factors (surrounding solution concentration and the membrane-protein interactions such as scaffolding and insertion). Membrane pearling from a large vesicle is simulated by the model and compared with the result of constant curvature and also with empirical observations. It can be shown that accurate simulation of some membrane deformation mechanisms depends on careful consideration of key factors such as the SC variations. In addition, the importance of different uniform and non-uniform distributions of spontaneous curvature is discussed with reference to specific cases.

Clathrin polymerization exhibits high mechano-geometric sensitivity.

How tension modulates cellular transport has become a topic of interest in the recent past. However, the effect of tension on clathrin assembly and vesicle growth remains less understood. Here, we use the classical Helfrich theory to predict the energetic cost that clathrin is required to pay to remodel the membrane at different stages of vesicle formation. Our study reveals that this energetic cost is highly sensitive to not only the tension in the membrane but also to the instantaneous geometry of the membrane during shape evolution. Our study predicts a sharp reduction in clathrin coat size in the intermediate tension regime (0.01-0.1 mN m-1). Remarkably, the natural propensity of the membrane to undergo bending beyond the Ω shape causes a significant decrease in the energy needed from clathrin to drive vesicle growth. Our studies in mammalian cells confirm a reduction in clathrin coat size in an increased tension environment. In addition, our findings suggest that the two apparently distinct clathrin assembly modes, namely coated pits and coated plaques, observed in experimental investigations might be a consequence of varied tensions in the plasma membrane. Overall, the mechano-geometric sensitivity revealed in this study might also be at play during the polymerization of other membrane remodeling proteins.

Anisotropic spontaneous curvatures in lipid membranes.

Symmetry restrictions due to fluidity require the strain energy in the Helfrich theory of lipid membranes to be locally isotropic in nature. Although this framework is suitable for modeling the interaction of membranes with proteins that generate spherical curvature such as clathrin, there are other important membrane-bending proteins such as BIN-amphiphysin-Rvs proteins that form a cylindrical coat with different curvatures in the longitudinal and the circumferential directions. In this work, we present a detailed mathematical treatment of the theory of lipid membranes incorporating anisotropic spontaneous curvatures. We derive the associated Euler-Lagrange equations and the edge conditions in a generalized setting that allows spatial heterogeneities in the properties of the membrane-protein system. We employ this theory to model the constriction of a membrane tubule by a cylindrical scaffold. In particular, we highlight the role of the equilibrium equation in the tangential plane in regulating the spatial variation of the surface tension field.

Challenges in Nanoparticle Delivery: Cellular Uptake and Intracellular Escape

Control of cellular uptake of nanoparticles is crucial step in drug delivery and nanomedicine. However, the conditions under which passive endocytosis, i.e. not ATP driven, occurs are not well understood. using Molecular Dynamics simulations with coarse grained model we investigated the conditions of the passive uptake and consequential cytosol escape of the ligand-coated nanoparticles with varying size, shape, coverage, and receptor-binding strength. In qualitative agreement with Helfrich theory we have found that larger spherical particles undergo endocytosis easier than smaller ones due to a more favorable compromise between bending rigidity and surface adhesive energy. Moreover, our simulations and elastic analysis suggest that the prolate shape of spherocylinders can lead to more efficient delivery than spheres of the same diameter as both shapes seems to have the same kinetic barrier for uptake across a lipid membrane but the spherocylinders have a larger volume. In addition, we observed that the sharp edges on nanoparticles can hinder the uptake process. The cytosol escape of the uptaken nanoparticles can occur upon a decrease of ligand-receptor interaction and a sufficient deformation of the membrane vesicle from its equilibrium state. The deformation can be administered by a non-spherical shape or a pressure exerted by the nanoparticle. Large rigid spheres were not observed to be able of the escape during the time of simulations. The insight should be helpful in rational design of drug delivery nanocarriers.

Shapes of pored membranes

We study the shapes of pored membranes within the framework of the Helfrich theory under the constraints of fixed area and pore size. We show that the mean curvature term leads to a budding- like structure, while the Gaussian curvature term tends to flatten the membrane near the pore; this is corroborated by simulation. We propose a scheme to deduce the ratio of the Gaussian rigidity to the bending rigidity simply by observing the shape of the pored membrane. This ratio is usually difficult to measure experimentally. In addition, we briefly discuss the stability of a pore by relaxing the constraint of a fixed pore size and adding the line tension. Finally, the flattening effect due to the Gaussian curvature as found in studying pored membranes is extended to two-component membranes. We find that sufficiently high contrast between the components' Gaussian rigidities leads to budding which is distinct from that due to the line tension.

Thermodynamics of Charged Nanoparticle Adsorption on Charge-Neutral Membranes: A Simulation Study

The interactions between charged nanoparticles (NPs) and charge-neutral phospholipid membranes are investigated by coarse-grained molecular dynamics simulations. Three kinds of nanoparticles are modeled with different surface charge densities: the uncharged one, the positively charged one, and the negatively charged one. We find that the electrostatic attraction improves the adhesion of a charged nanoparticle to the membrane. With the increase of electrostatic energy, a charged NP can be almost fully wrapped by the membrane. In addition, analyses of structural variations suggest that the adhesion of a charged NP induces a local transition in fluid bilayers. Some thermodynamic quantities such as free energy, entropy, and enthalpy are also obtained to explain the process of NPs binding. Furthermore, the bending energy of wrapping of NPs against the electrostatic potential energy is also discussed based on the Helfrich theory, indicating that the driving force of the wrap originates from the gain in electrostatic energy at the cost of the elastic energy of biomembranes. Our observations shed light on the origin of experiments of the wrap as well as the mechanism of structural transitions of membranes due to the electrostatic binding.

Converse Flexoelectric Effect in Bent-Core Nematic Liquid Crystals

We report on the converse flexoelectric effect in two bent-core nematic liquid crystals with opposite dielectric anisotropies. The results are based on electro-optic investigations of inplane field-driven distortions in homeotropic samples (the Helfrich method). They are interpreted by an extension of the Helfrich theory that takes into account the higher order distortions. The bend flexocoefficient for both the compounds is of the usual order of magnitude as in calamitics, unlike in a previously investigated bent-core nematic for which giant values of the bend flexocoefficient are reported. In order to resolve this discrepancy, we propose a molecular model with nonpolar clusters showing quadrupolar flexoelectricity. The study also includes measurements on surface polarization instabilities in the dielectrically positive material; the splay flexocoefficient thereby deduced is also of the conventional order.