Elastic properties and line tension of self-assembled bilayer membranes

Stability and Elasticity of Ultrathin Sphere-Patterned Block Copolymer Films

Sphere-patterned ultrathin block copolymer films are potentially interesting for a variety of applications in nanotechnology. We use self-consistent field theory to investigate the elastic response of sphere monolayer films with respect to in-plane shear, in-plane extension, compression deformations, and bending. The relations between the in-plane elastic moduli are roughly compatible with the expectations for two-dimensional elastic systems with hexagonal symmetry, with one notable exception: The pure shear and the simple shear moduli differ from each other by roughly 20%. Even more importantly, the bending constants are found to be negative, indicating that free-standing block copolymer membranes made of only a sphere monolayer are inherently unstable above the glass transition. Our results are discussed in view of the experimental findings.

Membrane fission via transmembrane contact

Division of intracellular organelles often correlates with additional membrane wrapping, e.g., by the endoplasmic reticulum or the outer mitochondrial membrane. Such wrapping plays a vital role in proteome and lipidome organization. However, how an extra membrane impacts the mechanics of the division has not been investigated. Here we combine fluorescence and cryo-electron microscopy experiments with self-consistent field theory to explore the stress-induced instabilities imposed by membrane wrapping in a simple double-membrane tubular system. We find that, at physiologically relevant conditions, the outer membrane facilitates an alternative pathway for the inner-tube fission through the formation of a transient contact (hemi-fusion) between both membranes. A detailed molecular theory of the fission pathways in the double membrane system reveals the topological complexity of the process, resulting both in leaky and leakless intermediates, with energies and topologies predicting physiological events.

Mimicking effects of cholesterol in lipid bilayer membranes by self-assembled amphiphilic block copolymers

The effect of cholesterol on biological membranes is important in biochemistry. In this study, a polymer system is used to simulate the consequences of varying cholesterol content in membranes. The system consists of an AB-diblock copolymer, a hydrophilic homopolymer hA, and a hydrophobic rigid homopolymer C, corresponding to phospholipid, water, and cholesterol, respectively. The effect of the C-polymer content on the membrane is studied within the framework of a self-consistent field model. The results show that the liquid-crystal behavior of B and C has a great influence on the chemical potential of cholesterol in bilayer membranes. The effects of the interaction strength between components, characterized by the Flory-Huggins parameters and the Maier-Saupe parameter, were studied. Some consequences of adding a coil headgroup to the C-rod are presented. Results of our model are compared to experimental findings for cholesterol-containing lipid bilayer membranes.

Self-Assembly of Lipid Mixtures in Solutions: Structures, Dynamics Processes and Mechanical Properties

The self-assembly of lipid mixtures in aqueous solution was investigated by dissipative particle dynamics simulation. Two types of lipid molecules were modelled, where three mixed structures, i.e., the membrane, perforated membrane and vesicle, were determined in the self-assembly processes. Phase behaviour was investigated by using the phase diagrams based on the tail chain lengths for the two types of lipids. Several parameters, such as chain number and average radius of gyration, were employed to explore the structural formations of the membrane and perforated membrane in the dynamic processes. Interface tension was used to demonstrate the mechanical properties of the membrane and perforated membrane in the equilibrium state and dynamics processes. Results help us to understand the self-assembly mechanism of the biomolecule mixtures, which has a potential application for designing the lipid molecule-based bio-membranes in solutions.

Tilt Modulus of Bilayer Membranes Self-Assembled from Rod-Coil Diblock Copolymers

Quantitatively understanding membrane fission and fusion requires a mathematical model taking their underlying elastic degrees of freedom, such as the molecule's tilt, into account. Hamm-Kozlov's model is such a framework that includes a tilt modulus along with the bending modulus and Gaussian modulus. This paper investigates the tilt modulus of liquid-crystalline bilayer membranes by applying self-consistent field theory. Unlike the widely used method in molecular dynamics simulation which extracts the tilt modulus by simulating bilayer buckles with various single modes, we introduce a tilt constrain term in the free energy to stabilize bilayers with various tilt angles. Fitting the energy curve as a function of the tilt angle to Hamm-Kozlov's elastic energy allows us to extract the tilt modulus directly. Based on this novel scheme and focused on the bilayers self-assembled from rod-coil diblock copolymers, we carry out a systematic study of the dependence of the tensionless A-phase bilayer's tilt modulus on the microscopic parameters.

Analytical Calculation of the Elastic Moduli of Self-Assembled Liquid-Crystalline Bilayer Membranes

Liquid-crystalline orders are ubiquitous in membranes and could significantly affect the elastic properties of the self-assembled bilayers. Calculating the free energy of bilayer membranes with different geometries and fitting them to their theoretical expressions allow us to extract the elastic moduli, such as the bending modulus and Gaussian modulus. However, this procedure is time-consuming for liquid-crystalline bilayers. In paper reports a novel method to calculate the elastic moduli of the self-assembled liquid-crystalline bilayers within the self-consistent field theory framework. Based on the asymptotic expansion method, we derive the analytical expression of the elastic moduli, which reduces the computational cost significantly. Numerical simulations illustrate the validity and efficiency of the proposed method.

Free energy of hydrophilic and hydrophobic pores in lipid bilayers by free energy perturbation of a restraint

The free energy of pore formation in lipid bilayers has been previously calculated using a variety of reaction coordinates. Here, we use free energy perturbation of a cylindrical lipid exclusion restraint to compute the free energy profile as a function of pore radius in dimyristoylphosphatidylcholine (DMPC) and dioleoylphosphatidylcholine (DOPC) bilayers. Additionally restraining the headgroups to lie on the membrane surface allows us to also calculate the free energy profile of hydrophobic pores, i.e., cylindrical pores lined by acyl chains. For certain pore radii, the free energy of wetting of hydrophobic pores is calculated using the density bias method. It is found that wetting of hydrophobic pores becomes thermodynamically favorable at 5.0 Å for DMPC and 6.5 Å for DOPC, although significant barriers prevent spontaneous wetting of the latter on a nanosecond time scale. The free energy of transformation of hydrophilic pores to hydrophobic ones is also calculated using free energy perturbation of headgroup restraints along the bilayer normal. This quantity, along with wetting and pore growth free energies, provides complete free energy profiles as a function of radius. Pore line tension values for the hydrophilic pores obtained from the slope of the free energy profiles are 37.6 pN for DMPC and 53.7 pN for DOPC. The free energy profiles for the hydrophobic pores are analyzed in terms of elementary interfacial tensions. It is found that a positive three-phase line tension is required to explain the results. The estimated value for this three-phase line tension (51.2 pN) lies within the expected range.

Regulation Mechanism of Bubbling Deformation and Fracture Toughness of the Membrane by Asymmetric Phospholipids: A Model System Study

Dynamic regulation of the deformation modulus and fracture toughness of a membrane is critical to organelles and cells for matching their conflicting needs of resilient and fractured behaviors. These properties implement the protection of the function in the normal condition and the fission function in the endocytosis condition of a membrane. Naturally, a membrane contains phospholipids that have different hydrophilic and hydrophobic group length. The diffusion and aggregation of the phospholipids with asymmetry of the hydrophilic-hydrophobic ratio on the membrane play a key role in regulating the mechanical behaviors passively to the external force. In present work, the effects of the asymmetry of phospholipids on the bubbling deformation and fracture toughness of the membrane to external stretching are investigated in a model system. A disk-shaped micelle formed from the blend of symmetric and asymmetric diblock copolymers in a selective solvent is considered as the membrane sheet. Its mechanically responsive behaviors are investigated by self-consistent field theory. By analyzing the evolution of different components during the stretching process, the mechanism of formation of the bubbling structure is revealed. Moreover, the fracture toughness depending on the asymmetry of the phospholipids is determined quantitatively.

Elastic properties of self-assembled bilayer membranes: Analytic expressions via asymptotic expansion

Bilayer membranes self-assembled from amphiphilic molecules are ubiquitous in biological and soft matter systems. The elastic properties of bilayer membranes are essential in determining the shape and structure of bilayers. A novel method to calculate the elastic moduli of the self-assembled bilayers within the framework of the self-consistent field theory is developed based on an asymptotic expansion of the order parameters in terms of the bilayer curvature. In particular, the asymptotic expansion method is used to derive analytic expressions of the elastic moduli, which allows us to design more efficient numerical schemes. The efficiency of the proposed method is illustrated by a model system composed of flexible amphiphilic chains dissolved in hydrophilic polymeric solvents.

The responsive behaviors of bilayer membrane under uniaxial mechanical probe

In experiments, atomic force microscopy technology was used to measure the modulus of the membrane. However, these studies mainly focus on the linear responsive behavior. In the present work, a theoretical study is performed to show the nonlinear responsive behavior, which includes the stretching induced structural transitions. It demonstrates that the structural transition of the bilayer membrane takes place during the stretching process of the mechanical probe. A vertical cylindrical micelle can be obtained by stretching the membrane under deep compression conditions, and the cylindrical micelle can grow continuously along the axial direction. Moreover, under shallow compression conditions, the probe pulls a spherical micelle from the membrane, and then, the membrane returns to flatness. A comprehensive study is performed to show the mechanism of the responsive behaviors of the structural transition during the compression and stretching processes. When the probe acts on the B-rich layer, it is more likely to pull out a regular micelle. However, when the probe acts on the bottom A-rich layer, complex vesicles are more likely to be pulled out from the bilayer membrane. This study provides a comprehensive diagram of the mechanical responsive behavior of the membrane, which would be a guide for an experiment of biomembranes and the design of new self-assembled structures.

Pathways connecting two opposed bilayers with a fusion pore: a molecularly-informed phase field approach

A phase field model with two phase fields, representing the concentration and the head-tail separation of amphiphilic molecules, respectively, has been constructed using an extension of the Ohta-Kawasaki model (Macromolecules, 1986, 19, 2621-2632). It is shown that this molecularly-informed phase field model is capable of producing various self-assembled amphiphilic aggregates, such as bilayers, vesicles and micelles. Furthermore, pathways connecting two opposed bilayers with a fusion pore are obtained by using a combination of the phase field model and the string method. Multiple fusion pathways, including a classical pathway and a leaky pathway, have been obtained depending on the initial separation of the two bilayers. The study shed light on the understanding of the membrane fusion pathways and, more importantly, laid a foundation for further investigation of more complex membrane morphologies and transitions.

Elastic properties of liquid-crystalline bilayers self-assembled from semiflexible-flexible diblock copolymers

The mechanical response and shape of self-assembled bilayer membranes depend crucially on their elastic properties. Most of the studies focused on the elastic properties of fluid membranes, despite the ubiquitous presence of membranes with liquid-crystalline order. Here the elastic properties of liquid-crystalline bilayers self-assembled from diblock copolymers composed of a semiflexible block are studied theoretically. Specifically, the self-consistent field theory (SCFT) is applied to a model system composed of semiflexible-flexible diblock copolymers dissolved in flexible homopolymers that act as solvents. The free energy of self-assembled tensionless bilayer membranes in three different geometries, i.e. planar, cylindrical and spherical, is obtained by solving the SCFT equations using a hybrid method, in which the orientation-dependent functions are treated using the spherical harmonics, whereas the position-dependent operators are treated using the compact difference schemes. The bending modulus κ_M and Gaussian modulus κ_G of the bilayer are extracted from the free energies. The effects of the molecular parameters of the system, such as the chain rigidity and the orientational interaction, are systematically examined.

Elastic property of membranes self-assembled from diblock and triblock copolymers

The elastic property of membranes self-assembled from AB diblock and ABA triblock copolymers, as coarse-grained model of lipids and the bolalipids, are studied using the self-consistent field theory (SCFT). Specifically, solutions of the SCFT equations, corresponding to membranes in different geometries (planar, cylindrical, spherical, and pore) have been obtained for a model system composed of amphiphilic AB diblock copolymers and ABA triblock copolymers dissolved in A homopolymers. The free energy of the membranes with different geometries is then used to extract the bending modulus, Gaussian modulus, and line tension of the membranes. The results reveal that the bending modulus of the triblock membrane is greater than that of the diblock membrane. Furthermore, the Gaussian modulus and line tension of the triblock membrane indicate that the triblock membranes have higher pore formation energy than that of the diblock membranes. The equilibrium bridging and looping fractions of the triblock copolymers are also obtained. Implications of the theoretical results on the elastic properties of biologically equivalent lipid bilayers and the bolalipid membranes are discussed.

Dissipative particle dynamics simulation of critical pore size in a lipid bilayer membrane

We investigate with computer simulations the critical radius of pores in a lipid bilayer membrane. Ilton et al. (Ilton et al. 2016 Phys. Rev. Lett. 117, 257801 (doi:10.1103/PhysRevLett.117.257801)) recently showed that nucleated pores in a homopolymer film can increase or decrease in size, depending on whether they are larger or smaller than a critical size which scales linearly with film thickness. Using dissipative particle dynamics, a particle-based simulation method, we investigate the same scenario for a lipid bilayer membrane whose structure is determined by lipid-water interactions. We simulate a perforated membrane in which holes larger than a critical radius grow, while holes smaller than the critical radius close, as in the experiment of Ilton et al. (Ilton et al. 2016 Phys. Rev. Lett. 117, 257801 (doi:10.1103/PhysRevLett.117.257801)). By altering key system parameters such as the number of particles per lipid and the periodicity, we also describe scenarios in which pores of any initial size can seal or even remain stable, showing a fundamental difference in the behaviour of lipid membranes from polymer films.

Liquid crystalline bilayers self-assembled from rod-coil diblock copolymers

The structure and phase behaviour of bilayer membranes self-assembled from rod-coil diblock copolymers are studied using the self-consistent field theory, focusing on the occurrence and relative stability of liquid crystalline phases induced by the geometric shape and orientational interaction of the rod-blocks. A variety of liquid crystalline bilayers, corresponding to the smectic phases in bulk systems, are predicted to occur as equilibrium phases of the system. The ordered morphologies and phase behaviour of the system are analyzed. Phase diagrams of the self-assembled bilayers are constructed. The theoretical results provide an understanding of the formation mechanisms of these intricate phases.

Formation and size distribution of self-assembled vesicles

When detergents and phospholipid membranes are dispersed in aqueous solutions, they tend to self-assemble into vesicles of various shapes and sizes by virtue of their hydrophobic and hydrophilic segments. A clearer understanding of such vesiculation processes holds promise for better elucidation of human physiology and disease, and paves the way to improved diagnostics, drug development, and drug delivery. Here we present a detailed analysis of the energetics and thermodynamics of vesiculation by recourse to nonlinear elasticity, taking into account large deformation that may arise during the vesiculation process. The effects of membrane size, spontaneous curvature, and membrane stiffness on vesiculation and vesicle size distribution were investigated, and the critical size for vesicle formation was determined and found to compare favorably with available experimental evidence. Our analysis also showed that the critical membrane size for spontaneous vesiculation was correlated with membrane thickness, and further illustrated how the combined effects of membrane thickness and physical properties influenced the size, shape, and distribution of vesicles. These findings shed light on the formation of physiological extracellular vesicles, such as exosomes. The findings also suggest pathways for manipulating the size, shape, distribution, and physical properties of synthetic vesicles, with potential applications in vesicle physiology, the pathobiology of cancer and other diseases, diagnostics using in vivo liquid biopsy, and drug delivery methods.

Direct Measurement of the Critical Pore Size in a Model Membrane

We study pore nucleation in a model membrane system, a freestanding polymer film. Nucleated pores smaller than a critical size close, while pores larger than the critical size grow. Holes of varying size were purposefully prepared in liquid polymer films, and their evolution in time was monitored using optical and atomic force microscopy to extract a critical radius. The critical radius scales linearly with film thickness for a homopolymer film. The results agree with a simple model which takes into account the energy cost due to surface area at the edge of the pore. The energy cost at the edge of the pore is experimentally varied by using a lamellar-forming diblock copolymer membrane. The underlying molecular architecture causes increased frustration at the pore edge resulting in an enhanced cost of pore formation.

Physical mechanisms of micro- and nanodomain formation in multicomponent lipid membranes

This article summarizes a variety of physical mechanisms proposed in the literature, which can generate micro- and nanodomains in multicomponent lipid bilayers and biomembranes. It mainly focusses on lipid-driven mechanisms that do not involve direct protein-protein interactions. Specifically, it considers (i) equilibrium mechanisms based on lipid-lipid phase separation such as critical cluster formation close to critical points, and multiple domain formation in curved geometries, (ii) equilibrium mechanisms that stabilize two-dimensional microemulsions, such as the effect of linactants and the effect of curvature-composition coupling in bilayers and monolayers, and (iii) non-equilibrium mechanisms induced by the interaction of a biomembrane with the cellular environment, such as membrane recycling and the pinning effects of the cytoplasm. Theoretical predictions are discussed together with simulations and experiments. The presentation is guided by the theory of phase transitions and critical phenomena, and the appendix summarizes the mathematical background in a concise way within the framework of the Ginzburg-Landau theory. This article is part of a Special Issue entitled: Lipid order/lipid defects and lipid-control of protein activity edited by Dirk Schneider.

Calculation of membrane bending rigidity using field-theoretic umbrella sampling

The free-energy change of membrane shape transformations can be small, e.g., as in the case of membrane bending. Therefore, the calculation of the free-energy difference between different membrane morphologies is a challenge. Here, we discuss a computational method - field-theoretic umbrella sampling - to compute the local chemical potential of a non-equilibrium configuration and illustrate how one can apply it to study free-energy changes of membrane transformations using simulations. Specifically, the chemical potential profile of the bent membrane and the bending rigidity of membrane are calculated for a soft, coarse-grained amphiphile model and the MARTINI model of a dioleoylphosphatidylcholine (DOPC) membrane.

Thermal fluctuations of vesicles and nonlinear curvature elasticity--implications for size-dependent renormalized bending rigidity and vesicle size distribution

Both closed and open biological membranes noticeably undulate at physiological temperatures. These thermal fluctuations influence a broad range of biophysical phenomena, ranging from self-assembly to adhesion. In particular, the experimentally measured thermal fluctuation spectra also provide a facile route to the assessment of mechanical and certain other physical properties of biological membranes. The theoretical assessment of thermal fluctuations, be it for closed vesicles or the simpler case of flat open lipid bilayers, is predicated upon assuming that the elastic curvature energy is a quadratic functional of the curvature tensor. However, a qualitatively correct description of several phenomena such as binding-unbinding transition, vesicle-to-bicelle transition, appearance of hats and saddles among others, appears to require consideration of constitutively nonlinear elasticity that includes fourth order curvature contributions rather than just quadratic. In particular, such nonlinear considerations are relevant in the context of large-curvature or small-sized vesicles. In this work we discuss the statistical mechanics of closed membranes (vesicles) incorporating both constitutive and geometrical nonlinearities. We derive results for the renormalized bending rigidity of small vesicles and show that significant stiffening may occur for sub-20 nm vesicle sizes. Our closed-form results may also be used to determine nonlinear curvature elasticity properties from either experimentally measured fluctuation spectra or microscopic calculations such as molecular dynamics. Finally, in the context of our results on thermal fluctuations of vesicles and nonlinear curvature elasticity, we reexamine the problem of determining the size distribution of vesicles and obtain results that reconcile well with experimental observations. However, our results are somewhat paradoxical. Specifically, the molecular dynamics predictions for the thermo-mechanical behavior of small vesicles of prior studies appear to be inconsistent with the nonlinear elastic properties that we estimate by fitting to the experimentally determined vesicle size-distribution trends and data.

An engineered membrane to measure electroporation: effect of tethers and bioelectronic interface

This article reports on the construction and predictive models for a platform comprised of an engineered tethered membrane. The platform provides a controllable and physiologically relevant environment for the study of the electroporation process. The mixed self-assembled membrane is formed via a rapid solvent exchange technique. The membrane is tethered to the gold electrode and includes an ionic reservoir separating the membrane and gold surface. Above the membrane, there is an electrolyte solution, and a gold counterelectrode. A voltage is applied between the gold electrodes and the current measured. The current is dependent on the energy required to form aqueous pores and the conductance of each pore. A two-level predictive model, consisting of a macroscopic and a continuum model, is developed to relate the pore dynamics to the measured current. The macroscopic model consists of an equivalent circuit model of the tethered membrane, and asymptotic approximations to the Smoluchowski-Einstein equation of electroporation that is dependent on the pore conductance and the energy required to form aqueous pores. The continuum model is a generalized Poisson-Nernst-Planck (GPNP) system where an activity coefficient to account for steric effects of ions is added to the standard PNP system. The GPNP is used to evaluate the conductance of aqueous pores, and the electrical energy required to form the pores. As an outcome of the setup of the device and the two-level model, biologically important variables can be estimated from experimental measurements. To validate the accuracy of the two-level model, the predicted current is compared with experimentally measured current for different tethering densities.

Line tension of multicomponent bilayer membranes

The line tension or edge energy of bilayer membranes self-assembled from binary amphiphilic molecules is studied using self-consistent-field theory (SCFT). Specifically, solutions of the SCFT equations corresponding to an infinite membrane with a circular pore, or an open membrane, are obtained for a coarse-grained model in which the amphiphilic species and hydrophilic solvents are represented by ABandED diblock copolymers and C homopolymers, respectively. The edge energy of the membrane is extracted from the free energy of the open membranes. Results for membranes composed of mixtures of symmetric and cone- or inverse cone-shaped amphiphilic molecules with neutral and/or repulsive interactions are obtained and analyzed. It is observed that an increase in the concentration of the cone-shaped species leads to a decrease of the line tension. In contrast, adding inverse cone-shaped copolymers results in an increase of the line tension. Furthermore, the density profile of the copolymers reveals that the line tension is regulated by the distribution of the amphiphiles at the bilayer edge.

Emergence and stability of intermediate open vesicles in disk-to-vesicle transitions

The transition between two basic structures, a disk and an enclosed vesicle, of a finite membrane is studied by examining the minimum energy path (MEP) connecting these two states. The MEP is constructed using the string method applied to continuum elastic membrane models. The results reveal that, besides the commonly observed disk and vesicle, open vesicles (bowl-shaped vesicles or vesicles with a pore) can become stable or metastable shapes. The emergence, stability, and probability distribution of these open vesicles are analyzed. It is demonstrated that open vesicles can be stabilized by higher-order elastic energies. The estimated probability distribution of the different structures is in good agreement with available experiments.