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

Periodic cylindrical bilayers self-assembled from biblock polymers

Amphiphilic polymers in aqueous solutions can self-assemble to form bilayer membranes, and their elastic properties can be captured using the well-known Helfrich model involving several elastic constants. In this paper, we employ the self-consistent field model to simulate sinusoidal bilayers self-assembled from diblock copolymers where an appropriate constraint term is introduced to stabilize periodic bilayers with prescribed amplitudes. Then, we devise several methods to extract the shape of these bilayers and examine the accuracy of the free energy predicted by the Helfrich model. Numerical results show that when the bilayer curvature is small, the Helfrich model predicts the excess free energy more accurately. However, when the curvature is large, the accuracy heavily depends on the method used to determine the shape of the bilayer. In addition, the dependence of free energy on interaction strength, constraint amplitude, and constraint period are systematically studied. Moreover, we have devised a method for attaining equilibrium states through the adjustment of constraints. Within the self-consistent field model, these equilibrium states manifest as distinct periodic cylindrical bilayers, which are consonant with the theoretical predictions formulated using the shape equations.

Morphological Influence on a Nonionic Bilayer Bending Rigidity and Compression Modulus

The mechanical properties of multilamellar vesicles and their relevance to soft matter physics and material science are of significant interest. The bending rigidity, κ, and compression modulus, B, of three-dimensional (3D) finite nonspontaneous multilamellar vesicles, formed by a nonionic surfactant, are linked to nanoscale bilayer thickness, δ, estimated via small-angle X-ray scattering, and macroscopic elastic modulus measured through small-amplitude oscillatory shear experiments. κ and B significantly differ from the same system in the two-dimensional (2D) infinite nanostructured planar lamellar phase. Particularly, κ3D was found to be much smaller than κ2D, while an opposite behavior was seen for B. The 2D-to-3D morphology transition occurs under a transient mechanical field, resulting in rheopectic behavior. κ scales quadratically with δ, consistent with bilayer membrane theories, and linearly with vesicle radius in the densely packed state. These findings have implications for understanding and designing soft interfaces due to the influence of bending rigidity on transport properties.

Pure measures of bending for soft plates

This paper, originally motivated by a question raised by Wood and Hanna [Soft Matter, 2019, 15, 2411], shows that pure measures of bending for soft plates can be defined by introducing the class of bending-neutral deformations, which represent finite incremental changes in the plate's shape that do not induce any additional bending. This class of deformations is subject to a geometric compatibility condition, which is fully characterized. A tensorial pure measure of bending, which is invariant under bending-neutral deformations, is described in detail. As shown by an illustrative class of examples, the general notion of a pure measure of bending could be useful in formulating direct theories for soft plates, where stretching and bending energies are treated separately.

Forceful patterning: theoretical principles of mechanochemical pattern formation

Biological pattern formation is essential for generating and maintaining spatial structures from the scale of a single cell to tissues and even collections of organisms. Besides biochemical interactions, there is an important role for mechanical and geometrical features in the generation of patterns. We review the theoretical principles underlying different types of mechanochemical pattern formation across spatial scales and levels of biological organization.

The effects of internal forces and membrane heterogeneity on three-dimensional cell shapes

The shape of cells and the control thereof plays a central role in a variety of cellular processes, including endo- and exocytosis, cell division and cell movement. Intra- and extracellular forces control the shapes, and while the shape changes in some processes such as exocytosis are intracellularly-controlled and localized in the cell, movement requires force transmission to the environment, and the feedback from it can affect the cell shape and mode of movement used. The shape of a cell is determined by its cytoskeleton (CSK), and thus shape changes involved in various processes involve controlled remodeling of the CSK. While much is known about individual components involved in these processes, an integrated understanding of how intra- and extracellular signals are coupled to the control of the mechanical changes involved is not at hand for any of them. As a first step toward understanding the interaction between intracellular forces imposed on the membrane and cell shape, we investigate the role of distributed surrogates for cortical forces in producing the observed three-dimensional shapes. We show how different balances of applied forces lead to such shapes, that there are different routes to the same end state, and that state transitions between axisymmetric shapes need not all be axisymmetric. Examples of the force distributions that lead to protrusions are given, and the shape changes induced by adhesion of a cell to a surface are studied. The results provide a reference framework for developing detailed models of intracellular force distributions observed experimentally, and provide a basis for studying how movement of a cell in a tissue or fluid is influenced by its shape.

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.

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.

Newton-Cartan submanifolds and fluid membranes

We develop the geometric description of submanifolds in Newton-Cartan spacetime. This provides the necessary starting point for a covariant spacetime formulation of Galilean-invariant hydrodynamics on curved surfaces. We argue that this is the natural geometrical framework to study fluid membranes in thermal equilibrium and their dynamics out of equilibrium. A simple model of fluid membranes that only depends on the surface tension is presented and, extracting the resulting stresses, we show that perturbations away from equilibrium yield the standard result for the dispersion of elastic waves. We also find a generalization of the Canham-Helfrich bending energy for lipid vesicles that takes into account the requirements of thermal equilibrium.

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.

Construction of Nuclear Envelope Shape by a High-Genus Vesicle with Pore-Size Constraint

Nuclear pores have an approximately uniform distribution in the nuclear envelope of most living cells. Hence, the morphology of the nuclear envelope is a spherical stomatocyte with a high genus. We have investigated the morphology of high-genus vesicles under pore-size constraint using dynamically triangulated membrane simulations. Bending-energy minimization without volume or other constraints produces a circular-cage stomatocyte, where the pores are aligned in a circular line on an oblate bud. As the pore radius is reduced, the circular-pore alignment is more stabilized than a random pore distribution on a spherical bud. However, we have clarified the conditions for the formation of a spherical stomatocyte: a small perinuclear volume, osmotic pressure within nucleoplasm, and repulsion between the pores. When area-difference elasticity is taken into account, the formation of cylindrical or budded tubules from the stomatocyte and discoidal stomatocyte is found.

General neck condition for the limit shape of budding vesicles

The shape equation and linking conditions for a vesicle with two phase domains are derived. We refine the conjecture on the general neck condition for the limit shape of a budding vesicle proposed by Jülicher and Lipowsky [Phys. Rev. Lett. 70, 2964 (1993)PRLTAO0031-900710.1103/PhysRevLett.70.2964; Phys. Rev. E 53, 2670 (1996)1063-651X10.1103/PhysRevE.53.2670], and then we use the shape equation and linking conditions to prove that this conjecture holds not only for axisymmetric budding vesicles, but also for asymmetric ones. Our study reveals that the mean curvature at any point on the membrane segments adjacent to the neck satisfies the general neck condition for the limit shape of a budding vesicle when the length scale of the membrane segments is much larger than the characteristic size of the neck but still much smaller than the characteristic size of the vesicle.

Bilayers directly scrolling up to form nanotubes via self-assembly of an achiral small molecule

The self-assembly behavior of a molecule composed of an azobenzene segment, carboxylic-acid group and flexible alkyl chains (denoted by ABA11) was studied as an extension of our proceeding work. We have previously reported that in DMSO solution ABA11 self-assembled into nanotubes starting from nanosheets and experiencing a meta-state of helical ribbons. Herein, we found that changing the solvent can also affect the self-assembly pathway of ABA11 to nanotubes. When DMSO was replaced by ethanol, the nanosheets formed by ABA11 bilayers either directly scrolled up to form nanotubes without a meta-state of helical ribbons at low concentrations, or stacked up to form nanobricks at higher concentrations. In addition, increasing the water content in ethanol can significantly facilitate the formation of the assembled nanostructures.

Quantitative Profiling of Nanoscale Liposome Deformation by a Localized Surface Plasmon Resonance Sensor

Characterizing the shape of sub-100 nm, biological soft-matter particulates (e.g., liposomes and exosomes) adsorbed at a solid-liquid interface remains a challenging task. Here, we introduce a localized surface plasmon resonance (LSPR) sensing approach to quantitatively profile the deformation of nanoscale, fluid-phase 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) liposomes contacting a titanium dioxide substrate. Experimental and theoretical results validate that, due to its high sensitivity to the spatial proximity of phospholipid molecules near the sensor surface, the LSPR sensor can discriminate fine differences in the extent of ionic strength-modulated liposome deformation at both low and high surface coverages. By contrast, quartz crystal microbalance-dissipation (QCM-D) measurements performed with equivalent samples were qualitatively sensitive to liposome deformation only at saturation coverage. Control experiments with stiffer, gel-phase 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) liposomes verified that the LSPR measurement discrimination arises from the extent of liposome deformation, while the QCM-D measurements yield a more complex response that is also sensitive to the motion of adsorbed liposomes and coupled solvent along with lateral interactions between liposomes. Collectively, our findings demonstrate the unique measurement capabilities of LSPR sensors in the area of biological surface science, including competitive advantages for probing the shape properties of adsorbed, nanoscale biological particulates.

Polymorphic transformation towards formation of nanotubes by self-assembly of an achiral molecule

In this paper, nanotubes with a uniform diameter were prepared by self-assembly of an achiral azobenzene-containing fatty acid. The polymorphic transformation of the assemblies during the cooling process was systematically studied. By controlling the incubation temperature, different morphologies, such as membranes, stripes, helical ribbons and tubes, were all obtained in our experiment. These elements were all predicted by Selinger et al. in the theoretical model of the formation of nanotubes. To the best of our knowledge, this is the first experimental example to fully support their theory.