Analytical calculation of the lipid bilayer bending modulus

On the Gaussian modulus of lipid membranes

The Gaussian modulus is a crucial property that influences topological transformations in lipid membranes. However, unlike the bending modulus, estimating the Gaussian modulus has been particularly challenging due to the constraints imposed by the Gauss-Bonnet theorem. Despite this, various theoretical, computational, and experimental approaches have been developed to estimate the Gaussian modulus, though they are often complex, and analytical estimates remain rare. In this work, we present a minimalist model inspired by the folding of a sheet of paper, which provides an exact calculation of the Gaussian modulus. Remarkably, the induced deformation does not affect the Gaussian curvature or alter the system's topology, yet it yields the modulus that governs these geometric properties.

Curvature expansion method for polymer self-assembled membranes

Polymer vesicles consist of a thin bilayer membrane, the shape of which can be precisely controlled by adjusting the composition of block copolymers. These self-assembled structures provide a powerful method for mimicking biological cell membranes. A key characteristic of these membranes is their elastic moduli, which play a crucial role in determining their mechanical properties. Traditionally, these moduli are derived by fitting the free energy of different membrane shapes to Helfrich's model. In this paper, we present a curvature expansion method that yields an analytical expression for the moduli within the framework of self-consistent field theory. A significant advantage of our approach is its ability to easily extract higher-order moduli, a task that presents substantial challenges for existing methods.

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.

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.

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.

Biophysical Principles of Ion-Channel-Mediated Mechanosensory Transduction

Recent rapid progress in the field of mechanobiology has been driven by novel emerging tools and methodologies and growing interest from different scientific disciplines. Specific progress has been made toward understanding how cell mechanics is linked to intracellular signaling and the regulation of gene expression in response to a variety of mechanical stimuli. There is a direct link between the mechanoreceptors at the cell surface and intracellular biochemical signaling, which in turn controls downstream effector molecules. Among the mechanoreceptors in the cell membrane, mechanosensitive (MS) ion channels are essential for the ultra-rapid (millisecond) transduction of mechanical stimuli into biologically relevant signals. The three decades of research on mechanosensitive channels resulted in the formulation of two basic principles of mechanosensitive channel gating: force-from-lipids and force-from-filament. In this review, we revisit the biophysical principles that underlie the innate force-sensing ability of mechanosensitive channels as contributors to the force-dependent evolution of life forms.

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.

Origin of lipid tilt in flat monolayers and bilayers

This paper continues the series of our works devoted to the liquid-gel phase transition in lipid membranes. Previously we described a variation of area per lipid, membrane thickness, and diffusion coefficient at the temperature-driven liquid-gel phase transition in bilayers. Here we expand the application of our analytic model approach to include a description of the lipid tilt and also extend the investigation to include Langmuir and self-assembled monolayers. The theory describes tilt formation at the temperature-driven liquid-gel phase transition in bilayers and the pressure-driven phase transition in Langmuir monolayers. Neither uniform tilt nor liquid-gel phase transition is found in self-assembled monolayers chemically bonded to the substrate.

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.

Remembering Mechanosensitivity of NMDA Receptors

An increase in post-synaptic Ca²⁺ conductance through activation of the ionotropic N-methyl-D-aspartate receptor (NMDAR) and concomitant structural changes are essential for the initiation of long-term potentiation (LTP) and memory formation. Memories can be initiated by coincident events, as occurs in classical conditioning, where the NMDAR can act as a molecular coincidence detector. Binding of glutamate and glycine, together with depolarization of the postsynaptic cell membrane to remove the Mg²⁺ channel pore block, results in NMDAR opening for Ca²⁺ conductance. Accumulating evidence has implicated both force-from-lipids and protein tethering mechanisms for mechanosensory transduction in NMDAR, which has been demonstrated by both, membrane stretch and application of amphipathic molecules such as arachidonic acid (AA). The contribution of mechanosensitivity to memory formation and consolidation may be to increase activity of the NMDAR leading to facilitated memory formation. In this review we look back at the progress made toward understanding the physiological and pathological role of NMDA receptor channels in mechanobiology of the nervous system and consider these findings in like of their potential functional implications for memory formation. We examine recent studies identifying mechanisms of both NMDAR and other mechanosensitive channels and discuss functional implications including gain control of NMDA opening probability. Mechanobiology is a rapidly growing area of biology with many important implications for understanding form, function and pathology in the nervous system.

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.

Coupled Interfacial Tension and Phase Behavior Model Based on Micellar Curvatures

This article introduces a consistent and robust model that predicts interfacial tensions for all microemulsion Winsor types and overall compositions. The model incorporates film bending arguments and Huh's equation and is coupled to phase behavior so that simultaneous tuning of both interfacial tension (IFT) and phase behavior is possible. The oil-water interfacial tension and characteristic length are shown to be related to each other through the hydrophilic-lipophilic deviation (HLD). The phase behavior is tied to the micelle curvatures, without the need for using the net average curvature (NAC). The interfacial tension model is related to solubilization ratios in order to introduce a coupled interfacial tension-phase behavior model for all phase environments. The approach predicts two- and three-phase interfacial tensions and phase behavior (i.e., tie lines and tie triangles) for changes in composition and HLD input parameters, such as temperature, pressure, surfactant structure, and oil equivalent alkane carbon number. Comparisons to experimental data show excellent fits and predictive capability.

Theory and algorithms to compute Helfrich bending forces: a review

Cell membranes are vital to shield a cell's interior from the environment. At the same time they determine to a large extent the cell's mechanical resistance to external forces. In recent years there has been considerable interest in the accurate computational modeling of such membranes, driven mainly by the amazing variety of shapes that red blood cells and model systems such as vesicles can assume in external flows. Given that the typical height of a membrane is only a few nanometers while the surface of the cell extends over many micrometers, physical modeling approaches mostly consider the interface as a two-dimensional elastic continuum. Here we review recent modeling efforts focusing on one of the computationally most intricate components, namely the membrane's bending resistance. We start with a short background on the most widely used bending model due to Helfrich. While the Helfrich bending energy by itself is an extremely simple model equation, the computation of the resulting forces is far from trivial. At the heart of these difficulties lies the fact that the forces involve second order derivatives of the local surface curvature which by itself is the second derivative of the membrane geometry. We systematically derive and compare the different routes to obtain bending forces from the Helfrich energy, namely the variational approach and the thin-shell theory. While both routes lead to mathematically identical expressions, so-called linear bending models are shown to reproduce only the leading order term while higher orders differ. The main part of the review contains a description of various computational strategies which we classify into three categories: the force, the strong and the weak formulation. We finally give some examples for the application of these strategies in actual simulations.