Membrane Boundary Conditions Research Articles

Flexible membrane boundary condition DEM-FEM for drained and undrained monotonic and cyclic triaxial tests

Flexible membrane boundary condition DEM-FEM for drained and undrained monotonic and cyclic triaxial tests

Practical computation of the diffusion MRI signal based on Laplace eigenfunctions: permeable interfaces.

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal, called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the tissue geometry and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. In a previous publication, we presented a simulation framework that we implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for biological cells subject to impermeable membrane boundary conditions. In this work, we extend our simulation framework to include geometries that contain permeable cell membranes. We describe the new computational techniques that allowed this generalization and we analyze the effects of the magnitude of the permeability coefficient on the eigendecomposition of the diffusion and Bloch-Torrey operators. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

Biomembranes undergo complex, non-axisymmetric deformations governed by Kirchhoff-Love kinematicsand revealed by a three-dimensional computational framework.

Biomembranes play a central role in various phenomena like locomotion of cells, cell-cell interactions, packaging and transport of nutrients, transmission of nerve impulses, and in maintaining organelle morphology and functionality. During these processes, the membranes undergo significant morphological changes through deformation, scission, and fusion. Modelling the underlying mechanics of such morphological changes has traditionally relied on reduced order axisymmetric representations of membrane geometry and deformation. Axisymmetric representations, while robust and extensively deployed, suffer from their inability to model-symmetry breaking deformations and structural bifurcations. To address this limitation, a three-dimensional computational mechanics framework for high fidelity modelling of biomembrane deformation is presented. The proposed framework brings together Kirchhoff–Love thin-shell kinematics, Helfrich-energy-based mechanics, and state-of-the-art numerical techniques for modelling deformation of surface geometries. Lipid bilayers are represented as spline-based surface discretizations immersed in a three-dimensional space; this enables modelling of a wide spectrum of membrane geometries, boundary conditions, and deformations that are physically admissible in a three-dimensional space. The mathematical basis of the framework and its numerical machinery are presented, and their utility is demonstrated by modelling three classical, yet non-trivial, membrane deformation problems: formation of tubular shapes and their lateral constriction, Piezo1-induced membrane footprint generation and gating response, and the budding of membranes by protein coats during endocytosis. For each problem, the full three-dimensional membrane deformation is captured, potential symmetry-breaking deformation paths identified, and various case studies of boundary and load conditions are presented. Using the endocytic vesicle budding as a case study, we also present a ‘phase diagram’ for its symmetric and broken-symmetry states.

Study of Shear Behavior of Binary Mixtures by DEM Simulation of Biaxial Test in the Membrane Boundary Condition

This paper aims to study the shear behavior of binary particles with irregular shapes by discrete element method simulations of the biaxial test in the membrane boundary condition. Binary particle samples are generated according to different volume fractions of coarse and fine particles. The deviatoric stress and volumetric strain curves are plotted to describe the contracting-dilatancy relationship of binary samples under shearing conditions. The anisotropy of the normal and tangential contact forces are explored by visualization of the orientation of contact forces to describe the evolution of micro structures of samples during the shearing process. Besides, the formations of the shear band are observed by the visualization of the newly generated contact force chains between particles. The research shows that the volume fraction of coarse particles and particle size ratio have significant influences on the shear behavior of binary particles both in macroscopic and microscopic points of view. Moreover, the increased volume fraction of coarse particles leads to a more difficult formation of a shear band.

Contact strength and deformation of straining free-standing borophene

We apply nanoindentation using molecular dynamics simulation to investigate the contact strength, deformation behaviour, anisotropic cracking propagation and self-reforming mechanism of a single-layer borophene (SLB). The effects of indenter shape, indenter size, borophene membrane size, boundary conditions, temperature, and loading rate on the indentation force, strain and stress distributions, crack directions, and the self-reform ratio of SLB are considered. Especially a cyclic indentation process on the borophene monolayer is also analysed to evaluate its long-term dynamic reliability. The single/multi loading–unloading processes show that the SLB has a perfect elastic deformation when the indentation depth is below or equal to the critical indentation depth. We found that the crack direction is not influenced when changing spherical indenter diameter from 1 to 4 nm or loading rate from 1 to 15 m/s. However, with a conical indenter, the smallest substrate size, and FTE(y) boundary condition, the cracks develop not only in the armchair direction but also in the zigzag direction. Furthermore, the borophene sheet has a high self-reform ratio of 90% (with v = 15 m/s) after being cracked along the armchair direction. Meanwhile, the cracks in the zigzag direction have no possibility of self-reform. At 100 K, after the tenth cycle indentation, the borophene sheet can still preserve its shape while it is broken at 300 K. In summary, this study further highlights the mechanical, long-term dynamic behaviour and potentially speeds up the promising applications of the SLB.

Study of shear behavior of granular materials by 3D DEM simulation of the triaxial test in the membrane boundary condition

This paper studies the shear behavior of granular materials by using a three-dimensional (3D) discrete element method (DEM) simulation of the triaxial test. The experimental triaxial tests were conducted on glass beads samples for verification. DEM simulations of the triaxial test were carried out in the membrane boundary condition consisting of 37,989 membrane particles. A new method that divides the irregular sample shape into two parts of cones and parts of three-dimensional simplexes is used to follow the volume change of irregular deformation of samples. The free rotatable upper platen is considered during the shearing process, which influences the shear behavior of samples especially in the residual stage and formations of a single shear band or X-shape shear band. The confining pressures have been demonstrated to influence the rotation angle and angular velocity of the upper platen. Moreover, the timing of replacing a rigid wall boundary condition with the membrane boundary condition is investigated, which affects the porosity of samples before shearing and the mechanical strength. The DEM model in the membrane boundary condition reflects well the evolution of irregular sample deformation and shear band in the shearing process. From the perspective of micro structures, the normal force decreases and the tangential stress increases during the shearing stage. This study greatly improves the accuracy of DEM simulations of the triaxial test in the membrane boundary condition.

Existence of a global weak solution for a reaction–diffusion problem with membrane conditions

Several problems, issued from physics, biology or the medical science, lead to parabolic equations set in two sub-domains separated by a membrane with selective permeability to specific molecules. The corresponding boundary conditions, describing the flow through the membrane, are compatible with mass conservation and energy dissipation, and are called the Kedem-Katchalsky conditions. Additionally, in these models, written as reaction-diffusion systems, the reaction terms have a quadratic behaviour. M. Pierre and his collaborators have developed a complete $L^1$ theory for reaction-diffusion systems with different diffusions. Here, we adapt this theory to the membrane boundary conditions and prove the existence of weak solutions when the initial data has only $L^1$ regularity using the truncation method for the nonlinearities. In particular, we establish several estimates as the $W^{1,1}$ regularity of the solutions. Also, a crucial step is to adapt the fundamental $L^2$ (space, time) integrability lemma to our situation.

Lattice Boltzmann method for simulation of diffusion magnetic resonance imaging physics in multiphase tissue models.

We report an implementation of the lattice Boltzmann method (LBM) to integrate the Bloch-Torrey equation, which describes the evolution of the transverse magnetization vector and the fate of the signal of diffusion magnetic resonance imaging (dMRI). Motivated by the need to interpret dMRI experiments in biological tissues, and to offset the small time-step limitation of classical LBM, a hybrid LBM scheme is introduced and implemented to solve the Bloch-Torrey equation. A membrane boundary condition is presented which is able to accurately represent the effects of thin curvilinear membranes typically found in biological tissues. As implemented, the hybrid LBM scheme accommodates piece-wise uniform transport, dMRI parameters, periodic and mirroring outer boundary conditions, and finite membrane permeabilities on non-boundary-conforming inner boundaries. By comparing with analytical solutions of limiting cases, we demonstrate that the hybrid LBM scheme is more accurate than the classical LBM scheme. The proposed explicit LBM scheme maintains second-order spatial accuracy, stability, and first-order temporal accuracy for a wide range of parameters. The parallel implementation of the hybrid LBM code in a multi-CPU computer system, as well as on GPUs, is straightforward and efficient. Along with offering certain advantages over finite element or Monte Carlo schemes, the proposed hybrid LBM constitutes a flexible scheme that can by easily adapted to model more complex interfacial conditions and physics in heterogeneous multiphase tissue models and to accommodate sophisticated dMRI sequences.

DEM study of the shear behavior and formation of shear band in biaxial test

This paper aims to study the shear behavior of granular matter by DEM simulations. Granular samples are prepared by automatic clump generation algorithm to create particles of irregular shapes. Simulations of the biaxial test with membrane boundary condition are used to test the shear behavior of samples. A new method for computing sample volume in membrane boundary condition is proposed. Deviatoric stress and volumetric strain curves are plotted to describe contracting-dilatancy of granular materials during the shearing stage. Formation of the shear band is studied from particle rotation and particle displacement fields. The influence of confining pressure, initial porosity, and friction coefficient on the development of shear band are studied. Lower confining pressure, higher initial porosity can be resulted in later formation of shear bands.

Dynamic boundary conditions for membranes whose surface energy depends on the mean and Gaussian curvatures

Membranes are an important subject of study in physical chemistry and biology. They can be considered as material surfaces with a surface energy depending on the curvature tensor. Usually, mathematical models developed in the literature consider the dependence of surface energy only on mean curvature with an added linear term for Gauss curvature. Therefore, for closed surfaces the Gauss curvature term can be eliminated because of the Gauss-Bonnet theorem. In [18], the dependence on the mean and Gaussian curvatures was considered in statics. The authors derived the shape equation as well as two scalar boundary conditions on the contact line. In this paper-thanks to the principle of virtual working-the equations of motion and boundary conditions governing the fluid membranes subject to general dynamical bending are derived. We obtain the dynamic 'shape equa-tion' (equation for the membrane surface) and the dynamic conditions on the contact line generalizing the classical Young-Dupr{\'e} condition.

Global shape optimization of fixtures to suppress wrinkles in large-displacement membrane structures

Membrane fixtures provide external tensile forces and boundary conditions for thin membranes; thus, they play an important role in the minor principal stress distribution and wrinkling patterns. Here, we propose a global optimization approach for designing the shape of membrane fixtures to suppress stress-related wrinkles under large membrane deformations. The objective of the optimization model is to maximize the minimal value of the in-plane minor principal stress over the entire membrane. This model incorporates the Kriging surrogate method, which is a robust optimization technique, to make the membrane in a taut state and to find out the global optimal solution. Several examples with different stretching displacements are presented to demonstrate the effectiveness of this proposed optimization method. Moreover, the method provides an efficient way to obtain wrinkle-free membrane structures without reducing the membrane area.

An efficient flexible membrane boundary condition for DEM simulation of axisymmetric element tests

SummaryThe mechanical response of an assembly of particles depends on the applied boundary conditions. Robust calibration of numerical discrete systems to laboratory results is also a primary step in many studies of granular materials. In this study, a new membrane model was developed for simulating axisymmetric element tests. This membrane model uses a simple algorithm of an array of independently controlled walls and is computationally efficient. The effect of boundary flexibility on the system response was investigated by simulating a series of triaxial tests on dense and loose specimens. At the specimen scale, differences in shear strength and volume change of specimens were observed. It was shown that localization pattern depends on the applied boundary conditions. At the particle scale, particle‐membrane contact forces, coordination number, local void ratio, and anisotropy of fabric were all affected by the boundary flexibility.

Efficient CFD-based method for designing cross-flow nanofiltration small devices

Small devices of cross-flow nanofiltration (NF) may become a tool to concentrate small samples of valuable substances, if the concentration polarization can be minimized by designing new feed channels with bas-reliefs similar to the ones that have been developed for efficient passive micromixers. These bas-reliefs may have a complex 3D shape and the recourse to intensive CFD simulations is necessary to optimize the geometry. To speed up the simulations, a hybrid, semi empirical approach is used where the average and local mass transfer coefficients are first computed by CFD, using a dissolving wall instead of a semi-permeable membrane boundary condition. The computed mass transfer coefficients are then used to predict the local and average concentration polarization, using a mass transfer correction factor correlation. This computational approach was validated using detailed CFD simulations and experimental concentration distributions obtained by holographic interferometry. A NF ribbed-wall rectangular channel with 2mm height was used, with potassium sulfate solutions, transmembrane pressures lower than 8bar and Reynolds numbers between 1 and 125. Using this computational approach, local and average concentration polarization and permeate flux were predicted with an error less than 15%. With these findings, it is possible to speed up the simulation and design of small nanofiltration cross-flow devices.

Structures and pathways for clathrin self-assembly in the bulk and on membranes

We present a coarse-grained model of clathrin that is simple enough to be computationally tractable yet includes key observed qualitative features: a triskelion structure with excluded volume between legs; assembly of polymorphic cages in the bulk; formation of buds on a membrane. We investigate the assembly of our model using both Monte Carlo simulations and molecular dynamics with hydrodynamic interactions, in the latter employing a new membrane boundary condition. In the bulk, a range of known clathrin structures are assembled. A membrane budding pathway involving the coalescence of multiple small clusters is identified.

Electro-optical BLM chips enabling dynamic imaging of ordered lipid domains

Studies of lipid rafts, ordered microdomains of sphingolipids and cholesterol within cell membranes, are essential in probing the relationships between membrane organization and cellular function. While in vitro studies of lipid phase separation are commonly performed using spherical vesicles as model membranes, the utility of these models is limited by a number of factors. Here we present a microfluidic device that supports simultaneous electrical measurements and confocal imaging of on-chip bilayer lipid membranes (BLMs), enabling real-time multi-domain imaging of membrane organization. The chips further support closed microfluidic access to both sides of the membrane, allowing the membrane boundary conditions to be rapidly changed and providing a mechanism for dynamically adjusting membrane curvature through application of a transmembrane pressure gradient. Here we demonstrate the platform through the study of dynamic generation and dissolution of ordered lipid domains as membrane components are transported to and from the supporting annulus containing solvated lipids and cholesterol.

Rapid Microfluidic Perfusion Enabling Kinetic Studies of Lipid Ion Channels in a Bilayer Lipid Membrane Chip

There is growing recognition that lipids play key roles in ion channel physiology, both through the dynamic formation and dissolution of lipid ion channels and by indirect regulation of protein ion channels. Because existing technologies cannot rapidly modulate the local (bio)chemical conditions at artificial bilayer lipid membranes used in ion channel studies, the ability to elucidate the dynamics of these lipid-lipid and lipid-protein interactions has been limited. Here we demonstrate a microfluidic system supporting exceptionally rapid perfusion of reagents to an on-chip bilayer lipid membrane, enabling the responses of lipid ion channels to dynamic changes in membrane boundary conditions to be probed. The thermoplastic microfluidic system allows initial perfusion of reagents to the membrane in less than 1s, and enables kinetic behaviors with time constants below 10s to be directly measured. Application of the platform is demonstrated toward kinetic studies of ceramide, a biologically important lipid known to self-assemble into transmembrane ion channels, in response to dynamic treatments of small ions (La(3+)) and proteins (Bcl-x(L) mutant). The results reveal the broader potential of the technology for studies of membrane biophysics, including lipid ion channel dynamics, lipid-protein interactions, and the regulation of protein ion channels by lipid micro domains.

Approximate large-deflection analysis of simply supported rectangular plates under transverse loading using plate post-buckling solutions

This paper describes an approximate analytical method to determine the large-deflection behaviour of rectangular simply supported thin plates under transverse loading. The method is based on a simple trial function to describe the shape of the initial and total deflections of the plate, corresponding to the first buckling mode shape of the plate when subjected to uniform in-plane compression. Therefore, the large-deflection behaviour of the plate under transverse loading can be expressed as a function of the ratio of the pre- to post-buckling in-plane stiffness of the plate, which is given in literature for plates with various membrane boundary conditions of the plate edges. Due to the simplicity of the trial function flexural and membrane actions of the plate are decoupled. It was found that the approximate method gives good insight into the large-deflection behaviour. The results of the approximate method have been compared to the results of finite element solutions and solutions available in literature. For uniformly distributed loads the approximate method gives deflections which differ less than about 10% from the finite element simulation values.

Phospholipid membrane bending as assessed by the shape sequence of giant oblate phospholipid vesicles

Vesicle shape transformations caused by decreasing the difference between the equilibrium areas of membrane monolayers were studied on phospholipid vesicles with small volume to membrane area ratios. Slow transformations of the vesicle shape were induced by lowering of the concentration of lipid monomers in the solution outside the vesicle. The complete sequence of shapes consisted of a string of pearls, and wormlike, starfish, discocyte and stomatocyte shapes. The transformation from discocyte to stomatocyte vesicle shapes was analyzed theoretically to see whether these observations accord with the area difference elasticity (ADE) model. The membrane shape equation and boundary conditions were derived for axisymmetrical shapes for low volume vesicles, part of whose membranes are in contact. Calculated shapes were arranged into a phase diagram. The theory predicts that the transition between discocyte and stomatocyte shapes is discontinuous for relatively high volumes and continuous for low volumes. The calculated shape sequences matched well with the observed ones. By assuming a linear decrease of the equilibrium area difference with time, the ratio between the nonlocal and local bending constants is in agreement with reported values.

DYNAMIC ANALYSIS OF SHALLOW SHELLS OF RECTANGULAR BASE

In this investigation a systematic analytical procedure for the dynamic analysis and response of thin shallow shells with a rectangular layout is presented. The shell types examined are the elliptic and hyperbolic paraboloid, the hypar, the conoidal parabolic and the soap-bubble shell, although in principle any shell geometry expressed by a continuous surface equation can be treated. The eigenvalue problem solution is based on the one hand on the consideration of the shell as a system of two interdependent plates whose boundary conditions comply with the prevailing bending and membrane boundary conditions of the shell, and on the other hand on the consistent use of beam eigenfunctions, in the context of a Galerkin solution procedure. The series solution obtained in this way converges rapidly and provides practically acceptable results even in cases with one or more free edges, where the boundary conditions cannot be strictly satisfied. The whole analysis is carried out on the basis of a few non-dimensionalized geometrical parameters, which are the only input required for the computer program specially written for that purpose.

Boundary conditions and heterotic construction in topological membrane theory

Using the topological membrane approach to string theory, we suggest a geometric origin for the heterotic string. We show how different membrane boundary conditions lead to different string theories. We discuss the construction of closed oriented strings and superstrings, and demonstrate how the heterotic construction naturally arises from a specific choice of boundary conditions on the left and right boundaries of a cylindrical topological membrane.