Geometrical Influence on Particle Transport in Cross-Flow Ultrafiltration: Cylindrical and Flat Sheet Membranes

Cross-flow membrane ultrafiltration (UF) is used for the enrichment and purification of small colloidal particles and proteins. We explore the influence of different membrane geometries on the particle transport in, and the efficiency of, inside-out cross-flow UF. For this purpose, we generalize the accurate and numerically efficient modified boundary layer approximation (mBLA) method, developed in recent work by us for a hollow cylindrical membrane, to parallel flat sheet geometries with one or two solvent-permeable membrane sheets. Considering a reference dispersion of Brownian hard spheres where accurate expressions for its transport properties are available, the generalized mBLA method is used to analyze how particle transport and global UF process indicators are affected by varying operating parameters and the membrane geometry. We show that global process indicators including the mean permeate flux, the solvent recovery indicator, and the concentration factor are strongly dependent on the membrane geometry. A key finding is that irrespective of the many input parameters characterizing an UF experiment and its membrane geometry, the process indicators are determined by three independent dimensionless variables only. This finding can be very useful in the design, optimization, and scale-up of UF processes.

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.

Molecular landscape of etioplast inner membranes in higher plants

Etioplasts are photosynthetically inactive plastids that accumulate when light levels are too low for chloroplast maturation. The etioplast inner membrane consists of a paracrystalline tubular lattice and peripheral, disk-shaped membranes, respectively known as the prolamellar body and prothylakoids. These distinct membrane regions are connected into one continuous compartment. To date, no structures of protein complexes in or at etioplast membranes have been reported. Here, we used electron cryo-tomography to explore the molecular membrane landscape of pea and maize etioplasts. Our tomographic reconstructions show that ATP synthase monomers are enriched in the prothylakoids, and plastid ribosomes in the tubular lattice. The entire tubular lattice is covered by regular helical arrays of a membrane-associated protein, which we identified as the 37-kDa enzyme, light-dependent protochlorophyllide oxidoreductase (LPOR). LPOR is the most abundant protein in the etioplast, where it is responsible for chlorophyll biosynthesis, photoprotection and defining the membrane geometry of the prolamellar body. Based on the 9-Å-resolution volume of the subtomogram average, we propose a structural model of membrane-associated LPOR.

Biomembranes undergo complex, non-axisymmetric deformations governed by Kirchhoff-Love kinematics and 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. Modeling 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 modeling 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 modeling deformation of surface geometries. Lipid bilayers are represented as spline-based surface discretizations immersed in a three-dimensional space; this enables modeling of a wide spectrum of membrane geometries, boundary conditions, and deformations that are physically admissible in a 3D space. The mathematical basis of the framework and its numerical machinery are presented, and their utility is demonstrated by modeling 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.

Protein diffusion on membrane domes, tubes and pearling structures

Diffusion is a fundamental mechanism for protein distribution in cell membranes. These membranes often exhibit complex shapes, which range from shallow domes to elongated tubular or pearl-like structures. Shape complexity of the membrane influences the diffusive spreading of proteins and molecules. Despite the importance membrane geometry plays in these diffusive processes, it is challenging to establish the dependence between diffusion and membrane morphology. We solve the diffusion equation numerically on various curved shapes representative for experimentally observed membrane shapes. Our results show that membrane necks become diffusion barriers. We determine the diffusive half time, i.e., the time that is required to reduce the amount of proteins in the budded region by one half and find a quadratic relation between the diffusive half time and the averaged mean curvature of the membrane shape. Our findings thus help to estimate the characteristic diffusive time scale based on the simple measure for membrane morphology.Significance statementDiffusion is an integral process for distributing proteins throughout biological membranes. These membranes can have complex shapes and structures, often featuring elongated shapes such as tubes and like a necklace of pearls. The diffusion process on these shapes is significantly different from the well studied planar substrate. We use numerical simulations to understand how the characteristic diffusion time is a function of membrane shape, where we find the diffusion of proteins on strongly curved shapes is significantly slower than on planar membranes. Our results provide a simple relationship to estimate the characteristic diffusion time of proteins on membranes based on its mean and Gaussian curvature.

Modelling IR Spectra of Sulfonated Polyether Ether Ketone (SPEEK) Membranes for Fuel Cells

SPEEK (sulfonated polyether ether ketone) membranes have been prepared and characterized. The SPEEK membrane geometry and theoretical vibration spectra calculated using density functional theory (DFT) as depending from membrane chain length and polymer cross-linking. Analyzed the limitations of the method by comparing theoretical and experimental IR spectra.

Time matters for macroscopic membranes formed by alginate and cationic β-sheet peptides

The peptide age and membrane geometry affect the micro- and nano-structure of hierarchically ordered planar and spherical membranes constructed at the interface of cationic β-sheet peptides and alginate solution.

Geometric coupling of helicoidal ramps and curvature-inducing proteins in organelle membranes

Cellular membranes display an incredibly diverse range of shapes, both in the plasma membrane and at membrane bound organelles. These morphologies are intricately related to cellular functions, enabling and regulating fundamental membrane processes. However, the biophysical mechanisms at the origin of these complex geometries are not fully understood from the standpoint of membrane–protein coupling. In this study, we focused on a minimal model of helicoidal ramps representative of specialized endoplasmic reticulum compartments. Given a helicoidal membrane geometry, we asked what is the distribution of spontaneous curvature required to maintain this shape at mechanical equilibrium? Based on the Helfrich energy of elastic membranes with spontaneous curvature, we derived the shape equation for minimal surfaces, and applied it to helicoids. We showed the existence of switches in the sign of the spontaneous curvature associated with geometric variations of the membrane structures. Furthermore, for a prescribed gradient of spontaneous curvature along the exterior boundaries, we identified configurations of the helicoidal ramps that are confined between two infinitely large energy barriers. Overall our results suggest possible mechanisms for geometric control of helicoidal ramps in membrane organelles based on curvature-inducing proteins.