Lipid rafts are hypothesized to facilitate protein interaction, tension regulation, and trafficking in biological membranes, but the mechanisms responsible for their formation and maintenance are not clear. Insights into many other condensed matter phenomena have come from colloidal systems, whose micron-scale particles mimic basic properties of atoms and molecules but permit dynamic visualization with single-particle resolution. Recently, experiments showed that bidisperse mixtures of filamentous viruses can self-assemble into colloidal monolayers with thermodynamically stable rafts exhibiting chiral structure and repulsive interactions. We quantitatively explain these observations by modeling the membrane particles as chiral liquid crystals. Chiral twist promotes the formation of finite-sized rafts and mediates a repulsion that distributes them evenly throughout the membrane. Although this system is composed of filamentous viruses whose aggregation is entropically driven by dextran depletants instead of phospholipids and cholesterol with prominent electrostatic interactions, colloidal and biological membranes share many of the same physical symmetries. Chiral twist can contribute to the behavior of both systems and may account for certain stereospecific effects observed in molecular membranes.
Colloidal membranes are monolayer assemblies of rodlike particles that capture the long-wavelength properties of lipid bilayer membranes on the colloidal scale. Recent experiments on colloidal membranes formed by chiral rodlike viruses showed that introducing a second species of virus with different length and opposite chirality leads to the formation of rafts --- micron-sized domains of one virus species floating in a background of the other viruses [Sharma et al., Nature 513, 77 (2014)]. In this article we study the interaction of such rafts using liquid crystal elasticity theory. By numeric…
Membrane-mediated particle interactions depend both on the properties of the particles themselves and the membrane environment in which they are suspended. Experiments have shown that chiral rod-like inclusions dissolved in a colloidal membrane of opposite handedness assemble into colloidal rafts, which are finite-sized reconfigurable droplets consisting of a large but precisely defined number of rods. Here, we systematically tune the chirality of the background membrane, and find that in the achiral limit colloidal rafts acquire fundamentally new structural properties and interactions. In par…
Liquid-liquid phase separation is ubiquitous in suspensions of nanoparticles, proteins and colloids. With a few notable exceptions, surface-tension-minimizing liquid droplets in bulk suspensions continuously coalesce, increasing in size without bound until achieving macroscale phase separation. In comparison, the phase behavior of colloids, nanoparticles or proteins confined to interfaces, surfaces or membranes is significantly more complex. Inclusions distort the local interface structure leading to interactions that are fundamentally different from the well-studied interactions mediated by i…
Using a combination of theory and experiments we study the interface between two immiscible domains in a colloidal membrane composed of rigid rods of different lengths. Geometric considerations of rigid rod packing imply that a domain of sufficiently short rods in a background membrane of long rods is more susceptible to twist than the inverse structure, a long-rod domain in a short-rod membrane background. The tilt at the inter-domain edge forces splay, which in turn manifests as a spontaneous edge curvature whose energetics are controlled by the length asymmetry of constituent rods. A thermo…
We present a model for the kinetics of spontaneous membrane domain (raft) assembly that includes the effect of membrane recycling ubiquitous in living cells. We show that the domains have a broad power-law distribution with an average radius that scales with the 1/4 power of the domain lifetime when the line tension at the domain edges is large. For biologically reasonable recycling and diffusion rates the average domain radius is in the tens of nm range, consistent with observations. This represents one possible link between signaling (involving rafts) and traffic (recycling) in cells. Finall…
Though 1-alkanols are well known to have anesthetic properties, the mode of operation remains enigmatic. We perform extensive atomistic molecular dynamics simulation to study the partitioning of 1- alkanols of different chain-lengths in raft-like model membrane made up of mixture of unsaturated dioleoyl-phosphatidylcholine (DOPC) and saturated dipalmitoyl-phosphatidylcholine (DPPC) and choles- terol (Chol) exhibiting phase coexistence of liquid-ordered (lo) - liquid disordered (ld) phase domains. Our simulation shows that the effect of 1-alkanols on the mechanical properties of the membrane ha…
Inclusions dissolved in an anisotropic quasi-2D membrane acquire new types of interactions that can drive assembly of complex structures and patterns. We study colloidal membranes composed of a binary mixture of long and short rods, such that the length ratio of the long to short rods is approximately two. At very low volume fractions, short rods dissolve in the membrane of long rods by strongly anchoring to the membrane polymer interface. At higher fractions, the dissolved short rods phase separate from the background membrane, creating a composite structure comprised of bilayer droplets enri…
To understand mechanism of cell-membrane compartmentalization, we studied coarse-grained model systems consisting of lipid and cholesterol molecules. Cholesterol plays crucial role in lateral phase segregation in the bilayered lipid membrane to form cholesterol rich liquid ordered domains, termed as `rafts'. In this NVT Molecular Dynamics simulation study, we have investigated the role of cholesterol in raft-like domain formation. The study reveals that the strength of cholesterol interaction is an important factor behind the phase segregation to form cholesterol rich domains, whereas, the rel…
We elucidate the molecular forces leading to assembly of two dimensional membrane-like structures composed of a one rod-length thick monolayer of aligned rods from an immiscible suspension of hard rods and depleting polymers. We perform simulations which predict that monolayer membranes are thermodynamically stable above a critical rod aspect ratio and below a critical depletion interaction length scale. Outside of these conditions alternative structures such as stacked smectic columns or nematic droplets are thermodynamically stable. These predictions are confirmed using an experimental model…
We study the collective dynamics of self-propelled rods in an inhomogeneous motility field. At the interface between two regions of constant but different motility, a smectic rod layer is spontaneously created through aligning interactions between the active rods, reminiscent of an artificial, semi-permeable membrane. This "active membrane" engulfes rods which are locally trapped in low-motility regions and thereby further enhances the trapping efficiency by self-organization, an effect which we call "self-encapsulation". Our results are gained by computer simulations of self-propelled rod mod…
Clustering of ligand-binding receptors of different types on thickened isles of the cell membrane, namely lipid rafts, is an experimentally observed phenomenon. Although its influence on cell response is deeply investigated, the role of the coupling between mechanical processes and multiphysics involving the active receptors and the surrounding lipid membrane during ligand-binding has not yet been understood. Specifically, the focus of this work is on G-protein-coupled receptors (GPCRs), which regulate specific cell processes through chemical signalling pathways involving a synergistic balance…
Lipid rafts are heterogeneous dynamic lipid domains of the cell membranes that are involved in several biological processes, like protein and lipids specific transport and signaling. Our understanding of lipid raft formation is still limited, due to the transient and elusive nature of these domains {\it in vivo}, in contrast to the stable phase-separated domains observed in artificial membranes. Inspired by experimental findings highlighting the relevance of transmembrane proteins for lipid rafts, we investigate lipid domain nucleation by coarse-grained molecular dynamics and Ising model simul…
According to the lipid raft hypothesis, biological lipid membranes are laterally heterogeneous and filled with nanoscale ordered "raft" domains, which are believed to play an important role for the organization of proteins in membranes. However, the mechanisms stabilizing such small rafts are not clear, and even their existence is sometimes questioned. Here we report the observation of raft-like structures in a coarse-grained molecular model for multicomponent lipid bilayers. On small scales, our membranes demix into a liquid ordered (lo) and a liquid disordered (ld) phase. On large scales, ph…
Chiral rodlike colloids exposed to strong depletion attraction may self-assemble into chiral membranes whose twisted director field differs from that of a 3D bulk chiral nematic. We formulate a simple microscopic variational theory to determine the elastic moduli of rods assembled into a bi-dimensional smectic membrane. The approach is based on a simple Onsager-Straley theory for a non-uniform director field that we apply to describe rod twist within the membrane. A microscopic approach enables a detailed estimate of the individual Frank elastic moduli (splay, twist and bend) as well as the tw…
Rafts, or functional domains, are transient nano- or mesoscopic structures in the plasma membrane and are thought to be essential for many cellular processes such as signal transduction, adhesion, trafficking and lipid/protein sorting. Observations of these membrane heterogeneities have proven challenging, as they are thought to be both small and short-lived. With a combination of coarse-grained molecular dynamics simulations and neutron diffraction using deuterium labeled cholesterol molecules we observe raft-like structures and determine the ordering of the cholesterol molecules in binary ch…
In the presence of a non-adsorbing polymer, monodisperse rod-like colloids assemble into one-rod-length thick liquid-like monolayers, called colloidal membranes. The density of the rods within a colloidal membrane is determined by a balance between the osmotic pressure exerted by the enveloping polymer suspension and the repulsion between the colloidal rods. We developed a microfluidic device for continuously observing an isolated membrane while dynamically controlling the osmotic pressure of the polymer suspension. Using this technology we measured the membrane rod density over a range of osm…
Monolayer spontaneous curvatures for cholesterol, DOPE, POPE, DOPC, DPPC, DSPC, POPC, SOPC, and egg sphingomyelin were obtained using small-angle X-ray scattering (SAXS) on inverted hexagonal phases (HII). Spontaneous curvatures of bilayer forming lipids were estimated by adding controlled amounts to a HII forming template following previously established protocols. Spontanous curvatures of both phosphatidylethanolamines and cholesterol were found to be at least a factor of two more negative than those of phosphatidylcholines, whose J0 are closer to zero. Interestingly, a significant positive …
The assembly of curved protein rods on fluid membranes is studied using implicit-solvent meshless membrane simulations. As the rod curvature increases, the rods on a membrane tube assemble along the azimuthal direction first and subsequently along the longitudinal direction. Here, we show that both transition curvatures decrease with increasing rod stiffness. For comparison, curvature-inducing isotropic inclusions are also simulated. When the isotropic inclusions have the same bending rigidity as the other membrane regions, the inclusions are uniformly distributed on the membrane tubes and ves…
Motivated by experiments on colloidal membranes composed of chiral rod-like viruses, we use Monte Carlo methods to determine the phase diagram for the liquid crystalline order of the rods and the membrane shape. We generalize the Lebwohl-Lasher model for a nematic with a chiral coupling to a curved surface with edge tension and a resistance to bending, and include an energy cost for tilting of the rods relative to the local membrane normal. The membrane is represented by a triangular mesh of hard beads joined by bonds, where each bead is decorated by a director. The beads can move, the bonds c…
We formulate a density functional theory that describes the phase behavior of hard rods and depleting polymers, as realized in recent experiments on suspensions of \emph{fd} virus and non-adsorbing polymer. The theory predicts the relative stability of nematic droplets, stacked smectic columns, and a recently discovered phase of isolated monolayers of rods, or colloidal membranes. We find that a minimum rod aspect ratio is required for stability of colloidal membranes and that collective protrusion undulations are the dominant effect that stabilizes this phase. The theoretical predictions are …
In the presence of a non-adsorbing polymer, monodisperse rod-like particles assemble into colloidal membranes, which are one rod-length thick liquid-like monolayers of aligned rods. Unlike 3D edgeless bilayer vesicles, colloidal monolayer membranes form open structures with an exposed edge, thus presenting an opportunity to study physics of thin elastic sheets. Membranes assembled from single-component chiral rods form flat disks with uniform edge twist. In comparison, membranes comprised of mixture of rods with opposite chiralities can have the edge twist of either handedness. In this limit d…
We study the dynamics of extended rod-like bodies in (or associated with) membranes and films. We demonstrate a striking difference between the mobilities in films and bulk fluids, even when the dissipation is dominated by the fluid stress: for large inclusions we find that rotation and motion perpendicular to the rod axis exhibits purely local drag, in which the drag coefficient is algebraic in the rod dimensions. We also study the dynamics of the {\em internal} modes of a semiflexible inclusion and find two dynamical regimes in the relaxation spectrum.
Ever since the raft model for biomembranes has been proposed, the traditional view of biomembranes based on the fluid-mosaic model has been altered. In the raft model, dynamical heterogeneities in multi-component lipid bilayers play an essential role. Focusing on the lateral phase separation of biomembranes and vesicles, we review some of the most relevant research conducted over the last decade. We mainly refer to those experimental works that are based on physical chemistry approach, and to theoretical explanations given in terms of soft matter physics. In the first part, we describe the pha…
Proteins often form chiral assembly structures on a biomembrane. However, the role of the chirality in the interaction with an achiral membrane is poorly understood. Here, the differential behavior between chiral and achiral crescent-shaped protein rods was investigated using meshless membrane simulations. The achiral rods deformed the membrane tube into an elliptical shape by stabilizing the edges of the ellipse. In contrast, the chiral rods formed a helical assembly that generated a cylindrical membrane tube with a constant radius in addition to the elliptical tube. This helical assembly cou…
We study the mobility of extended objects (rods) on a spherical liquid-liquid interface to show how this quantity is modified in a striking manner by both the curvature and the topology of the interface. We present theoretical calculations and experimental measurements of the interfacial fluid velocity field around a moving rod bound to the crowded interface of a water-in-oil droplet. By using different droplet sizes, membrane viscosities, and rod lengths, we show that the viscosity mismatch between the interior and exterior fluids leads to a suppression of the fluid flow on small droplets tha…
Monodisperse suspensions of rodlike chiral $fd$ viruses are condensed into a rod-length thick colloidal monolayers of aligned rods by depletion forces. Twist deformations of the molecules are expelled to the monolayer edge as in a chiral smectic $A$ liquid crystal, and a cholesteric band forms at the edge. Coalescence of two such isolated membranes results in a twist wall sandwiched between two regions of aligned rods, dubbed $π$-walls. By modeling the membrane as a binary fluid of coexisting cholesteric and chiral smectic $A$ liquid-crystalline regions, we develop a unified theory of the $π$-…
The assembly of banana-shaped rodlike proteins on membranes, and the associated membrane shape transformations, are investigated by analytical theory and coarse-grained simulations. The membrane-mediated interactions between two banana-shaped inclusions are derived theoretically using a point-like formalism based on fixed anisotropic curvatures, both for zero surface tension and for finite surface tension. On a larger scale, the interactions between assemblies of such rodlike inclusions are determined analytically. Meshless membrane simulations are performed in the presence of a large number o…
Membrane phase-separation is a mechanism that biological membranes often use to locally concentrate specific lipid species in order to organize diverse membrane processes. Phase separation has also been explored as a tool for the design of liposomes with heterogeneous and spatially organized surfaces. These "patchy" liposomes are promising platforms for delivery purposes, however their design and optimization through experimentation can be expensive and time-consuming. We developed a computationally efficient method based on the surface Cahn-Hilliard phase-field model to complement experimenta…
Lipid bilayers forming biological membranes are known to behave as viscous 2D fluids on submicrometer scales; usually they contain a large number of active protein inclusions. Recently, it has been shown [Proc. Nat. Acad. Sci. USA 112, E3639 (2015)] that such active proteins should in- duce non-thermal fluctuating lipid flows leading to diffusion enhancement and chemotaxis-like drift for passive inclusions in biomembranes. Here, a detailed analytical and numerical investigation of such effects is performed. The attention is focused on the situations when proteins are concentrated within lipid …
Using both theoretical modeling and computer simulations we study a model system for DNA interactions in the vicinity of charged membranes. We focus on the polarization of the mobile charges in the membranes due to the nearby charged rods (DNA) and the resulting screening of their fields and inter-rod interactions. We find, both within a Debye-Huckel model and in Brownian dynamics simulations, that the confinement of the mobile charges to the surface leads to a qualitative reduction in their ability to screen the charged rods to the degree that the fields and resulting interactions are not fin…
Colloidal membranes, self assembled monolayers of aligned rod like molecules, offer a template for designing membranes with definite shapes and curvature, and possibly new functionalities in the future. Often the constituent rods, due to their molecular chirality, are tilted with respect to the membrane normal. Spatial patterns of this tilt on curved membranes result from a competition among depletion forces, nematic interaction, molecular chirality and boundary effects. We present a covariant theory for the tilt pattern on minimal surfaces, like helicoids and catenoids, which have been genera…
Membrane organization is essential for cellular functions such as signal transduction and membrane trafficking. A major challenge is to understand the lateral heterogeneous structures in membranes and membrane fluidity in the presence of inclusions. Based on a considerable amount of experimental evidence for lateral organization of lipid membranes which share astonishingly similar features in the presence of different inclusions, we first present a general model system of bilayer membranes embedded by nanosized inclusions, and explain experimental findings. Here, the hydrophobic inclusions are…
The fluctuation-induced, Casimir-like interaction between two parallel rods of length L adsorbed on a fluid membrane is calculated analytically at short separations d<<L. The rods are modeled as constraints imposed on the membrane curvature along a straight line. This allows to define four types of rods, according to whether the membrane can twist along the rod and/or curve across it. For stiff constraints, all the interaction potentials between the different types of rods are attractive and proportional to L/d. Two of the four types of rods are then equivalent, which yields six universal Casi…
The shape transformations of fluid membranes induced by curved protein rods are studied using meshless membrane simulations. The rod assembly at low rod density induces a flat membrane tube and oblate vesicle. It is found that the polyhedral shapes are stabilized at high rod densities. The discrete shape transition between triangular and buckled discoidal tubes is obtained and their curvature energies are analyzed by a simple geometric model. For vesicles, triangular hosohedron and elliptic-disk shapes are formed in equilibrium, whereas tetrahedral and triangular prism shapes are obtained as m…
In living cells, membrane morphology is regulated by various proteins. Many membrane reshaping proteins contain a Bin/Amphiphysin/Rvs (BAR) domain, which consists of a banana-shaped rod. The BAR domain bends the biomembrane along the rod axis and the features of this anisotropic bending have recently been studied. Here, we report on the role of the BAR protein rods in inducing membrane tubulation, using large-scale coarse-grained simulations. We reveal that a small spontaneous side curvature perpendicular to the rod can drastically alter the tubulation dynamics at high protein density, whereas…
Curvature-inducing proteins containing a Bin/Amphiphysin/Rvs domain often have intrinsically disordered domains. Recent experiments have shown that these disordered chains enhance curvature sensing and generation. Here, we report on the modification of protein-membrane interactions by disordered chains using meshless membrane simulations. The protein and bound membrane are modeled together as a chiral crescent protein rod with two excluded-volume chains. As the chain length increases, the repulsion between them reduces the cluster size of the proteins. It induces spindle-shaped vesicles and a …
We develop general methods to calculate the mobilities of extended bodies in (or associated with) membranes and films. We demonstrate a striking difference between in-plane motion of rod-like inclusions and the corresponding case of bulk (three-dimensional) fluids: for rotations and motion perpendicular to the rod axis, we find purely local drag, in which the drag coefficient is purely algebraic in the rod dimensions. These results, as well as the calculational methods are applicable to such problems as the diffusion of objects in or associated with Langmuir films and lipid membranes. The meth…
Critical lateral pressure for a pore formation and phase diagram of porous membrane are derived analytically as functions of the microscopic parameters of the lipid chains. The derivation exploits path-integral calculation of the free energy of the ensembles of semi-flexible strings and rigid rods that mimic the hydrophobic tails of lipids in the lipid bilayers and bolalipid membranes respectively. Analytical expressions for the area stretch/compressibility moduli of the membranes are derived in both models.
The depletion interaction mediated by non-adsorbing polymers promotes condensation and assembly of repulsive colloidal particles into diverse higher-order structures and materials. One example, with particularly rich emergent behaviors, is the formation of two-dimensional colloidal membranes from a suspension of filamentous $\it{fd}$ viruses, which act as rods with effective repulsive interactions, and dextran, which acts as a condensing, depletion-inducing agent. Colloidal membranes exhibit chiral twist even when the constituent virus mixture lacks macroscopic chirality, change from a circula…
Filamentous bacteriophages such as fd-like viruses are monodisperse rod-like colloids that have well defined properties: diameter, length, rigidity, charge and chirality. Engineering those viruses leads to a library of colloidal rods which can be used as building blocks for reconfigurable and hierarchical self-assembly. Their condensation in aqueous solution þ{with additive polymers which act as depletants to induce} attraction between the rods leads to a myriad of fluid-like micronic structures ranging from isotropic/nematic droplets, colloid membranes, achiral membrane seeds, twisted ribbons…
In experiments on model membranes, a formation of large domains of different lipid composition is readily observed. However, no such phase separation is observed in the membranes of intact cells. Instead, a structure of small transient inhomogeneities called lipid rafts are expected in these systems. One of the numerous attempts to explain small domains refers to the coupling of the membrane to its surroundings, which leads to the immobilization of some of the membrane molecules. These immobilized molecules then act as static obstacles for the remaining mobile ones. We present detailed Molecul…
This review presents the elastic theory of low-dimensional (one- and two-dimensional) continua and its applications in bio- and nano-structures. First, the curve and surface theory, as the geometric representation of the low-dimensional continua, is briefly described through Cartan moving frame method. The elastic theory of Kirchhoff rod, Helfrich rod, bending-soften rod, fluid membrane, and solid shell is revisited. Secondly, the application and availability of the elastic theory of low-dimensional continua in bio-structures, including short DNA rings, lipid membranes, and cell membranes, are…
We characterize the role of charge correlations in the adsorption of a short, rod-like anionic polyelectrolyte onto a similarly charged membrane. Our theory reveals two different mechanisms driving the like-charge polyelectrolyte-membrane complexation: in weakly charged membranes, repulsive polyelectrolyte-membrane interactions lead to the interfacial depletion and a parallel orientation of the polyelectrolyte with respect to the membrane; while in the intermediate membrane charge regime, the interfacial counterion excess gives rise to an attractive "salt-induced" image force. This furthermore…
Coarse-grained molecular-dynamics simulations were used to study the morphological changes induced in a Nafion$^{\tiny \textregistered}$-like ionomer by the imposition of a strong electric field. We observe the formation of novel structures aligned along the direction of the applied field. The polar head groups of the ionomer side chains aggregate into clusters, which then form rod-like formations which assemble into a hexatic array aligned with the direction of the field. Occasionally these lines of sulfonates and protons form a helical structure. Upon removal of the electric field, the hexat…
The fluctuation-induced interaction between two rod-like, rigid inclusions in a fluid vesicle is studied by means of canonical ensemble Monte Carlo simulations. The vesicle membrane is represented by a triangulated network of hard spheres. Five rigidly connected hard spheres form rod-like inclusions that can leap between sites of the triangular network. Their effective interaction potential is computed as a function of mutual distance and angle of the inclusions. On account of the hard-core potential among these, the nature of the potential is purely entropic. Special precaution is taken to re…
We discuss a realistic scenario, accounting for the existence of sub-micrometric protein domains in cell membranes. At the biological level, such membrane domains have been shown to be specialized, in order to perform a determined biological task, in the sense that they gather one or a few protein species out of the hundreds of different ones that a cell membrane may contain. By analyzing the balance between mixing entropy and protein affinities, we propose that such protein sorting in distinct domains can be explained without appealing to pre-existing lipidic micro-phase separations, as in th…
We present a Ginzburg-Landau theory of micro phase separation in a bidisperse chiral membrane consisting of rods of opposite handendness. This model system undergoes a phase transition from an equilibrium state where the two components are completely phase separated to a microphase separated state composed of domains of a finite size comparable to the twist penetration depth. Characterizing the phenomenology using linear stability analysis and numerical studies, we trace the origin of the discontinuous change in domain size that occurs during this to a competition between the cost of creating …
We carry out Monte Carlo simulations of a colloidal fluid membrane composed of chiral rod-like viruses. The membrane is modeled by a triangular mesh of beads connected by bonds in which the bonds and beads are free to move at each Monte Carlo step. Since the constituent viruses are experimentally observed to twist only near the membrane edge, we use an effective energy that favors a particular sign of the geodesic torsion of the edge. The effective energy also includes membrane bending stiffness, edge bending stiffness, and edge tension. We find three classes of membrane shapes resulting from …
We show that lateral fluidity in membranes containing quenched protein obstacles belongs to the universality class of the two-dimensional random-field Ising model. The main feature of this class is the absence of a phase transition: there is no critical point, and macroscopic domain formation does not occur. Instead, there is only one phase. This phase is highly heterogeneous, with a structure consisting of micro-domains. The presence of quenched protein obstacles thus provides a mechanism to stabilize lipid rafts in equilibrium. Crucial for two-dimensional random-field Ising universality is t…
