Using a continuum bead-spring Monte Carlo model, we study the anomalous diffusion dynamics of a self-avoiding tethered membrane by means of extensive computer simulations. We focus on the subdiffusive stochastic motion of the membrane's central node in the regime of flat membranes at temperatures above the membrane folding transition. While at times, larger than the characteristic membrane relaxation time $τ_R$, the mean-square displacement of the center of mass of the sheet, $<R_c^2>$, as well as that of its central node, $<R_n^2>$, show the normal Rouse diffusive behavior with a diffusion coefficient $D_N$ scaling as $D_N \propto N^{-1}$ with respect to the number of segments $N$ in the membrane, for short times $t\le τ_R$ we observe a {\em multiscale dynamics} of the central node, $<R_n^2> \propto t^α$, where the anomalous diffusion exponent $α$ changes from $α\approx 0.86$ to $α\approx 0.27$, and then to $α\approx 0.5$, before diffusion turns eventually to normal. By means of simple scaling arguments we show that our main result, $α\approx 0.27$, can be related to particular mechanisms of membrane dynamics which involve different groups of segments in the membrane sheet. A comparative study involving also linear polymers demonstrates that the diffusion coefficient of self-avoiding tethered membranes, containing $N$ segments, is three times smaller than that of linear polymer chains with the same number of segments.
The thermal fluctuation spectrum of a fluid membrane coupled harmonically to a solid support by an array of tethers is calculated. For strong tethers, this spectrum exhibits non-monotonic, anisotropic behavior with a relative maximum at a wavelength about twice the tether distance. The root mean square displacement is evaluated to estimate typical membrane displacements. Possible applications cover pillar-supported or polymer-tethered membranes.
A first-order phase transition separating the smooth phase from the crumpled one is found in a fixed connectivity surface model defined on a disk. The Hamiltonian contains the Gaussian term and an intrinsic curvature term.
We report results concerning the destabilisation of supported phospholipid bilayers in a well-defined geometry. When heating up supported phospholipid membranes deposited on highly hydrophilic glass slides from room temperature (i.e. with lipids in the gel phase), unbinding was observed around the main gel to fluid transition temperature of the lipids. It lead to the formation of relatively monodisperse vesicles, of which most remained tethered to the supported bilayer. We interpret these observations in terms of a sharp decrease of the bending rigidity modulus $κ$ in the transition region, co…
We show that inversion-asymmetric tethered membranes exhibit a new double-spiral phase with long range orientational order not present in symmetric membranes. We calculate the universal algebraic spiral shapes of these membranes in this phase. Asymmetry can trigger the crumpling of these membranes as well. In-vitro experiments on lipid, red blood cell membrane extracts, and on graphene coated on one side, could test these predictions.
Biologically important membrane channels are gated by force at attached tethers. Here, we generically characterize the non-trivial interplay of force, membrane tension, and channel deformations that can affect gating. A central finding is that minute conical channel deformation under force leads to significant energy release during opening. We also calculate channel-channel interactions and show that they can amplify force sensitivity of tethered channels.
A tethered surface model is investigated by using the canonical Monte Carlo simulation technique on a torus with an intrinsic curvature. We find that the model undergoes a first-order phase transition between the smooth phase and the crumpled one.
first-principles numerical simulation model for crumpling of a stiff tethered membrane is introduced. In our model membranes, wrinkles, ridge formation, ridge collapse, as well as the initiation of stiffness divergence, are observed. The ratio of the amplitude and wave length of the wrinkles, and the scaling exponent of the stiffness divergence, are consistent with both theory and experiment. We observe that close to the stiffness divergence there appears a crossover beyond which the elastic behavior of a tethered membrane becomes similar to that of dry granular media. This suggests that ridge…
We found that three types of tethered surface model undergo a first-order phase transition between the smooth and the crumpled phase. The first and the third are discrete models of Helfrich, Polyakov, and Kleinert, and the second is that of Nambu and Goto. These are curvature models for biological membranes including artificial vesicles. The results obtained in this paper indicate that the first-order phase transition is universal in the sense that the order of the transition is independent of discretization of the Hamiltonian for the tethered surface model.
We set up the statistical mechanics for a nearly flat, thermally equilibrated fluid membrane, attached to an elastic network through one of its sides. We predict that the resulting structural (inversion) asymmetry of the membrane, notably due to the elastic network attached to one of its sides, can generate a local spontaneous curvature $C_0$, that may in turn destabilize the otherwise flat membrane. As $C_0$ rises above a threshold at a fixed temperature, a flat tethered membrane in the thermodynamic limit becomes structurally unstable, signaling {\em crumpling} of the flat membrane. In-vitro…
We analyse the effect of thermal fluctuations on the elastic constants of the decoupled lamellar phase of tethered, crystalline membranes. Using a momentum-shell renormalization group technique, we show that the smectic-A -like compressional elastic constant, the in-plane Lamé coefficients, and the cross-coupling elastic constant vanish as (\ln q)^{- a} whereas the bend elastic constant diverges as (\ln q)^{b}, with a, b > 0, as the wavenumber q tends to zero. The exponents a and b satisfy the relation a + 3b = 2.
We study a model of phantom tethered membranes, embedded in three-dimensional space, by extensive Monte Carlo simulations. The membranes have hexagonal lattice structure where each monomer is interacting with six nearest-neighbors (NN). Tethering interaction between NN, as well as curvature penalty between NN triangles are taken into account. This model is new in the sense that NN interactions are taken into account by a truncated Lennard-Jones potential including both repulsive and attractive parts. The main result of our study is that the system undergoes a first-order crumpling transition f…
A tethered vesicle, which consists of a cylindrical membrane tube and a spherical vesicle, is produced by a mechanical force that is experimentally imposed by optical tweezers and a micropipette. This tethered vesicle is employed for examining the curvature sensing of curvature-inducing proteins. In this study, we clarify how the binding of proteins with a laterally isotropic spontaneous curvature senses and generates the membrane curvatures of the tethered vesicle using mean-field theory and meshless membrane simulation. The force-dependence curves of the protein density in the membrane tube …
We found that the order for the crumpling transition of an intrinsic curvature model changes depending on the distance between two boundary vertices fixed on the surface of spherical topology. The model is a curvature one governed by an intrinsic curvature energy, which is defined on triangulated surfaces. It was already reported that the model undergoes a first-order crumpling transition without the boundary conditions on the surface. However, the dependence of the transition on such boundary condition is yet to be studied. We have studied in this paper this problem by using the Monte Carlo s…
If a catenoid is inverted in any interior point, a deflated compact geometry is obtained which touches at two points (its poles). The catenoid is a minimal surface and, as such, is an equilibrium shape of a symmetric fluid membrane. The conformal symmetry of the Hamiltonian implies that inverted minimal surfaces are also equilibrium shapes. However, they exhibit curvature singularities at their poles. These singularities are associated with external forces pulling the poles together. Unlike the catenoid which is free of stress, there will be stress within the inverted shapes. If the surface ar…
We study the shape, elasticity and fluctuations of the recently predicted (cond-mat/9510172) and subsequently observed (in numerical simulations) (cond-mat/9705059) tubule phase of anisotropic membranes, as well as the phase transitions into and out of it. This novel phase lies between the previously predicted flat and crumpled phases, both in temperature and in its physical properties: it is crumpled in one direction, and extended in the other. Its shape and elastic properties are characterized by a radius of gyration exponent $ν$ and an anisotropy exponent $z$. We derive scaling laws for the…
We report a numerical evidence of the discontinuous transition of a tethered membrane model which is defined within a framework of the membrane elasticity of Helfrich. Two kinds of phantom tethered membrane models are studied via the canonical Monte Carlo simulation on triangulated fixed connectivity surfaces of spherical topology. A surface model is defined by the Gaussian term and the bending energy term, and the other, which is tensionless, is defined by the bending energy term and a hard wall potential. The bending energy is defined by using the normal vector at each vertex. Both of the mo…
We use non-perturbative renormalization group techniques to calculate the momentum dependence of thermal fluctuations of graphene, based on a self-consistent calculation of the momentum dependent elastic constants of a tethered membrane. We find a sharp crossover from the perturbative to the anomalous regime, in excellent agreement with Monte Carlo results for graphene, and give an accurate value for the crossover scale. Our work strongly supports the notion that graphene is well described as a tethered membrane. Ripples emerge naturally from our analysis.
Using numerical simulations, we characterized the behavior of an elastic membrane immersed in an active fluid. Our findings reveal a nontrivial folding and re-expansion of the membrane that is controlled by the interplay of its resistance to bending and the self-propulsion strength of the active components in solution. We show how flexible membranes tend to collapse into multi-folded states, whereas stiff membranes oscillates between an extended configuration and a singly folded state. This study provides a simple example of how to exploit the random motion of active particles to perform mecha…
We investigate the lateral dynamics in a purely viscous lipid membrane surrounded by viscoelastic media such as polymeric solutions. We first obtain the generalized frequency-dependent mobility tensor and focus on the case when the solvent is sandwiched by hard walls. Due to the viscoelasticity of the solvent, the mean square displacement of a disk embedded in the membrane exhibits an anomalous diffusion. An useful relation which connects the mean square displacement and the solvent modulus is provided. We also calculate the cross-correlation of the particle displacements which can be applied …
I introduce an alternative interpretation for the short-time anomalous diffusion observed by Gupta et al. using neutron spin echo. The interpretation involves the Brochard-Lennon "red blood cell" mode to describe membrane thickness fluctuations, and yields an anomalous diffusion exponent 1/3 that is consistent with the experimental results.
We use coarse grained molecular dynamics simulations to investigate diffusion properties of sheared lipid membranes with embedded transmembrane proteins. In membranes without proteins, we find normal in-plane diffusion of lipids in all flow conditions. Protein embedded membranes behave quite differently: by imposing a simple shear flow and sliding the monolayers of the membrane over each other, the motion of protein clusters becomes strongly superdiffusive in the shear direction. In such a circumstance, subdiffusion regime is predominant perpendicular to the flow. We show that superdiffusion i…
The physical approach of a small particle (virus, medical drug) to the cell membrane represents the crucial first step before active internalization and is governed by thermal diffusion. Using a fully analytical theory we show that the stretching and bending of the elastic membrane by the approaching particle induces a memory in the system which leads to anomalous diffusion, even though the particle is immersed in a purely Newtonian liquid. For typical cell membranes the transient subdiffusive regime extends beyond 10 ms and can enhance residence times and possibly binding rates up to 50\%. Ou…
In cell membranes, proteins and lipids diffuse in a highly crowded and heterogeneous landscape, where aggregates and dense domains of proteins or lipids obstruct the path of diffusing molecules. In general, hindered motion gives rise to anomalous transport, though the nature of the onset of this behavior is still under debate and difficult to investigate experimentally. Here, we present a systematic study where proteins bound to supported lipid membranes diffuse freely in two dimensions, but are increasingly hindered by the presence of other like proteins. In our model system, the surface cove…
The effect of asymmetry in the transbilayer lipid distribution on the dynamics of phase separation in fluid vesicles is investigated numerically for the first time. This asymmetry is shown to set a spontaneous curvature for the domains that alter the morphology and dynamics considerably. For moderate tension, the domains are capped and the spontaneous curvature leads to anomalously slow dynamics, as compared to the case of symmetric bilayers. In contrast, in the limiting cases of high and low tensions, the dynamics proceeds towards full phase separation.
We study anomalous elasticity in the tubule phases of nematic and smectic elastomer membranes, which are flat in one direction and crumpled in another. These phases share the same macroscopic symmetry properties including spontaneously-broken in-plane isotropy and hence belong to the same universality class. Below an upper critical value $D_c =3$ of the membranes' intrinsic dimension D, thermal fluctuations renormalize the elasticity with respect to elastic displacements along the tubule axis so that elastic moduli for compression along the tubule axis and for bending the tubule axis become le…
The dynamics of membrane undulations inside a viscous solvent is governed by distinctive, anomalous, power laws. Inside a viscoelastic continuous medium these universal behaviors are modified by the specific bulk viscoelastic spectrum. Yet, in structured fluids the continuum limit is reached only beyond a characteristic correlation length. We study the crossover to this asymptotic bulk dynamics. The analysis relies on a recent generalization of the hydrodynamic interaction in structured fluids, which shows a slow spatial decay of the interaction toward the bulk limit. For membranes which are w…
Motivated by a freely suspended graphene and polymerized membranes in soft and biological matter we present a detailed study of a tensionless elastic sheet in the presence of thermal fluctuations and quenched disorder. The manuscript is based on an extensive draft dating back to 1993, that was circulated privately. It presents the general theoretical framework and calculational details of numerous results, partial forms of which have been published in brief Letters (Le Doussal and Radzihovsky 1992). The experimental realization of atom-thin graphene sheets has driven a resurgence in this fasci…
We investigate the properties of crystalline phantom membranes, at the crumpling transition and in the flat phase, using a nonperturbative renormalization group approach. We avoid a derivative expansion of the effective average action and instead analyse the full momentum dependence of the elastic coupling functions. This leads to a more accurate determination of the critical exponents and further yields the full momentum dependence of the correlation functions of the in-plane and out-of-plane fluctuation. The flow equations are solved numerically for D=2 dimensional membranes embedded in a d=…
Mathematically modeling complex transport phenomena at the molecular level can be a powerful tool for identifying transport mechanisms and predicting macroscopic properties. We use two different stochastic time series models, parameterized from long molecular dynamics (MD) simulation trajectories of a cross-linked HII phase lyotropic liquid crystal (LLC) membrane, in order to predict solute mean squared displacements (MSDs) and solute flux, and thus solute selectivity, in macroscopic length pores. First, using anomalous diffusion theory, we show how solute dynamics can be modeled as a fraction…
A ubiquitous observation in cell biology is that diffusion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarising their densely packed and heterogeneous structures. The most familiar phenomenon is a power-law increase of the MSD, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations, non-…
A ubiquitous observation in crowded cell membranes is that molecular transport does not follow Fickian diffusion but exhibits subdiffusion. The microscopic origin of such a behaviour is not understood and highly debated. Here we discuss the spatio-temporal dynamics for two models of subdiffusion: fractional Brownian motion and hindered motion due to immobile obstacles. We show that the different microscopic mechanisms can be distinguished using fluorescence correlation spectroscopy (FCS) by systematic variation of the confocal detection area. We provide a theoretical framework for space-resolv…
Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) which converges back to a Brownian motion with reduced diffusion coefficient at long times, after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed-down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the…
For reproducing the anomalous -- i.e., sub- or super-diffusive -- behavior in some stochastic dynamical systems, the Generalized Langevin Equation (GLE) has gained considerable popularity in recent years. Motivated by the question whether or not a system with anomalous dynamics can have the GLE formulation, here I consider polymer physics, where sub-diffusive behavior is commonplace. I provide an exact derivation of the GLE for phantom Rouse polymers, andby identifying polymeric response to local strains, I argue the case for the GLE formulation for self-avoiding polymers and polymer transloca…
We consider the passage of long polymers of length N through a hole in a membrane. If the process is slow, it is in principle possible to focus on the dynamics of the number of monomers s on one side of the membrane, assuming that the two segments are in equilibrium. The dynamics of s(t) in such a limit would be diffusive, with a mean translocation time scaling as N^2 in the absence of a force, and proportional to N when a force is applied. We demonstrate that the assumption of equilibrium must break down for sufficiently long polymers (more easily when forced), and provide lower bounds for th…
It has been recently shown that the phase space trajectories for the anomalous dynamics of a tagged monomer of a polymer --- for single polymeric systems such as phantom Rouse, self-avoiding Rouse, Zimm, reptation, and translocation through a narrow pore in a membrane; as well as for many-polymeric system such as polymer melts in the entangled regime --- is robustly described by the Generalized Langevin Equation (GLE). Here I show that the probability distribution of phase space trajectories for all these classical anomalous dynamics for single polymers is that of a fractional Brownian motion …
During the synthesis of integral membrane proteins (IMPs), the hydrophobic amino acids of the polypeptide sequence are partitioned mostly into the membrane interior and hydrophilic amino acids mostly into the aqueous exterior. We analyze the minimum free energy state of polypeptide sequences partitioned into alpha-helical transmembrane (TM) segments and the role of thermal fluctuations using a many-body statistical mechanics model. Results suggest that IMP TM segment partitioning shares important features with general theories of protein folding. For random polypeptide sequences, the minimum f…
The dynamics of particles interacting by key-lock binding of attached biomolecules are studied theoretically. Experimental realizations of such systems include colloids grafted with complementary single-stranded DNA (ssDNA), and particles grafted with antibodies to cell-membrane proteins. Depending on the coverage of the functional groups, we predict two distinct regimes. In the low coverage localized regime, there is an exponential distribution of departure times. As the coverage is increased the system enters a diffusive regime resulting from the interplay of particle desorption and diffusio…
Combining extensive molecular dynamics simulations of lipid bilayer systems of varying chemical composition with single-trajectory analyses we systematically elucidate the stochastic nature of the lipid motion. We observe subdiffusion over more than four orders of magnitude in time, clearly stretching into the sub-microsecond domain. The lipid motion delicately depends on the lipid chemistry, the lipid phase, and especially on the presence of cholesterol. We demonstrate that fractional Langevin equation motion universally describes the lipid motion in all phases including the gel phase, and in…
From the spread of pollutants in the atmosphere to the transmission of nutrients across cell membranes, anomalous diffusion processes are ubiquitous in natural systems. The ability to understand and control the mechanisms guiding such processes across various scales has important application to research in materials science, finance, medicine, and energetics. Here we present a numerical study of anomalous diffusion of light through a semi-crystalline polymer structure where transport is guided by random disorder and nonlocal interactions. The numerical technique examines diffusion properties i…
We investigate the dynamics of water confined in soft ionic nano-assemblies, an issue critical for a general understanding of the multi-scale structure-function interplay in advanced materials. We focus in particular on hydrated perfluoro-sulfonic acid compounds employed as electrolytes in fuel cells. These materials form phase-separated morphologies that show outstanding proton-conducting properties, directly related to the state and dynamics of the absorbed water. We have quantified water motion and ion transport by combining Quasi Elastic Neutron Scattering, Pulsed Field Gradient Nuclear Ma…
We consider a model system in which anomalous diffusion is generated by superposition of underlying linear modes with a broad range of relaxation times. In the language of Gaussian polymers, our model corresponds to Rouse (Fourier) modes whose friction coefficients scale as wavenumber to the power $2-z$. A single (tagged) monomer then executes subdiffusion over a broad range of time scales, and its mean square displacement increases as $t^α$ with $α=1/z$. To demonstrate non-trivial aspects of the model, we numerically study the absorption of the tagged particle in one dimension near an absorbi…
Any first course on polymer physics teaches that the dynamics of a tagged monomer of a polymer is anomalously subdiffusive, i.e., the mean-square displacement of a tagged monomer increases as $t^α$ for some $α<1$ until the terminal relaxation time $τ$ of the polymer. Beyond time $τ$ the motion of the tagged monomer becomes diffusive. Classical examples of anomalous dynamics in polymer physics are single polymeric systems, such as phantom Rouse, self-avoiding Rouse, self-avoiding Zimm, reptation, translocation through a narrow pore in a membrane, and many-polymeric systems such as polymer melts…
A topic of intense current investigation pursues the question how the highly crowded environment of biological cells affects the dynamic properties of passively diffusing particles. Motivated by recent experiments we report results of extensive simulations of the motion of a finite sized tracer particle in a heterogeneously crowded environment. For given spatial distributions of monodisperse crowders we demonstrate how anomalous diffusion with strongly non-Gaussian features arises in this model system. We investigate both biologically relevant situations of particles released either at the sur…
We study the dynamics of the passage of a polymer through a membrane pore (translocation), focusing on the scaling properties with the number of monomers $N$. The natural coordinate for translocation is the number of monomers on one side of the hole at a given time. Commonly used models which assume Brownian dynamics for this variable predict a mean (unforced) passage time $τ$ that scales as $N^2$, even in the presence of an entropic barrier. However, the time it takes for a free polymer to diffuse a distance of the order of its radius by Rouse dynamics scales with an exponent larger than 2, a…
