Correlated diffusion of membrane proteins and their effect on membrane viscosity

Hydrodynamically induced aggregation of two dimensional oriented active particles

We investigate a system of co-oriented active particles interacting only via hydrodynamic and steric interactions in a two-dimensional fluid. We offer a new method of calculating the flow created by any active particle in such a fluid, focusing on the dynamics of flow fields with a high-order spatial decay, which we analyze using a geometric Hamiltonian. We show that when the particles are oriented and the flow has a single, odd power decay, such systems lead to stable, fractal-like aggregation, with the only exception being the force dipole. We discuss how our results can easily be generalized to more complicated force distributions and to other effective two-dimensional systems.

Physical limits on galvanotaxis

Eukaryotic cells can polarize and migrate in response to electric fields via "galvanotaxis," which aids wound healing. Experimental evidence suggests cells sense electric fields via molecules on the cell's surface redistributing via electrophoresis and electroosmosis, though the sensing species has not yet been conclusively identified. We develop a model that links sensor redistribution and galvanotaxis using maximum likelihood estimation. Our model predicts a single universal curve for how galvanotactic directionality depends on field strength. We can collapse measurements of galvanotaxis in keratocytes, neural crest cells, and granulocytes to this curve, suggesting that stochasticity due to the finite number of sensors may limit galvanotactic accuracy. We find cells can achieve experimentally observed directionalities with either a few (∼100) highly polarized sensors or many (∼10^{4}) sensors with an ∼6-10% change in concentration across the cell. We also identify additional signatures of galvanotaxis via sensor redistribution, including the presence of a tradeoff between accuracy and variance in cells being controlled by rapidly switching fields. Our approach shows how the physics of noise at the molecular scale can limit cell-scale galvanotaxis, providing important constraints on sensor properties and allowing for new tests to determine the specific molecules underlying galvanotaxis.

Dynamic correlations in lipid bilayer membranes over finite time intervals

Recent single-molecule measurements [Schoch et al., Proc. Natl. Acad. Sci. U. S. A. 118, e2113202118 (2021)] have observed dynamic lipid-lipid correlations in membranes with submicrometer spatial resolution and submillisecond temporal resolution. While short from an instrumentation standpoint, these length and time scales remain long compared to microscopic molecular motions. Theoretical expressions are derived to infer experimentally measurable correlations from the two-body diffusion matrix appropriate for membrane-bound bodies coupled by hydrodynamic interactions. The temporal (and associated spatial) averaging resulting from finite acquisition times has the effect of washing out correlations as compared to naive predictions (i.e., the bare elements of the diffusion matrix), which would be expected to hold for instantaneous measurements. The theoretical predictions are shown to be in excellent agreement with Brownian dynamics simulations of experimental measurements. Numerical results suggest that the experimental measurement of membrane protein diffusion, in complement to lipid diffusion measurements, might help to resolve the experimental ambiguities encountered for certain black lipid membranes.

Protein drift-diffusion dynamics and phase separation in curved cell membranes and dendritic spines: Hybrid discrete-continuum methods

We develop methods for investigating protein drift-diffusion dynamics in heterogeneous cell membranes and the roles played by geometry, diffusion, chemical kinetics, and phase separation. Our hybrid stochastic numerical methods combine discrete particle descriptions with continuum-level models for tracking the individual protein drift-diffusion dynamics when coupled to continuum fields. We show how our approaches can be used to investigate phenomena motivated by protein kinetics within dendritic spines. The spine geometry is hypothesized to play an important biological role regulating synaptic strength, protein kinetics, and self-assembly of clusters. We perform simulation studies for model spine geometries varying the neck size to investigate how phase-separation and protein organization is influenced by different shapes. We also show how our methods can be used to study the roles of geometry in reaction-diffusion systems including Turing instabilities. Our methods provide general approaches for investigating protein kinetics and drift-diffusion dynamics within curved membrane structures.

Modeling Receptor Motility along Advecting Lipid Membranes

This work aims to overview multiphysics mechanobiological computational models for receptor dynamics along advecting cell membranes. Continuum and statistical models of receptor motility are the two main modeling methodologies identified in reviewing the state of the art. Within the former modeling class, a further subdivision based on different biological purposes and processes of proteins’ motion is recognized; cell adhesion, cell contractility, endocytosis, and receptor relocations on advecting membranes are the most relevant biological processes identified in which receptor motility is pivotal. Numerical and/or experimental methods and approaches are highlighted in the exposure of the reviewed works provided by the literature, pertinent to the topic of the present manuscript. With a main focus on the continuum models of receptor motility, we discuss appropriate multiphyisics laws to model the mass flux of receptor proteins in the reproduction of receptor relocation and recruitment along cell membranes to describe receptor–ligand chemical interactions, and the cell’s structural response. The mass flux of receptor modeling is further supported by a discussion on the methodology utilized to evaluate the protein diffusion coefficient developed over the years.

Hyperuniformity and phase enrichment in vortex and rotor assemblies

Ensembles of particles rotating in a two-dimensional fluid can exhibit chaotic dynamics yet develop signatures of hidden order. Such rotors are found in the natural world spanning vastly disparate length scales — from the rotor proteins in cellular membranes to models of atmospheric dynamics. Here we show that an initially random distribution of either driven rotors in a viscous membrane, or ideal vortices with minute perturbations, spontaneously self assemble into a distinct arrangement. Despite arising from drastically different physics, these systems share a Hamiltonian structure that sets geometrical conservation laws resulting in prominent structural states. We find that the rotationally invariant interactions isotropically suppress long-wavelength fluctuations — a hallmark of a disordered hyperuniform material. With increasing area fraction, the system orders into a hexagonal lattice. In mixtures of two co-rotating populations, the stronger population will gain order from the other and both will become phase enriched. Finally, we show that classical 2D point vortex systems arise as exact limits of the experimentally accessible microscopic membrane rotors, yielding a new system through which to study topological defects.

Rotor-like dynamics is observed in many natural systems, from the rotor proteins in cellular membranes to atmospheric models. Here, the authors uncover geometrical conservation laws that limit distribution of driven rotors in a membrane or a soap film and allow to predict their structural states.

Correlated diffusion in lipid bilayers

Dynamic processes on membrane surfaces are essential for biological function. Traditionally, quantitative measurements of lipid/protein motion have been interpreted in the framework of membrane hydrodynamics. However, some recent single-molecule tracking studies have proven difficult to interpret via hydrodynamic arguments. Does this suggest a failure of hydrodynamic theory or simply highlight the dangers in attempting to extend hydrodynamic arguments down to molecular scales? Intermolecular correlations are superior to single-molecule observations for studying hydrodynamics due to the longer length scales involved. The current work reports dynamic pair correlations of lipids in model membranes. Submicron distance-dependent correlations are well resolved, and complementary numerical calculations indicate that hydrodynamic theory can predict membrane dynamics over distances of tens of nanometers and longer.

Lipid membranes are complex quasi–two-dimensional fluids, whose importance in biology and unique physical/materials properties have made them a major target for biophysical research. Recent single-molecule tracking experiments in membranes have caused some controversy, calling the venerable Saffman–Delbrück model into question and suggesting that, perhaps, current understanding of membrane hydrodynamics is imperfect. However, single-molecule tracking is not well suited to resolving the details of hydrodynamic flows; observations involving correlations between multiple molecules are superior for this purpose. Here dual-color molecular tracking with submillisecond time resolution and submicron spatial resolution is employed to reveal correlations in the Brownian motion of pairs of fluorescently labeled lipids in membranes. These correlations extend hundreds of nanometers in freely floating bilayers (black lipid membranes) but are severely suppressed in supported lipid bilayers. The measurements are consistent with hydrodynamic predictions based on an extended Saffman–Delbrück theory that explicitly accounts for the two-leaflet bilayer structure of lipid membranes.

Active gels, heavy tails, and the cytoskeleton

The eukaryotic cell's cytoskeleton is a prototypical example of an active material: objects embedded within it are driven by molecular motors acting on the cytoskeleton, leading to anomalous diffusive behavior. Experiments tracking the behavior of cell-attached objects have observed anomalous diffusion with a distribution of displacements that is non-Gaussian, with heavy tails. This has been attributed to "cytoquakes" or other spatially extended collective effects. We show, using simulations and analytical theory, that a simple continuum active gel model driven by fluctuating force dipoles naturally creates heavy power-law tails in cytoskeletal displacements. We predict that this power law exponent should depend on the geometry and dimensionality of where force dipoles are distributed through the cell; we find qualitatively different results for force dipoles in a 3D cytoskeleton and a quasi-two-dimensional cortex. We then discuss potential applications of this model both in cells and in synthetic active gels.

Avidity and surface mobility in multivalent ligand-receptor binding

Targeted drug delivery relies on two physical processes: the selective binding of a therapeutic particle to receptors on a specific cell membrane, followed by transport of the particle across the membrane. In this article, we address some of the challenges in controlling the thermodynamics and dynamics of these two processes by combining a simple experimental system with a statistical mechanical model. Specifically, we characterize and model multivalent ligand-receptor binding between colloidal particles and fluid lipid bilayers, as well as the surface mobility of membrane-bound particles. We show that the mobility of the receptors within the fluid membrane is key to both the thermodynamics and dynamics of binding. First, we find that the particle-membrane binding free energy-or avidity-is a strongly nonlinear function of the ligand-receptor affinity. We attribute the nonlinearity to a combination of multivalency and recruitment of fluid receptors to the binding site. Our results also suggest that partial wrapping of the bound particles by the membrane enhances avidity further. Second, we demonstrate that the lateral mobility of membrane-bound particles is also strongly influenced by the recruitment of receptors. Specifically, we find that the lateral diffusion coefficient of a membrane-bound particle is dominated by the hydrodynamic drag against the aggregate of receptors within the membrane. These results provide one of the first direct validations of the working theoretical framework for multivalent interactions. They also highlight that the fluidity and elasticity of the membrane are as important as the ligand-receptor affinity in determining the binding and transport of small particles attached to membranes.

Nonreciprocal response of a two-dimensional fluid with odd viscosity

We discuss the linear hydrodynamic response of a two-dimensional active chiral compressible fluid with odd viscosity. The viscosity coefficient represents broken time-reversal and parity symmetries in the 2D fluid and characterizes the deviation of the system from a passive fluid. Taking into account the hydrodynamic coupling to the underlying bulk fluid, we obtain the odd viscosity-dependent mobility tensor, which is responsible for the nonreciprocal hydrodynamic response to a point force. Furthermore, we consider a finite-size disk moving laterally in the 2D fluid and demonstrate that the disk experiences a nondissipative lift force in addition to the dissipative drag one.

Persistent collective motion of a dispersing membrane domain

We study the Brownian motion of an assembly of mobile inclusions embedded in a fluid membrane. The motion includes the dispersal of the assembly, accompanied by the diffusion of its center of mass. Usually, the former process is much faster than the latter because the diffusion coefficient of the center of mass is inversely proportional to the number of particles. However, in the case of membrane inclusions, we find that the two processes occur on the same timescale, thus significantly prolonging the lifetime of the assembly as a collectively moving object. This effect is caused by the quasi-two-dimensional membrane flows, which couple the motions of even the most remote inclusions in the assembly. The same correlations also cause the diffusion coefficient of the center of mass to decay slowly with time, resulting in weak subdiffusion. We confirm our analytical results by Brownian dynamics simulations with flow-mediated correlations. The effect reported here should have implications for the stability of nanoscale membrane heterogeneities.

Tunable Collective Dynamics of Active Inclusions in Viscous Membranes

We study hydrodynamic interactions and clustering mechanisms of active membrane inclusions within lipid bilayers. Pairs of inclusions display unique oscillatory dynamics that disappear when the 3D fluid adjacent to the membrane is confined. We reduce the governing equations to a coupled dynamical system whose phase behavior reveals the striking role of bulk confinement in enhancing cluster formation within the membrane. Using numerical simulations, we then extend this finding to demonstrate the role of confinement in controlling large-scale aggregation of membrane inclusions.

Surfactant dynamics: hidden variables controlling fluid flows

Surfactants - molecules and particles that preferentially adsorb to fluid interfaces - play a ubiquitous role in the fluids of industry, of nature, and of life. Since most surfactants cannot be seen directly, their behavior must be inferred from their impact on observed flows, like the buoyant rise of a bubble, or the thickness of a coating film. In so doing, however, a difficulty arises: physically distinct surfactant processes can affect measurable flows in qualitatively identical ways, raising the specter of confusion or even misinterpretation. This Perspective describes, in one coherent piece, both the equilibrium properties and dynamic processes of surfactants, to better enable the fluid mechanics community to understand, interpret, and design surfactant/fluid systems. Specifically, §2 treats the equilibrium thermodynamics of surfactants at interfaces, including surface pressure, isotherms of soluble and insoluble surfactants, and surface dilatational moduli (Gibbs and Marangoni). §3 describes surfactant dynamics in fluid systems, including surfactant transport and interfacial stress boundary conditions, the competition between surface diffusion, advection, and adsorption/desorption, Marangoni stresses and flows, and surface excess rheology. §4 discusses paradigmatic problems from fluid mechanics that are impacted by surfactants, including translating drops and bubbles, surfactant adsorption to clean and oscillating interfaces; capillary wave damping, thin film dynamics, foam drainage, and the dynamics of particles and probes at surfactant-laden interfaces. Finally, §5 discusses the additional richness and complexity that frequently arise in 'real' surfactants, including phase transitions, phase coexistence, and polycrystalline phases within surfactant monolayers, and their impact on non-Newtonian surface rheology.

Surface Response of a Polymer Network: Semi-infinite Network

We study theoretically the surface response of a semi-infinite viscoelastic polymer network using the two-fluid model. We focus on the overdamped limit and on the effect of the network's intrinsic length scales. We calculate the decay rate of slow surface fluctuations, and the surface displacement in response to a localized force. Deviations from the large-scale continuum response are found at length scales much larger than the network's mesh size. We discuss implications for surface scattering and microrheology. We provide closed-form expressions that can be used for surface microrheology: the extraction of viscoelastic moduli and intrinsic length scales from the motions of tracer particles lying on the surface without doping the bulk material.

Shear viscosity of two-state enzyme solutions

We discuss the shear viscosity of a Newtonian solution of catalytic enzymes and substrate molecules. The enzyme is modeled as a two-state dimer consisting of two spherical domains connected with an elastic spring. The enzymatic conformational dynamics is induced by the substrate binding and such a process is represented by an additional elastic spring. Employing the Boltzmann distribution weighted by the waiting times of enzymatic species in each catalytic cycle, we obtain the shear viscosity of dilute enzyme solutions as a function of substrate concentration and its physical properties. The substrate affinity distinguishes between fast and slow enzymes, and the corresponding viscosity expressions are obtained. Furthermore, we connect the obtained viscosity with the diffusion coefficient of a tracer particle in enzyme solutions.

Motion of objects embedded in lipid bilayer membranes: Advection and effective viscosity

An interfacial regularized Stokeslet scheme is presented to predict the motion of solid bodies (e.g., proteins or gel-phase domains) embedded within flowing lipid bilayer membranes. The approach provides a numerical route to calculate velocities and angular velocities in complex flow fields that are not amenable to simple Faxén-like approximations. Additionally, when applied to shearing motions, the calculations yield predictions for the effective surface viscosity of dilute rigid-body-laden membranes. In the case of cylindrical proteins, effective viscosity calculations are compared to two prior analytical predictions from the literature. Effective viscosity predictions for a dilute suspension of rod-shaped objects in the membrane are also presented.

Flow Arrest in the Plasma Membrane

The arrangement of receptors in the plasma membrane strongly affects the ability of a cell to sense its environment both in terms of sensitivity and in terms of spatial resolution. The spatial and temporal arrangement of the receptors is affected in turn by the mechanical properties and the structure of the cell membrane. Here, we focus on characterizing the flow of the membrane in response to the motion of a protein embedded in it. We do so by measuring the correlated diffusion of extracellularly tagged transmembrane neurotrophin receptors TrkB and p75 on transfected neuronal cells. In accord with previous reports, we find that the motion of single receptors exhibits transient confinement to submicron domains. We confirm predictions based on hydrodynamics of fluid membranes, finding long-range correlations in the motion of the receptors in the plasma membrane. However, we discover that these correlations do not persist for long ranges, as predicted, but decay exponentially, with a typical decay length on the scale of the average confining domain size.

Finite-Size-Corrected Rotational Diffusion Coefficients of Membrane Proteins and Carbon Nanotubes from Molecular Dynamics Simulations

We investigate system-size effects on the rotational diffusion of membrane proteins and other membrane-embedded molecules in molecular dynamics simulations. We find that the rotational diffusion coefficient slows down relative to the infinite-system value by a factor of one minus the ratio of protein and box areas. This correction factor follows from the hydrodynamics of rotational flows under periodic boundary conditions and is rationalized in terms of Taylor–Couette flow. For membrane proteins like transporters, channels, or receptors in typical simulation setups, the protein-covered area tends to be relatively large, requiring a significant finite-size correction. Molecular dynamics simulations of the protein adenine nucleotide translocase (ANT1) and of a carbon nanotube porin in lipid membranes show that the hydrodynamic finite-size correction for rotational diffusion is accurate in standard-use cases. The dependence of the rotational diffusion on box size can be used to determine the membrane viscosity.

Spatial Cross-Correlated Diffusion of Colloids under Shear Flow

The spatial cross-correlated diffusion of colloidal particles is often used as an essential tool to study the dynamic properties of a fluid because it directly describes the hydrodynamic interaction between two particles in a fluid. However, the experimental measurement of cross-correlated diffusion can be substantially modified by even a weak shear flow. In this work, the effect of a shear flow on spatial cross-correlated diffusion is demonstrated using experimental measurements that show a clear dependence on pair angles. An analytical solution is proposed to explain the experimental observations. A numerical simulation is performed to systemically demonstrate the influence of shear flow on spatial cross-correlated diffusion. The results of the experiment, theoretical analysis, and numerical simulation agree well with each other. Therefore, this research provides a sensitive experimental method to determine the weak shear flow in any quasi-two-dimensional fluid systems.

From Dynamics to Membrane Organization: Experimental Breakthroughs Occasion a "Modeling Manifesto"

New experimental techniques, especially in the context of observing molecular dynamics, reveal the plasma membrane to be heterogeneous and "scale rich," from nanometers to microns and from microseconds to seconds. This is critical information, which shows that scale-dependent transport governs the molecular encounters that underlie cellular signaling. The data are rich and reaffirm the importance of the cortical cytoskeleton, protein aggregates, and lipidomic complexity on the statistics of molecular encounters. Moreover, the data demand simulation approaches with a particular set of features, hence the "manifesto." Together with the experimental data, simulations that satisfy these requirements hold the promise of a deeper understanding of membrane spatiotemporal organization. Several experimental breakthroughs in measuring molecular membrane dynamics are reviewed, the constraints that they place on simulations are discussed, and the status of simulation approaches that aim to meet them are detailed.

Many-particle mobility and diffusion tensors for objects in viscous sheets

We derive a mobility tensor for many cylindrical objects embedded in a viscous sheet. This tensor guarantees a positive dissipation rate for any configuration of particles and forces, analogous to the Rotne-Prager-Yamakawa tensor for spherical particles in a three-dimensional viscous fluid. We test our result for a ring of radially driven particles, demonstrating the positive-definite property at all particle densities. The derived tensor can be utilized in Brownian dynamics simulations with hydrodynamic interactions for such systems as proteins in biomembranes and inclusions in free-standing liquid films.

Brownian Motion at Lipid Membranes: A Comparison of Hydrodynamic Models Describing and Experiments Quantifying Diffusion within Lipid Bilayers

The capability of lipid bilayers to exhibit fluid-phase behavior is a fascinating property, which enables, for example, membrane-associated components, such as lipids (domains) and transmembrane proteins, to diffuse within the membrane. These diffusion processes are of paramount importance for cells, as they are for example involved in cell signaling processes or the recycling of membrane components, but also for recently developed analytical approaches, which use differences in the mobility for certain analytical purposes, such as in-membrane purification of membrane proteins or the analysis of multivalent interactions. Here, models describing the Brownian motion of membrane inclusions (lipids, peptides, proteins, and complexes thereof) in model bilayers (giant unilamellar vesicles, black lipid membranes, supported lipid bilayers) are summarized and model predictions are compared with the available experimental data, thereby allowing for evaluating the validity of the introduced models. It will be shown that models describing the diffusion in freestanding (Saffman-Delbrück and Hughes-Pailthorpe-White model) and supported bilayers (the Evans-Sackmann model) are well supported by experiments, though only few experimental studies have been published so far for the latter case, calling for additional tests to reach the same level of experimental confirmation that is currently available for the case of freestanding bilayers.

Noise-induced drift in two-dimensional anisotropic systems

We study the isothermal Brownian dynamics of a particle in a system with spatially varying diffusivity. Due to the heterogeneity of the system, the particle's mean displacement does not vanish even if it does not experience any physical force. This phenomenon has been termed "noise-induced drift," and has been extensively studied for one-dimensional systems. Here, we examine the noise-induced drift in a two-dimensional anisotropic system, characterized by a symmetric diffusion tensor with unequal diagonal elements. A general expression for the mean displacement vector is derived and presented as a sum of two vectors, depicting two distinct drifting effects. The first vector describes the tendency of the particle to drift toward the high diffusivity side in each orthogonal principal diffusion direction. This is a generalization of the well-known expression for the noise-induced drift in one-dimensional systems. The second vector represents a novel drifting effect, not found in one-dimensional systems, originating from the spatial rotation in the directions of the principal axes. The validity of the derived expressions is verified by using Langevin dynamics simulations. As a specific example, we consider the relative diffusion of two transmembrane proteins, and demonstrate that the average distance between them increases at a surprisingly fast rate of several tens of micrometers per second.

Effect of Hydrodynamic Interactions on Reaction Rates in Membranes

The Brownian motion of two particles in three dimensions serves as a model for predicting the diffusion-limited reaction rate, as first discussed by von Smoluchowski. Deutch and Felderhof extended the calculation to account for hydrodynamic interactions between the particles and the target, which results in a reduction of the rate coefficient by about half. Many chemical reactions take place in quasi-two-dimensional systems, such as on the membrane or surface of a cell. We perform a Smoluchowski-like calculation in a quasi-two-dimensional geometry, i.e., a membrane surrounded by fluid, and account for hydrodynamic interactions between the particles. We show that rate coefficients are reduced relative to the case of no interactions. The reduction is more pronounced than the three-dimensional case due to the long-range nature of two-dimensional flows.

Lateral diffusion induced by active proteins in a biomembrane

We discuss the hydrodynamic collective effects due to active protein molecules that are immersed in lipid bilayer membranes and modeled as stochastic force dipoles. We specifically take into account the presence of the bulk solvent that surrounds the two-dimensional fluid membrane. Two membrane geometries are considered: the free membrane case and the confined membrane case. Using the generalized membrane mobility tensors, we estimate the active diffusion coefficient and the drift velocity as a function of the size of a diffusing object. The hydrodynamic screening lengths distinguish the two asymptotic regimes of these quantities. Furthermore, the competition between the thermal and nonthermal contributions in the total diffusion coefficient is characterized by two length scales corresponding to the two membrane geometries. These characteristic lengths describe the crossover between different asymptotic behaviors when they are larger than the hydrodynamic screening lengths.

Lipid and Peptide Diffusion in Bilayers: The Saffman-Delbrück Model and Periodic Boundary Conditions

The periodic Saffman-Delbrück (PSD) model, an extension of the Saffman-Delbrück model developed to describe the effects of periodic boundary conditions on the diffusion constants of lipids and proteins obtained from simulation, is tested using the coarse-grained Martini and all-atom CHARMM36 (C36) force fields. Simulations of pure Martini dipalmitoylphosphatidylcholine (DPPC) bilayers and those with one embedded gramicidin A (gA) dimer or one gA monomer with sizes ranging from 512 to 2048 lipids support the PSD model. Underestimates of D^∞ (the value of the diffusion constant for an infinite system) from the 512-lipid system are 35% for DPPC, 45% for the gA monomer, and 70% for the gA dimer. Simulations of all-atom DPPC and dioleoylphosphatidylcholine (DOPC) bilayers yield diffusion constants not far from experiment. However, the PSD model predicts that diffusion constants at the sizes of the simulation should underestimate experiment by approximately a factor of 3 for DPPC and 2 for DOPC. This likely implies a deficiency in the C36 force field. A Bayesian method for extrapolating diffusion constants of lipids and proteins in membranes obtained from simulation to infinite system size is provided.

Hydrodynamic interactions in freely suspended liquid crystal films

Hydrodynamic interactions play an important role in biological processes in cellular membranes, a large separation of length scales often allowing such membranes to be treated as continuous, two-dimensional (2D) fluids. We study experimentally and theoretically the hydrodynamic interaction of pairs of inclusions in two-dimensional, fluid smectic liquid crystal films suspended in air. Such smectic membranes are ideal systems for performing controlled experiments as they are mechanically stable, of highly uniform structure, and have well-defined, variable thickness, enabling experimental investigation of the crossover from 2D to 3D hydrodynamics. Our theoretical model generalizes the Levine-MacKintosh theory of point-force response functions and uses a boundary-element approach to calculate the mobility matrix for inclusions of finite extent. We describe in detail the theoretical and computational approach previously outlined in Z. Qi et al., Phys. Rev. Lett. 113, 128304 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.128304 and extend the method to study the mutual mobilities of inclusions with asymmetric shapes. The model predicts well the observed mutual mobilities of pairs of circular inclusions in films and the self-mobility of a circular inclusion in the vicinity of a linear boundary.

Strong influence of periodic boundary conditions on lateral diffusion in lipid bilayer membranes

The Saffman-Delbrück hydrodynamic model for lipid-bilayer membranes is modified to account for the periodic boundary conditions commonly imposed in molecular simulations. Predicted lateral diffusion coefficients for membrane-embedded solid bodies are sensitive to box shape and converge slowly to the limit of infinite box size, raising serious doubts for the prospects of using detailed simulations to accurately predict membrane-protein diffusivities and related transport properties. Estimates for the relative error associated with periodic boundary artifacts are 50% and higher for fully atomistic models in currently feasible simulation boxes. MARTINI simulations of LacY membrane protein diffusion and LacY dimer diffusion in DPPC membranes and lipid diffusion in pure DPPC bilayers support the underlying hydrodynamic model.

Hydrodynamic coupling of particle inclusions embedded in curved lipid bilayer membranes

We develop theory and computational methods to investigate particle inclusions embedded within curved lipid bilayer membranes. We consider the case of spherical lipid vesicles where inclusion particles are coupled through (i) intramembrane hydrodynamics, (ii) traction stresses with the external and trapped solvent fluid, and (iii) intermonolayer slip between the two leaflets of the bilayer. We investigate relative to flat membranes how the membrane curvature and topology augment hydrodynamic responses. We show how both the translational and rotational mobility of protein inclusions are effected by the membrane curvature, ratio of intramembrane viscosity to solvent viscosity, and intermonolayer slip. For general investigations of many-particle dynamics, we also discuss how our approaches can be used to treat the collective diffusion and hydrodynamic coupling within spherical bilayers.

Hydrodynamics of bilayer membranes with diffusing transmembrane proteins

We consider the hydrodynamics of lipid bilayers containing transmembrane proteins of arbitrary shape. This biologically-motivated problem is relevant to the cell membrane, whose fluctuating dynamics play a key role in phenomena ranging from cell migration, intercellular transport, and cell communication. Using Onsager's variational principle, we derive the equations that govern the relaxation dynamics of the membrane shape, of the mass densities of the bilayer leaflets, and of the diffusing proteins' concentration. With our generic formalism, we obtain several results on membrane dynamics. We find that proteins that span the bilayer increase the intermonolayer friction coefficient. The renormalization, which can be significant, is in inverse proportion to the protein's mobility. Second, we find that asymmetric proteins couple to the membrane curvature and to the difference in monolayer densities. For practically all accessible membrane tensions (σ > 10(-8) N m(-1)) we show that the protein density is the slowest relaxing variable. Furthermore, its relaxation rate decreases at small wavelengths due to the coupling to curvature. We apply our formalism to the large-scale diffusion of a concentrated protein patch. We find that the diffusion profile is not self-similar, owing to the wavevector dependence of the effective diffusion coefficient.

Fluctuating hydrodynamics of multicomponent membranes with embedded proteins

A simulation method for the dynamics of inhomogeneous lipid bilayer membranes is presented. The membrane is treated using stochastic Saffman-Delbrück hydrodynamics, coupled to a phase-field description of lipid composition and discrete membrane proteins. Multiple applications are considered to validate and parameterize the model. The dynamics of membrane composition fluctuations above the critical point and phase separation dynamics below the critical point are studied in some detail, including the effects of adding proteins to the mixture.

Calculating hydrodynamic interactions for membrane-embedded objects

A recently introduced numerical scheme for calculating self-diffusion coefficients of solid objects embedded in lipid bilayer membranes is extended to enable calculation of hydrodynamic interactions between multiple objects. The method is used to validate recent analytical predictions by Oppenheimer and Diamant [Biophys. J. 96, 3041 2009] related to the coupled diffusion of membrane embedded proteins and is shown to converge to known near-field lubrication results as objects closely approach one another; however, the present methodology also applies outside of the limiting regimes where analytical results are available. Multiple different examples involving pairs of disk-like objects with various constraints imposed on their relative motions demonstrate the importance of hydrodynamic interactions in the dynamics of proteins and lipid domains on membrane surfaces. It is demonstrated that the relative change in self-diffusion of a membrane embedded object upon perturbation by a similar proximal solid object displays a maximum for object sizes comparable to the Saffman-Delbrück length of the membrane.

6-Arylcoumarins: versatile scaffolds for fluorescent sensors

6-Arylcoumarins are available as versatile scaffolds for various types of fluorescent sensors like those for cation and viscosity.

Physical aspects of heterogeneities in multi-component lipid membranes

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 phase behavior and the conformation of multi-component lipid bilayers. After formulating the hydrodynamics of fluid membranes in the presence of the surrounding solvent, we discuss the domain growth-law and decay rate of concentration fluctuations. Finally, we review several attempts to describe membrane rafts as two-dimensional microemulsion.

Hydrodynamic interaction between overlapping domains during recurrence of registration within planar lipid bilayer membranes

Due to a thermodynamic coupling between the two leaflets comprising a lipid bilayer, compositional lipid domains residing within opposing leaflets are often found in registry. If the system is perturbed by displacing one domain relative to the other, diffusive and advective lipid fluxes are established to restore equilibrium and reestablish domain overlap. In this work, we focus on the advective part of the process, and first derive an analytical expression for the hydrodynamic drag coefficient associated with the advective flow for the special case of perfect domain overlap. The resulting expression identifies parameter regions where sliding friction between the leaflets dominates over viscous dissipation within the leaflets or vice versa. It is shown that in all practically relevant cases, sliding friction between the leaflets is the dominant factor. Finally, we investigate the domain separation dependence of the hydrodynamic drag coefficient via direct simulations of a continuum diffuse interface model, and provide useful empirical expressions for its behavior.

Correlated diffusion of colloidal particles near a liquid-liquid interface

Optical microscopy and multi-particle tracking are used to investigate the cross-correlated diffusion of quasi two-dimensional colloidal particles near an oil-water interface. The behaviors of the correlated diffusion along longitudinal and transverse direction are asymmetric. It is shown that the characteristic length for longitudinal and transverse correlated diffusion are particle diameter and the distance from particle center to the interface, respectively, for large particle separation . The longitudinal and transverse correlated diffusion coefficient and are independent of the colloidal area fraction when , which indicates that the hydrodynamic interactions(HIs) among the particles are dominated by HIs through the surrounding fluid for small . For high area fraction , the power law exponent for the spatial decay of begins to decrease, which suggests the HIs are more contributed from the 2D particle monolayer self for large .

Universal scaling of correlated diffusion in colloidal monolayers

Using the techniques of optical microscopy and particle tracking, we measure the correlated diffusion in a monolayer of uniform silica spheres dispersed at a water-air interface. It is found that the correlated motion of the interfacial particles can be well described by two universal response functions, the normalized longitudinal and transverse diffusion coefficients D(∥)(r/r0) and D(⊥)(r/r0), where r is the interparticle distance and r0=a(λS/a)(3/2) is a new scaling length, which depends on both the Saffman length λS and particle radius a. The obtained response functions characterize the crossover behavior of the colloidal monolayers from the subphase-dominated three-dimensional hydrodynamics at low surface coverage to the monolayer-dominated 2D hydrodynamics at high concentrations. The surface viscosity ηs(2) of the colloidal monolayer obtained by two-particle rheology compares well with the one-particle measurements.

Dynamic scaling in phase separation kinetics for quasi-two-dimensional membranes

We consider the dynamics of phase separation in lipid bilayer membranes, modeled as flat two-dimensional liquid sheets within a bulk fluid, both in the creeping flow approximation. We present scaling arguments that suggest asymptotic coarsening in these systems is characterized by a length scale R(t) ~ t(1/2) for critical (bicontinuous) phase separation and R(t) ~t(1/3) for off-critical concentrations (droplet morphology). In this limit, the bulk fluid is the primary source of dissipation. We also address these questions with continuum stochastic hydrodynamic simulations. We see evidence of scaling violation in critical phase separation, where isolated circular domains coarsen slower than elongated ones. However, we also find a region of apparent scaling where R(t) ~ t(1/2) is observed. This appears to be due to the competition of thermal and hydrodynamic effects. We argue that the diversity of scaling exponents measured in experiment and prior simulations can in part be attributed to certain measurements lying outside the asymptotic long-length-scale regime, and provide a framework to help understand these results. We also discuss a few simple generalizations to confined membranes and membranes in which inertia is relevant.

In-plane dynamics of membranes with immobile inclusions

Cell membranes are anchored to the cytoskeleton via immobile inclusions. We investigate the effect of such anchors on the in-plane dynamics of a fluid membrane and mobile inclusions (proteins) embedded in it. The immobile particles lead to a decreased diffusion coefficient of mobile ones and suppress the correlated diffusion of particle pairs. Because of the long-range, quasi-two-dimensional nature of membrane flows, these effects become significant at a low area fraction (below 1%) of immobile inclusions.

Diffusion coefficient of an inclusion in a liquid membrane supported by a solvent of arbitrary thickness

The diffusion coefficient of an inclusion in a liquid membrane is investigated by taking into account the interaction between membranes and bulk solvents of arbitrary thickness. As illustrative examples, the diffusion coefficients of two types of inclusions, a circular domain composed of fluid with the same viscosity as the host membrane and that of a polymer chain embedded in the membrane, are studied. The diffusion coefficients are expressed in terms of the hydrodynamic screening lengths, which vary according to the solvent thickness. When the membrane fluid is dragged by the solvent of finite thickness, via stick boundary conditions, multiple hydrodynamic screening lengths together with the weight factors to the diffusion coefficients are obtained from the characteristic equation. The conditions for which the diffusion coefficients can be approximated by the expression including only a single hydrodynamic screening length are also shown.

Beyond the creeping viscous flow limit for lipid bilayer membranes: theory of single-particle microrheology, domain flicker spectroscopy, and long-time tails

Recent experiments suggest that lipid bilayer membranes may be viscoelastic. We present a generalized "Saffman-Einstein" relation that may be used to determine the linear viscoelastic shear modulus from single-bead microrheology experiments on membranes. We show that viscoelastic parameters can also be extracted from membrane domain flicker spectroscopy experiments. Contributions from fluid inertia are expected to be negligible in both microrheology and domain flicker spectroscopy experiments, but can create a "long-time tail" in the membrane velocity autocorrelation function. In a viscous membrane, this tail crosses over from t(-1) at intermediate times, as in a two-dimensional fluid, to t(-3/2) at long times, as in a three-dimensional fluid. If the membrane is viscoelastic, the velocity autocorrelation function may be negative at intermediate times.

Continuum simulations of biomembrane dynamics and the importance of hydrodynamic effects

Traditional particle-based simulation strategies are impractical for the study of lipid bilayers and biological membranes over the longest length and time scales (microns, seconds and longer) relevant to cellular biology. Continuum-based models developed within the frameworks of elasticity theory, fluid dynamics and statistical mechanics provide a framework for studying membrane biophysics over a range of mesoscopic to macroscopic length and time regimes, but the application of such ideas to simulation studies has occurred only relatively recently. We review some of our efforts in this direction with emphasis on the dynamics in model membrane systems. Several examples are presented that highlight the prominent role of hydrodynamics in membrane dynamics and we argue that careful consideration of fluid dynamics is key to understanding membrane biophysics at the cellular scale.

Synthesis and evaluation of self-calibrating ratiometric viscosity sensors

We describe the design, synthesis and fluorescent profile of a family of self-calibrating dyes that provide ratiometric measurements of fluid viscosity. The design is based on covalently linking a primary fluorophore (reference) that displays a viscosity-independent fluorescence emission with a secondary fluorophore (sensor) that exhibits a viscosity-sensitive fluorescence emission. Characterization of fluorescent properties was made with separate excitation of the units and through Resonance Energy Transfer from the reference to the sensor dye. The chemical structures of both fluorophores and the linker length have been evaluated in order to optimize the overall brightness and sensitivity of the viscosity measurements. We also present an application of such ratiometric dyes for the detection of membrane viscosity changes in a liposome model.

Dynamics of a polymer chain confined in a membrane

We present a Brownian dynamics theory with full hydrodynamics (Stokesian dynamics) for a Gaussian polymer chain embedded in a liquid membrane which is surrounded by bulk solvent and walls. The mobility tensors are derived in Fourier space for the two geometries, namely, a free membrane embedded in a bulk fluid, and a membrane sandwiched by the two walls. Within the preaveraging approximation, a new expression for the diffusion coefficient of the polymer is obtained for the free-membrane geometry. We also carry out a Rouse normal mode analysis to obtain the relaxation time and the dynamical structure factor. For large polymer size, both quantities show Zimm-like behavior in the free-membrane case, whereas they are Rouse-like for the sandwiched membrane geometry. We use the scaling argument to discuss the effect of excluded-volume interactions on the polymer relaxation time.

Hydrodynamic coupling between two fluid membranes

The coupled in-plane diffusion dynamics between point-particles embedded in stacked fluid membranes is investigated. We calculate the contributions to the coupling longitudinal and transverse diffusion coefficients due to particle motion within the different as well as the same membranes. The stacked geometry leads to a hydrodynamic coupling between the two membranes.

Correlated dynamics of inclusions in a supported membrane

The hydrodynamic theory of heterogeneous fluid membranes is extended to the case of a membrane adjacent to a solid substrate. We derive the coupling diffusion coefficients of pairs of membrane inclusions in the limit of large separation compared to the inclusion size. Two-dimensional compressive stresses in the membrane make the coupling coefficients decay asymptotically as 1/r(2) with interparticle distance r. For the common case, where the distance to the substrate is of submicrometer scale, we present expressions for the coupling between distant disklike inclusions, which are valid for arbitrary inclusion size. We calculate the effect of inclusions on the response of the membrane and the associated corrections to the coupling diffusion coefficients to leading order in the concentration of inclusions. While at short distances the response is modified as if the membrane were a two-dimensional suspension, the large-distance response is not renormalized by the inclusions.