The human red blood cell (RBC) membrane, a fluid lipid bilayer tethered to an elastic 2D spectrin network, provides the principal control of the cell's morphology and mechanics. These properties, in turn, influence the ability of RBCs to transport oxygen in circulation. Current mechanical measurements of RBCs rely on external loads. Here we apply a noncontact optical interferometric technique to quantify the thermal fluctuations of RBC membranes with 3 nm accuracy over a broad range of spatial and temporal frequencies. Combining this technique with a new mathematical model describing RBC membrane undulations, we measure the mechanical changes of RBCs as they undergo a transition from the normal discoid shape to the abnormal echinocyte and spherical shapes. These measurements indicate that, coincident with this morphological transition, there is a significant increase in the membrane's shear, area, and bending moduli. This mechanical transition can alter cell circulation and impede oxygen delivery. promises more sensitive probes of their structure at the nanoscale and suggests new insights into the etiology of a number of human diseases (1, 2). In the healthy individual, these cells withstand repeated, large-amplitude mechanical deformations as they circulate through the microvasculature. Certain pathological conditions such as spherocytosis, malaria, and Sickle cell disease cause changes in both the equilibrium shape and mechanics of RBCs, which impact their transport function. Here we communicate measurements of RBC mechanics that rely on unique experimental and theoretical techniques to characterize the mechanics/rheology of normal and pathological RBCs over a range of length and time scales.Lacking a 3D cytoskeleton, RBCs maintain their shape and mechanical integrity through a spectrin-dominated, triangular 2D network attached to the cytosolic side of their plasma membrane. This semiflexible filament network, along with the surface tension of the bilayer, contributes to the elastic moduli of the composite membrane (3). The fluid lipid bilayer is thought to be the principal contributor to its bending or curvature modulus. Little is known about the molecular and structural transformations that take place in the membrane and spectrin network during the cell's morphological transitions from discocyte (DC, normal shape) to echinocyte (EC, spiculated shape) to spherocyte (SC, nearly spherical) (Figs. 1 A-C), which are accompanied by changes in RBC mechanics.A number of techniques have been used to study the rheology of live cells (2). Micropipette aspiration (4), electric field deformation (5), and optical tweezers (2) provide quantitative information about the shear and bending moduli of RBC membranes in static conditions. However, dynamic, frequencydependent knowledge of RBC mechanics is currently very limited with the notable exception of ref. 6. RBC thermal fluctuations ("flickering") have been studied for more than a century (7) to better understand the interaction between the lipid bilayer and the cytoskeleto...

The membranes of human red blood cells (RBCs) are a composite of a fluid lipid bilayer and a triangular network of semiflexible filaments (spectrin). We perform cellular microrheology using the dynamic membrane fluctuations of the RBCs to extract the elastic moduli of this composite membrane. By applying known osmotic stresses we measure the changes in the elastic constants under imposed strain and thereby determine the nonlinear elastic properties of the membrane. We find that the elastic nonlinearities of the shear modulus in tensed RBC membranes can be well-understood in terms of a simple worm-like chain model. Our results show that the elasticity of the spectrin network can mostly account for the area compression modulus at physiological osmolality, suggesting that the lipid bilayer has significant excess area. As the cell swells, the elastic contribution from the now tensed lipid membrane becomes dominant.

We determine the particulate transport properties of fluid membranes with nontrivial geometries that are surrounded by viscous Newtonian solvents. Previously, this problem in membrane hydrodynamics was discussed for the case of flat membranes by Saffman and Delbrück [P. G. Saffman and M. Delbrück, Proc. Natl. Acad. Sci. U.S.A. 72, 3111 (1975)]. We review and develop the formalism necessary to consider the hydrodynamics of membranes with arbitrary curvature and show that the effect of local geometry is twofold. First, local Gaussian curvature introduces in-plane viscous stresses even for situations in which the velocity field is coordinate-independent. Secondly, even in the absence of Gaussian curvature, the geometry of the membrane modifies the momentum transport between the bulk fluids and the membrane. We illustrate these effects by examining in detail the mobilities of particles bound to spherical and cylindrical membranes. These two examples provide experimentally testable predictions for particulate mobilities and membrane velocity fields on giant unilamellar vesicles and membrane tethers. Finally, we use the examples of spherical and cylindrical membranes to demonstrate how the global geometry and topology of the membrane influences the membrane velocities and the particle mobilities.

Multivalent counterions can induce an effective attraction between like-charged rodlike polyelectrolytes, leading to the formation of polelectrolyte bundles. In this paper, we calculate the equilibrium bundle size using a simple model in which the attraction between polyelectrolytes (assumed to be pairwise additive) is treated phenomenologically. If the counterions are point-like, they almost completely neutralize the charge of the bundle, and the equilibrium bundle size diverges. When the counterions are large, however, steric and short-range electrostatic interactions prevent charge neutralization of the bundle, thus forcing the equilibrium bundle size to be finite. We also consider the possibility that increasing the number of nearest neighbors for each rod in the bundle frustrates the attractive interaction between the rods. Such a frustration leads to the formation of finite size bundles as well, even when the counterions are small. The mean-field Poisson-Boltzmann (PB) theory of electrostatic interactions predicts that two identical macromolecules in any salt solution will repel each other [1]. However, the presence of multivalent counterions can actually induce an attraction between like-charged polyelectrolytes (PEs). This has been experimentally observed for several different PEs, including doublestranded DNA [2, 3], F-actin [4,5], microtubules [4,6], and the fd, M13, and tobacco mosaic viruses [4,7]. Computer simulations of both homogeneously charged rods [8,9,10,11] and realistic DNA molecules [11,12,13] unambiguously show that attractive interactions can arise solely from counterion correlations not included in PB theory. Several theories that take these correlations into account -including perturbative expansions of PB theory [14,15], structural-correlation theory [16,17,18], and strong-coupling theory [19] -obtain an attractive interaction between two rods. It is still a matter of discussion, however, as to which of these theories is the most appropriate description of the correlation-induced attraction seen in experiments and simulations. Furthermore, it is unknown whether the interactions between multiple rods is pairwise additive or not [17,20,21].Under experimental conditions in which the interaction between PEs is attractive, the PEs typically form dense, ordered bundles of a well-defined size, rather than precipitating into a PE-rich phase [2,3,4,5,6,7]. In this paper, we theoretically investigate the thermodynamic stability of these bundles (if bundle growth is not limited thermodynamically, then it must be limited by kinetic barriers [21,22,23]). We assume that the attractive interactions are pairwise additive, but do not specify the precise nature of the counterion correlations. Rather, we simply introduce a phenomenological parameter γ to characterize the attractive energy between two PEs in a bundle.Consider, then, an aqueous solution of volume V with N identical rodlike PEs of length L, radius a 0 , and a uniform linear charge density −eλ 0 (the aggregation of flexible PEs has been cons...

Abstract. -Realistic charged macromolecules are characterized by discrete (rather than homogeneous) charge distributions. We investigate the effects of surface charge discretization on the counterion distribution at the level of mean-field theory using a two-state model. Both planar and cylindrical geometries are considered; for the latter case, we compare our results to numerical solutions of the full Poisson-Boltzmann equation. We find that the discretization of the surface charge can cause enhanced localization of the counterions near the surface; for charged cylinders, counterion condensation can exceed Oosawa-Manning condensation.Introduction. -The interaction between charged macromolecules and counterions is a crucial component of the physics of charged systems [1]. Although realistic macromolecules are often composed of discrete charges, theoretical models typically assume a homogeneous surface charge distribution. In the case of cylindrical polyelectrolytes (PEs), this assumption leads to Oosawa-Manning (OM) condensation [2,3], in which counterions become closely associated with the macromolecule, effectively lowering its overall charge. A similar effect also occurs for spherical macromolecules [4]. Theoretically, the effects of surface charge inhomogeneities on the counterion distribution has been studied for planar [5,6], cylindrical [7][8][9][10], and spherical [11,12] geometries. In particular, both Moreira et al. [5] and Lukatsky et al. [6] have shown analytically that the heterogeneity of a planar charge distribution leads to an enhanced localization of the counterions near the macromolecular surface. Thus, theoretical models that assume homogeneous surface charge distributions -including OM condensation theory -must be modified for realistic charge distributions.In this letter, we explore the effects of surface charge discretization on the counterion distribution in the absence of added salt. For small charge modulation, it is appropriate to use a perturbative expansion of the Poisson-Boltzmann (PB) equation to describe the meanfield behavior of the counterion distribution. It has already been shown that such an expansion

We study the capillary wave dynamics of a single viscoelastic supported film and of a double layer of immiscible viscoelastic supported films. Using both simple scaling arguments and a continuum hydrodynamic theory, we investigate the effects of viscoelasticity and interfacial slip on the relaxation dynamics of these capillary waves. Our results account for the recent observation of a wavelength-independent decay rate for capillary waves in a supported polystyrene/brominated polystyrene double layer [X. Hu et al., Phys. Rev. E 74, 010602 (R) (2006)].

When particulate suspensions are sheared, perturbations in the shear flows around the rigid particles increase the local energy dissipation so that the viscosity of the suspension is effectively higher than that of the solvent. For bulk ͑three-dimensional͒ fluids, understanding this viscosity enhancement is a classic problem in hydrodynamics that originated over a century ago with Einstein's study of a dilute suspension of spherical particles ͓A. Einstein, Ann. Phys. 19, 289 ͑1906͔͒. In this paper, we investigate the analogous problem of the effective viscosity of a suspension of disks embedded in a two-dimensional membrane or interface. Unlike the hydrodynamics of bulk fluids, low-Reynolds number membrane hydrodynamics is characterized by an inherent length scale generated by the coupling of the membrane to the bulk fluids that surround it. As a result, we find that the size of the particles in the suspension relative to this hydrodynamic length scale has a dramatic effect on the effective viscosity of the suspension. Our study also helps elucidate the mathematical tools needed to solve the mixed boundary value problems that generically arise when considering the motion of rigid inclusions in fluid membranes.

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 that cannot be captured by the flat interface predictions.The dynamics of mobile inclusions in lipid membranes are fundamental to a variety of biological processes, including signal transduction [1] and the endocytosis of bacterial toxins [2]. Membrane inclusions, such as proteins [3] or lipid "rafts" [4], are in many cases significantly larger than the lipids making up the membrane, so their dynamics can be studied in terms of macroscopic objects moving in a continuum fluid environment. Additionally, elucidating the mobilities and hydrodynamic interactions of colloidal particles at a fluid-fluid interface has important technological ramifications for the design and formation of membranes and capsules composed of particles assembled on droplets [5].Low Reynolds number hydrodynamics in viscous membranes or interfaces differs substantially from the betterknown problem of hydrodynamics in three dimensions. Because of the coupling between the two-dimensional membrane and its surrounding viscous solvent, in-plane momentum in the membrane is lost to the surrounding fluid. The flows induced in the surrounding fluid generate nonlocal couplings between the membrane velocity and stress. The net result of these effects is to introduce an inherent length scale 0 -the Saffman-Delbrück (SD) length -which is set by the ratio of the (2D) membrane viscosity η m to the (3D) fluid viscosity η, 0 ∼ η m /η [6]. For cellular plasma membranes 0 1µm [7]. In contrast, low-Reynolds number hydrodynamics in 3D fluids is a scale-invariant theory. The existence of an inherent length scale in membrane and interfacial hydrodynamics has complex and rather subtle effects on a variety of problems, including protein diffusion in cell membranes [6,8], the flow of monolayers through channels [9], the dynamics of monolayer domains [10], microrheology of fluid-fluid interfaces [11], and the mobilities of rigid and flexible extended objects in membranes [12].In this letter we explore the effect of nontrivial interfacial geometry and topology on the hydrodynamics of viscous interfaces. We find two principal results with broad implications for particulate transport on curved or spherical interfaces. The first is that the compact topology of a spherical interface fundamentally alters the nature of the 2D interfacial velocity field. On a sphere, any vector field must include at least two vortices [13]. In contrast, there are no such singularities in the velocity field on...