Cytoskeleton confinement and tension of red blood cell membranes

We analyze theoretically both the static and dynamic fluctuation spectra of the red blood cell in a unified manner, using a simple model of the composite membrane. In this model, the two-dimensional spectrin network that forms the cytoskeleton is treated as a rigid shell, located at a fixed, average distance from the lipid bilayer. The cytoskeleton thereby confines both the static and dynamic fluctuations of the lipid bilayer. The sparse connections of the cytoskeleton and bilayer induce a surface tension, for wavelengths larger than the bilayer persistence length. The predictions of the model give a consistent account for both the wave vector and frequency dependence of the experimental data.     

Nonequilibrium mechanics and dynamics of motor-activated gels

The mechanics of cells is strongly affected by molecular motors that generate forces in the cellular cytoskeleton. We develop a model for cytoskeletal networks driven out of equilibrium by molecular motors exerting transient contractile stresses. Using this model we show how motor activity can dramatically increase the network's bulk elastic moduli. We also show how motor binding kinetics naturally leads to enhanced low-frequency stress fluctuations that result in nonequilibrium diffusive motion within an elastic network, as seen in recent in vitro and in vivo experiments.     

Configurations of fluid membranes and vesicles

Vesicles consisting of a bilayer membrane of amphiphilic lipid molecules are remarkably flexible surfaces that show an amazing variety of shapes of different symmetry and topology. Owing to the fluidity of the membrane, shape transitions such as budding can be induced by temperature changes or the action of optical tweezers. Thermally excited shape fluctuations are both strong and slow enough to be visible by video microscopy. Depending on the physical conditions, vesicles adhere to and unbind from each other or a substrate.

This article describes the systematic physical theory developed to understand the static and dynamic aspects of membrane and vesicle configurations. The preferred shapes arise from a competition between curvature energy, which derives from the bending elasticity of the membrane, geometrical constraints such as fixed surface area and fixed enclosed volume, and a signature of the bilayer aspect. These shapes of lowest energy are arranged into phase diagrams, which separate regions of different symmetry by continuous or discontinuous transitions. The geometrical constraints affect the fluctuations around these shapes by creating an effective tension.

For vesicles of non-spherical topology, the conformal invariance of the curvature energy leads to conformal diffusion, which signifies a one-fold degeneracy of the ground state. Unbinding and adhesion transitions arise from the balance between attractive interactions and entropic repulsion or a cost in bending energy, respectively. Both the dynamics of equilibrium fluctuations and the dynamics of shape transformations are governed not only by viscous damping in the surrounding liquid but also by internal friction if the two monolayers slip over each other. More complex membranes such as that of the red blood cell exhibit a variety of new phenomena because of coupling between internal degrees of freedom and external geometry.     

Cytoskeleton influence on normal and tangent fluctuation modes in the red blood cells

We argue that the paradoxal softness of the red blood cells (RBC) in fluctuation experiments is apparent. We show that the effective surface shear modulus mus of the RBC obtained from fluctuation data and that measured in static deformation experiments have the same order of magnitude. In the RBC model developed for this purpose the spectrin network cytoskeleton with the bulk shear modulus estimated as mu approximately 105-165 Pa contributes to both normal and tangent fluctuations of the system and confines the membrane fluctuations. The calculated ratio of the mean-square amplitudes / is 2-3 orders of magnitude smaller than it is in the free membrane with the same bending and shear moduli.     

Fluctuation-induced forces between inclusions in a fluid membrane under tension

We develop an exact method to calculate thermal Casimir forces between inclusions of arbitrary shapes and separation, embedded in a fluid membrane whose fluctuations are governed by the combined action of surface tension, bending modulus, and Gaussian rigidity. Each object's shape and mechanical properties enter only through a characteristic matrix, a static analog of the scattering matrix. We calculate the Casimir interaction between two elastic disks embedded in a membrane. In particular, we find that at short separations the interaction is strong and independent of surface tension.     

Brownian dynamics in Fourier space: membrane simulations over long length and time scales

A simulation algorithm for elastic membrane sheets is presented. Overdamped stochastic dynamics including hydrodynamic coupling to surrounding solvent and arbitrary external forces are generated by employing Fourier modes of the sheet as the primary dynamic variables. Simulations over the micron length scale and second time scale are easily achieved. The dynamics of a lipid bilayer attached to an underlying network of cytoskeletal filaments is used to estimate the diffusion constant of membrane-bound proteins on the surface of the red blood cell.     

Optical measurement of cell membrane tension

Using a novel noncontact technique based on optical interferometry, we quantify the nanoscale thermal fluctuations of red blood cells (RBCs) and giant unilamellar vesicles (GUVs). The measurements reveal a nonvanishing tension coefficient for RBCs, which increases as cells transition from a discocytic shape to a spherical shape. The tension coefficient measured for GUVs is, however, a factor of 4-24 smaller. By contrast, the bending moduli for cells and vesicles have similar values. This is consistent with the cytoskeleton confinement model, in which the cytoskeleton inhibits membrane fluctuations [Gov et al., Phys. Rev. Lett. 90, 228101, (2003).     

Force balance and membrane shedding at the red-blood-cell surface

During the aging of the red-blood cell, or under conditions of extreme echinocytosis, membrane is shed from the cell plasma membrane in the form of nanovesicles. We propose that this process is the result of the self-adaptation of the membrane surface area to the elastic stress imposed by the spectrin cytoskeleton, via the local buckling of membrane under increasing cytoskeleton stiffness. This model introduces the concept of force balance as a regulatory process at the cell membrane and quantitatively reproduces the rate of area loss in aging red-blood cells.     

The moving boundary node method: A level set-based,finite volume algorithm with applications to cell motility

Eukaryotic cell crawling is a highly complex biophysical and biochemical process, where deformation and motion of a cell are driven by internal, biochemical regulation of a poroelastic cytoskeleton. One challenge to built quantitative models that describe crawling cells is solving the reaction–diffusion–advection dynamics for the biochemical and cytoskeletal components of the cell inside its moving and deforming geometry. Here we develop an algorithm that uses the level set method to move the cell boundary and uses information stored in the distance map to construct a finite volume representation of the cell. Our method preserves Cartesian connectivity of nodes in the finite volume representation while resolving the distorted cell geometry. Derivatives approximated using a Taylor series expansion at finite volume interfaces lead to second order accuracy even on highly distorted quadrilateral elements. A modified, Laplacian-based interpolation scheme is developed that conserves mass while interpolating values onto nodes that join the cell interior as the boundary moves. An implicit time stepping algorithm is used to maintain stability. We use the algorithm to simulate two simple models for cellular crawling. The first model uses depolymerization of the cytoskeleton to drive cell motility and suggests that the shape of a steady crawling cell is strongly dependent on the adhesion between the cell and the substrate. In the second model, we use a model for chemical signalling during chemotaxis to determine the shape of a crawling cell in a constant gradient and to show cellular response upon gradient reversal.     

The concept of effective tension for fluctuating vesicles

Vesicles are closed surfaces of bilayer membranes. Their mean shapes and fluctuations are governed by the competition of curvature energy and geometrical constraints on the enclosed volume and total surface area. A scheme to calculate these fluctuations to lowest order in the ratio of temperature to bending rigidity is developed. It is shown that for fluctuations that break a symmetry of the mean shape the area constraint indeed acts like a tension whose value is given by the Lagrange multiplier used to enforce the area constraint in the first place. As a consequence, these fluctuations are also insensitive to the specific variants of the curvature model. For fluctuations that preserve the symmetry of the mean shape the role of the area constraint is more subtle. The low temperature expansion breaks down in the spherical limit where with the excess area another small parameter enters. By incorporating the area constraint in this limit exactly, the validity of the conventional approach using an effective tension for fluctuations of quasi-spherical vesicles can be assessed.Dedicated to Prof. Herbert Wagner on the occasion of his 60th birthday     

Influence of erythrocyte aggregation on leukocyte margination in postcapillary expansions: A lattice Boltzmann analysis

Leukocyte rolling on the vascular endothelium requires initial contact between the circulating leukocytes in the blood and the vessel wall. Although specific adhesion mechanisms are involved in leukocyte–endothelium interactions, adhesion patterns in vivo suggest other rheological mechanisms are involved as well. Previous studies have proposed that the abundance of leukocyte rolling in postcapillary venules is due to interactions between red blood cells and leukocytes as they enter capillary expansions as well as red blood cell (RBC) aggregation. We have established a lattice Boltzmann approach to analyze the interactions of RBC aggregates and leukocytes as they flow through a postcapillary expansion. The lattice Boltzmann technique provides the complete solution of the flow field and quantification of the particle–particle forces. Our results show that RBC aggregation strongly influences leukocyte–endothelium interactions.     

Fluctuations, dynamics, and the stretch-coil transition of single actin filaments in extensional flows

Semiflexible polymers subject to hydrodynamic forcing play an important role in cytoskeletal motions in the cell, particularly when filaments guide molecular motors whose motions create flows. Near hyperbolic stagnation points, filaments experience a competition between bending elasticity and tension and are predicted to display suppressed thermal fluctuations in the extensional regime and a buckling instability under compression. Using a microfluidic cross-flow geometry, we verify these predictions in detail, including a fluctuation-rounded stretch-coil transition of actin filaments.     

Morphological phase diagram for lipid membrane domains with entropic tension

Circular domains in phase-separated lipid vesicles with symmetric leaflet composition commonly exhibit three stable morphologies: flat, dimpled, and budded. However, stable dimples (i.e., partially budded domains) present a puzzle since simple elastic theories of domain shape predict that only flat and spherical budded domains are mechanically stable in the absence of spontaneous curvature. We argue that this inconsistency arises from the failure of the constant surface tension ensemble to properly account for the effect of entropic bending fluctuations. Formulating membrane elasticity within an entropic tension ensemble, wherein tension represents the free energy cost of extracting membrane area from thermal bending of the membrane, we calculate a morphological phase diagram that contains regions of mechanical stability for each of the flat, dimpled, and budded domain morphologies.     

Nonequilibrium microtubule fluctuations in a model cytoskeleton

Biological activity gives rise to nonequilibrium fluctuations in the cytoplasm of cells; however, there are few methods to directly measure these fluctuations. Using a reconstituted actin cytoskeleton, we show that the bending dynamics of embedded microtubules can be used to probe local stress fluctuations. We add myosin motors that drive the network out of equilibrium, resulting in an increased amplitude and modified time dependence of microtubule bending fluctuations. We show that this behavior results from steplike forces on the order of 10 pN driven by collective motor dynamics.     

Johnson-Kendall-Roberts theory applied to living cells

Johnson-Kendall-Roberts (JKR) theory is an accurate model for strong adhesion energies of soft slightly deformable material. Little is known about the validity of this theory on complex systems such as living cells. We have addressed this problem using a depletion controlled cell adhesion and measured the force necessary to separate the cells with a micropipette technique. We show that the cytoskeleton can provide the cells with a 3D structure that is sufficiently elastic and has a sufficiently low deformability for JKR theory to be valid. When the cytoskeleton is disrupted, JKR theory is no longer applicable.     

Cell sorting in three dimensions: topology, fluctuations, and fluidlike instabilities

Previous 2D and 3D models concluded that cell sorting requires cytoskeletal fluctuations and is stalled by high tension at heterotypic interfaces. New deterministic and stochastic models show that this is not true in 3D. Sorting in 3D involves both topological untangling and domain coalescence. Coalescence requires fluctuations and low tension, but untangling does not. It occurs by a Plateau-Rayleigh instability of cell threads-deterministically driven by high tension. At high minority-cell fractions, untangling dominates and significant partial sorting can occur without fluctuations.     

Minimal bending energies of bilayer polyhedra

Motivated by recent experiments on bilayer polyhedra composed of amphiphilic molecules, we study the elastic bending energies of bilayer vesicles forming polyhedral shapes. Allowing for segregation of excess amphiphiles along the ridges of polyhedra, we find that bilayer polyhedra can indeed have lower bending energies than spherical bilayer vesicles. However, our analysis also implies that, contrary to what has been suggested on the basis of experiments, the snub dodecahedron, rather than the icosahedron, generally represents the energetically favorable shape of bilayer polyhedra.     

Adhesion and membrane tension of single vesicles and living cells using a micropipette-based technique

The fundamental study of the adhesion of cells to each other or to a substrate is a key research topic in cellular biophysics because cell adhesion is important to many biological processes. We report on the adhesion of a model cell, a liposome, and a living HeLa cell to a substrate measured with a novel experimental technique. The cells are held at the end of a micropipette mounted on a micromanipulator and brought into contact with a surface. The adhesion energy and membrane tension are measured directly using the deflection of the micropipette when binding or unbinding the cell from the substrate. Since the force applied on the cells is known throughout the experiment, the technique presented enables the measurement of dynamics such as changes in the adhesion, elasticity, and membrane tension with time.