Spectrin properties and the elasticity of the red blood cell membrane skeleton

Two models of spectrin elasticity are developed and compared to experimental measurements of the red blood cell (RBC) membrane shear modulus through the use of an elastic finite element model of the RBC membrane skeleton. The two molecular models of spectrin are: (i) An entropic spring model of spectrin as a flexible chain. This is a model proposed by several previous authors. (ii) An elastic model of a helical coiled-coil which expands by increasing helical pitch. In previous papers, we have computed the relationship between the stiffness of a single spectrin molecule (K) and the shear modulus of a network (μ), and have shown that this behavior is strongly dependent upon network topology. For realistic network models of the RBC membrane skeleton, we equate μ to micropipette measurements of RBCs and predict K for spectrin that is consistent with the coiled-coil molecular model. The value of spectrin stiffness derived from the entropic molecular model would need to be at least 30 times greater to match the experimental results. Thus, the conclusion of this study is that a helical coiled-coil model for spectrin is more realistic than a purely entropic model.     

An elastic network model based on the structure of the red blood cell membrane skeleton.

A finite element network model has been developed to predict the macroscopic elastic shear modulus and the area expansion modulus of the red blood cell (RBC) membrane skeleton on the basis of its microstructure. The topological organization of connections between spectrin molecules is represented by the edges of a random Delaunay triangulation, and the elasticity of an individual spectrin molecule is represented by the spring constant, K, for a linear spring element. The model network is subjected to deformations by prescribing nodal displacements on the boundary. The positions of internal nodes are computed by the finite element program. The average response of the network is used to compute the shear modulus (mu) and area expansion modulus (kappa) for the corresponding effective continuum. For networks with a moderate degree of randomness, this model predicts mu/K = 0.45 and kappa/K = 0.90 in small deformations. These results are consistent with previous computational models and experimental estimates of the ratio mu/kappa. This model also predicts that the elastic moduli vary by 20% or more in networks with varying degrees of randomness. In large deformations, mu increases as a cubic function of the extension ratio lambda 1, with mu/K = 0.62 when lambda 1 = 1.5.     

Simulations of the erythrocyte cytoskeleton at large deformation. I. Microscopic models.

Three variations of a polymer chain model for the human erythrocyte cytoskeleton are used in large deformation simulations of microscopic membrane patches. Each model satisfies an experimental observation that the contour length of the spectrin tetramers making up the erythrocyte cytoskeleton is roughly square root of 7 times the end-to-end distance of the tetramer in vivo. Up to modest stress, each brushy cytoskeletal network behaves, consistently, like a low-temperature, planar network of Hookean springs, with a model-dependent effective spring constant, keff, in the range of 20-40 kBT/s(o)2, where T is the temperature and s(o) is the force-free spring length. However, several features observed at large deformation distinguish these models from spring networks: 1) Network dimensions do not expand without bound in approaching a critical isotropic tension (square root of 3 keff) that is a characteristic limit of Hookean spring nets. 2) In surface compression, steric interactions among the chain elements prevent a network collapse that is otherwise observed in compression of planar triangulated networks of springs. 3) Under uniaxial surface tension, isotropy of the network disappears only as the network is stretched by more than 50% of its equilibrium dimensions. Also found are definitively non-Hookean regimes in the stress dependence of the elastic moduli. Lastly, determinations of elastic moduli from both fluctuations and stress/strain relations prove to be consistent, implying that consistency should be expected among experimental determinations of these quantities.     

Spectrin, human erythrocyte shapes, and mechanochemical properties.

Physical studies of human erythrocyte spectrin indicate that isolated spectrin dimers and tetramers in solution are worm-like coils with a persistence length of approximately 20 nm. This finding, the known polyelectrolytic nature of spectrin, and other structural information about spectrin and the membrane skeleton molecular organization have lead us to the hypothesis that the human erythrocyte membrane skeleton constitutes a two-dimensional ionic gel (swollen ionic elastomer). This concept is incorporated in what we refer to as the protein gel-lipid bilayer membrane model. The model accounts quantitatively for red elastic shear modulus and the maximum elastic extension ratio reported for the human erythrocytes membrane. Gel theory further predicts that depending on the environmental conditions, the membrane skeleton modulus of area compression may be small or large relative to the membrane elastic shear modulus. Our analyses show that the ratio between these two parameters affects both the geometry and the stability of the favored cell shapes and that the higher the membrane skeleton compressibility the smaller the values of the gel tension needed to induce cell shape transformations. The main virtue of the protein gel-lipid bilayer membrane model is that it offers a novel theoretical and molecular basis for the various mechanical properties of the membrane skeleton such as the membrane skeleton modulus of area compression and osmotic tension, and the effects of these properties on local membrane skeleton density, cell shape, and shape transformations.     

Properties of the spectrin-like structural element of smooth-muscle alpha-actinin.

The fragment of smooth muscle alpha-actinin, comprising the four spectrin-like structural repeating units, has a high alpha-helix content, similar to that of spectrin, and a hydrodynamic frictional coefficient, indicative of an elongated, probably bent or kinked rod-like structure, as found for spectrin dimer and tetramer. The fragment exists in solution as an extremely stable dimer, which is dissociated only under denaturing conditions and is much more resistant to dissociation by urea than is the spectrin heterodimer. High-resolution proton magnetic resonance spectra reveal that a part of the polypeptide chain gives rise to sharp resonances; this is also true of spectrin and it implies that the individual structural repeating units contain segmentally mobile elements, which may be required to generate the elastic properties of the spectrin family of proteins. Again like spectrin, the alpha-actinin fragment contains multiple binding sites for long-chain fatty acids, as revealed by quenching of tryptophan fluorescence by 2-bromostearate (though not by 9(10)-bromostearate). The results point to extensive structural and functional similarities between the repeating units of all the proteins of the spectrin family.     

Is the spring quality of muscle plastic?

During locomotion, major muscle groups are often activated cyclically. This alternate stretch-shorten pattern of activity could enable muscle to function as a spring, storing and recovering elastic recoil potential energy. Because the ability to store and recover elastic recoil energy could profoundly affect the energetics of locomotion, one might expect this to be an adaptable feature of skeletal muscle. This study tests the hypothesis that chronic eccentric (Ecc) training results in a change in the spring properties of skeletal muscle. Nine female Sprague-Dawley rats underwent chronic Ecc training for 8 wk on a motorized treadmill. The spring properties of muscle were characterized by both active and passive lengthening force productions. A single "spring constant (Deltaforce/Deltalength) from the passive length-tension curves was calculated for each muscle. Results from measurements on long heads of triceps brachii muscle indicate that the trained group produced significantly more passive lengthening force (P = 0.0001) as well as more active lengthening force (P = 0.0001) at all lengths of muscle stretch. In addition, the spring constants were significantly different between the Ecc (1.71 N/mm) and the control (1.31 N/mm) groups. A stiffer spring is capable of storing more energy per unit length stretched, which is of functional importance during locomotion.     

Viscoelastic properties of single attached cells under compression

The viscoelastic properties of single, attached C2C12 myoblasts were measured using a recently developed cell loading device. The device allows global compression of an attached cell, while simultaneously measuring the associated forces. The viscoelastic properties were examined by performing a series of dynamic experiments over two frequency decades (0.1-10 Hz) and at a range of axial strains (approximately 10-40%). Confocal laser scanning microscopy was used to visualize the cell during these experiments. To analyze the experimentally obtained force-deformation curves, a nonlinear viscoelastic model was developed. The nonlinear viscoelastic model was able to describe the complete series of dynamic experiments using only a single set of parameters, yielding an elastic modulus of 2120 +/- 900 Pa for the elastic spring, an elastic modulus of 1960 +/- 1350 for the nonlinear spring, and a relaxation time constant of 0.3 +/- 0.12 s. To our knowledge, it is the first time that the global viscoelastic properties of attached cells have been quantified over such a wide range of strains. Furthermore, the experiments were performed under optimal environmental conditions and the results are, therefore, believed to reflect the viscoelastic mechanical behavior of cells, such as would be present in vivo.     

Elasticity of the human red blood cell skeleton

We have measured by optical tweezers micromanipulations the area expansion and the shear moduli of spectrin skeletons freshly extracted from human red blood cells, in different controlled salinity conditions. At medium osmolarity (150 mOsm/kg), we measure KC=9.7+/-3.4 microN/m, muC=5.7+/-2.3 microN/m, KC/muC=2.1+/-0.7. When decreasing the osmolarity, both KC and muC decrease, while KC/muC is nearly constant and equal to about 2. This result is consistent with the predictions made when modeling the spectrin skeleton by a two-dimensional triangular lattice of springs. From the measured elastic moduli we estimate the persistence length of a spectrin filament: xi approximately 2.5 nm at 150 mOsm/kg.     

States and transitions during forced unfolding of a single spectrin repeat

Spectrin is a vital and abundant protein of the cytoskeleton. It has an elongated structure that is made by a chain of so-called spectrin repeats. Each repeat contains three antiparallel alpha-helices that form a coiled-coil structure. Spectrin forms an oligomeric structure that is able to cross-link actin filaments. In red cells, the spectrin/actin meshwork underlying cell membrane is thought to be responsible for special elastic properties of the cell. In order to determine mechanical unfolding properties of the spectrin repeat, we have used single molecule force spectroscopy to study the states of unfolding of an engineered polymeric protein consisting of identical spectrin domains. We demonstrate that the unfolding of spectrin domains can occur in a stepwise fashion during stretching. The force-extension patterns exhibit features that are compatible with the existence of at least one intermediate between the folded and the completely unfolded conformation. Only those polypeptides that still contain multiple intact repeats display intermediates, indicating a stabilisation effect. Precise force spectroscopy measurements on single molecules using engineered protein constructs reveal states and transitions during the mechanical unfolding of spectrin. Single molecule force spectroscopy appears to open a new window for the analysis of transition probabilities between different conformational states.     

Binding of a denatured heme protein and ATP to erythroid spectrin

Spectrin is a large, worm-like cytoskeletal protein that is abundant in all cell types. The denatured heme enzyme, horseradish peroxidase showed significant decrease in the reactivation yield, after 30 min of refolding, in presence of increasing concentrations of spectrin from that in the absence. This indicated that spectrin could bind denatured HRP and inhibit their refolding. In presence of 1 mM ATP and 10 mM MgCl(2) the spectrin binding of denatured HRP is abolished. This activity of decreasing the reactivation yield was found to be ATP-dependent and the denatured enzyme after 30 min refolding in the presence of spectrin, pretreated with Mg/ATP, showed about 40% increase in the reactivation yield compared to the same in absence of spectrin. Fluorescence spectroscopic studies indicated binding of ATP to native spectrin showing concentration-dependent quenching of tryptophan fluorescence by ATP. The apparent dissociation constant of binding of ATP to spectrin was estimated to be 1.1 mM. A high affinity binding of spectrin with denatured HRP has been characterized (K(d) = 16 nM). Since these properties are similar to those of established molecular chaperone proteins, these data indicate that spectrin might have a chaperone-like function in erythrocytes.     

Elastic properties of bacterial flagellar filaments: I. Free rotation case

Elongation of a helical bacterial flagellar filament in a fluid flow with one end attached to a slide glass is calculated. The flagellar filament is regarded as a coil spring. In this case, the spring constant is a function of the elastic constants of the flagellar filament. Relations between the elongation and the elastic constants are discussed.     

Spring constants for channel-induced lipid bilayer deformations. Estimates using gramicidin channels.

Hydrophobic interactions between a bilayer and its embedded membrane proteins couple protein conformational changes to changes in the packing of the surrounding lipids. The energetic cost of a protein conformational change therefore includes a contribution from the associated bilayer deformation energy (DeltaGdef0), which provides a mechanism for how membrane protein function depends on the bilayer material properties. Theoretical studies based on an elastic liquid-crystal model of the bilayer deformation show that DeltaGdef0 should be quantifiable by a phenomenological linear spring model, in which the bilayer mechanical characteristics are lumped into a single spring constant. The spring constant scales with the protein radius, meaning that one can use suitable reporter proteins for in situ measurements of the spring constant and thereby evaluate quantitatively the DeltaGdef0 associated with protein conformational changes. Gramicidin channels can be used as such reporter proteins because the channels form by the transmembrane assembly of two nonconducting monomers. The monomerleft arrow over right arrow dimer reaction thus constitutes a well characterized conformational transition, and it should be possible to determine the phenomenological spring constant describing the channel-induced bilayer deformation by examining how DeltaGdef0 varies as a function of a mismatch between the hydrophobic channel length and the unperturbed bilayer thickness. We show this is possible by analyzing experimental studies on the relation between bilayer thickness and gramicidin channel duration. The spring constant in nominally hydrocarbon-free bilayers agrees well with estimates based on a continuum analysis of inclusion-induced bilayer deformations using independently measured material constants.     

Spectrin-level modeling of the cytoskeleton and optical tweezers stretching of the erythrocyte

We present a three-dimensional computational study of whole-cell equilibrium shape and deformation of human red blood cell (RBC) using spectrin-level energetics. Random network models consisting of degree-2, 3, ..., 9 junction complexes and spectrin links are used to populate spherical and biconcave surfaces and intermediate shapes, and coarse-grained molecular dynamics simulations are then performed with spectrin connectivities fixed. A sphere is first filled with cytosol and gradually deflated while preserving its total surface area, until cytosol volume consistent with the real RBC is reached. The equilibrium shape is determined through energy minimization by assuming that the spectrin tetramer links satisfy the worm-like chain free-energy model. Subsequently, direct stretching by optical tweezers of the initial equilibrium shape is simulated to extract the variation of axial and transverse diameters with the stretch force. At persistence length p = 7.5 nm for the spectrin tetramer molecule and corresponding in-plane shear modulus mu(0) approximately 8.3 microN/m, our models show reasonable agreement with recent experimental measurements on the large deformation of RBC with optical tweezers. We find that the choice of the reference state used for the in-plane elastic energy is critical for determining the equilibrium shape. If a position-independent material reference state such as a full sphere is used in defining the in-plane energy, then the bending modulus kappa needs to be at least a decade larger than the widely accepted value of 2 x 10(-19) J to stabilize the biconcave shape against the cup shape. We demonstrate through detailed computations that this paradox can be avoided by invoking the physical hypothesis that the spectrin network undergoes constant remodeling to always relax the in-plane shear elastic energy to zero at any macroscopic shape, at some slow characteristic timescale. We have devised and implemented a liquefied network structure evolution algorithm that relaxes shear stress everywhere in the network and generates cytoskeleton structures that mimic experimental observations.     

Extending a spectrin repeat unit. II: rupture behavior

A spectrin repeat unit was subject to extension using cyclic expansion nonequilibrium molecular dynamics. Periodic boundary conditions were used to examine the effects of the contiguous alpha-helical linker on the force response. The measured force-extension curve shows a linear increase in the force response when the spectrin repeat unit is extended by approximately 0.4 nm. After that point, the force response peaks and subsequently declines. The peak in the force response marks the point where the spectrin repeat unit undergoes a change in its material properties from a strongly elastic material to a mostly viscous one, on the timescales of the simulations. The force peak is also correlated with rupture of the alpha-helical linker, and is likely the event responsible for the peaks in the sawtooth-pattern force-extension curves measured by atomic force microscopy experiments. Rupture of the linker involves simultaneously breaking approximately four hydrogen bonds that maintain the alpha-helical linker. After this initial rupture, the linker undergoes simple helix-to-coil transitions as the spectrin repeat unit continues to be extended. The implications of linker rupture in the interpretation of unfolding and atomic force microscopy experiments are also discussed.     

The structural basis of ankyrin function. II. Identification of two functional domains

Human erythrocyte ankyrin was cleaved by restricted proteolysis at 0 degrees C into two distinct chemical domains. The site on ankyrin that binds spectrin was found to be within a 55,000-dalton domain by spectrin affinity chromatography and co-sedimentation with spectrin in a sucrose gradient. A 32,000-dalton fragment of this domain was prepared (tryptic digest, 0 degrees C, 24 h), separated by gel filtration, and shown to inhibit spectrin binding to the membrane. By comparison with previous two-dimensional peptide maps, the spectrin-binding site was located within this 32,000-dalton fragment near the end of the molecule. The band 3-binding site was identified within an 82,000-dalton domain by binding to a band 3 affinity column. Gel electrophoresis in the absence of detergents confirmed these results and demonstrated that a peptide from the cytoplasmic portion of band 3 retained the capacity to bind the 82,000-dalton domain. The binding properties of the structural domains of ankyrin were correlated with a determination of the affinity constant of the intact molecule. Ankyrin bound with a high affinity to the cytoplasmic portion of band 3 (KD = 8 X 10(-8) M) and to spectrin tetramer (KD = 1 X 10(-7) M) but less so to spectrin dimer (KD = 1 X 10(-6) M). These findings are summarized in a preliminary structural and functional model of ankyrin's role in linking spectrin to the membrane.     

Scaling behaviour of different polymer models in dissipative particle dynamics of unentangled melts

We present a dissipative particle dynamics (DPD) study of scaling behaviour for three polymer models. The scaling behaviour is explored for the conformational and dynamic properties of unentangled polymer melts. DPD employs a bead–spring model together with an aggressive coarse-graining to represent polymers at the mesoscale. The first model studied utilises a simple soft repulsion potential for the bead–bead interactions together with a harmonic spring potential to connect beads into a polymer chain. The second model differs from the first model by replacing the harmonic spring with a finitely extensible nonlinear elastic spring. The third model uses realistic coarse-grain potentials for the bead–bead, spring and bending interactions based on the iterative Boltzmann inversion procedure and it corresponds to a mesoscopic model of polyethylene. We systematically vary the chain length and spring constant (in the case of the first and second models), and simulate the conformational properties such as the end-to-end distance or radius of gyration, and dynamic properties such as the centre-of-mass self-diffusion coefficient or viscosity. The scaling of the conformational and dynamic properties with chain length (scaling laws) is compared with the Rouse theory, which is considered as a standard theory for unentangled polymer melts. The comparison shows that simulated scaling laws typically agree with the Rouse scaling laws for the DPD polymer models with more than 10 DPD beads. For the shorter DPD polymers, deviations from the Rouse theory exist and become significant for the dynamic properties, especially for the viscosity of the polymer melts.     

Elastic properties of the cell wall of Aspergillus nidulans studied with atomic force microscopy

Currently, little is known about the mechanical properties of filamentous fungal hyphae. To study this topic, atomic force microscopy (AFM) was used to measure cell wall mechanical properties of the model fungus Aspergillus nidulans. Wild type and a mutant strain (deltacsmA), lacking one of the chitin synthase genes, were grown in shake flasks. Hyphae were immobilized on polylysine-coated coverslips and AFM force--displacement curves were collected. When grown in complete medium, wild-type hyphae had a cell wall spring constant of 0.29 +/- 0.02 N/m. When wild-type and mutant hyphae were grown in the same medium with added KCl (0.6 M), hyphae were significantly less rigid with spring constants of 0.17 +/- 0.01 and 0.18 +/- 0.02 N/m, respectively. Electron microscopy was used to measure the cell wall thickness and hyphal radius. By use of finite element analysis (FEMLAB v 3.0, Burlington, MA) to simulate AFM indentation, the elastic modulus of wild-type hyphae grown in complete medium was determined to be 110 +/- 10 MPa. This decreased to 64 +/- 4 MPa for hyphae grown in 0.6 M KCl, implying growth medium osmotic conditions have significant effects on cell wall elasticity. Mutant hyphae grown in KCl-supplemented medium were found to have an elastic modulus of 67 +/- 6 MPa. These values are comparable with other microbial systems (e.g., yeast and bacteria). It was also found that under these growth conditions axial variation in elastic modulus along fungal hyphae was small. To determine the relationship between composition and mechanical properties, cell wall composition was measured by anion-exchange liquid chromatography and pulsed electrochemical detection. Results show similar composition between wild-type and mutant strains. Together, these data imply differences in mechanical properties may be dependent on varying molecular structure of hyphal cell walls as opposed to wall composition.     

The effects of p-mercuribenzenesulfonate on purified spectrin and actin

The compound p-mercuribenzenefulfonate was found to affect the self-association behavior of both spectrin and actin. The reagent brings about the depolymerization of F-actin, as judged from the decrease in the fluorescence of an attached pyrene label, with a second-order rate constant an order of magnitude less than that for the disruption of isolated erythrocyte cytoskeletons. Therefore, it is unlikely that the depolymerization of actin is the rate-determining step in the mercurial-dependent disruption of the erythrocyte cytoskeleton. Low reagent concentrations caused an initial rapid dissociation of spectrin tetramers at a rate comparable with that of cytoskeleton disruption. Prolonged incubation, or higher reagent concentrations, resulted in subsequent aggregation of spectrin. The reagent also prevented the interaction between spectrin and actin, presumably through its depolymerization of actin and its effects on spectrin. The early event in the disruption of isolated erythrocyte cytoskeletons by p-mercuribenzenesulfonate thus appears to be the dissociation of spectrin oligomers. Subsequent depolymerization of actin brought about by the reagent then results in total disruption of the cytoskeleton.     

A cyclic AMP-dependent phosphorylation of spectrin dimer

In contrast to the properties of spectrin obtained from [³²P]phosphate-labeled red cells, purified spectrin dimer could be phosphorylated by a cAMP-dependent protein kinase from bovine heart. Both spectrin bands were phosphorylated. Spectrin band 2 contained in addition to autophosphorylated peptides several phosphopeptides that were distinct from autophosphorylated ones. The cAMP-dependent phosphorylation of spectrin band 1 was modulated by reducing agent and the concentration of spectrin. At high concentrations spectrin band 2 was predominantly labeled. The cAMP-dependent phosphoform of spectrin band 2 had a pI slightly higher than that of autophosphorylated spectrin band 2, but lower than that of ankyrin.