The membrane skeleton of erythrocytes. A percolation model.

The spectrin network on the cytoplasmic surface of the erythrocyte membrane is modeled as a triangular lattice of spectrin tetramers. This network obstructs lateral diffusion of proteins and provides mechanical reinforcement to the membrane. These effects are treated in a systematic and unified manner in terms of a percolation model. The diffusion coefficient is obtained as a function of the fraction of normal spectrin tetramers for both static and fluctuating barriers. The elasticity of the network is calculated as a function of the fraction of normal spectrin and the ratio of bending to stretching energies. For static barriers, elasticity and lateral diffusion are incompatible: if a network is connected enough to be elastic, it is connected enough to block long-range lateral diffusion. The elasticity and the force required for mechanical breakdown go to zero at the percolation threshold; experimental evidence suggests the existence of a stability threshold at or near the percolation threshold. The model is qualitatively applicable to other cells with membrane skeletons, such as epithelial cells, in which localization of membrane proteins is essential to differentiation.     

Gaps in the erythrocyte membrane skeleton: a stretched net model.

The geometry of spectrin-free regions in the erythrocyte membrane skeleton is modeled using Monte Carlo calculations for an incomplete triangular lattice of entropy springs under tension. Intact springs correspond to normal spectrin molecules, and cut springs correspond to spectrin that is missing or unable to associate normally. As springs are cut and the network is allowed to relax to mechanical equilibrium, gaps in the network appear. Geometrical properties of these gaps are obtained as a function of the fraction of springs cut. The most important property modeled is the area of the largest spectrin-free region; this area increases approximately exponentially as the fraction of normal spectrin decreases from 100% to approximately 50%. The effect of these gaps on lateral diffusion and vesiculation is discussed.     

Lateral diffusion and aggregation. A Monte Carlo study.

Aggregation in a lipid bilayer is modeled as cluster-cluster aggregation on a square lattice. In the model, clusters carry out a random walk on the lattice, with a diffusion coefficient inversely proportional to mass. On contact, they adhere with a prescribed probability, rigidly and irreversibly. Monte Carlo calculations show that, as expected, rotational diffusion of the aggregating species is highly sensitive to the initial stages of aggregation. Lateral diffusion of an inert tracer obstructed by the aggregate is a sensitive probe of the later stages of aggregation. Cluster-cluster aggregates are much more effective barriers to lateral diffusion of an inert tracer than the same area fraction of random point obstacles is, but random point obstacles are more effective barriers than the same area fraction of compact obstacles. The effectiveness of aggregates as obstacles is discussed in terms of particle-particle correlation functions and fractal dimensions. Results are applicable to aggregation of membrane proteins, and at least qualitatively to aggregation of gel-phase lipid during lateral phase separation.     

Lateral diffusion in an archipelago. Distance dependence of the diffusion coefficient.

An understanding of the distance dependence of the lateral diffusion coefficient is useful in comparing the results of diffusion measurements made over different length scales, and in analyzing the kinetics of mobile redox carriers in organelles. A distance-dependent, concentration-dependent diffusion coefficient is defined, and it is evaluated by Monte Carlo calculations of a random walk by mobile point tracers in the presence of immobile obstacles on a triangular lattice, representing the diffusion of a lipid or a small protein in the presence of immobile membrane proteins. This work confirms and extends the milling crowd model of Eisinger, J., J. Flores, and W. P. Petersen (1986. Biophys J. 49:987-1001). Similar calculations for diffusion of mobile particles interacting by a hard-core repulsion yield the distance dependence of the self-diffusion coefficient. An expression for the range of short-range diffusion is obtained, and the distance scales for various diffusion measurements are summarized.     

Lateral diffusion in a mixture of mobile and immobile particles. A Monte Carlo study.

The lateral diffusion coefficient for mixtures of mobile and immobile particles is obtained from Monte Carlo calculations of random walks by mobile tracers in the presence of immobile obstacles on a triangular lattice. The diffusion coefficient of the mobile species is obtained as a function of the area fractions of mobile and immobile species. The results are applied to diffusion of band 3 in the erythrocyte membrane, and indicate that obstruction of diffusion of mobile band 3 by band 3 and glycophorin attached to the membrane skeleton is not sufficient to explain the observed diffusion coefficient.     

Some viscoelastic properties of human erythrocyte spectrin networks end-linked in vitro

We have succeeded in making macroscopic networks of end-linked human erythrocyte spectrin. The network junctions were made using erythrocyte protein 4.1 irreversibly attached to 5 nm (diameter) colloidal gold particles. Rotary shadowing electron microscopy verifies that the protein 4.1-labelled colloidal gold particles bind only to the tail end of the spectrin molecules. Electron micrographs of protein 4.1-labelled colloidal gold particles incubated at 4 degrees C with spectrin dimers reveal that 1-5 spectrin dimers attach to each protein 4.1-labelled colloidal gold particle yielding a spider-like appearance of these complexes. Incubation with a low concentration of spectrin tetramers instead of dimers leads to extensive formation of spectrin microaggregates whereas use of spectrin concentrations higher than 3 mg/ml and a molar ratio between spectrin tetramers and protein 4.1/Au of 4 leads to formation of macroscopic spectrin networks. We have quantitated the viscoelastic properties of such end-linked macroscopic spectrin networks using a gravitational pendulum viscoelastometer. We find that in vitro end-linked spectrin networks can be described by linear viscoelastic theory. The dynamic storage modulus increases almost linearly with the spectrin-protein 4.1/gold particle concentration when the spectrin concentration exceeds about 3 mg/ml and the molar ratio between spectrin tetramers and protein 4.1/Au is 4. At a spectrin concentration of 6 mg/ml and the same ratio between spectrin and protein 4.1/Au, we find a dynamic storage modulus at low frequency of about 80 dyn/cm2. This is in adequate agreement with what is predicted by simple elastomer theory.     

Single-particle tracking: the distribution of diffusion coefficients.

In single-particle tracking experiments, the diffusion coefficient D may be measured from the trajectory of an individual particle in the cell membrane. The statistical distribution of single-trajectory diffusion coefficients is examined by Monte Carlo calculations. The width of this distribution may be useful as a measure of the heterogeneity of the membrane and as a test of models of hindered diffusion in the membrane. For some models, the distribution of the short-range diffusion coefficient is much narrower than the observed distribution for proteins diffusing in cell membranes. To aid in the analysis of single-particle tracking measurements, the distribution of D is examined for various definitions of D and for various trajectory lengths.     

Brain ankyrin. Purification of a 72,000 Mr spectrin-binding domain

Polypeptides of Mr = 190,000-220,000 that cross-react with erythrocyte ankyrin were detected in immunoblots of membranes from pig lens, pig brain, and rat liver. The cross-reacting polypeptides from brain were cleaved by chymotrypsin to fragments of Mr = 95,000 and 72,000 which are the same size as fragments obtained with erythrocyte ankyrin. The brain 72,000 Mr fragment associated with erythrocyte spectrin, and the binding occurred at the same site as that of erythrocyte ankyrin 72,000 Mr fragment since (a) brain 72,000 Mr fragment was adsorbed to erythrocyte spectrin-agarose and (b) 125I-labeled erythrocyte spectrin bound to brain 72,000 Mr fragment following transfer of the fragment from a sodium dodecyl sulfate gel to nitrocellulose paper, and this binding was displaced by erythrocyte ankyrin 72,000 Mr fragment. Brain 72,000 Mr fragment was purified about 400-fold by selective extraction and by continuous chromatography on columns attached in series containing DEAE-cellulose followed by erythrocyte spectrin coupled to agarose, and finally hydroxylapatite. The brain 72,000 Mr fragment was not derived from contaminating erythrocytes since peptide maps of pig brain and pig erythrocyte 72,000 Mr fragments were distinct. The amount of brain 72,000 Mr fragment was estimated as 0.28% of membrane protein or 39 pmol/mg based on radioimmunoassay with 125I-labeled brain fragment and antibody against erythrocyte ankyrin. Brain spectrin tetramer was present in about the same number of copies (30 pmol/mg of membrane protein) based on densitometry of Coomassie blue-stained sodium dodecyl sulfate gels. The binding site on brain spectrin for both brain and erythrocyte ankyrin 72,000 Mr fragments was localized by electron microscopy to the midregion of spectrin tetramers about 90 nM from the near end and 110 nM from the far end. These studies demonstrate the presence in brain membranes of a protein closely related to erythrocyte ankyrin, and are consistent with a function of the brain ankyrin as a membrane attachment site for brain spectrin.     

Restriction of the lateral motion of band 3 in the erythrocyte membrane by the cytoskeletal network: dependence on spectrin association state

The effects of incubation of erythrocyte ghosts under various conditions (ionic strength or addition of ankyrin, diamines, or ATP) on the lateral motion of band 3 in the membranes were studied by using the fluorescence photobleaching recovery technique. Incubation of ghosts with exogenous ankyrin increased the immobile fraction of band 3, from 0.6 in intact ghosts to 0.8-0.9 when an average of 0.2 mol of extra ankyrin was bound per mole of band 3. Ankyrin-free band 3 proteins were mobile, but their mobility was governed by the spectrin association state in the cytoskeletal network. The diffusion constant was 5.3 X 10(-11) cm2 s-1 at a spectrin tetramer mole fraction of 0.3-0.4 in 10 mM NaCl/5 mM sodium phosphate, pH 7.8, and decreased 1 order of magnitude when the tetramer fraction increased to 0.5 in higher NaCl concentration (150 mM NaCl). A similar decrease was observed when the spectrin tetramer fraction was increased by 0.2 mM spermine in 10 mM NaCl/10 mM tris(hydroxymethyl)aminomethane hydrochloride, pH 7.6. On the other hand, the rotational motion of band 3 in the membranes was not affected by the spectrin association state. Trypsin treatment of ghosts cleaved off the cytoplasmic domain of band 3 and caused a marked (8-fold) increase in the lateral mobility, D = 4.0 X 10(-10) cm2 s-1. These results indicate that the lateral mobility of ankyrin-free band 3 protein is restricted by interactions of their cytoplasmic domain with the cytoskeletal network. A model is presented that band 3 can pass the network when spectrins are in dissociated dimers and cannot pass when they are tetramers. The lateral diffusion constant is thus determined by the spectrin dimer population in the network.     

A biological interpretation of transient anomalous subdiffusion. I. Qualitative model

Anomalous subdiffusion has been reported for two-dimensional diffusion in the plasma membrane and three-dimensional diffusion in the nucleus and cytoplasm. If a particle diffuses in a suitable infinite hierarchy of binding sites, diffusion is well known to be anomalous at all times. But if the hierarchy is finite, diffusion is anomalous at short times and normal at long times. For a prescribed set of binding sites, Monte Carlo calculations yield the anomalous diffusion exponent and the average time over which diffusion is anomalous. If even a single binding site is present, there is a very short, almost artifactual, period of anomalous subdiffusion, but a hierarchy of binding sites extends the anomalous regime considerably. As is well known, an essential requirement for anomalous subdiffusion due to binding is that the diffusing particle cannot be in thermal equilibrium with the binding sites; an equilibrated particle diffuses normally at all times. Anomalous subdiffusion due to barriers, however, still occurs at thermal equilibrium, and anomalous subdiffusion due to a combination of binding sites and barriers is reduced but not eliminated on equilibration. This physical model is translated directly into a plausible biological model testable by single-particle tracking.     

Brain ankyrin. A membrane-associated protein with binding sites for spectrin, tubulin, and the cytoplasmic domain of the erythrocyte anion channel

Brain ankyrin was purified from pig brain membranes in milligram quantities by a procedure involving affinity chromatography on erythrocyte spectrinagarose. Brain ankyrin included two polypeptides of Mr = 210,000 and 220,000 that were nearly identical by peptide mapping and were monomers in solution. Brain ankyrin and erythrocyte ankyrin are closely related proteins with the following properties in common: 1) shared antigenic sites, 2) high-affinity binding to the spectrin beta subunit at the midregion of spectrin tetramers, 3) a binding site for the cytoplasmic domain of the erythrocyte anion channel, 4) a binding site for tubulin, 5) a similar domain structure with a protease-resistant domain of Mr = 72,000 that contains the spectrin-binding activity and domains of Mr = 95,000 (brain ankyrin) or 90,000 (erythrocyte ankyrin) that contain binding sites for both tubulin and the anion channel. Brain ankyrin is present at about 100 pmol/mg of membrane protein in demyelinated membranes based on radioimmunoassay with antibody raised against brain ankyrin and affinity purified on brain ankyrin-agarose. Brain spectrin tetramers are present at 30 pmol/mg of membrane protein. Brain ankyrin thus is present in sufficient amounts to attach spectrin to membranes. Brain ankyrin also may attach microtubules to membranes independently of spectrin and has the potential to interconnect microtubules and spectrin-associated actin filaments.     

Ankyrin and synapsin: spectrin-binding proteins associated with brain membranes

Brain membranes contain an actin-binding protein closely related in structure and function to erythrocyte spectrin. The proteins that attach brain spectrin to membranes are not established, but, by analogy with the erythrocyte membrane, may include ankyrin and protein 4.1. In support of this idea, proteins closely related to ankyrin and 4.1 have been purified from brain and have been demonstrated to associate with brain spectrin. Brain ankyrin binds with high affinity to the spectrin beta subunit at the midregion of spectrin tetramers. Brain ankyrin also has binding sites for the cytoplasmic domain of the erythrocyte anion channel (band 3), as well as for tubulin. Ankyrins from brain and erythrocytes have a similar domain structure with protease-resistant domains of Mr = 72,000 that contain spectrin-binding activity, and domains of Mr = 95,000 (brain ankyrin) or 90,000 (erythrocyte ankyrin) that contain binding sites for both tubulin and the anion channel. Brain ankyrin is present at about 100 pmol/mg membrane protein, or about twice the number of copies of spectrum beta chains. Brain ankyrin thus is present in sufficient amounts to attach spectrin to membranes, and it has the potential to attach microtubules to membranes as well as to interconnect microtubules with spectrin-associated actin filaments. Another spectrin-binding protein has been purified from brain membranes, and this protein cross-reacts with erythrocyte 4.1. Brain 4.1 is identical to the membrane protein synapsin, which is one of the brain's major substrates for cAMP-dependent and Ca/calmodulin-dependent protein kinases with equivalent physical properties, immunological cross-reaction, and peptide maps. Synapsin (4.1) is present at about 60 pmol/mg membrane protein, and thus is a logical candidate to regulate certain protein linkages involving spectrin.     

“Nanostructures in Thin-film Epitaxy: Exploring and Exploiting Substrate-mediated Interactions”

We review recent advances made in understanding the ramifications of substrate-mediated interactions for thin-film growth. Experimental studies and first-principles calculations with density-functional theory (DFT) indicate that substrate-mediated interactions can significantly influence thin-film growth. We review the findings from our kinetic Monte Carlo simulations used to model the growth of thin films, both with and without substrate-mediated interactions. For Ag heteroepitaxy on Pt(1?1?1), the pair interaction energies and adsorbate diffusion barriers were obtained from DFT calculations. Island densities for this system show significant deviations from what is predicted by classical nucleation theory. The electronic interactions created by the adsorbed atoms lead to the formation of repulsive barriers surrounding small islands and, as a result, sharp island-size distributions are produced. The island-size distributions can be manipulated by changing the growth conditions to yield desirable island sizes and shapes.     

The elasticity of spectrin-actin gels at high protein concentration

Human erythrocyte spectrin of high purity was studied alone and mixed with rabbit skeletal actin by dynamic rheometry as a function of protein concentration at pH 7.4 and 24 degrees C. Pure spectrin had a very low storage modulus, G', increasing slightly with increase in protein concentration (approximately 3 dynes/cm at 25 mg/ml). In contrast, unpurified cytoskeletal extracts containing spectrin, actin, and band 4.1 showed a marked concentration dependence for G', increasing to 150 dynes/cm at 20 mg/ml. Mixtures of purified spectrin and skeletal actin at a weight ratio of 4:1 also showed G' markedly dependent on concentration (approximately 150-200 dynes/cm at 20 mg/ml). Maximum elasticity of spectrin-actin gels occurred at a molar ratio of actin monomers to spectrin tetramers of 14:1. We conclude that the reconstituted in vitro spectrin-actin network consists of actin fibers cross-linked by spectrin tetramers at regular intervals. The gel is rapidly reformed after mechanical disruption or thermal collapse, indicating that the polymer fibers are in equilibrium with the constituent monomers.     

Visualization of the hexagonal lattice in the erythrocyte membrane skeleton

The isolated membrane skeleton of human erythrocytes was studied by high resolution negative staining electron microscopy. When the skeletal meshwork is spread onto a thin carbon film, clear images of a primarily hexagonal lattice of junctional F-actin complexes crosslinked by spectrin filaments are obtained. The regularly ordered network extends over the entire membrane skeleton. Some of the junctional complexes are arranged in the form of pentagons and septagons, approximately 3 and 8%, respectively. At least five forms of spectrin crosslinks are detected in the spread skeleton including a single spectrin tetramer linking two junctional complexes, three-armed Y-shaped spectrin molecules linking three junctional complexes, three-armed spectrin molecules connecting two junctional complexes with two arms bound to one complex and the third arm bound to the adjacent complex, double spectrin filaments linking two junctional complexes, and four-armed spectrin molecules linking two junctional complexes. Of these, the crosslinks of single spectrin tetramers and three-armed molecules are the most abundant and represent 84 and 11% of the total crosslinks, respectively. These observations are compatible with the presence of spectrin tetramers and oligomers in the erythrocyte membrane skeleton. Globular structures (9-12 nm in diameter) are attached to the majority of the spectrin tetramers or higher order oligomer-like molecules, approximately 80 nm from the distal ends of the spectrin tetramers. These globular structures are ankyrinor ankyrin/band 3-containing complexes, since they are absent when ankyrin and residual band 3 are extracted from the skeleton under hypertonic conditions.     

Spectrin-dependent and -independent association of F-actin with the erythrocyte membrane

Binding of F-actin to spectrin-actin-depleted erythrocyte membrane inside-out vesicles was measured using [3H]F-actin. F-actin binding to vesicles at 25 degrees C was stimulated 5-10 fold by addition of spectrin dimers or tetramers to vesicles. Spectrin tetramer was twice as effective as dimer in stimulating actin binding, but neither tetramer nor dimer stimulated binding at 4 degrees C. The addition of purified erythrocyte membrane protein band 4.1 to spectrin- reconstituted vesicles doubled their actin-binding capacity. Trypsinization of unreconstituted vesicles that contain < 10% of the spectrin but nearly all of the band 4.1, relative to ghosts, decreased their F-actin-binding capacity by 70%. Whereas little or none of the residual spectrin was affected by trypsinization, band 4.1 was significantly degraded. Our results show that spectrin can anchor actin filaments to the cytoplasmic surface of erythrocyte membranes and suggest that band 4.1 may be importantly involved in the association.     

Studies on protein folding, unfolding and fluctuations by computer simulation. III. Effect of short-range interactions.

The theoretical model of proteins on the two-dimensional square lattice, introduced previously, is extended to include the specific short-range interactions. Attractive long-range interactions with various specificities and non-specific repulsive long-range interactions in the form of self-avoidance of the polymer chain are also operative in the model. Dynamics of the model protein is studied by a Monte Carlo method. The short-range interactions are found to accelerate the folding and unfolding transitions. Non-specific part of the attractive long-range interactions have a competing effect of decelerating the transitions. When the short-range interactions are weighted beyond a certain extent over the attractive long-range interactions are weighted beyond a certain extent over the attractive long-range interactions, the all-or-none character of the folding and unfolding transitions is destroyed. How the destruction proceeds is quantitatively expressed in terms of the S-H curves. The limiting case of dominance of the specific short-range interactions over the attractive long-range interactions is studied in detail. The lattice polymer in this limit does not behave like a globular protein at all. This observation leads to a reexamination of the currently popular notion of the dominance of the short-range interactions. A new concept of consistency is proposed to replace it. Possible mechanisms of the acceleration of the transitions by the specific short-range interactions are discussed.     

The role of Se in the interconversion of polymeric states of spectrin from human erythrocytes

It has been shown that Se can markedly prevent the dissociation of spectrin from erythrocyte ghosts. We now report that the transformation of incubated spectrin oligomers to its tetramers and dimers was obviously decreased in the presence of trace amounts (0.5-2.0 p.p.m.) of Na2SeO3. The spectrin tetramers and dimers are in a reversible equilibrium and Se could alter this equilibrium in favour of tetramers. This Se effect is concentration dependent and an inverse result was obtained with higher Na2SeO3 concentrations (greater than 4.0 p.p.m.). We suggest that the equilibrium state of the spectrin tetramer-dimer may be governed by a conformation adjustment induced by Se and the difference in conformation in the presence of low and high Na2SeO3 concentration may lead to an alteration of the spectrin tetramer-dimer balance.     

Spectrin-actin associations studied by electron microscopy of shadowed preparations

By shadowing specimens dried onto mica sheets we have obtained clear images of actin crosslinked by spectrin, an actin-binding protein found in erythrocytes. We conclude that spectrin dimers possess a single binding site for F actin. Tetramers formed by head-to-head association of two dimers possess two actin binding sites, one at each tail. Polymerizing G actin in the presence of spectrin tetramers or mixing preformed F actin with spectrin tetramer plus band 4.1 results in an extensively crosslinked network of actin filaments. When G actin is polymerized in the presence of spectrin at spectrin:actin mole ratios close to that present on the erythrocyte membrane, large amorphous protein networks are formed. These networks are clusters of spectrin around 25 nm diameter structures which may be actin protofilaments. These networks are similar to the cytoskeletal network seen after erythrocyte membranes are extracted with detergent, and may represent the first in vitro assembly of a cytoskeletal complex resembling that of the native cell both biochemically and structurally.     

A biological interpretation of transient anomalous subdiffusion. II. Reaction kinetics

Reaction kinetics in a cell or cell membrane is modeled in terms of the first passage time for a random walker at a random initial position to reach an immobile target site in the presence of a hierarchy of nonreactive binding sites. Monte Carlo calculations are carried out for the triangular, square, and cubic lattices. The mean capture time is expressed as the product of three factors: the analytical expression of Montroll for the capture time in a system with a single target and no binding sites; an exact expression for the mean escape time from the set of lattice points; and a correction factor for the number of targets present. The correction factor, obtained from Monte Carlo calculations, is between one and two. Trapping may contribute significantly to noise in reaction rates. The statistical distribution of capture times is obtained from Monte Carlo calculations and shows a crossover from power-law to exponential behavior. The distribution is analyzed using probability generating functions; this analysis resolves the contributions of the different sources of randomness to the distribution of capture times. This analysis predicts the distribution function for a lattice with perfect mixing; deviations reflect imperfect mixing in an ordinary random walk.