David H. Boal

Simulations of the Erythrocyte Cytoskeleton at Large Deformation. II. Micropipette Aspiration

Coarse-grained molecular models of the erythrocyte membranes spectrin cytoskeleton are presented in Monte Carlo simulations of whole cells in micropipette aspiration. The nonlinear chain elasticity and sterics revealed in more microscopic cytoskeleton models (developed in a companion paper; Boey et al., 1998. Biophys. J. 75:1573-1583) are faithfully represented here by two- and three-body effective potentials. The number of degrees of freedom of the system are thereby reduced to a range that is computationally tractable. Three effective models for the triangulated cytoskeleton are developed: two models in which the cytoskeleton is stress-free and does or does not have internal attractive interactions, and a third model in which the cytoskeleton is prestressed in situ. These are employed in direct, finite-temperature simulations of erythrocyte deformation in a micropipette. All three models show reasonable agreement with aspiration measurements made on flaccid human erythrocytes, but the prestressed model alone yields optimal agreement with fluorescence imaging experiments. Ensemble-averaging of nonaxisymmetrical, deformed structures exhibiting anisotropic strain are thus shown to provide an answer to the basic question of how a triangulated mesh such as that of the red cell cytoskeleton deforms in experiment.

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.

Computer simulation of a model network for the erythrocyte cytoskeleton

The geometry and mechanical properties of the human erythrocyte membrane cytoskeleton are investigated by a computer simulation in which the cytoskeleton is represented by a network of polymer chains. Four elastic moduli as well as the area and thickness are predicted for the chain network as a function of temperature and the number of segments in each chain. Comparisons are made with mean field arguments to examine the importance of steric interactions in determining network properties. Applied to the red blood cell, the simulation predicts that in the bilayer plane the membrane cytoskeleton has a shear modulus of 10 +/- 2 x 10(-6) J/m2 and an areal compression modulus of 17 +/- 2 x 10(-6) J/m2. The volume compression modulus and the transverse Youngs modulus of the cytoskeleton are predicted to be 1.2 +/- 0.1 x 10(3) J/m3 and 2.0 +/- 0.1 x 10(3) J/m3, respectively. Elements of the cytoskeleton are predicted to have a mean displacement from the bilayer plane of 15 nm. The simulation agrees with some, but not all, of the shear modulus measurements. The other predicted moduli have not been measured.

Entropy-driven instability and rupture of fluid membranes.

A computer simulation is used to investigate hole formation in a model membrane. The model parameters are the stress applied to the membrane, and the edge energy per unit length along the hole boundary (edge tension). Even at zero stress, the membrane has an entropically driven instability against hole formation. Within the model, the minimum edge tension required for the stability of a typical biological membrane is in the region of 1 x 10(-11) J/m, which is similar to the edge tension obtained in many measurements of biomembranes. At the zero-stress instability threshold, the hole shape is the same as a self-avoiding ring, but under compression, the hole shape assumes a branched polymer form. In the presence of large holes at zero stress, the membrane itself behaves like a branched polymer. The boundaries of the phase diagram for membrane stability are obtained, and general features of the rate of membrane rupture under stress are investigated. A model in which the entropy of hole formation is proportional to the hole perimeter is used to interpret the simulation results at small stress near the instability threshold.

Dynamical and statistical aspects of intermediate energy nucleus-nucleus collisions

Abstract Recent experimental and theoretical results for intermediate energy nucleus-nucleus collisions ( E / A =20–200 MeV ) will be reviewed. The experimental topics include incomplete fusion reactions, linear momentum transfer measurements, non-equilibrium light and complex particle emission, sub-threshold pion production, high-energy photon emission, and particle correlations at large and small relative momenta. The theoretical discussion covers thermal models, liquid-gas phase transitions, transport equations, numerical simulations, two-particle correlation functions and cluster formation.

Barrier-free paths of directed protein motion in the erythrocyte plasma membrane.

A model is presented for the steric interaction between a plasma membrane protein and the membrane cytoskeleton in the human erythrocyte. The cytoskeleton is treated as a network of polymer chains attached to a flat bilayer, and the membrane protein is a hemisphere of effective radius R(e) with center on the bilayer edge. The simulation is used to investigate the barrier-free path L for linear guided motion of a protein in the bilayer plane. It is shown that the barrier-free paths of small proteins can be used to extract the effective in-plane diameter of cytoskeletal components. For example, the in-plane diameter of an ankyrin attachment site is found to be approximately 12 nm in the simulation, or twice the computational spectrin diameter. The barrier-free paths of large proteins (R(e) > 23 nm) vanish when the proteins are corralled by the cytoskeleton. For intermediate size proteins, L decreases approximately as L is directly proportional to S-1.4 where S is proportional to the sum of the protein and cytoskeleton chain radii.

Energy dependence of source radii and emission temperatures for 14N induced reactions at EA = 35 MeV

Abstract Correlations between coincident alpha particles and deuterons emitted in 14 N induced reactions on 197 Au at E A = 35 MeV are measured. Source radii ( r 0 ≈ 3−4 fm) and emission temperatures ( T ≈ 9−3 MeV) are extracted and shown to depend on the kinetic energy of the emitted particles.

Large-angle proton emission in the 9Be(p, 2p) reaction at 300 MeV

Abstract A 9 Be(p, 2p) coincidence experiment performed to further elucidate the reaction mechanism for the production of energetic wide-angle protons in intermediate-energy proton-induced reac- tions is reported. Detectors in a coplanar geometry were used to measure coincidences between trigger protons at 90° to the beam and forward-angle protons on the opposite side of the beam. The incident proton energy was 300 MeV. We report both the inclusive spectra for the trigger protons and the differential mean multiplicities for the coincidence events. The outgoing proton energies were measured using NaI detectors. Trigger protons were grouped into 10 MeV bins covering the kinetic energy range from 55 to 155 MeV. The forward protons were measured over a kinetic energy range of 65–280 MeV and an angular range of 14–60° with respect to the beam. The present results are compared with two previous experiments which covered a more restrictive kinematical range. Calculations are performed with both phase-space and direct knockout models, and compared with experiment. Observation of angle and energy correlation effects suggested by knockout models indicate that such direct mechanisms provide a significant contribution to energetic wide-angle inclusive proton spectra.

Shape analysis of filamentous Precambrian microfossils and modern cyanobacteria

Abstract Variations in the orientation and cross-sectional shape of filamentous microfossils provide quantitative measures for characterizing them and probing their native mechanical structure. Here, we determine the tangent correlation length, which is the characteristic length scale for the variation in direction of a sinuous curve, for both a suite of Precambrian filamentous microfossils and six strains of modern filamentous cyanobacteria, all with diameters of a few microns. Among 1.9–2-Ga microfossils, Gunflintia grandis, Gunflintia minuta and Eomycetopsis filiformis possess, respectively, correlation lengths of 360 ± 40 µm, 670 ± 40 µm and 700 ± 100 µm in two dimensions. Hundreds of times larger than the filament diameters, these values lie in the same range as the cyanobacteria Geitlerinema and Pseudanabaena, but are smaller than several strains of Oscillatoria. In contrast, the 2-Ga microfossil trichome Halythrix, is found to have a short correlation length of 29 ± 4 µm in two dimensions. Micron-wide pyritic replacement filaments observed in 3.23-Ga volcanogenic deposits also display a modest correlation length of 100 ± 15 µm in two dimensions. Sequences of species in two genera of our modern cyanobacteria possess tangent correlation lengths that rise as a power of the filament diameter D—D3.3 ± 1 for Oscillatoria and D5.1 ± 1 for Geitlerinema. These results can be compared with power-law scaling of D3 for hollow tubes and D4 for solid cylinders that is expected from continuum mechanics. Extrapolating the observed scaling behavior to smaller filament diameters, the measured correlation length of the pyrite filaments is consistent with modern Geitlerinema whereas that of Halythrix lies not far from modern Oscillatoria, suggesting that there may be structural similarities among these genera.

Two-Dimensional Cytoskeletons Under Stress

Planar triangular networks under stress are predicted to have several interesting properties: a first-order transition to a collapsed state for a range of compressive stresses, and a negative Poisson ratio for a range of tensions (i.e., they expand transversely when stretched longitudinally). When these two-dimensional nets are allowed to fluctuate in three dimensions, they are predicted to be asymptotically rigid at long length scales and to have a universally negative Poisson ratio, even at zero stress (reviewed in Boal, 1996). There are many examples of two-dimensional networks in nature: auditory outer hair cells (Tolomeo et al., 1996) and bacterial cell walls (Ghuysen, 1968) contain few or many layers of networks with square or honeycomb symmetry. Further, not all networks are isotropic: the peptidoglycan network of the bacterial cell wall is anisotropic in the network plane, being stiff in one direction but soft in the other. One well-studied network is the membrane-associated cytoskeleton of the human red blood cell-a two-dimensional network whose elements are tetramers of the protein spectrin. Although the contour length of a spectrin tetramer is approximately 200 nm, the average separation between the sixfold junctions linking the tetramers is closer to 70 nm (Steck, 1989). Thus, one picture of the erythrocyte cytoskeleton is that of a triangular network of convoluted chains, as shown by the simulation in Figure 1. By mechanically manipulating the erythrocyte, measurements can be made of the shear modulus ,U and compression modulus K, of its cytoskeleton in the lipid bilayer plane to which the network is attached (Discher et al., 1994).