The translational friction of toroids

An approximate analytic expression for the translational friction coefficient of a toroid modeled as a continuous shell of frictional elements is derived using the Kirkwood approximation. The accuracy of this expression was determined by comparing the friction coefficients predicted by it to those predicted by extrapolated shell-model calculations using the modified Oseen tensor. To show that these calculations do indeed yield the correct friction coefficients, actual translational friction coefficients were determined by observing settling rates of macroscopic model rings or toroids in a high-viscosity silicone fluid. Our conclusion is that the approximate expression yields friction coefficients that are about 1.5–3% low for finite rings. For thin rings, a comparison is also made with the exact result of Yamakawa and Yamaki [J. Chem. Phys. 57 , 1572 (1972); 58 , 2049 (1973)] for the translational friction of plane polygonal rings. This comparison shows that the approximate expression yields results which are low by 2–3% unless the rings are extremely thin, in which case the error is larger. In the limit of an infinitely thin ring the approximate expression reduces to the Kirkwood result [J. Polym. Sci. 12 , 1 (1954)], which is low by 8.3%. We discuss briefly how this work may be useful in determining the structure of DNA compacted by various solvent–electrolyte systems and polyamines.     

Transport properties of oligomeric subunit structures

The translational friction coefficients, rotational friction coefficient, and intrinsic viscosity of rigid regular structures composed of up to eight identical spherical subunits have been accurately calculated. The aim of this calculation is to interpret the hydrodynamic properties of oligomeric subunit proteins. To avoid the well-known failure of the theory in the evaluation of rotational coefficients and intrinsic viscosities, each subunit is hydrodynamically modeled as a polyhedral array of smaller spheres. The analysis of several alternatives suggests that a cubic array is the best choice. The reliability of this strategy is checked by comparison of the calculated values for all the transport properties of a sphere and the translational friction coefficients of a dimer with their exact values. Finally, the hydrodynamic properties of a number of subunit structures with varying number of subunits and different geometries are tabulated.     

Calculation of translational friction and intrinsic viscosity. II. Application to globular proteins.

The translational friction coefficients and intrinsic viscosities of four proteins (ribonuclease A, lysozyme, myoglobin, and chymotrypsinogen A) are calculated using atomic-level structural details. Inclusion of a 0.9-A-thick hydration shell allows calculated results for both hydrodynamic properties of each protein to reproduce experimental data. The use of detailed protein structures is made possible by relating translational friction and intrinsic viscosity to capacitance and polarizability, which can be calculated easily. The 0.9-A hydration shell corresponds to a hydration level of 0.3-0.4 g water/g protein. Hydration levels within this narrow range are also found by a number of other techniques such as nuclear magnetic resonance spectroscopy, infrared spectroscopy, calorimetry, and computer simulation. The use of detailed protein structures in predicting hydrodynamic properties thus allows hydrodynamic measurement to join the other techniques in leading to a unified picture of protein hydration. In contrast, earlier interpretations of hydrodynamic data based on modeling proteins as ellipsoids gave hydration levels that varied widely from protein to protein and thus challenged the existence of a unified picture of protein hydration.     

Translational friction of rigid assemblies of spheres: Derivation and application of new hydrodynamic interaction tensors

In connection with our goal of calculating by practical methods the frictional properties of biopolymers from surface shells composed of spheres, we have investigated by the method of reflections the low-Reynolds-number hydrodynamic interaction between two unequal-sized spheres in translation. Previous results, in which the velocities were used as independent variables and which have the form of truncated infinite power series, were substantially extended. By inversion of the power series, new power series with better convergence properties were obtained. Equivalence of these inverted power series with those previously reported based on the method of reflections, when forces are used as independent variables, was demonstrated, and the solutions were again substantially extended. Applying the Lagrange interpolation to data generated from exact theories for the hydrodynamic inteaction between two spheres, it was demonstrated that the various forms of the method of reflections do not just give reasonable power series, but actually yield optimal ones. These findings constitute a unification of diverse approaches and show methods of interconversion of results. On the basis of the power series obtained, a set of new hydrodynamic interaction tensors for two unequal spheres were derived. While the new tensors described the case of two unequal spheres with considerably more accuracy than those previously reported, direct application of these tensors to objects composed of more than two spheres revealed some unexpected problems resulting from overcorrection in the fourth-order term. However, when the tensors were preaveraged over all orientations of the multisphere object, a formula for the scalar translational friction coefficient was obtained that outperformed all but the most involved earlier approaches. It thus constitutes an improved and practical solution to the problem of computing translational friction coefficients of objects describable by a surface shell of many spheres, such as proteins.     

Hydrodynamic properties of rigid particles: comparison of different modeling and computational procedures.

The hydrodynamic properties of rigid particles are calculated from models composed of spherical elements (beads) using theories developed by Kirkwood, Bloomfield, and their coworkers. Bead models have usually been built in such a way that the beads fill the volume occupied by the particles. Sometimes the beads are few and of varying sizes (bead models in the strict sense), and other times there are many small beads (filling models). Because hydrodynamic friction takes place at the molecular surface, another possibility is to use shell models, as originally proposed by Bloomfield. In this work, we have developed procedures to build models of the various kinds, and we describe the theory and methods for calculating their hydrodynamic properties, including approximate methods that may be needed to treat models with a very large number of elements. By combining the various possibilities of model building and hydrodynamic calculation, several strategies can be designed. We have made a quantitative comparison of the performance of the various strategies by applying them to some test cases, for which the properties are known a priori. We provide guidelines and computational tools for bead modeling.     

Frictional coefficients of multisubunit structures. I. Theory

The theory of Kirkwood for the translational frictional coefficients of structures composed of subunits has been generalized in two ways in order to consider aggregates of nonidentical subunits. One of these generalizations fails when the sizes of subunits are too disparate; the other, derived from a surface shell distribution of frictional elements, is effective over the whole range of relative sizes. It is shown that, in the limit of a continuous surface distribution, a shell model reproduces Stoke's law for a sphere. Comparison is made between the frictional coefficients of spheres, ellipsoids, and rods modeled by finite numbers of subunits and by continuous shells of frictional elements, and those calculated from other theories. Agreement is generally good, though the shell model for prolate ellipsoids of revolution deviates by a few per cent from the Perrin value.     

Frictional models for stochastic simulations of proteins

As a first step toward a systematic parametrization of friction constants of atoms in proteins, a model in which frictional resistance is placed explicitly on each atom accessible to solvent is used to calculate overall translational and rotational diffusion constants. It is found that these quantities are relatively insensitive to the precise value of the atomic friction constant, as long as the effective hydrodynamic radius of the surface atoms is approximately 1 Å. However, if only protein atoms are included in the calculation, no reasonable range atomic of radii can reproduce the experimental translational diffusion constant to better than 20% for lysozyme and 5% for ribonuclease. When a hydration shell of approximately 70% coverage for lysozyme and 20% for ribonuclease is included, there is quantitative agreement with experimental results. The sensitivity of peptide diffusion to levels of hydration is also investigated; it is found that for glycine, two bound waters are required to provide agreement with experiment. These findings imply that the effects of solvent damping will be underestimated in stochastic simulations of proteins and peptides unless bound waters are taken into account.     

Protein loading,elution, and resolution behavior in a novel device that integrates ultrafiltration and chromatographic separation

Hollow fiber membranes and chromatographic resin beads are commonly employed in a variety of bioseparation processes. A new class of integrated separation devices is being studied in which the shell side of a hollow fiber device is filled with adsorbents/chromatographic resin beads. Such devices and the corresponding separation methods integrate feed broth clarification by the microfiltration/ultrafiltration membrane with bioproduct purification by the shell-side resin beads either as an adsorbent or as beads in elution chromatography. A mathematical model has been developed for the prediction of the chromatographic behavior of such an integrated device. Simulations have been done to study the effects of axial dispersion, feed flow rate, water permeation rate, fiber packing density, and void fraction. Numerical solutions were obtained by solving the governing equations. This model can reasonably describe the concentration profiles as well as the breakthrough and elution behaviors in the integrated device.     

Nassarius kraussianus shell beads from Blombos Cave: evidence for symbolic behaviour in the Middle Stone Age

Since 1991, excavations at Blombos Cave have yielded a well-preserved sample of faunal and cultural material in Middle Stone Age (MSA) levels. The uppermost MSA phase, M1, is dated to c. 75 ka by optically stimulated luminescence (OSL) and thermoluminescence, and the middle M2 phase to a provisional c. 78 ka. Artefacts unusual in a MSA context from these phases include bifacial points, bone tools, engraved ochre and engraved bone. In this paper, we describe forty-one marine tick shell beads recovered from these MSA phases and tick shell beads from Later Stone Age (LSA) levels at Blombos Cave and the Die Kelders site. Thirty-nine shell beads come from the upper M1 phase and two from M2. Morphometric, taphonomic and microscopic analysis of modern assemblages of living and dead tick shell demonstrate that the presence of perforated Nassarius kraussianus shells in the Blombos MSA levels cannot be due to natural processes or accidental transport by humans. The types of perforation seen on the MSA shells are absent on modern accumulations of dead shells and not attributable to post-depositional damage. Their location, size, and microscopic features are similar to those obtained experimentally by piercing the shell wall, through the aperture, with a sharp bone point. Use-wear, recorded on the perforation edge, the outer lip, and the parietal wall of the aperture indicates the shells having being strung and worn. MSA shell beads differ significantly in size, perforation type, wear pattern and shade compared to LSA beads and this eliminates the possibility of mixing across respective levels. Thirty-one beads were found in four groups of five to twelve beads, each group being recovered in a single square or in two adjacent sub-squares during a single excavation day. Within a group, shells display a similar shade, use-wear pattern and perforation size suggesting their provenance from the same beadwork item, lost or disposed during a single event. The likely symbolic significance of these finds suggests levels of cognitively modern behaviour not previously associated with MSA people.     

Transient relative motion of two cells in a channel flow

The hydrodynamic interaction of a red blood cell and a white blood cell in microvessels is studied, by use of a two-dimensional numerical model. The red blood cell, modeled as a small rigid circular cylinder, and the white blood cell, modeled as a larger rigid circular cylinder, are immersed in an incompressible Newtonian fluid in a two-dimensional channel. It is assumed that no external force or moment acts on the model cells, and the effect of inertia forces on the motion of the fluid and the cells is neglected. The velocity field of the suspending fluid and the instantaneous velocities of the two model cells are computed by the finite element method. Using the translational velocities of the model cells obtained, the trajectories of their relative motion are determined, for various initial positions. It is shown that the cells may or may not pass each other or separate, depending on the initial positions. The present results compare well to the experimental results.     

Hydrodynamic shielding of spherical subunits in macromolecular complexes

The Garcia de la Torre-Bloomfield hydrodynamic interaction tensor was used to calculate the shielding coefficient matrices for each spherical friction bead in the rigid arrays of a rod and helix. Negative values for the average shielding coefficient result for small beads adjacent to large beads, suggesting premature truncation in the series expansions employed in the hydrodynamic interaction tensor. Numerical analysis also suggests that the magnitude of geometric asymmetry is not a good measure of the extent of friction asymmetry of the molecule due to extensive hydrodynamic shielding by other subunit beads.     

Fibronectin-coated beads are endocytosed by cells and align with microfilament bundles

Carboxylated latex beads coated with fibronectin bind to the surfaces of NIL8 hamster cells. Similar beads coated with bovine serum albumin, gelatin or normal rabbit immunoglobulin do not bind to the cells, whereas beads coated with polylysine or rabbit anti-hamster immunoglobulin bind, as do fibronectin-coated beads. Surface-bound beads are endocytosed by the cells. This endocytosis is inhibited by N-ethyl maleimide or low temperatures. The endocytosed beads form arrays aligned with the microfilament bundles inside the cells and after extended incubations (18–24 h) become concentrated in the perinuclear region. These results suggest that, after phagocytosis of large particles, the endocytic vesicles align with microfilament bundles, as previously reported for coated pits involved in receptor-mediated endocytosis. The system described here offers possibilities for the analysis of membrane and transmembrane interactions involved in cell-substratum binding and phagocytosis.     

Freeze-thaw immobilization of liposomes in chromatographic gel beads: evaluation by confocal microscopy and effects of freezing rate

Biological membranes immobilized in chromatographic gel beads constitute a multifunctional affinity matrix. Membrane protein-solute interactions and drug partitioning into the lipid bilayers can conveniently be studied. By the use of confocal laser-scanning microscopy (CLSM) the distribution of immobilized model membranes in the beads has been visualized for the first time. Freeze-thaw-immobilized liposomes in Superdex 200 gel beads were situated in a thick shell surrounding a liposome-free core. The amount of phospholipids immobilized by freeze-thawing was dependent on the temperature in the cooling bath and the type of test tube used. A bath temperature of -25 degrees C gave higher immobilization yield than freezing at -75 or -8 degrees C did. Freeze-thawing in the presence of liposomes did not affect the gel bead shape or the refractive index homogeneity of the agarose network of the beads, as shown by confocal microscopy.     

Modeling the electrophoresis of rigid polyions: application to lysozyme.

An algorithm is developed to determine the electrophoretic mobility of a rigid polyion modeled as a low dielectric volume element of arbitrary shape containing an arbitrary charge distribution. The solvent is modeled as a high dielectric continuum with salt distributed according to the linearized Poisson Boltzmann equation. Account is also taken of a Stern layer that separates the molecular surface and the surface of hydrodynamic shear, or Stern surface. Relaxation of the ion atmosphere because of the presence of the external field is ignored. The electrostatic and hydrodynamic problems are both solved by boundary element methods. The procedure is first applied to spherical polyions containing monopolar, dipolar, and quadrupolar charge distributions, and calculated mobilities are found to be in excellent agreement with the theory of Yoon and Kim. It is then applied to lysozyme by using models that account for the detailed shape and charge distribution of the enzyme. For reasonable choices of the molecular and Stern surfaces, calculated and experimental mobilities are found to be in fair agreement with each other. However, if a pH independent Stern layer (or, equivalently, translational diffusion constant, Dt) is assumed, the calculated mobilities exhibit a stronger pH dependence than is observed experimentally. A small increase in Dt with increasing pH could correct this discrepancy.     

Static equilibrium configurations of a model red blood cell

Summary The membrane of the red blood cell is modeled as a fluid shell which resists bending and changes in area. The differential equations governing the mechanical equilibrium of such a membrane are derived and axisymmetric solutions are obtained numerically.     

Energy transformations in human movement by contact.

This study focuses on the transformation of energy in multilinkage systems by a deliberate use of the contact with the ground, leading to a derivation of the directional change of translational velocity of the body's center of mass. The coefficient of friction on the surface on which the impact occurs, and its effect on the overall movement, is studied for general multilinkage systems undergoing impact. The effect of surface friction is made apparent via simulation studies for a two-link example, where two interesting conditions arise: slippage or no slippage on the surface at impact. It is found that once the system stops on the surface, the translational energy increases as the angular velocity increases. Likewise, it is seen that the rotational energy after impact increases as the angular velocity of the first link increases, but the rate of increase of energy is less in the case where the system stops on the ground with no slippage.     

Dynamic response of intraocular pressure and biomechanical effects of the eye considering fluid-structure interaction

The vibration characteristics of shell structures such as eyes have been shown to vary with intraocular pressure (IOP). Therefore, vibration characteristics of the eye have the potential to provide improved correlation to IOP over traditional IOP measurements. As background to examine an improved IOP correlation, this paper develops a finite element model of an eye subject to vibration. The eye is modeled as a shell structure filled with inviscid pressurized fluid in which there is no mean flow. This model solves a problem of a fluid with coupled structural interactions of a generally spherically shaped shell system. The model is verified by comparing its vibrational characteristics with an experimental modal analysis of an elastic spherical shell filled with water. The structural dynamic effects due to change in pressure of the fluid are examined. It is shown that the frequency response of this fluid-solid coupled system has a clear increase in natural frequency as the fluid pressure rises. The fluid and structure interaction is important for accurate prediction of system dynamics. This model is then extended to improve its accuracy in modeling the eye by including the effect of the lens to study corneal vibration. The effect of biomechanical parameters such as the thicknesses of different parts of the eye and eye dimensions in altering measured natural frequencies is investigated and compared to the influence of biomechanical parameters in Goldmann applanation tonometry models. The dynamic response of the eye is found to be less sensitive to biomechanical parameters than the applanation tonometry model.     

Imaging the distribution and secondary structure of immobilized enzymes using infrared microspectroscopy

Synchrotron infrared microspectroscopy (SIRMS) was used for the first time to image the distribution and secondary structure of an enzyme (lipase B from Candida antarctica, CALB) immobilized within a macroporous polymer matrix (poly(methyl methacrylate)) at 10 microm resolution. The beads of this catalyst (Novozyme435) were cut into thin sections (12 microm). SIRMS imaging of these thin sections revealed that the enzyme is localized in an external shell of the bead with a thickness of 80-100 microm. Also, the enzyme was unevenly distributed throughout this shell. Furthermore, by SIRMS-generated spectra, it was found that CALB secondary structure was not altered by immobilization. Unlike CALB, polystyrene molecules of similar molecular weight diffuse easily throughout Novozyme435 beads. Scanning electron micrograph (SEM) images of the Novozyme435 beads showed that the average pore size is 10 times larger than CALB or polystyrene molecules, implying that there is no physical barrier to enzyme or substrate diffusion throughout the bead. Thus, the difference between polystyrene and enzyme diffusivity suggests that protein-matrix and protein-protein interactions govern the distribution of the enzyme within the macroporous resin.