Frictional coefficients of multisubunit structures. II. Application to proteins and viruses

The theory of Kirkwood for the translational frictional coefficients of structures composed of identical subunits has been extended in the previous paper to the case where nonidentical subunits are involved. In this paper, the theory is applied to particular proteins and viruses. It is found that the calculated sedimentation coefficients of various states of aggregation of the reversibly associating proteins hemocyanin and phycocyanin are in excellent agreement with experiment. The dimensions of the fibrinogen molecule obtained from electron micrographs do not give good agreement between calculated and experimental frictional coefficients. The frictional coefficient of tobacco mosaic virus calculated without explicit consideration of end effects is in good agreement with experiment if a hydrodynamic diameter of 18O A., corresponding to the maximum diameter from x-ray studies, is used. Agreement is also good for the fast sedimenting form of bacteriophage T2 and the protein ghosts of bacteriophage λ but the slow form of T2 and whole λ have frictional coefficients considerably in excess of those calculated. Tail fiber configuration or head porosity are unable to account for the difference in sedimentation coefficients between the fast and slow forms of T2.     

Surface structure and frictional properties of the skin of the Amazon tree boa <Emphasis Type="Italic">Corallus hortulanus</Emphasis> (Squamata,Boidae)

The legless locomotion of snakes requires specific adaptations of their ventral scales to maintain friction force in different directions. The skin microornamentation of the snake Corallus hortulanus was studied by means of scanning electron microscopy and the friction properties of the skin were tested on substrates of different roughness. Skin samples from various parts of the body (dorsal, lateral, ventral) were compared. Dorsal and lateral scales showed similar, net-like microornamentation and similar friction coefficients. Average friction coefficients for dorsal and lateral scales on the epoxy resin surfaces were 0.331 and 0.323, respectively. In contrast, ventral scales possess ridges running parallel to the longitudinal body axis. They demonstrated a significantly lower friction coefficient compared to both dorsal and lateral scales (0.191 on average). In addition, ventral scales showed frictional anisotropy comparing longitudinal and perpendicular direction of the ridges. This study clearly demonstrates that different skin microstructure is responsible for different frictional properties in different body regions.     

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.     

Equations for membrane transport. Experimental and theoretical tests of the frictional model.

Frictional models for membrane transport are tested experimentally and theoretically for the simple case of a solution consisting of a mixture of two perfect gases and a membrane consisting of a porous graphite septum. Serious disagreement is found, which is traced to a missing viscous term. Kinetic theory is then used as a guide in formulating a corrected set of transport equations, and in giving a physical interpretation to the frictional coefficients. Sieving effects are found to be attributable to entrance effects rather than to true frictional effects within the body of the membrane. The results are shown to be compatible with nonequilibrium thermodynamics. Some correlations and predictions are made of the behavior of various transport coefficients for general solutions.     

The effect of membrane heterogeneity on the predictability of fluxes,with application to the cornea

Many phenomenological treatments of biological membrane transport are based on the assumption that the membrane consists of a single class of passive transport paths; i.e. that the membrane is simple. Simplicity is assumed both in the measurement of the membrane's transport coefficients and in the use of these coefficients to predict membrane fluxes. Such a procedure will in general lead to an error in the prediction of solute flux across parallel arrays. The error depends on the distribution of the reflection coefficients of the parallel paths and upper and lower bounds on it are given in terms of conventionally measured transport coefficients. The frictional representation of electrolyte transport across membranes possessing metabolic pumps is generalized to take this structure effect into account. The flux error resulting from the neglect of membrane heteroreflectivity is essentially the same for nonelectrolytes and electrolytes, irrespective of whether phenomenological or frictional membrane transport properties are used. It is shown that some information about transport structure can be obtained from global measurements made without regard for the organization of pathways across the membrane.The values of the measured transport coefficients of the corneal epithelium and endothelium imply that these cell layers are heteroreflective. Analysis of corneal transport, taking structure effects into account, shows that the corneal thickness may be intrinsically insensitive to tear tonicity, by a mechanism which may be of more general homeostatic significance.     

Subunits of the alkaline phosphatase of Bacillus licheniformis: chemical, physicochemical, and dissociation studies.

The alkaline phosphatase (orthophosphoric monoester phosphydrolase, EC 3.1.3.1) of Bacillus licheniformis MC14 was studied in an attempt to determine the number of subunits contained in the 120,000-molecular-weight native enzyme. Two moles of arginine was liberated per mole of native enzyme by carboxypeptidases A and B in the presence of sodium dodecyl sulfate. The effect on the native enzyme of progressively lowering the solvent buffer pH was monitored by determining the molecular weight by sedimentation equilibrium analysis, the sedimentation coefficient, the frictional coefficient, and the percent alpha-helix content of the enzyme. The alkaline phosphatase dissociates into two subunits around pH 4. At pH 2.8 a further decrease in S value, but no change in molecular weight, is observed, indicating a change in conformation. The frictional coefficients and percent alpha-helix content agree with this interpretation. A subunit molecular weight of 59,000 was calculated from sodium dodecyl sulfate gels.     

Heterogeneity in the physicochemical properties of deoxycholate-solubilized benzodiazepine receptors from calf cerebral cortex

The hydrodynamic behaviour of benzodiazepine receptors solubilized by deoxycholate from calf cerebral cortex reveals two molecular forms. The Stokes radii are 46.5 A and 67.2 A, and the sedimentation coefficients are 10.9 S and 14.6 S. The calculated apparent molecular weights and frictional ratios suggest either two nearly globular proteins of ca. 200K and 400K daltons each, or two ca. 300K daltons proteins which differ significantly in their degree of asymmetry. The benzodiazepine binding site is located on ca. 51K daltons component(s) in both forms.     

Forces and stresses acting on fusion pore membrane during secretion

To assess the forces and stresses present in fusion pore during secretion the stationary convective flux of lipid through a fusion pore connecting two planar membranes under different tensions was investigated through computer simulations. The physics of the problem is described by Navier-Stokes equations, and the convective flux of lipid was evaluated using finite element method. Each of the membrane monolayer is considered separately as an isotropic, homogeneous and incompressible viscous medium with the same viscosity. The difference in membrane tensions, which is simulated as the pressure difference at two ends of each monolayer, is the driving force of the lipid flow. The two monolayers interact by sliding past each other with inter-monolayer frictional viscosity. Fluid velocity, pressure, shear and normal stresses, viscous and frictional dissipations and forces were calculated to evaluate where the fusion pore will deform, extend (or compress) and dilate. The pressure changes little in the planar sections, whereas in the toroidal section the change is rapid. The magnitude of lipid velocity peaks at the pore neck. The radial lipid velocity is zero at the neck, has two peaks one on each side of the pore neck, and diminishes without going to zero in planar parts of two monolayers. The peaks are of opposite signs due to the change of direction of lipid flow. The axial velocity is confined to the toroidal section, peaks at the neck and is clearly greater in the outer monolayer. As a result of the spatially highly uneven lipid flow the membrane is under a significant stress, shear and normal. The shear stress, which indicates where the membrane will deform without changing the volume, has two peaks placed symmetrically about the neck. The normal stress shows where the membrane may extend or compress. Both, the radial and axial normal stresses are negative (extensive) in the upper toroidal section and positive (compressive) in the lower toroidal section. The pressure difference determines lipid velocity and velocity dependent variables (shear as well as normal axial and radial stresses), but also contributes directly to the force on the membranes and critically influences where and to what extent the membrane will deform, extend or dilate. The viscosity coefficient (due to friction of one element of lipid against neighboring ones), and frictional coefficient (due to friction between two monolayers sliding past each other) further modulate some variables. Lipid velocity rises as pressure difference increases, diminishes as the viscosity coefficient rises but is unaffected by the frictional coefficient. The shear and normal stresses rise as pressure difference increases, but the change of the viscosity coefficients has no effect. Both the viscous dissipation (which has two peaks placed symmetrically about the neck) and much smaller frictional dissipation (which peaks at the pore neck) rise with pressure and diminish if the viscosity coefficient rises, but only the frictional dissipation increases if the frictional coefficient increases. Finally, the radial force causing pore dilatation, and which is significant only in the planar section of the vesicular membrane, is governed almost entirely by the pressure, whereas the viscosity and frictional coefficients have only a marginal effect. Many variables are altered during pore dilatation. The lipid velocity and dissipations (viscous and frictional) rise approximately linearly with pore radius, whereas the lipid mass flow increases supra-linearly owing to the combined effects of the changes in pore radius and greater lipid velocity. Interestingly the radial force on the vesicular membrane increases only marginally.     

The Electro-osmotic Theory of Phloem Transport in the Light of Recent Measurements on Heracleum Phloem

The bearing, on the electro-osmotic theory, of work on the Onsagercoefficients for Heracleum phloem, currently reported by Tyreeand Fensom (1970), is briefly discussed. As an aid to the discussionthe Onsager resistance coefficients are evaluated in terms ofSpiegler's frictional model. It is concluded that the experimentalevidence, while it does not support the theory, does not invalidateit. Some further suggestions regarding the electro-osmotic theoryare put forward.     

Hydrodynamic properties of macromolecular complexes. I. Translation

We have developed an improved theory for calculating the translational frictional coefficients of rigid macromolecular complexes composed of unequal spherical subunits. The Yamakawa hydrodynamic interaction tensor, which improves on the Oseen tensor by taking account of the finite sizes of the frictional subunits, has been generalized to accomodate nonidentical subunits. Iterative numerical methods are described for solving the set of simultaneous hydrodynamic interaction equations, thus avoiding preaveraging. The theory is applied to prolate ellipsoids of revolution, to lollipops, and to dumbbells, and comparison is made with earlier, more approximate theories.     

A local friction model to develop microscopic transport parameters for membrane-soluble ions in bilayer membranes

Charged membrane-soluble ions in bilayer membranes move between opposing interfacial states on application of an electric field. The kinetics of this transition is modeled as discrete state electrophoresis using a Langevin formulation that focuses on local velocities at points along the spatial trajectory of the ion. These local velocities are used to establish both the decay time for the transition and the activation energy profile along the trajectory. The local velocities depend on local frictional coefficients. These coefficients are developed from the molecular structure along the trajectory by introducing a microscopic frictional force. This frictional force is produced by the lateral displacement of segments of the surrounding membrane molecules as the ion passes. The displaced molecular chains produce a non-linear resistive force that translates into a frictional force directly proportional to both ion radius and velocity.     

Physical properties of moloney murine leukemia virus high-molecular-weight RNA: a two subunit structure.

The high-molecular-weight RNA of Moloney murine leukemia virus (MuLV) was analyzed by sedimentation equilibrium ultracentrifugation. Molecular weights of 7.2 x 10(6) and 3.4 x 10(6) were found for the native and subunit forms, respectively, indicating that the native structure is a dimer. S20,w and frictional coefficients were determined for MuLV RNA by analytical velocity centrifugation as a function of ionic strength. The apparent S20,w of native MuLV RNA was 47.3, 57.4, and 66.5 in 0.01, 0.1, and 0.20 M Na+, respectively; the corresponding frictional coefficients were 5.44, 4.48, and 3.87. Native RNA was estimated by circular dichroism to be 85% helical, whereas denatured RNA was 54% helical. Thermal denaturation profiles were obtained from uv absorbance scans. Melting temperatures of 57 and 68 C were obtained for high-molecular-weight RNA in 0.01 M Na+ and 0.122 M Na+, 1mM Mg2+, respectively. van't Hoff plots of the thermal denaturation data gave enthalpies for the helix-coil transition of 21,600 cal (ca. 90,500 J) per mol of cooperatively melting unit in high salt and 19,600 cal (ca. 82,100 J) per mol in low salt, consistent with both base stacking and pairing. The melting of Mu LV RNA occurred over a broad temprange and van't Hoff plots were linear over most of the melting range, indicating a noncooperative process of helix stabilization.     

Comparison of the electrophoretic and hydrodynamic properties of DNA and RNA oligonucleotide duplexes.

The electrophoretic behavior of defined DNA and RNA oligonucleotide duplexes from 10 to 20 bp in length has been investigated as a function of salt conditions, gel concentration, and temperature. The RNA oligomers migrated much more slowly than the DNA oligomers of the same sequence under all conditions. From sedimentation equilibrium and velocity measurements, the apparent partial specific volume in 0.1 M KCI, 20 mM NaPi, pH 7, was determined as 0.56 +/- 0.015 ml g(-1) for DNA and 0.508 ml g(-1) for RNA. The translational friction coefficients were determined and compared with the values calculated for cylinders. Taking into account the shape factors, the solution density, and partial specific volumes, the effective degree of hydration was estimated as 0.8-1 g g(-1) DNA. There was no significant difference in the frictional coefficients of the DNA and RNA oligomers, indicating that the effective sizes of DNA and RNA are very similar in solution. The differential electrophoretic mobility of DNA and RNA must arise from the differences in interaction with counterions, which is probably a global property of the oligonucleotides.     

Effects of surface roughness on the coefficients of friction in model orthodontic systems

Orthodontists, like others (Engel, P.A. (1976) Impact Wear of Materials. Elsevier Scientific, New York.), often equate the smoothness of surfaces with the absence of friction. To investigate whether the surface roughness of opposing materials influence the coefficients of friction and ultimately the movement of teeth, arch wires were slid between contact flats to simulate orthodontic arch wire-bracket appliances. From laser specular reflectance measurements, the RMS surface roughness of these arch wires varied from 0.04 microns for stainless steel to 0.23 microns for nickel titanium. Using the same technique, the roughnesses of the contact flats varied from 0.03 microns for the 1 micron lapped stainless steel, to 0.26 microns for the as-received alumina. After each of the arch wire-contact flat couples was placed in a friction tester, fifteen normal forces were systemically applied at 34 degrees C. From plots of the static and kinetic frictional forces vs the normal forces, dry coefficients of friction was obtained that were greater than those reported in the dental literature. The all-stainless steel couples had lower kinetic coefficients (0.120-0.148) than the stainless steel-polycrystalline alumina couple (0.187). When pressed against the various flats, the beta-titanium arch wire (RMS = 0.14 microns) had the highest coefficients of friction (0.445-0.658), although the nickel titanium arch wire was the roughest (RMS = 0.23 microns). Scanning electron microscopy (SEM) and energy dispersive X-ray analysis (EDX) verified that mass transfer of the beta-titanium arch wire occurred by adhesion onto the stainless steel flats or by abrasion from the sharply faceted polycrystalline alumina flats.     

A Continuum Mechanical Approach to the Flow Equations for Membrane Transport: I. Water Flow

A concept is presented for modeling flows through membranes using continuum mechanics. Viscous interactions (due to velocity gradients) are explicitly incorporated and position-dependent local water-membrane interactions are taken into account before obtaining slab averages. This is in distinction to other treatments where strictly one-dimensional force balance equations are written using slab average friction coefficients which are really composite functions of local interactions. It is shown that the viscous and other frictional interactions do not simply form linear combinations in the solutions to the equations of motion. Flow profiles for pressure-driven flows ranging from Poiseuille's flow to “diffusion” flow are obtained depending on the strength and extent of the water-membrane interaction. The model is also applied to self-diffusion flows and the measurement of “equivalent pore size.” It is shown that for a fixed pore size the ratio of filtration flow to self-diffusion flow for equal driving forces is able to vary over a wide range depending on the water-membrane interaction.     

Hydrodynamic theory of swimming of flagellated microorganisms.

A theory of the type commonly used in polymer hydrodynamics is developed to calculate swimming properties of flagellated microorganisms. The overall shape of the particle is modeled as an array of spherical beads which act, at the same time, as frictional elements. The fluid velocity field is obtained as a function of the forces acting at each bead through Oseen-type, hydrodynamic interaction tensors. From the force and torque equilibrium conditions, such quantities as swimming velocity, angular velocity, and efficiency can be calculated. Application is made to a spherical body propelled by a helical flagellum. A recent theory by Lighthill, and earlier formulations based on tangential and normal frictional coefficients of a curved cylinder, CT and CN, are analyzed along with our theory. Although all the theories predict similar qualitative characteristics, such as optimal efficiency and the effect of fluid viscosity, they lead to rather different numerical values. In agreement with Lighthill, we found the formalisms based on CN and CT coefficients to be somewhat inaccurate, and head-flagellum interactions are shown to play an important role.     

Some remarks on the Kedem-Katchalsky equations for non-electrolytes

The Kedem-Katchalsky equation for the flow of a non-electrolyte through a homogeneous membrane is shown to be a first order expansion of an exact integral of the Spiegler-Bearman-Kirkwood frictional equations under the assumption that the partial frictional coefficients, ζ _ij , are concentration independent. The equations are solved in terms of volume flow; there are no water-to-volume flow correction terms for the permeability, ω, or the reflection coefficient, σ. The precision of the expansion depends upon the magnitude of the water flow. The frictional coefficientsf _sm andf _sw are given as functions of the experimentally determined parameters ω and σ; the frictions, are shown to be independent ofL _p .     

Hydrodynamics of macromolecular complexes. III. Bacterial viruses

We have calculated the translational and rotational frictional coefficients of structures related to T2 and T4 bacteriophage, using the theoretical framework developed in the preceding two papers. The structures considered were models for tail-fiberless phage, and for whole phage with fibers in the extended and retracted portions. We also computed and compared with the experiment the changes in translational frictional coefficient produced by successive addition of 1–6 fibers to the fiberless particle. Agreement with experimental results is markedly improved over previous theoretical efforts, especially with respect to the effect of tail-fiber extension. Some significant discrepancies remain, however, in the comparison of fiber-retracted and fiberless phage.     

Physical and chemical properties of the NH2-terminal glutamic acid and lysine forms of human plasminogen and their derived plasmins with an NH2-terminal lysine heavy (A) chain.

Comparative physical and chemical data are described for the human NH2-terminal Glu-plasminogen and Lys-plasminogen forms in order to determine the exact relationship between these two types of the zymogen. The molecular weights of Glu-plasminogen and Lys-plasminogen were similar and were determined to be 83, 800 plus or minus 4, 500 and 82, 400 plus or minus 3, 300, respectively, by sedimentation equilibrium methods. The molecular weights were identical in dodecyl sulfate solutions, approximately 83, 000, by sedimentation equilibrium methods. The sedimentation coefficients, s-020, w of Glu-plasminogen and Lys-plasminogen were determined to be 5.0 S, and 4.4 S, respectively. These two plasminogen forms had different partial specific volumes, and calculations of the frictional coefficients from sedimentation coefficients and molecular weights indicated conformation differences. Glu-plasminogen appeared to be larger in size than Lys-plasminogen in acrylamide gel-dodecyl sulfate electrophoresis. The amino acid compositions of Glu-plasminogen and Lys-plasminogen, and their major isolated isoelectric forms, were found to be similar, but several amino acid residues (glutamic acid, alanine, isoleucine, phenylalanine, and lysine) were found to be significantly higher in the Glu-plasminogen forms. The derived plasmins from both the Glu- and Lys-plasminogens with an nh2-terminal Lys- heavy (A) chain were found to have identical molecular weights of 76, 500 plus or minus 2, 500, and sedimentation coefficients, s-020, w of 4.3 S.     

Measuring principles of frictional coefficients in cartilaginous tissues and its substitutes

The frictional properties of cartilaginous tissues, such as the hydraulic permeability, the electro-osmotic permeability, the diffusion coefficients of various ions and solutes, and the electrical conductance, are vital data to characterise the extracellular environment in which chondrocytes reside. This paper analyses one-dimensional measurement principles of these coefficients. Particular attention is given to the deformation dependence of them and the highly deformable nature of the tissues. A suggested strategy is the combination of a diffusion experiment using radiotracer methods, an electro-osmotic flow experiment and an electro-osmotic pressure experiment at low electric current.