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.     

Hydrodynamic properties of macromolecular complexes. IV. Intrinsic viscosity theory,with applications to once-broken rods and multisubunit proteins

We have extended our previous theories of the translational and rotational frictional properties of multisubunit complexes to calculate the intrinsic viscosity of such structures. Our theory is similar to those recently construced by McCammon and Deutch, and by Nakajima and Wada, in that it uses a modified hydrodynamic interaction tensor and solves the system of simultaneous interaction equations by digital computation rather than by successive approximations. However, there are some differences in the formulation and averaging of these equations. Extensive numerical comparison is made between this theory and others that are available—associated with the names of Hearst and Tagami, Abdel-Khalik and Bird, and Tsuda—using as a basis exact results for prolate ellipsoids of revolution. For large axial ratios, only our theory asymptotically approaches the correct limit; but for small axial ratios, only the Tsuda “shell-model” theory is adequate, because the other theories neglect the preponderant influence of the sphere located at the center of rotation. Intrinsic viscosities, translational frictional coefficients, and Scheraga-Mandelkern β parameters, are tabulated for a large number of polygonal and polyhedral subunit structures, with up to eight elements, using both our theory and Tsuda's. Particular application is made to hemerythrin and aspartate transcarbamylase. Finally, the viscosities and friction coefficients o once-broken rods are calculated and compared with an approximate theory by Wilenski.     

Hydrodynamic properties of rigid macromolecules composed of ellipsoidal and cylindrical subunits.

A procedure is devised for the calculation of hydrodynamic properties of rigid macromolecules composed subunits that are modeled as ellipsoids of revolution and cylinders. Owing to the axial symmetry of these shapes, smooth shell models can be constructured for the subunit structure. The bead shell model so constructed is employed for the calculation of the properties. A computer program, HYDROSUB, has been written implementing both the model building and the hydrodynamic calculation. A detailed example of the use of this methodology is presented for the case of the solution properties of the human antibody molecule immunoglobulin G3 (IgG3). Finally, hints are given on other uses and applications of the procedure.     

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.     

Hydrodynamic properties of macromolecular complexes. V. Improved calculation of rotational diffusion coefficient and intrinsic viscosity

In our previous calculations of rotational diffusion coefficients and intrinsic viscosities of macromolecular complexes modeled by arrays of spherical subunits [J. G. de la Torre & V. A. Bloomfield, Biopolymers 16 , 1765, 1779 (1977); 17 , 1605 (1978)], results were poor when the dominant subunit was located near the center of frictional resistance. A simple means of correcting this flaw, which gives satisfactorily accurate results with little increase in computation time, is to replace the single large subunit with an array of smaller ones. We have examined trigonal bipyramidal, octahedral, and cubic arrays of spheres whose radii were chosen to give the same total volume or the same rotational diffusion coefficient as the parent sphere. These all give similar results, so the details of the modeling are not important. Results obtained using this stratagem are in much better agreement with the theories of Perrin and Simha for short prolate ellipsoids of revolution, and with experimental measurements of rotational diffusion coefficients of T-even bacteriophage without fibers or with fibers retracted. We have also extended our previous calculations to consider phage with various numbers of fibers attached.     

Physical characterization of lumazine proteins from Photobacterium

The physicochemical properties of Photobacterium lumazine proteins have been investigated. The molecular weights obtained by several physical techniques are in good agreement, and the averages are 2% and 8% higher than the minimum molecular weights from amino acid and ligand content. The average molecular weights, sedimentation coefficients, and molecular radii are respectively the following: Photobacterium leiognathi lumazine protein, 21 200 +/- 300, 2.18 S, and 22.9 A; Photobacterium phosphoreum lumazine protein, 21 300 +/- 500, 2.16 S, and 23.0 A. The hydrations of the lumazine proteins, estimated in several ways, indicate less hydration for P. leiognathi than for P. phosphoreum. The frictional ratios corrected for hydration give axial ratios less than 1.3 for both lumazine proteins. These values agree with those obtained by a combination of rotational and translational frictional parameters and elimination of the common hydrated volume terms. There is insufficient area on the exterior surface to accommodate hydration when the lumzine proteins are considered as smooth-surfaced ellipsoids. The required surface area can be accommodated however by surface roughness with a minimum of 30% internal water.     

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.     

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.     

Dynamic light scattering from polydisperse suspensions of thin ellipsoidal shells of revolution with submicron diameters.

Osmotic swelling of membrane vesicles has been studied in combination with dynamic light scattering, to obtain information about the elastic properties of biomembranes. In such studies, there arise some technical problems specific to dynamic light scattering, which include the effects on the light-scattering results of the size distribution and nonsphericity of the vesicles with submicron sizes. Even for highly monodisperse suspensions of spherical vesicles (sigma/dn = [(the mean of d2)/d2n-1]1/2 = 0.1; dn being the number-average diameter of vesicles), the average diameter d obtained from dynamic light scattering is shown to be strongly dependent on dnK, where K is the length of the scattering vector. This is solely due to the shell structure of the vesicles. For ellipsoidal vesicles, another complication appears which is due to the rotational motion of ellipsoids.     

Trace elements in the atmosphere.

The distribution and behaviour of particulate trace elements in the atmosphere have been studied by continuous measurements for 5 years at seven non-urban sites in the United Kingdom. Samples have been taken regularly of airborne dust, rainwater and dry deposition: these have been analysed for up to 36 elements. Concentrations of trace elements vary considerably between sites but the relative concentrations are among uniform: this suggests similarity of origin or good atmospheric mixing. By comparing the relative concentrations with those in soil it is possible to differentiate between trace elements that are derived from soil and those that may be attributed to industrial activity. This classification is supported by estimates of the particle sizes in air. The deposition of trace elements can be related to the concentrations presnet in soil and to the annual removal by crops. Retrospective analyses of stored samples from one site describe the history of trace element concentrations in air since 1957. The sea surface is considered as a possible source of atmospheric trace elements.     

Low-resolution structure of the tetrameric phenylalanyl-tRNA synthetase from Escherichia coli. A neutron small-angle scattering study of hybrids composed of protonated and deuterated protomers

Escherichia coli phenylalanyl-tRNA synthetase is a tetrameric protein composed of two types of protomers. In order to resolve the subunit organization, neutron small-angle scattering experiments have been performed in different contrasts with all types of isotope hybrids that could be obtained by reconstituting the alpha 2 beta 2 enzyme from the protonated and deuterated forms of the alpha and beta subunits. Experiments have been also made with the isolated alpha promoter. A model for the alpha 2 beta 2 tetramer is deduced where the two alpha promoters are elongated ellipsoids (45 x 45 x 160 A3) lying side by side with an angle of about 40 degrees between their long axes and where the two beta subunits are also elongated ellipsoids (31 x 31 x 130 A3) with an angle of 30 degrees between their axes. This model was obtained by assuming that the two pairs of subunits are in contact in an orthogonal manner and by taking advantage of the measured distance between the centers of mass of the alpha 2 and beta 2 pairs (d = 23 +/- 2 A).     

Crystallohydrodynamics for solving the hydration problem for multi-domain proteins: open physiological conformations for human IgG.

Hydrodynamic methods provide a route for studying the low-resolution conformation--in terms of time-averaged spatial orientation of the Fab' and Fc domains relative to each other--of the human IgG subclasses, IgG1, IgG2, IgG3 and IgG4 in the environment in which many exist naturally---a solution. Representative modelling strategies are now available using 'shell-bead' or 'shell' modelling of the surface of the molecules with the size-independent programme SOLPRO [J. Garcia de la Torre, S.E. Harding, B. Carrasco, Eur. Biophys. J. 28 (1999) 119-132]. The shell model fits to the equivalent inertial surface ellipsoids of the published crystal structures for the Fab' and Fc domains of IgG are made and an apparent hydration delta(app) of 0.51g/g for Fab' and 0.70 g/g for the glycoprotein Fc are obtained, which yield an average value of (0.59+/-0.07) g/g for the intact antibody (2 Fab'+1 Fc). The relative orientations of these domains for each of the IgG subclasses is then found (using where appropriate a cylindrical hinge) from SOLPRO by modelling the Perrin function, P (i.e. 'frictional ratio due to shape') using this delta(app) and experimentally measured sedimentation coefficients. All the IgG subclasses appear as open, rather than compact structures with the degree of openness IgG3>IgG1>(IgG2, IgG4), with IgG3 and IgG1 non-coplanar. The hingeless mutant IgGMcg, with s degrees (20,w) approximately 6.8 S yields a coplanar structure rather similar to IgG2 and IgG4 and consistent with its crystallographic structure. The extension of this procedure for representing solution conformations of other antibody classes and other multi-domain proteins is indicated.     

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.     

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.     

Structure and assembly of turnip crinkle virus. II. Mechanism of reassembly in vitro

Dissociation of turnip crinkle virus (TCV) at elevated pH and ionic strength produces free dimers of the coat protein and a ribonucleoprotein complex that contains the viral RNA, six coat-protein subunits, and the minor protein species, p80 (a covalently linked coat-protein dimer). This "rp-complex" is stable for several days in high salt at pH 8.5. Reassembly of TCV can be accomplished under physiological conditions, using isolated coat protein and either rp-complex or protein-free RNA. If rp-complex is used in reassembly, the same subunits remain bound to RNA on subsequent dissociation; if free RNA is used, rp-complex is regenerated. In both cases, the assembly is selective for viral RNA in competition experiments with heterologous RNA. Electron microscopy shows that assembly proceeds by continuous growth of a shell from an initiating structure, rather than by formation of distinct intermediates. We suggest that rp-complex is the initiating structure, suggest a model based on the organization of the TCV particle, and propose a mechanism for TCV assembly.     

Distribution of bacteria in the velocity gradient centrifuge.

Cells in different parts of the cell cycle can be separated by brief centrifugation in a density stabilized gradient: the Mitchison-Vincent technique. The position of a cell in the tube depends upon its size, shape, and density, upon the gradients of density, viscosity, and centrifugal force through which it sediments, and upon time. A program to compute the velocities and integrate the velocity profile for particles of a particular size class is presented. Because enteric bacteria are a form intermediate between right cylinders and prolate ellipsoids of revolution, the program uses values for the frictional coefficient intermediate between those calculated for ellipsoids and for cylinders. The formula f=6pietab(a/b)1/2 possesses this property and because of its simplicity greatly speeds the calculations. A second program computes the distribution of masses and then of sedimentation constants for a bacterial population, expressed either as a frequency distribution or as total mass per s-class. The effect of the known variation in cell size at division is included in these calculations, which apply to organisms undergoing balanced, asynchronous growth in which mass increase is proportional to cell size. The two programs in conjunction compute the mass or cell-number profile in an arbitrary gradient. The programs have been used to design gradients to maximize the resolution of the technique.     

Generalization of the Spiegler-Kedem-Katchalsky frictional model equations of the transmembrane transport for multicomponent non-electrolyte solutions.

The Spiegler-Kedem-Katchalsky frictional model equations of the transmembrane transport for systems containing n-component, non-ionic solutions is presented. The frictional interpretation of the phenomenological coefficients of membrane and the expressions connecting the practical coefficients (Lp, sigma i, omega ij) with frictional coefficients (fij) are presented.     

Ice friction during speed skating.

During speed skating, the external power output delivered by the athlete is predominantly used to overcome the air and ice frictional forces. Special skates were developed and used to measure the ice frictional forces during actual speed skating. The mean coefficients of friction for the straights and curves were, respectively, 0.0046 and 0.0059. The minimum value of the coefficient of ice friction was measured at an ice surface temperature of about -7 degrees C. It was found that the coefficient of friction increases with increasing speed. In the literature, it is suggested that the relatively low friction in skating results from a thin film of liquid water on the ice surface. Theories about the presence of water between the rubbing surfaces are focused on the formation of water by pressure-melting, melting due to frictional heating and on the 'liquid-like' properties of the ice surface. From our measurements and calculations, it is concluded that the liquid-like surface properties of ice seem to be a reasonable explanation for the low friction during speed skating.     

A micelle model for the sedimentation behavior of bovine beta-casein

The monomer-single polymer model of G.A. Gilbert (Disc. Faraday Soc. 20 (1955) 68) for moving boundary sedimentation has been used by Payens and colleagues to explain the observed results for bovine caseins, and by Harrington and colleagues to explain the observed results for myosin fibrils. Electron microscope pictures of Buchheim and Schmidt have subsequently revealed micellar beta-casein in the form of slightly elongated or spherical particles having a bimodal size distribution, but with a broad range of particle sizes, at concentrations not too far above the critical micelle concentration. The equilibrium properties of a broadly distributed micellar system can be fitted by the shell model developed by one of us, and in the present article, the shell model is extended to predict the moving boundary sedimentation behavior of such a system. The observed sedimentation patterns, as well as the critical concentration predictions of the monomer-single polymer Gilbert sedimentation model, are satisfactorily described with the present model, based on a continuous distribution of intermediates between monomers and the largest possible spherical micelles. For one example considered, the predicted frequency distribution of molecular weight is in qualitative agreement with the frequency distribution of particle volume found by Buchheim and Schmidt.     

Atomic force microscopy investigation of isolated virions of murine leukemia virus

Virions of mouse leukemia virus spread on glass substrates were visualized by atomic force microscopy. The size distribution mode was 145 nm, significantly larger than that for human immunodeficiency virus particles. The distribution of particle sizes is broad, indicating that no two particles are likely identical in content or surface features. Virions possess knoblike protrusions, which may represent vestiges of budding from cell membranes. Particles which split open allowed imaging of intact cores with diameters of 65 nm. They also permitted estimation of viral shell thickness (35 to 40 nm) and showed the presence of a distinct trough between the shell and the core surface.