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.     

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.     

Transport properties of rigid, symmetrical oligomeric structures composed of prolate, ellipsoidal subunits

We have calculated translational and rotational diffusion coefficients and intrinsic viscosities of oligomeric structures composed of n identical subunits having a prolate ellipsoidal shape with axial ratio p. Results are presented for p = 1-6 for a variety of structures with n = 1-6. We compare our results with those obtained by a different modeling procedure, proposed by other workers, in which the monomeric subunit is represented as a string of touching, colinear spheres. If n and an estimate of p are known, the structure of the oligomer can be. in most cases, unambiguously determined by comparison of the experimental oligomer-to-monomer ratios of a given property with the numerical results of this work. As examples of the applicability of our results, we examine the relationship between structure and properties for neurophysin. bovine serum albumin, hemoglobin and phycocyanin.     

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.     

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.     

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.     

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.     

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.     

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.     

Frictional properties of nonspherical multisubunit structures. Application to tubules and cylinders

Methods are described for numerical calculation of the anisotropic components of the translational and rotational friction coefficient tensors and of the intrinsic viscosity for rigid multisubunit structures in dilute solution. The methods apply to assemblies of any shape, provided that translation–rotation coupling is negligible. Application is made to short cylindrical and tubular structures. Anomalous results arise when the Oseen tensor is used to describe the hydrodynamic interaction of the subunits, but these are corrected by use of a modified tensor. Transport coefficients for hollow tubules with typical supramolecular dimensions are found to be nearly the same as those for the corresponding solid cylinders. The Scheraga–Mandelkern equation is found to be useful for the determination of the molecular weights of such structures. For long hollow structures such as microtubules, use of the corresponding solid cylinder or wormlike chain equations should be adequate for interpreting hydrodynamic studies.     

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.     

Frictional properties of multisubunit structures

The translational drag, rotational drag, and intrinsic viscosity of spherical multisubunit structures have been calculated analytically using the Felderhof–Deutch theory of polymer frictional properties. The structures considered were hollow shells, spheres with uniform subunit density, and spheres covered with a subunit layer of different density. Changes in the transport coefficients resulting from the random removal of subunits and from the variation of subunit size are calculated. For the case of the shell, the results agree with the numerical computations of Bloomfield, Dalton, and Van Holde [Biopolymers 5 , 135, 149 (1967)].     

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.     

Urease oligomerizes in a linear pattern: further hydrodynamic evidence from intrinsic viscosity theories and measurement.

Measurements of the intrinsic viscosity of the urease “monomer” and of a mixture of oligomers are shown to lead to the conclusion that urease oligomerizes in a linear fashion, when viewed in the light of several available theories of intrinsic viscosity of rigid protein macromolecules. This conclusion is also consistent with previously reported work on the sedimentation coefficients of the urease oligomers. The use of a detailed subunit structure for the urease “monomer” and oligomers is explored and found to give quantitatively good results for all hydrodynamic properties of all urease oligomers. The realm of validity of the intrinsic viscosity theories proposed by Garcia de la Torre and Bloomfield, by Tsuda, and by Abdel-Khalik and Bird is explored in this context.     

Urea-induced structural changes in chromatin obtained by sedimentation.

Sedimentation coefficients have been determined for fractionated preparations of whole and stripped (depleted of very lysine-rich histones and non-histone proteins) chicken erythrocyte chromatin fragments in 0-10 M urea. Significant differences in urea effects are observed between these preparations; differences which can be interpreted structurally by use of Kirkwood's dynamical theory of the translational frictional coefficient. This type of analysis implies that urea-induced chain-swelling in stripped chromatin is due largely to the urea effect upon the constituent nu-bodies, whereas the much larger swelling observed in whole chromatin appears to involve also the effect of urea upon the region between adjacent nu-bodies.     

The use of sedimentation coefficients to distinguish between models for protein oligomers.

The sedimentation coefficients of proteins are dependent on their sizes, shapes and densities and on the density and viscosity of the solvent. However, when the sedimentation coefficients of an oligomeric protein and its protomer are measured under the same experimental conditions, the ratio of the two coefficients depends only on the protomer shape and the mode of aggregation. This property, which we shall call the sedimentation ratio, therefore provides a way of distinguishing between models for oligomeric proteins. To allow examination of the behaviour of the sedimentation ratio, sedimentation coefficients are calculated for a comprehensive range of protomer shapes and modes of aggregation in hexameric systems using equations derived by Kirkwood. As illustrations of the method the resulting sedimentation ratios are compared with experimental values for insulin and arthroped hemocyanin, which eliminates many of the possible structures for these proteins. When experimental estimates of degree of hydration and molecular dimensions are also considered, all but a group of virtually identical structures are eliminated for the insulin hexamer and a single most likely structure remains for arthropod hemocyanin. The insulin structure is in good agreement with that determined by X-ray crystallography while the hemocyanin hexameric structure is a hexagonal prism formed by the cyclic aggregation of prolate ellipsoids of axial ratio about 2.5 : 1.     

Hydrodynamic properties of a rigid molecule: Rotational and linear diffusion and fluorescence anisotropy

The fluorescence anisotropy of a general rigid body is formally the sum of five exponentials. We show that, to a high degree of approximation, there are relationships between the five time constants. As we define the time constants here, τ₁ ? τ₅, τ₂ ? τ₃, and τ₁^?1 + 3τ₄^?1 ? 4τ₂^?1. In practical cases, at most only three exponentials will be observed, and, of these, only two are independent. Using a numerical integration procedure, Perrin's equations for the rotational and translational diffusion of a general ellipsoid are solved. Rotational friction coefficients, frictional ratio, rotational relaxation times, and the five exponential terms in the fluorescence anisotropy are tabulated as functions of the axial ratios of the ellipsoid. In principle, the three axes of a general ellipsoid may be determined by a simultaneous measurement of the anisotropy and the linear diffusion constant. We examine, and illustrate, the effect of experimental error on such a determination.     

Transport properties of rigid bent-rod macromolecules and of semiflexible broken rods in the rigid-body treatment. Analysis of the flexibility of myosin rod.

The translational diffusion coefficients, rotational relaxation times and intrinsic viscosities of rigid bent rods, composed by two rodlike arms joined rigidly at an angle alpha, have been evaluated for varying conformation using the latest advances in hydrodynamic theory. We have considered semiflexible rods in which the joint is an elastic hinge or swivel, with a potential V(alpha) = 1/2Q alpha 2 with constant Q. Accepting the rigid-body treatment, we calculate properties of broken rods by averaging alpha-dependent values for rigid rods. The results are finally used to interpret literature values of the properties of myosin rod. Q is regarded as an adjustable parameter, and the value fitted is such that the average bending angle of myosin rod is approximately 60 degrees.     

Influence of resonant charge exchange on the viscosity of partially ionized plasma in a magnetic field

The influence of resonant charge exchange for ion-atom interaction on the viscosity of partially ionized plasma embedded in the magnetic field is investigated. The general system of equations used to derive the viscosity coefficients for an arbitrary plasma component in the 21-moment approximation of Grad’s method is presented. The expressions for the coefficients of total and partial viscosities of a multicomponent partially ionized plasma in the magnetic field are obtained. As an example, the coefficients of the parallel and transverse viscosities for the ionic and neutral components of the partially ionized hydrogen plasma are calculated. It is shown that the account for resonant charge exchange can lead to a substantial change of the parallel and transverse viscosity of the plasma components in the region of low degrees of ionization on the order of 0.1.