Hydrodynamics of macromolecular complexes. II. Rotation

We have used the modified Oseen hydrodynamic interaction tensor along with iterative numerical solution of the coupled hydrodynamic interaction equations to calculate the rotational diffusion coefficients of macromolecular complexes composed of nonidentical spherical subunits. For the one structure, a prolate ellipsoid of revolution, for which exact solutions are available, a subunit model with the same length and volume gives asymptotic agreement with the Perrin equations. Other structures considered include plane polygonal rings, lollipops, and dumbbells.     

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. 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.     

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.     

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 particles with segmental flexibility. I. Hydrodynamic resistance and diffusion coefficients of a freely hinged particle

Expressions are derived for the hydrodynamic resistance tensor and the diffusion tensor of a particle consisting of two rigid subunits connected by a free hinge. No restrictions are placed on the shapes of the subunits. The resistance tensor is obtained by using two independent approaches: first, from the Rayleigh dissipation function and, second, from an examination of the generalized forces for the appropriate seven-dimensional coordinate system. For the derivation of the generalized Einstein equation connecting the diffusion and resistance tensors, the Brownian motion is treated as a stochastic process. That derivation is based on the assumption that the restoring force for bending is negligible, and the Einstein relation holds instantaneously only if that assumption is true. The relationship between these tensors and the macroscopically observable parameters is discussed, and it is shown that the separate measurement of resistance and diffusion coefficients can be used to detect macromolecular flexibility. One example is treated, the diffusion of a particle composed of two long rods joined at a free hinge. Those calculations are carried out with the first-order assumption of negligible hydrodynamic interactions between the subunits. For the hinged rod, the bending degree of freedom produces a 34% increase in the translational diffusion coefficient over that of a stiff rod of the same total length, while the rotational diffusion coefficient about the axis perpendicular to the plane of bending is increased by 125%.     

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.     

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.     

Diffusion coefficients of segmentally flexible macromolecules with two spherical subunits

The translational and rotational diffusion coefficients have been calculated for a simple, segmentally flexible model: the hinged dumbbell (HD). In the HD, two spherical subunits are attached to an universal joint by means of frictionless connectors. In addition to the case in which hydrodynamic interactions are neglected (NI), we have also considered two more cases, including hydrodynamic interaction by means of the Kirkwood-Riseman approximate treatment (KR) and using accurate procedure based in the series expansions for the two-sphere diffusion tensor (SE). Expressions for the friction coefficients of the HD are given for the three cases, and the diffusion coefficients are evaluted inverting the 9 × 9 resistance matrix, for two HDs with different dimensions. The KR treatment, which includes a contribution from the finite volume of the subunits, is shown to be an excellent approximation to the more rigorous procedure. In the NI case for rotation, the various coefficients present different deviations with respect to the SE results. A rough estimate of the errors of the NI relaxation times indicates that they may be smaller than 15% for a HD with identical beads. However, the influence of hydrodynamic interaction should be more important for the rotational diffusivity of a small sphere attached to a larger one. The error of the NI result for the translational diffusion coefficient is of about 25% for the two HDs.     

Calculation of hydrodynamic properties of macromolecular bead models with overlapping spheres

For the calculation of hydrodynamic properties of rigid macromolecules using bead modelling, models with overlapping beads of different sizes are used in some applications. The hydrodynamic interaction tensor between unequal overlapping beads is unknown, and an oversimplified treatment with the Oseen tensor may introduce important errors. Here we discuss some aspects of the overlapping problem, and explore an ad hoc form of the interaction tensor, proposed by Zipper and Durchschlag. We carry out a systematic numerical study of the hydrodynamic properties of a two-spheres model, showing how the Zipper-Durchschlag correction removes efficiently the numerical instabilities, and predicts the correct limits. Received: 15 February 1999 / Revised version: 29 April 1999 / Accepted: 11 May 1999     

A physical model of ATP-induced actin-myosin movement in vitro.

The nature of the mechanism limiting the velocity of ATP-induced unidirectional movements of actin-myosin filaments in vitro is considered. In the sliding process two types of "cyclic" interactions between myosin heads and actin are involved, i.e., productive and nonproductive. In the productive interaction, myosin heads split ATP and generate a force which produces sliding between actin and myosin. In the nonproductive interaction "cycle," on the other hand, myosin heads rapidly attach to and detach from actin "reversibly," i.e., without splitting ATP or generating an active force. Such a nonproductive interaction "cycle" causes irreversible dissipation of sliding energy into heat, because the myosin cross-bridges during this interaction are passive elastic structures. This consideration has led us to postulate that such cross-bridges, in effect, exert viscous-like frictional drag on moving elements. Energetic considerations suggest that this frictional drag is much greater than the hydrodynamic viscous drag. We present a model in which the sliding velocity is limited by the balance between the force generated by myosin cross-bridges in the productive interaction and the frictional drag exerted by other myosin cross-bridges in the nonproductive interaction. The model is consistent with experimental findings of in vitro sliding, including the dependence of velocity on ATP concentration, as well as the sliding velocity of co-polymers of skeletal muscle myosin and phosphorylated and unphosphorylated smooth muscle myosins.     

Hydrodynamic properties of human cervical-mucus glycoproteins in 6M-guanidinium chloride.

Cervical mucins and fragments thereof were studied by sedimentation-velocity, rotatory viscometry and laser light-scattering performed as photon-correlation spectroscopy as well as low-angle total-intensity measurements. The Mr of the whole mucins is 10 X 10(6)-15 X 10(6), whereas fragments obtained after reduction of disulphide bonds ('subunits') have Mr 2.1 X 10(6)-2.9 X 10(6), depending on the method used. Subsequent trypsin digestion of subunits afforded glycopeptides with Mr approx. 0.4 X 10(6). The high frictional ratio for the whole mucins is interpreted as a large degree of expansion. The Stokes radius calculated from the diffusion coefficient is approx. 110nm for the whole mucins, which is in agreement with that estimated from the radius of gyration (130nm) by using the concept of the equivalent hydrodynamic sphere. The ratio of the concentration-dependence parameter for the reciprocal sedimentation coefficient (Ks) to the intrinsic viscosity ( [eta] ) for the whole mucins is 1.42, suggesting that the individual macromolecule occupies a spheroidal domain in solution. The relationship between [eta] and Mr for whole mucins, subunits and T-domains suggests that they are linear flexible macromolecules behaving as somewhat 'stiff' random coils. This conclusion is supported by the relationships between the sedimentation coefficients, the diffusion coefficients and the Mr. The hydrodynamic behaviour of the mucins is thus close to that expected for coiling macromolecules entrapping a lot of solvent, which is consistent with the postulated polymeric structure.     

Dielectrophoretic forces and potentials induced on pairs of cells in an electric field.

A combined numerical/experimental study is reported of the membrane potentials and dielectrophoretically induced forces between cells, membrane pressures, and velocity of attraction of cells under the influence of an electric field. This study was designed to explore electrical and mechanical effects produced by a field on cells in close proximity or undergoing electrically induced fusion. Laplace's equation for pairs of membrane-covered spheres in close proximity was solved numerically by the boundary element method, and the electrically induced forces on the cells and between cells were obtained by evaluating the Maxwell stress tensor. The velocity of approach of erythrocyte ghosts or fused ghosts in a 60-Hz field of 6 V/mm was measured experimentally, and the data were interpreted by using Batchelor's theory for hydrodynamic interaction of hard spheres. The numerical results show clearly the origin of the dielectrophoretic pressures and forces in fused and unfused cells and the effects of a nearby cell on the induced membrane potentials. The experimental results agree well with predictions based on the simple electrical model of the cell. The analysis shows the strong effect of hydrodynamic interactions between the cells in determining their velocity of approach.     

Precursors of distinct size for chicken αA, αD and β globin mRNAs

The reconstitution of bovine cardiac troponin from its subunits has been investigated using hydrodynamic techniques. Gel filtration (Sephacryl S-300) and sedimentation velocity experiments indicate that troponin-C and troponin-I from a stable binary complex (1:1 mole ratio) with an apparent Stokes' radius of 36 Å (frictional ratio = 1.6). Troponin-C and troponin-T do not interact significantly while troponin-I and troponin-T undergo partial complex formation. The effect of subunit ratio on the reconstitution of whole troponin has been examined by SDS—polyacrylamide gel electrophoresis and gel filtration and the results suggest that native troponin contains the subunits in an equimolar ratio.     

Resistance to the Brownian movement of red blood cells on flat horizontal surfaces

The movements of red blood cells (RBC), suspended in plasma, on plastic, glass, rhodium metal plate, siliconized glass, and siliconized rhodium were recorded on cinéfilm and analyzed. Values for the drag coefficient were calculated, using Einstein's theory of Brownian movement, and compared with the theoretical Stokes' hydrodynamic drag. The difference between the computed and Stokes' values gave the frictional coefficient or resistance resulting from the interaction of the cells, with the test surface. Of the three uncoated test surfaces, plastic was found to have the least interaction with the RBC. The frictional coefficient for plastic was found to be 1.75×10⁻⁷ N s m⁻¹ compared with a value of 2.82×10⁻⁷ N s m⁻¹ for rhodium metal, which had the largest interaction. Upon siliconization of the test surfaces, the interaction decreased by 40%. Reduction in the pH of the suspending plasma increased the interaction between the cells and the uncoated test surfaces, but the pH effect of diminished when the surfaces were siliconized.     

Protein friction exerted by motor enzymes through a weak-binding interaction.

Recently Vale et al. (1989, Cell 59, 915-925.) reported an observation of the one-dimensional Brownian movement of microtubules bound to flagellar dynein through a weak-binding interaction. In this study, we propose a theoretical model of this phenomenon. Our model consists of a rigid microtubule associated with a number of elastic dynein heads through a weak-binding interaction at equilibrium. The model implies that (1) the Brownian motion of the microtubule is not directly driven by the atomic collision of the solvent particles, but is driven by the thermally-generated structural fluctuations of the dynein heads which interact with the microtubule; (2) dynein heads through a weak-binding interaction exert a frictional drag force on the sliding motion of the microtubule and the drag force is proportional to the sliding velocity the same as in hydrodynamic viscous friction. This protein friction, with such viscous-like characteristics, may well play a role as a velocity-limiting factor in the normal ATP-induced sliding movement of motile proteins.     

Interaction between immobilized and soluble protein subunits. Analysis and applications

A distinctive property of oligomeric and self-associating proteins is the high specificity of the subunit recognition process. Protein subunits immobilized covalently on a solid matrix maintain this characteristic and are able to bind soluble subunits of the same or a closely related protein under conditions that allow the establishment of a finite association/dissociation equilibrium. The basic theory for studying the immobilized-soluble subunit interaction is presented together with the methodology for a proper protein immobilization. Specific examples are discussed to illustrate on the one hand benefits and caveats of using immobilized protein subunits to measure interaction constants, and on the other preparative applications of subunit affinity columns.     

Diffusion coefficients of segmentally flexible macromolecules with two subunits: a study of broken rods

A general treatment for the solution dynamics of segmentally flexible macromolecules having two subunits is presented. Bead modeling allows for a complete inclusion of hydrodynamic interactions in this treatment. The finite size of the beads is also considered, so that it is therefore possible to account properly for torsional motions of the subunits. Expressions for the components of the resistance matrix are derived. From them, the translational and rotational diffusion coefficients can be calculated. Distinction is made between hinged macromolecules, whose only internal motion is bending, and swivel-jointed macromolecules, for which torsions of the subunits are also allowed. Numerical results are presented for broken rods with the two types of flexibility. The effects of hydrodynamic interaction between arms of broken rods are about 25% for translation and under 10% for rotation. These findings give support to the treatments of Harvey, Wegener, and co-workers in which interactions were neglected. The rotational dynamics of hinged and swivel-jointed rods are compared. Although there are differences in the short-time behavior, the longest relaxation time is the same for the two cases. Finally, the validity of Wegener's rotational diffusion constants is discussed.     

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.     

Influence of hydrodynamic stress on the frictional drag of biofouling communities

The role of hydrodynamic wall shear stresses on the development of the fouling community structure and resulting frictional drag were examined using a commercially available fouling release coating. Immersed test panels were exposed to three different hydrodynamic treatments, one static and two dynamic (corresponding to an estimated wall shear stress of 7.0 and 25.5 Pa). The drag of the panels was measured in a hydrodynamic test chamber at discrete time intervals over 35 days. The fouling community composition on the static panels was significantly different from the organisms observed on the dynamic panels. Despite different fouling community composition, the drag forces measured on the panels were very similar. This suggests that the frictional drag of low form and soft fouling communities are similar and that there may be a stepwise increase in frictional drag associated with the presence of mature calcareous organisms.