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.     

Orientational exchange approach to fluorescence anisotropy decay.

Fluorescence depolarization is a powerful technique in resolving dynamics of molecular systems. Data obtained in fluorescence depolarization experiments are highly complex. Mathematical models for analyzing data from depolarization due to rotational motion have been largely based on the rotational diffusion equation. These results have been verified by Monte Carlo simulations. It has been implicitly stated that a 90 degrees jump model between predefined orientations such as presented by G. Weber (1971. J. Chem. Phys. 55:2399-2411) should, for the specific case of fluorescence depolarization, give the same answer as the diffusion equation. Since the highly symmetric cases considered by G. Weber gave the same result as the diffusion equation, it has been desirable to use this method in cases where depolarization arises from both discrete processes and rotational diffusion. We have derived, in a compartmental formalism, the general result for excitation and emission dipoles not necessarily coincident with any of the principal rotational axes of the fluorophore from this exchange model, and have found it to be different from that of the diffusion equation approach. We have also verified this difference with a Monte Carlo simulation of our exchange model. This derivation allows us to define the limits of validity of the 90 degrees exchanges to model rotational diffusion. Also, for systems where movements may be jumps between a few preferred orientations, the actual physical mechanism of depolarization may not be accurately represented by continuous diffusion. The compartmental formalism developed here can be used to easily combine rotational motions with discrete position jumps or other level kinetics.(ABSTRACT TRUNCATED AT 250 WORDS)     

Effect of hydrodynamic interactions on the diffusion of integral membrane proteins: diffusion in plasma membranes.

Tracer diffusion coefficients of integral membrane proteins (IMPs) in intact plasma membranes are often much lower than those found in blebbed, organelle, and reconstituted membranes. We calculate the contribution of hydrodynamic interactions to the tracer, gradient, and rotational diffusion of IMPs in plasma membranes. Because of the presence of immobile IMPs, Brinkman's equation governs the hydrodynamics in plasma membranes. Solutions of Brinkman's equation enable the calculation of short-time diffusion coefficients of IMPs. There is a large reduction in particle mobilities when a fraction of them is immobile, and as the fraction increases, the mobilities of the mobile particles continue to decrease. Combination of the hydrodynamic mobilities with Monte Carlo simulation results, which incorporate excluded area effects, enable the calculation of long-time diffusion coefficients. We use our calculations to analyze results for tracer diffusivities in several different systems. In erythrocytes, we find that the hydrodynamic theory, when combined with excluded area effects, closes the gap between existing theory and experiment for the mobility of band 3, with the remaining discrepancy likely due to direct obstruction of band 3 lateral mobility by the spectrin network. In lymphocytes, the combined hydrodynamic-excluded area theory provides a plausible explanation for the reduced mobility of sIg molecules induced by binding concanavalin A-coated platelets. However, the theory does not explain all reported cases of "anchorage modulation" in all cell types in which receptor mobilities are reduced after binding by concanavalin A-coated platelets. The hydrodynamic theory provides an explanation of why protein lateral mobilities are restricted in plasma membranes and why, in many systems, deletion of the cytoplasmic tail of a receptor has little effect on diffusion rates. However, much more data are needed to test the theory definitively. We also predict that gradient and tracer diffusivities are the same to leading order. Finally, we have calculated rotational diffusion coefficients in plasma membranes. They decrease less rapidly than translational diffusion coefficients with increasing protein immobilization, and the results agree qualitatively with the limited experimental data available.     

Intensity-fluctuation spectroscopy and tRNA conformation. II. Changes of size and shape of tRNA in the melting process

We have measured the translational (D^T) and rotational (D^R) diffusion coefficients of bulk tRNA from baker's yeast during the thermal unfolding process by means of photon-correlation spectroscopy. It should be noted that our estimate of the rotational diffusion coefficient represented, for the first time, measurements on a small macromolecule in solution by the photoelectron time-of-arrival technique with a delay-time resolution of 1 nsec. The melting curves expressed in terms of δD^T vs temperature were consistent with the literature data in revealing the melting steps and their dependence on NaCl concentration. Additionally, it was possible to prove the existence of an intermediate, more compact structure during the initial steps of the thermal unfolding process. We found that the temperature ranges over which this intermediate structure appears depend strongly on salt concentration. By utilizing both translational and rotational diffusion coefficients and Perrin's equations for ellipsoids of revolution, we have computed the values of the equivalent length and width of tRNA molecules in solution at four different temperatures for NaCl concentrations of 0.2, 0.5, and 1M. The approximate model of ellipsoids of revolution also permits us to obtain an estimate of the radius of gyration, which is in very good agreement with literature data measured by means of small-angle x-ray scattering. Furthermore, we have measured the shape and size changes of tRNA with varying NaCl concentrations at room temperatures (25°C). The molecule becomes smaller and more spherical when NaCl concentration increases. As a result of partial melting at 70°C, the macromolecule is surprisingly elongated with an approximate axial ratio of 8:1 and has dimensions of about 180/22Å. Such information on conformational changes by a simultaneous determination of rotational and translational diffusion coefficients illustrates the potential of this approach, not available by other methods.     

Rotational diffusion coefficients of a small,spherical subunit flexibly tethered to a larger sphere

The rotational diffusion coefficients of a small spherical particle, which is flexibly anchored to the surface of a much larger sphere, are calculated using the hydrodynamic theory of segmentally flexible particles. The model is intended for representing the rotational mobility of a small residue or chromophore in the surface of a globular macromolecule. The coefficients are found to be essentially independent, or to vary slowly with the relative dispositions of the spheres. They are also insensitive to the size ratio when this ratio is high enough. These findings support the use of an approximative treatment proposed by Wegener in which the small conformation dependence is averaged out. The resulting averages are tentatively used in the Lipari-Szabo model for restricted rotational diffusion in a cone. It is concluded that the rotational relaxation of the small sphere has three components: (i) a torsional rotation with the same diffusion coefficient as the free sphere; (ii) a perpendicular wobbling with a diffusion coefficient several (five in a typical case) times smaller; and (iii) an overall rotation of the whole macromolecule, that will appear in a much longer time scale if the two spheres have quite distinct sizes.     

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.     

Time-dependent rotational rates of excited fluorophores. A linkage between fluorescence depolarization and solvent relaxation

It is generally assumed that the rotational diffusion coefficients of fluorophores are independent of time subsequent to excitation, and that the rotational diffusion coefficients of the ground and the excited states are the same. We now describe a linkage between the extent of solvent relaxation and the rate of fluorescence depolarization. Specifically, if a fluorophore displays time-dependent solvent relaxation it may also show a time-dependent decrease in its rotational rate. A decreased rate of rotation could result from the increased interaction with polar solvent molecules which occurs as a result of solvent relaxation. The decays of anisotropy predicted from our model closely mimic those often observed for fluorophores which are bound to macromolecules. For example, the decays are more complex than a single exponential, and the time-resolved anisotropy can display a limiting value which does not decay to zero. The effect of solvent relaxation upon the rates of rotational diffusion is expected to be most dramatic for solvent-sensitive fluorophores in a viscous environment. These conditions are frequently encountered for fluorophore-macromolecule complexes. Consideration of the linkage between solvent relaxation and rotational diffusion leads to two unusual predictions. First even spherical fluorophores in an isotropic environment could display multi- or nonexponential decays of fluorescence anisotropy. Secondly, for the special case in which the fluorophore dipole moment decreases upon excitation, the theory predicts that the anisotropy decay rate may increase with time subsequent to pulsed excitation. The predictions of this theory are consistent with published data on the effects of red-edge excitation upon the apparent rotational rates of fluorophores in polar solvents.     

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.     

Transient electric birefringence of T-even bacteriophages. I. T4B in the absence of tryptophan and fiberless T4 particles

Measurements of the electric birefringence of suspensions of T4B in the absence of tryptophan and of fiberless T4 particles show that both kinds of particles are hydrodynamically equivalent. Their rotational diffusion coefficients corrected to 25°C and water viscosity (D_25,w) are 280 ± 9 sec^?1 and 295 ± 10 sec^?1, respectively. These corrected rotational diffusion coefficients are almost independent of buffer concentration and temperature. The sedimentation coefficient (s_20,w) of T4 B is equal to 1023 ± 12 S, a value which is likewise independent of buffer concentration. By analysis of the field strength dependence of the steady-state birefringence and by reversing pulse experiments it could be shown that the orientation in an electric field is largely due to a permanent dipole moment. This dipole moment is somewhat dependent on buffer concentration and amounts to about 24,000 debye for T4B and 95,000 debye for fiberless T4. An approximate calculation shows that the difference in dipole moment may be ascribed to positive charges on the fiber tip (at least ten per fiber), to negative charges along the fiber or (and) positive charges on the fiberless particle at those places where the fibers are attached in normal particles.     

Intensity fluctuation spectroscopy and transfer RNA conformation. III. Influence of NaCl concentration on the size and shape of the initially salt-free tRNA in solution.

The influence of sodium ion concentration in solution on the initially salt-free conformation of bulk tRNA from baker's yeast has been investigated by means of photon correlation spectroscopy. From the measured values of translational (D^T) and rotational (D^R) diffusion coefficients, the semiaxes of an ellipsoid of revolution, which are hydrodynamically equivalent to the tRNA molecule, were calculated for tRNA solutions in pure H₂O as well as in 0.005, 0.1, 0.5M NaCl and 0.01M MgCl₂ solutions at pH 4.2 and 7.5. These data, combined with our previous studies, suggested a model which describes the formation of an ordered tRNA structure due to increasing NaCl concentrations. Furthermore, we have obtained information concerning intermolecular interactions between tRNA molecules in solution. In low-salt or salt-free tRNA solutions, we detected in the linewidth distribution function an extra-fast component which can be attributed as possibly due to charge fluctuations related to the reaction of ionization of organic bases. In our light-scattering linewidth measurements, we do not see fluctuations of charged and uncharged states directly as concentration fluctuations. Rather, we postulate a modulation of long-range intermolecular electrostatic interactions between the tRNA molecules due to such charge fluctuations. It is this modulation which is related to the fast component of the time correlation function at finite concentrations. A quantitative theory is needed to provide a more definitive explanation of the dynamical behavior of tRNA in salt-free or low-salt solutions.     

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.     

Comparative study on fibers isolated from four R-type pyocins, phage-tail-like bacteriocins of Pseudomonas aeruginosa

Five R-type pyocins have been reported which are almost identical with one another in their morphology and subunit composition, though distinct in receptor-binding specificity. We isolated fibers from pyocin R2, R3, and R4 by essentially the same procedure as used in our previous isolation of pyocin R1 fiber with unimpaired receptor-binding ability. All the isolated fibers including R1 fiber were indistinguishable from one another, in terms of electron microscopic observation and subunit composition analysis by SDS gel electrophoresis. They consisted mainly of Subunit No. 2 (Mw 71,000) and No. 9 (31,000) proteins. Although Subunit No. 9 protein in every fiber was susceptible to trypsin and afforded a fragment with the same molecular weight (about 19,000) detectable in the SDS gel, Subunit No. 2 protein was cleaved with trypsin only after the fiber had been treated with an organomercurial, 4-(p-sulfophenylazo)-2-mercuriphenol. The cleavage of Subunit No. 2 protein proceeded to give several fragments with molecular weights ranging from 64,000 to 34,000, and the fragmentation patterns were electrophoretically distinct at least among R1 fiber, R3 fiber, and others (R2 and R4 fibers). The results indicate that Subunit No. 2 proteins of these fibers are different from one another in the structure surrounding trypsin-susceptible peptide bonds. Immunological investigations with anti-R1 fiber antibodies provided some additional information on the difference among R-type pyocins at the fiber level.     

Transient electric birefringence of T-even bacteriophages. IV. T2L0 and T6 with extended tail fibers

Transient electric birefringence measurements of the bacteriophages T2L0 and T6 were performed under such conditions that the tail fibers are extended. The data obtained are compared to previously reported data for T4B. For all T-even phages the degree of extension of the tail fibers is a function of pH, ionic strength, and temperature. For T4B, much higher ionic strengths are needed than for T2L0 and T6 to accomplish complete tail-fiber extension. The rotational diffusion coefficients of the phages with fully extended fibers are equal to 120 ± 3 sec^?1, 132 ± 5 sec^?1, 157 ± 4 sec^?1 for T2L0, T4B, and T6, respectively. The respective optical anistropies are ? (2.66 ± 0.05) × 10^?4, and ? (3.07 ± 0.15) × 10^?4. The differences in the rotational diffusion coefficient and optical anisotropy arise because the conformation of the fully extended tail fibers is different for the three phages. The tail fibers of T2L0 project further into the solution (away from the head) than do those of T4B and T6. The apparent permanent dipole moments of T2L0 and T6 decrease with increasing ionic strength. This decrease is caused by the screening of the surface charges on the phage body by the counter-ions in the solution. The biological relevance of this decrease is illustrated by the fact that the adsorption rate of T6 phages to E. coli B bacteria shows a similar dependence of ionic strength. Evidence is pressented that the tail fibers may move more or less independently of the phage body when an electric field is applied to the suspension.     

Structural analysis of G-DNA in solution: A combination of polarized and depolarized dynamic light scattering with hydrodynamic model calculations.

Specific intra- and intermolecular quadruplex conformations of model G-DNA oligonucleotides have been identified from their translational and rotational diffusion coefficients in aqueous solution. The transport properties were determined by polarized and depolarized dynamic light scattering. A comparison with hydrodynamic model calculations provides detailed information about the size and shape of the molecules and allows one to distinguish between alternative intra- and intermolecular association. The potential of this combination of methods to elucidate biomolecular structures in solution, to characterize conformational changes, and follow intermolecular interaction processes due to a response to external stimuli has been discussed.     

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)].     

Hydrodynamic properties of proteoglycan subunit from bovine nasal cartilage. Self-association behavior and interaction with hyaluronate studied by laser light scattering

Measurements of translational diffusion coefficients by quasielastic laser light scattering, sedimentation coefficients, and intrinsic viscosities at zero shear of proteoglycan subunit fraction A1-D1-D1 isolated from bovine nasal septa are reported. Molecular weights and hydrodynamic dimensions are compared with those expected on the basis of structural models previously proposed. Comparison of the concentration dependence of the diffusion coefficient in the presence of NaCl and GdnHCl leads to the conclusion that significant self-association behaviour of subunit occurs in the absence of GdnHCl. In the absence of added salt, anomalous nonlinear concentration dependence of D_t estimated from wide-angle light-scattering experiments is observed. In addition, D_t apparently becomes angle dependent. These results are interpreted in terms of the perturbation of normal translational diffusion of the monomer by strong repulsive intermolecular interactions due to the combined effects of long-range electrostatic forces and macromolecular congestion at higher concentrations. By carrying out experiments at small scattering angles, it is possible to determine D for proteoglycan subunit in the absence of supporting electrolyte. Titration of a dilute solution of subunit with hyaluronic acid results in a sigmoidal behaviour of the Stokes radius, indicating the formation of complexes of higher molecular weight results from the noncovalent association of proteoglycan subunits with hyaluronate. Observation of D_t appears to provide a useful method for studying the proteoglycan subunit–hyaluronate interactions.     

Kinetics of filament bundling with attractive interactions

We study the kinetics of filament bundling by variable time-step Brownian-dynamics simulations employing a simplified attractive potential based on earlier atomic-level calculations for actin filaments. Our results show that collisions often cluster in time, due to memory in the random walk. The clustering increases the bundling opportunities. Small-angle collisions and collisions with short center-to-center distance are more likely to lead to bundling. Increasing the monomer-monomer attraction decreases the bundling time to a diffusional limit, which is determined by the capture cross-section and diffusion coefficients. The simulations clearly show that the bundling process consists of two sequential phases: rotation, by which two filaments align parallel to each other; and sliding, by which they maximize their contact length. Whether two filaments bundle or not is determined by the competition between rotation to a parallel state and escape. Increasing the rotational diffusion coefficient and attraction enhances rotation; decreasing attraction and increasing the translational diffusion coefficients enhance escape. Because of several competing effects, the filament length only affects the bundling time weakly.     

Monte Carlo approach to the analysis of the rotational diffusion of wormlike chains

A Monte Carlo analysis is presented which establishes a relationship between the rotational diffusion coefficients and the flexibility (persistence length, P) of short, wormlike chains. The results of this analysis are presented in terms of experimentally observable quantities; namely, the rotational relaxation times for the field-free decay of optical anisotropy. The pertinent theoretical quantity is R, defined as the ratio of the longest rotational relaxation time of a wormlike chain to the transverse rotational relaxation time of a rigid cylinder having the same axial length (L) and segmental volume. R, so defined, is essentially independent of the axial ratio of the cylinder for any value of L/P within the range of validity of the present analysis (axial ratio > 20; 0.1 < L/P < 5). It is pointed out that P can be determined with reasonable accuracy even in the absence of a precise knowledge of the local hydrodynamic radius of the chain.