Hydrodynamic resistance and diffusion coefficients of a freely hinged rod

The seven-dimensional hydrodynamic resistance and diffusion tensors are evaluated for a rod which is freely hinged at its center and immersed in a viscous fluid. The hydrodynamic resistance tensor is first determined at the hinge, then transformed to other points and inverted to obtain the diffusion tensor. Hydrodynamic interactions between rod halves are neglected, which is asymptotically correct for long rods. In the long-rod limit, the diffusion coefficient characterizing translations over macroscopic distances is decreased by 3–6% from that for a rigid straight rod of same total length, while the average end-over-end rotational diffusion coefficient for each rod half is increased 4.67 times.     

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.     

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.     

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.     

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.     

Diffusion coefficients for rigid macromolecules with irregular shapes that allow rotational-translational coupling

We consider six-dimensional diffusion and frictional tensors for a rigid macromolecule immersed in a viscous fluid at low Reynolds number. Our treatment allows for screwlike properties which couple rotational and translational movements. We show that the center of diffusion of a screwlike body can be distinct from its hydrodynamic center of reaction. Symmetry conditions which ensure coincidence are examined. The center of diffusion is found to be the point of a body with the slowest diffusive movements, while rotations about the center of reaction encounter the least average resistance. The macroscopic translational diffusion coefficient is evaluated from a perturbation analysis of the six-dimensional diffusion equation. We show that methodologies which ignore translational–rotational coupling will necessarily underestimate the diffusion rate of screwlike particles. A procedural framework is presented to calculate diffusion coefficients of complicated bodies. As an example we treat a long bent rod.     

A swivel-jointed formalism for segmentally flexible macromolecules and its application to the rotational behavior of myosin

A formalism is presented to represent the hydrodynamic and diffusion behavior of segmentally flexible macromolecules with a high degree of flexibility. The development specifically treats open assemblies of cylindrically symmetric segments joined at their end points by frictionless swivels. Three-dimensional diffusion and frictional tensors that govern the rotations of each segment and the translations of some chosen swivel are established and related by a Stokes–Einstein relation. The dependence of these tensors on the choice of swivel is examined, and simplified expressions are obtained by omitting hydrodynamic interactions between segments. These simplified expressions are used to analyze the rotational behavior of a Y-shaped body in detail. If there is complete flexibility, the rotational motions of each segment reduce, for all practical purposes, to those of a cylindrically symmetric rigid body. Estimates are provided to relate segment rotations to internal motions and restrictions on flexibility. A procedural framework for generating rotational diffusion coefficients for segments of more complicated bodies is applied to a four-segment two-swivel myosin model. Comparison of calculated and experimental relaxation times for myosin and its fragments indicates the head–tail junction is highly flexible, with at least a ±60° angular extent of free bending between heads allowed, while the tail is neither straight and rigid nor highly flexible.     

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.     

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.     

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.     

A rotational diffusion coefficient of the 70S ribosome determined by depolarized laser light scattering.

We have obtained a rotational diffusion coefficient of the 70S ribosome isolated from Escherichia-coli (MRE-600), from the depolarized light scattering spectrum measured by photon correlation spectroscopy. The intensity correlation function of depolarized scattered light contains contributions due to multiple scattered and anisotropy scattered light from the ribosomal particle. We discuss extensively the subtraction procedure used to obtain the rotational correlation from the time from the experimental correlation function. We have also obtained the translational diffusion coefficient from the same sample by determining the polarized correlation function. The hydrodynamic radius determined from the rotational diffusion coefficient is only slightly larger than the radius obtained from the translational diffusion coefficient. Therefore the ribosomal particle has a non-spherical shape. This conclusion, however, could be impaired by the effect of free draining of the ribosome.     

A rotational diffusion coefficient of the 70S ribosome determined by depolarized laser light scattering

We have obtained a rotational diffusion coefficient of the 70S ribosome isolated from Escherichia-coli (MRE-600), from the depolarized light scattering spectrum measured by photon correlation spectroscopy. The intensity correlation function of depolarized scattered light contains contributions due to multiple scattered and anisotropy scattered light from the ribosomal particle. We discuss extensively the subtraction procedure used to obtain the rotational correlation time from the experimental correlation function. We have also obtained the translational diffusion coefficient from the same sample by determining the polarized correlation function. The hydrodynamic radius determined from the rotational diffusion coefficient is only slightly larger than the radius obtained from the translational diffusion coefficient. Therefore the ribosomal particle has a non-spherical shape. This conclusion, however, could be impaired by the effect of free draining of the ribosome.     

Mass Transfer in the Cornea: I. Interacting Ion Flows in an Arbitrarily Charged Membrane

The formalisms of irreversible thermodynamics are used to describe multi-ionic nonconvective flow through an arbitrarily charged membrane. Interactions between oppositely charged ions are included and are measured by a single phenomenological coefficient. The consequent generalized Nernst-Planck flux equations are integrated to yield a relation between the species fluxes and the composition of the solutions bounding the membrane. It is assumed in the derivation that activity coefficient gradients within the membrane and direct interactions between ions of like charge are negligible. Some special cases are examined. To illustrate the use of the final equations, a single membrane separating solutions of differing composition is modeled, and the effect of ion-ion interactions on the membrane potential and the ion fluxes is demonstrated for several values of diffusion current density and membrane charge density.     

Flexibility of duplex DNA on the submicrosecond timescale

Using a site-specific, Electron Paramagnetic Resonance (EPR)-active spin probe that is more rigidly locked to the DNA than any previously reported, the internal dynamics of duplex DNAs in solution were studied. EPR spectra of linear duplex DNAs containing 14-100 base pairs were acquired and simulated by the stochastic Liouville equation for anisotropic rotational diffusion using the diffusion tensor for a right circular cylinder. Internal motions have previously been assumed to be on a rapid enough time scale that they caused an averaging of the spin interactions. This assumption, however, was found to be inconsistent with the experimental data. The weakly bending rod model is modified to take into account the finite relaxation times of the internal modes and applied to analyze the EPR spectra. With this modification, the dependence of the oscillation amplitude of the probe on position along the DNA was in good agreement with the predictions of the weakly bending rod theory. From the length and position dependence of the internal flexibility of the DNA, a submicrosecond dynamic bending persistence length of around 1500 to 1700 A was found. Schellman and Harvey (Biophys. Chem. 55:95-114, 1995) have estimated that, out of the total persistence length of duplex DNA, believed to be about 500 A, approximately 1500 A is accounted for by static bends and 750 A by fluctuating bends. A measured dynamic persistence length of around 1500 A leads to the suggestion that there are additional conformations of the DNA that relax on a longer time scale than that accessible by linear CW-EPR. These measurements are the first direct determination of the dynamic flexibility of duplex DNA in 0.1 M salt.     

Normal mode analysis of human lysozyme: study of the relative motion of the two domains and characterization of the harmonic motion

A normal mode analysis of human lysozyme has been carried out at room temperature. Human lysozyme is an enzyme constituted of two domains separated by an active site cleft, the motion of which is thought to be relevant for biological function. This motion has been described as a hinge bending motion. McCammon et al. have determined the characteristics of the hinge bending motion but they assumed a prior knowledge of the hinge axis. In this work we propose a method which is free from this assumption and determines the hinge axis and root mean square (rms) rotation angle which give the best agreement with the pattern of changes in all the distances between nonhydrogen atoms in the two domains, obtained by the normal mode analysis. The hinge axis we found is notably different from the one previously determined and goes, roughly, through the C alpha 55 and C alpha 76, i.e., it is located at the base of the beta-sheet of the second domain. The rms value for the rotation angle is also twice as large as the previous one: 3.37 degrees. It is shown that this hinge bending motion provides a fairly good approximation of the dynamics of human lysozyme and that the normal mode with the lowest frequency has a dominating contribution to this hinge bending motion. A study of the accessible surface area of the residues within the cleft reveals that the motion does not result in a better exposure to the solvent of these residues. A characterization of the thermally excited state (under the hypothesis of the harmonicity of the potential energy surface) has been done using the concept of topology of atom packing. Under this hypothesis the thermal fluctuations result only in a small change of the topology of atom packing, leading therefore to nearly elastic deformations of the protein.     

The length dependence of translational diffusion, free solution electrophoretic mobility, and electrophoretic tether force of rigid rod-like model duplex DNA

In this work, boundary element modeling is used to study the transport of highly charged rod-like model polyions of various length under a variety of different aqueous salt conditions. Transport properties considered include free solution electrophoretic mobility, translational diffusion, and the components of the "tether force" tensor. The model parameters are chosen to coincide with transport measurements of duplex DNA carried out under six different salt/temperature conditions. The focus of the analysis is on the length dependence of the free solution electrophoretic mobility. In a solution containing 0.04 M Tris-acetate buffer at 25 degrees C, calculated mobilities using straight rod models show a stronger dependence on fragment length than that observed experimentally. By carrying out model studies on curved rod models, it is concluded that the "leveling off" of mobility with fragment length is due, in part at least, to the finite curvature of DNA. Experimental mobilities of long duplex DNA in monovalent alkali salts are reasonably well explained once account is taken of long-range bending and the simplifying assumptions of the model studies.     

Brownian dynamics simulation of rigid particles of arbitrary shape in external fields

We have developed a Brownian dynamics simulation algorithm to generate Brownian trajectories of an isolated, rigid particle of arbitrary shape in the presence of electric fields or any other external agents. Starting from the generalized diffusion tensor, which can be calculated with the existing HYDRO software, the new program BROWNRIG (including a case-specific subprogram for the external agent) carries out a simulation that is analyzed later to extract the observable dynamic properties. We provide a variety of examples of utilization of this method, which serve as tests of its performance, and also illustrate its applicability. Examples include free diffusion, transport in an electric field, and diffusion in a restricting environment.     

MR diffusion tensor spectroscopy and imaging.

This paper describes a new NMR imaging modality--MR diffusion tensor imaging. It consists of estimating an effective diffusion tensor, Deff, within a voxel, and then displaying useful quantities derived from it. We show how the phenomenon of anisotropic diffusion of water (or metabolites) in anisotropic tissues, measured noninvasively by these NMR methods, is exploited to determine fiber tract orientation and mean particle displacements. Once Deff is estimated from a series of NMR pulsed-gradient, spin-echo experiments, a tissue's three orthotropic axes can be determined. They coincide with the eigenvectors of Deff, while the effective diffusivities along these orthotropic directions are the eigenvalues of Deff. Diffusion ellipsoids, constructed in each voxel from Deff, depict both these orthotropic axes and the mean diffusion distances in these directions. Moreover, the three scalar invariants of Deff, which are independent of the tissue's orientation in the laboratory frame of reference, reveal useful information about molecular mobility reflective of local microstructure and anatomy. Inherently tensors (like Deff) describing transport processes in anisotropic media contain new information within a macroscopic voxel that scalars (such as the apparent diffusivity, proton density, T1, and T2) do not.     

Normal mode theory for the Brownian dynamics of a weakly bending rod: Comparison with Brownian dynamics simulations

A normal mode theory is developed for the Brownian dynamics of weakly bending rods with preset hydrodynamic interactions. The rod is replaced by a chain of contiguous spheres whose radius is chosen to yield the appropriate uniform translational and rotational diffusion coefficients. Despite the inclusion of preset hydrodynamic interactions in the dynamical operator, its normal modes are not coupled by the potential energy, so their amplitudes remain pairwise “orthogonal” under equilibrium averaging. The uniform translational and rotational diffusion coefficients obtained from Langevin theory are shown to be identical to those obtained from the Kirkwood algorithm, despite their rather different appearance. An expression is given for the mean squared angular displacement 〈Δ_xm(t)²〉 of the mth bond vector around the instantaneous x axis (perpendicular to the end-to-end vector z). Necessary algorithms are presented for the numerical evaluation of all quantities. The normal mode theory is compared with Brownian dynamics simulations for the same model by examining 3〈Δ_xm(t)²〉 for the central bond vector of rods comprising 10 and 30 subunits with various persistence lengths. The normal mode theory works very well for all times for L/P ? 0.6, where P = κ/k_BT is the persistence length and κ is the bending rigidity. With increasing flexibility, the domain of validity of the normal mode theory is restricted to shorter times, where violations of the weak bending approximation are less severe. However, increasing the length of the rod from 10 to 30 subunits yields improved agreement with the simulations for the same and even longer times. This latter effect is tentatively attributed to the greater fluctuating tension in the longer chains, which acts to retard the rotational relaxation in the simulations, but is not taken into account in the present normal mode theory.     

Investigation of the 50 S ribosomal subunit by electron microscopy and image analysis

In electron micrographs of 50 S (large) subunits from Escherichia coli ribosomes, the highly preferred crown view is inferred to represent the roughly hemispherical particle lying with its flat or concave face against the carbon film. Single particle averaging allows the reproducible details of the crown view particle to be recognized. Multivariate image analysis shows the most variable morphological features of this view to be the two side protrusions, the L7/L12 stalk and the L1 ridge, both of which show apparent positional variations. The invariance of the features of the particle body implies that the movements of the side protrusions are not merely a result of perspective changes produced by major rotations of the particle body out of its quasistable, flat-lying position. A bending point localized on the L7/L12 stalk is conjectured to represent a functional "hinge" that may be related to the secondary/tertiary structure of the L7/L12 dimeric protein.