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.     

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.     

Dynamic depolarization of interacting fluorophores. Effect of internal rotation and energy transfer.

Effects of internal rotation on the fluorescence decay functions and time-dependent anisotropies of fluorophores bound to a spherical macromolecule are theoretically investigated in the presence of the intramolecular energy transfer interaction by solving relevant rotational diffusion equations. The model system examined is one in which the energy donor is internally rotating around an axis fixed at the macromolecule and the acceptor is fixed at a definite position in the macromolecule. The effect of internal rotation in the system is described by Hill's functions with two cosine terms. The fluorescence decay function and anisotropy decay are functions of the ratio of energy-transfer probability averaged over the internal rotation angle to the rotary diffusion co-efficient. When the internal rotation is much faster than energy transfer, the decay function of the donor is predicted to be a single exponential, and the anisotropy decay is essentially described by the expression derived by Gotlieb and Wahl (1963. J. Chim. Phys. 60:849-856). However, deviation from it becomes pronounced as the rotation becomes slower. Methods of numerical analysis are presented for decay function and anisotropy decay, as well as relative quantum yield and polarization anisotropy under steady-state excitation, and examined for a simplified system under the variation of the diffusion coefficient.     

A general method for evaluation of diffusion constants,dilute-solution viscoelasticity,and the dielectric property of a rigid macromolecule with an arbitrary configuration. I

As the first part of the study on relaxational behavior of rigid macromolecules in solution, a general method for evaluating diffusion constants of a rigid macromolecule with an arbitrary configuration in a viscous solvent is presented. The evaluation is made by modeling the molecule with a rigid assembly of Stokes spheres. The diffusion constants are expressed by a 6 × 6 matrix which consists of translational, rotational, and cross constants, each being a 3 × 3 submatrix. Procedures of numerical calculation are presented for a given configuration of the molecule, together with some exemplifying results for the rigid rod, rigid sphere, and myosin molecule. The results for the rod and the sphere are compared with analytical results.     

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.     

Order and fluidity of lipid membranes as determined by fluorescence anisotropy decay

The fluorescence anisotropy decay of four different probes in bilayers of dimyristoylphosphatidylcholine was measured. The probes are diphenylhexatriene, diphenyloctatetraene, trimethylaminodiphenylhexatriene, and trans-parinaric acid. The data for each probe were analyzed in terms of two orientational order parameters, the ordinary order parameter and a higher one, and two rotational diffusion coefficients. The order parameters are largely independent of probe size, but depend on the position of the probes along the membrane normal, thus reflecting the profile of lipid order. If a probe is located in the plateau region of lipid order, its order parameters are interpreted as representing the rigid-body order of lipids. According to this interpretation, the total lipid order in the plateau region originates about equally from rigid-body order and conformational order. The two order parameters obtained for each probe are used to derive approximate angular distributions of the probe molecules. The diffusion coefficient for rotation about the long molecular axis is found to be infinitely large, indicating unhindered rotation about this axis. The diffusion coefficient for rotation about the short molecular axes is evaluated for a viscosity which results as 0.2 poise. This viscosity for rotational diffusion is an order of magnitude smaller than the viscosity for lateral diffusion indicating that at least two viscosities are required to characterize the fluidity of a lipid membrane.Abbreviations FAD fluorescence anisotropy decay - DMR deuterium magnetic resonance - ESR electron spin resonance - DMPC dimyristoylphosphatidylcholine - DPPC dipalmitoylphosphatidylcholine - DPH 1,6-diphenyl-1,3,5-hexatriene - DPO 1,6-diphenyl-1,3,5,7-octatetraene - TMA-DPH 1-[4-(trimethylamino)phenyl]-6-phenyl-1,3,5-hexatriene - tPnA trans-parinaric acid - NPN N-phenyl-1-naphthylamine - BBO 2,5-bis(4-biphenylyl)oxazole     

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.     

Solution structure of biopolymers: a new method of constructing a bead model

We propose a new, automated method of converting crystallographic data into a bead model used for the calculations of hydrodynamic properties of rigid macromolecules. Two types of molecules are considered: nucleic acids and small proteins. A bead model of short DNA fragments has been constructed in which each nucleotide is represented by two identical, partially overlapping spheres: one for the base and one for the sugar and phosphate group. The optimum radius sigma = 5.0 A was chosen on the basis of a comparison of the calculated translational diffusion coefficients (D(T)) and the rotational relaxation times (tau(R)) with the corresponding experimental data for B-DNA fragments of 8, 12, and 20 basepairs. This value was assumed for the calculation D(T) and tau(R) of tRNA(Phe). Better agreement with the experimental data was achieved for slightly larger sigma = 5.7 A. A similar procedure was applied to small proteins. Bead models were constructed such that each amino acid was represented by a single sphere or a pair of identical, partially overlapping spheres, depending on the amino acid's size. Experimental data of D(T) of small proteins were used to establish the optimum value of sigma = 4.5 A for amino acids. The lack of experimental data on tau(R) for proteins restricted the tests to the translational diffusion properties.     

Rotational dynamics of surface probes in lipid vesicles

Translational and rotational diffusion of fluorescent molecules on the surface of small biological systems such as vesicles, proteins and micelles depolarize the fluorescence. A recent study has treated the case of the translational dynamics of surface probes (M.M.G. Krishna, R. Das, N. Periasamy and R. Nityananda, J. Chem. Phys., 112 (2000) 8502-8514) using Monte Carlo and theoretical methods. Here we extend the application of the methodologies to apply the case of rotational dynamics of surface probes. The corresponding fluorescence anisotropy decays were obtained using the Monte Carlo simulation methods for the two cases: surface probes undergoing rotational dynamics on a plane and on a sphere. The results were consistent with the theoretical equations which show that Monte Carlo methods can be used to simulate the surface diffusion problems. The anisotropy decay for the rotational diffusion of a molecule on a planar surface is single exponential and the residual anisotropy is zero. However, residual anisotropy is finite for the case of rotational diffusion on a sphere because of the spatial averaging of the anisotropy function. The rotational correlation time in both the cases is (4Drot)(-1) with Drot being the rotational diffusion coefficient. Rotational dynamics of a surface bound dye in a single giant liposome and in sonicated vesicles were studied and the results were explained according to the theoretical equations. A fast component of fluorescence depolarization was also observed for sonicated vesicles which was interpreted as wobbling-in-cylinder dynamics of the surface-bound dye.     

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.     

Band centrifugation of macromolecules in self-generating density gradients. II. Sedimentation and diffusion of macromolecules in bands

Three approaches to the simultaneous sedimentation and diffusion of hands or zones of noninteracting homogeneous macromolecules are examined: (1) The authors' method of moments: (2) the transport me of Sehumaker and Rosenbloom; and (3) the stochastic solution of the Lamm equation due to Gehatia and Katehalski. All three methods indicate that the motion of the maximum of the hand may be used to evaluate the sedimentation coefficient. The moment, method provides relations which appear to be useful for measuring diffusion coefficients. Relations are given for the analysis of resolved components. The problem of measuring sedimentation coefficients of macromolecules with concentration-dependent sedimentation coefficients is examined. Methods are described for evaluating the sedimentation coefficient in these systems and for obtaining the sedimentation coefficient at infinite dilution. Methods are described for determining the weight-average sedimentation coefficient in Multi-component systems, and the differential and integral distribution of sedimentation coefficients of macromolecules with low-diffusion coefficients.     

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.     

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.     

Fluorescence quenching dynamics of tryptophan in proteins. Effect of internal rotation under potential barrier.

In many proteins fluorescence from single tryptophan exhibits a nonexponential decay function. To elucidate the origin of this nonexponential decay, we have examined the fluorescence decay function and time-resolved fluorescence anisotropy of a fluorophore covalently bound to a macromolecule by solving a rotational analogue of the Smoluchowski equation. An angular-dependent quenching constant and potential energy for the fluorophore undergoing internal rotation were introduced into the equation of motion for fluorophore. Results of numerical calculations using the equations thus obtained predict that both the fluorescence decay function and time-resolved anisotropy are dependent on rotational diffusion coefficients of fluorophore and potential energy for the internal rotation. The method was applied to the observed fluorescence decay curve of the single tryptophan in apocytochrome c from horse heart. The calculated decay curves fit the observed ones well.     

Movement of different-shaped particles in a pan-coating device using novel video-imaging techniques

The purpose of this study was to investigate the effects of particle shape on the movement of particles in a pan-coating device using novel video-imaging techniques. An area scan CCD camera was installed inside a 24-in pan coater at the same location as that of a spray nozzle, and the movement of particle was tracked using machine vision. A white tracer particle was introduced inside a bed of black-coated particles. The effects of pan loading, pan speed, and particle shape on the movement of particles was studied. The response variables were circulation time, surface time, projected area of particle per pass, dynamic angle of repose, cascading velocity, and dispersion coefficient. Experiments were conducted at 3 different pan speeds, 6, 9, and 12 rpm, and 2 fill levels (ratio of bed depth to pan diameter), one eighth and one quarter, and data were collected over a 30-minute time period. The differences in circulation times of spheres and tablets, with similar volume equivalent diameter as that of the sphere, were found to be insignificant at the 95% confidence interval. The circulation time ranged from 2.8 to 10.8 seconds depending on the operating condition and increased with increasing pan load and decreasing pan speed. The distributions of circulation time, surface time, and projected surface area were found to be nonnormal. The dynamic angle of repose for tablets was higher than for spheres. Also, the bed surface for spheres was much flatter in comparison with tablets where the bed shape attained a “wave-like” form. The average velocity of tablets in the cascading layer was found to be significantly higher than spheres. A linear model (R ²>0.98) best described the variation of velocity as a function of pan speed for all of the operating conditions. Published: October 6, 2005     

Transport properties and hydrodynamic centers of rigid macromolecules with arbitrary shapes

A general formalism, which includes translation–rotation coupling, is proposed for calculating translational and rotational transport properties, as well as intrinsic viscosities, of rigid macromolecules with an arbitrary shape. This formalism is based on Brenner's theory of translational–rotational dynamics and on methods for the calculation of hydrodynamic properties that have been already presented, and can be regarded as a generalization of the one proposed by Nakajima and Wada. The calculated transport properties depend on the origin as predicted by Brenner's theory, but in a disagreement with him, the center of resistance and the center of diffusion do not coincide. As one can define several hydrodynamic centers, which in practice turn out to be located at different points, the influence of the choice of the center on the calculated transport properties is discussed. An analysis of the translation–rotation coupling effects in translational diffusion reveals that they arise exclusively from hydrodynamic interactions and are rather small in some cases of interest. Finally, we present a study of the rotational diffusion of rigid bent rods with a fixed length-to-diameter ratio. The diffusion coefficients obtained can be useful to estimate changes with respect to a straight rod.     

Segmental flexibility in pig immunoglobulin G studied by neutron spin-echo technique

The dynamic behavior of pig immunoglobulin G in deuterium oxide solutions was investigated by the neutron spin-echo technique. This novel technique makes it possible to study intramolecular motion without introducing probes into the macromolecule. Using neutron spin-echo, the effective diffusion coefficient, D_eff, was obtained as a function of the transferred momentum, κ. For interpretation of the experimental data, two models were designed, and computed D_eff values were compared with experimental data. The rigid T-shape model was compatible with the experimental data only by assuming an unrealistically high rotational diffusion coefficient, and it was therefore unacceptable. Reasonable agreement with all available experimental data was obtained with a flexible model of immunoglobulin G molecule, which the Fab arms are assumed to wobble around the hinge region within an angle of 50°.     

Determinants of the translational mobility of a small solute in cell cytoplasm

The purposes of this study were: (a) to measure the translational mobility of a small solute in cell cytoplasm; (b) to define quantitatively the factors that determine solute translation; and (c) to compare and contrast solute rotation and translation. A small fluorescent probe, 2,7-bis-(2-carboxyethyl)-5-(and 6-)- carboxyfluorescein (BCECF), was introduced into the cytoplasm of Swiss 3T3 fibroblasts. BCECF translation was measured by fluorescence recovery after photo-bleaching; rotation was measured by Fourier transform polarization microscopy. Diffusion coefficients relative to those in water (D/D0) were determined by comparing mobility in cytoplasm with mobility in standard solutions of known viscosity. At isosmotic cell volume, the relative diffusion coefficients for BCECF translation and rotation in cytoplasm were 0.27 +/- 0.01 (SEM, n = 24, 23 degrees C) and 0.78 +/- 0.03 (n = 4), respectively. As cell volume increased from 0.33 to 2 times isosmotic volume, the relative translational diffusion coefficient increased from 0.047 to 0.32, while the relative rotational diffusion coefficient remained constant. The factors determining BCECF translation were evaluated by comparing rotation and translation in cytoplasm, and in artificial solutions containing dextrans (mobile barriers) and agarose gels (immobile barriers). It was concluded that the hindrance of BCECF translation in cytoplasm could be quantitatively attributed to three independent factors: (a) fluid-phase cytoplasmic viscosity is 28% greater than the viscosity of water (factor 1 = 0.78); (b) 19% of BCECF is transiently bound to intracellular components of low mobility (factor 2 = 0.81); and most importantly, (c) translation of unbound BCECF is hindered 2.5- fold by collisions with cell solids comprising 13% of isosmotic cell volume (factor 3 = 0.40). The product of the 3 factors is 0.25 +/- 0.03, in good agreement with the measured D/D0 of 0.27 +/- 0.01. These results provide the first measurement of the translational mobility of a small solute in cell cytoplasm and define quantitatively the factors that slow solute translation.     

A general method for the evaluation of diffusion constants,dilute-solution viscoelasticity,and the complex dielectric constant of a rigid macromolecule with an arbitrary configuration. II

As a continuation of a previous paper [Biopolymers 16 , (1977)], in which we described a general method for the evaluation of the diffusion constants of a rigid macromolecule with an arbitrary configuration, this paper presents a theory for evaluating the relaxational behavior of the molecule in solution. The diffusion equation of a molecule subjected to hydrodynamic or electric forces is solved and both a complex viscosity and a complex dielectric constant are obtained by assuming that the molecule is composed of Tokes spheres and, in the dielectric property, the molecule possesses a fixed charge distribution. The viscoelasticity of a rigid rod is calculated and compared with the analytical results.     

Macromolecular diffusion in crowded solutions.

The effects of crowding on the self or tracer diffusion of macromolecules in concentrated solutions is an important but difficult problem, for which, so far, there has been no rigorous treatment. Muramatsu and Minton suggested a simple model to calculate the diffusion coefficient of a hard sphere among other hard spheres. In this treatment, scaled particle theory is used to evaluate the probability that the target volume for a step in a random walk is free of any macromolecules. We have improved this approach by using a more appropriate target volume which also allows the calculation to be extended to the diffusion of a hard sphere among hard spherocylinders. We conclude that, to the extent that proteins can be approximated as hard particles, the hindrance of globular proteins by other proteins is reduced when the background proteins aggregate (the more so the greater the decrease in particle surface area), the hindrance due to rod-shaped background particles is reduced slightly if the rod-like particles are aligned, and the anisotropy of the diffusion of soluble proteins among cytoskeletal proteins will normally be small.