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 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.     

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.     

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.     

Hydrodynamics of segmentally flexible macromolecules

Segmentally flexible macromolecules are composed of a few rigid subunits linked by joints which are more or less flexible. The dynamics in solution of this type of macromolecule present special aspects that are reviewed here. Three alternative approaches are described. One is the rigid-body treatment, which is shown to be valid for overall dynamic properties such as translational diffusion and intrinsic viscosity. Another approach is the Harvey-Wegener treatment, which is particularly suited for rotational diffusion. The simplest version of this treatment, which ignores hydrodynamic interaction (HI) effects, is found to be quite accurate when compared to a more rigorous version including HI. A third approach is the Brownian dynamics simulation that, albeit at some computational cost, might describe rigorously cases of arbitrary complexity. This technique has been used to test the approximations in the rigid-body and Harvey-Wegener treatments, thus allowing a better understanding of their validity. Brownian trajectories of simplified models such as the trumbbell and the broken rod have been simulated. The comparison of the decay rates of some correlation functions with the predictions of the two treatments leads to a general conclusion: the Harvey-Wegener treatment determines the initial rate, while the long-time behavior is dominated by the rigid-body relaxation time. As an example of application to a specific biological macromolecule, we present a simulation of an immunoglobulin molecule, showing how Brownian Dynamics can be used to predict rotational and internal dynamics. Another typical example is myosin. Literature data of hydrodynamic properties of whole myosin and the myosin rod are compared with predictions from the Harvey-Wegener and rigid-body treatments. The present situation of the problem on myosin flexibility is analyzed, and some indications are given for future experimental and simulation work.     

Transport properties of particles with segmental flexibility. II. Decay of fluorescence polarization anisotropy from hinged macromolecules

An algorithm is presented for the Monte Carlo simulation of the decay of fluorescence polarization from segmentally flexible molecules. Based on the random walk model of Brownian motion, the treatment explicitly follows the stochastic changes in the diffusion coefficients as the molecule bends. It includes the effects of a linear restoring force opposing the bending and the effects of hydrodynamic coupling between the translational, rotational, and bending motions. One application is presented: the simulation of anisotropy decay curves for hinged rods. A variety of decay curves are obtained, including single- and multiexponential behavior, and the following conclusions are reached: (1) increasing the flexibility is usually, but not always, accompanied by a more rapid rate of depolarization; (2) reducing the size of the fluorescent subunit will usually, but not always, increase the rate of depolarization; and (3) the complex interplay between the effects of molecular shape, relative sizes of the subunits, restoring force, and orientation of the transition dipoles renders it unlikely that any simple method can be used to interpret anisotrophy data without simulation. In particular, it is not possible to determine the extent of bending by fitting the data with the two-exponential approximation used by some investigators in the past.     

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.     

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.     

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.     

Steady-state and transient electric birefringence of solutions of bent-rod macromolecules

We first calculate the steady-state birefringence, expressed in the form of specific Kerr constant, K_sp, of rigid, bent-rod macromolecules. Equations are derived for K_sp as a function of the geometric and electro-optical parameters. We also consider flexibly hinged rods and evaluate K_sp for them by averaging over the angle between the two arms, ?. Next, we turn to the time decay of the electric birefringence. The decay function for rigid bent rods is a sum of three exponential terms, and a procedure for their calculation is indicated. We observe that single-exponential decays can be found for ? > 90° or ? < 60°, in spite of the high electro-optical and hydrodynamic anisotropy of the macromolecule. Special attention is paid to the case of rods with equal arms.     

Diffusive Dynamics in a Detailed Potential: Application to Biological Macromolecules

Abstract

The local dynamics of macromolecules is obtained to second-order in the mode-coupling expansion of the Smoluchowski diffusion theory. The NMR spin-lattice relaxation times of different ¹³C or ¹⁵N nuclei along the chains are calculated and compared to experimental data from the literature. The macromolecules are considered as fluctuating 3D structures undergoing rotational diffusion. The fluctuations can be evaluated with any technique for sampling the configurational space. In the presented test cases Molecular Dynamics simulations have been applied to a DNA fragment and to the NK-2 homeodomain. In the case of the double-stranded DNA fragment d(TpCpGpCpG)₂, second and even first order theories are found to be in close agreement with experimental results. The major advantage of the diffusion technique is that only a good statistics is important as input while the solvent dynamic effects enter through hydrodynamic theory. Application based on Hybrid Monte Carlo schemes coupled with J-walking, are now in progress.     

Fluorescence anisotropy as a probe to study tracer proteins in crowded solutions

Fluorescence anisotropy has been widely used to study the dynamics and interactions of biomolecules in diluted solutions. Comparable studies on single tracer macromolecules at the cellular level are now feasible because of the recent development of non-invasive fluorescence markers, like the growing family of the green fluorescence proteins (GFPs), and the advances in time-resolved fluorescence microscopy instrumentation. The interpretation of fluorescence polarization data in terms of dynamics and biological function of the macromolecular complexes in these physiological environments requires a deep understanding of the tracer rotational diffusion in such complex media. In this work we have studied the rotational diffusion of a tracer protein, apomyoglobin labeled with 1-anilino-8-naphthalene sulfonate, in crowded solutions of an unrelated protein, ribonuclease A. We have evaluated the deviation of the different tracer rotational motions from the Stokes-Einstein-Debye diffusion behavior, and its relation to the properties of the transient molecular cavities where the tracer is rotating in the fluorescence lifetime window. Finally, we have analyzed the application of fluorescence polarization methods to determine the apparent equilibrium constants of homo and hetero-associations of macromolecules in crowded conditions.     

Smoluchowski dynamics of the vnd/NK-2 homeodomain from Drosophila melanogaster: second-order maximum correlation approximation

The mode coupling diffusion theory is applied to the derivation of local dynamics in proteins in solution. The rotational dynamics of the bonds along the protein sequence are calculated and compared to the experimentally measured nmr (15)N spin-lattice relaxation time T(1), at 36.5, 60.8, and 81.1 MHz of the vnd/NK-2 homeodomain from Drosophila melanogaster. The starting point for the calculations is the experimental three-dimensional solution structure of the homeodomain determined by multidimensional nmr spectroscopy. The higher order mode-coupling computations are compared also with the recently published first-order approximation calculations. The more accurate calculations improve substantially the first-order ORZLD calculations and show that the role of the strength of the hydrodynamic interactions becomes crucial to fix the order of magnitude of the rotational dynanics for these very compact molecules characterized by partial screening of the internal atoms to water. However, the relative mobility of the bonds along the sequence and the differential fluctuations depend only weakly on the hydrodynamic strength but strongly on the geometry of the three-dimensional structure and on the statistics incorporated into the theory. Both rigid and fluctuating dynamic models are examined, with fluctuations evaluated using molecular dynamics simulations. The comparison with nmr data shows that mode coupling diffusion accounts for the T(1) relaxation pattern at low frequency where the rotational tumbling dominates. An important contribution of internal motions in the nanosecond time scale is seen at high frequencies and is discussed in terms of diffusive concepts.     

Computer simulation of hydrodynamic properties of semiflexible macromolecules: randomly broken chains, wormlike chains, and analysis of properties of DNA

The translational and rotational diffusion coefficients and the intrinsic viscosity of semiflexible, randomly broken, and wormlike chains have been obtained by Monte Carlo simulation in the context of the rigid-body treatment. Both approximate and rigorous rigid-body hydrodynamics are used, so that the error introduced by the approximate methods can be evaluated. A randomly broken chain and a wormlike chain having the same contour length and persistence length have the same radius of gyration but different values for any of the hydrodynamic properties. The two types of chains are compared in this regard. Considering that the cross section of the chain is represented by a cylinder better than by a string of spheres, we devise a cylindrical correction to be applied to the results simulated for chains of beads. Application is made to the analysis of experimental data for the translational and rotational coefficients of DNA fragments with up to 10(3) base pairs, obtaining the persistence length for each model. The values for the wormlike chain agree well with model-independent values obtained from radii of gyration and with other literature data at varying ionic strength. The randomly broken chain is equally able to reproduce the experimental length dependence of the properties, but the resulting persistence length may be too high.     

Dynamic light scattering: a practical guide and applications in biomedical sciences

Dynamic light scattering (DLS), also known as photon correlation spectroscopy (PCS), is a very powerful tool for studying the diffusion behaviour of macromolecules in solution. The diffusion coefficient, and hence the hydrodynamic radii calculated from it, depends on the size and shape of macromolecules. In this review, we provide evidence of the usefulness of DLS to study the homogeneity of proteins, nucleic acids, and complexes of protein–protein or protein–nucleic acid preparations, as well as to study protein–small molecule interactions. Further, we provide examples of DLS’s application both as a complementary method to analytical ultracentrifugation studies and as a screening tool to validate solution scattering models using determined hydrodynamic radii.     

Macromolecular crowding in biological systems: hydrodynamics and NMR methods

Most biologically relevant environments involve highly concentrated macromolecular solutions and most biological processes involve macromolecules that diffuse and interact with other macromolecules. Macromolecular crowding is a general phenomenon that strongly affects the transport properties of macromolecules (rotational and translational diffusion) as well as the position of their equilibria. NMR methods can provide information on molecular interactions, as well as on translational and rotational diffusion. In fact, rotational diffusion, through its determinant role in NMR relaxation, places a practical limit on the systems that can be studied by NMR. While in dilute solutions of non-aggregating macromolecules this limit is set by macromolecular size, in crowded solutions excluded volume effects can have a strong effect on the observed diffusion rates. Hydrodynamic theory offers some insight into the magnitude of crowding effects on NMR observable parameters.     

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.     

Calculation of the solution properties of flexible macromolecules: methods and applications

While the prediction of hydrodynamic properties of rigid particles is nowadays feasible using simple and efficient computer programs, the calculation of such properties and, in general, the dynamic behavior of flexible macromolecules has not reached a similar situation. Although the theories are available, usually the computational work is done using solutions specific for each problem. We intend to develop computer programs that would greatly facilitate the task of predicting solution behavior of flexible macromolecules. In this paper, we first present an overview of the two approaches that are most practical: the Monte Carlo rigid-body treatment, and the Brownian dynamics simulation technique. The Monte Carlo procedure is based on the calculation of properties for instantaneous conformations of the macromolecule that are regarded as if they were instantaneously rigid. We describe how a Monte Carlo program can be interfaced to the programs in the HYDRO suite for rigid particles, and provide an example of such calculation, for a hypothetical particle: a protein with two domains connected by a flexible linker. We also describe briefly the essentials of Brownian dynamics, and propose a general mechanical model that includes several kinds of intramolecular interactions, such as bending, internal rotation, excluded volume effects, etc. We provide an example of the application of this methodology to the dynamics of a semiflexible, wormlike DNA.     

High field/high frequency saturation transfer electron paramagnetic resonance spectroscopy: increased sensitivity to very slow rotational motions

Saturation transfer electron paramagnetic resonance (ST-EPR) spectroscopy has been employed to characterize the very slow microsecond to millisecond rotational dynamics of a wide range of nitroxide spin-labeled proteins and other macromolecules in the past three decades. The vast majority of this previous work has been carried out on spectrometers that operate at X-band ( approximately 9 GHz) microwave frequency with a few investigations reported at Q-band ( approximately 34 GHz). EPR spectrometers that operate in the 94-250-GHz range and that are capable of making conventional linear EPR measurements on small aqueous samples have now been developed. This work addresses potential advantages of utilizing these same high frequencies for ST-EPR studies that seek to quantitatively analyze the very slow rotational dynamics of spin-labeled macromolecules. For example, the uniaxial rotational diffusion (URD) model has been shown to be particularly applicable to the study of the rotational dynamics of integral membrane proteins. Computational algorithms have been employed to define the sensitivity of ST-EPR signals at 94, 140, and 250 GHz to the correlation time for URD, to the amplitude of constrained URD, and to the orientation of the spin label relative to the URD axis. The calculations presented in this work demonstrate that these higher microwave frequencies provide substantial increases in sensitivity to the correlation time for URD, to small constraints in URD, and to the geometry of the spin label relative to the URD axis as compared with measurements made at X-band. Moreover, the calculations at these higher frequencies indicate sensitivity to rotational motions in the 1-100-ms time window, particularly at 250 GHz, thereby extending the slow motion limit for ST-EPR by two orders of magnitude relative to X- and Q-bands.