B. Carrasco

Calculation of Hydrodynamic Properties of Globular Proteins from Their Atomic-Level Structure

The solution properties, including hydrodynamic quantities and the radius of gyration, of globular proteins are calculated from their detailed, atomic-level structure, using bead-modeling methodologies described in our previous article (, Biophys. J. 76:3044-3057). We review how this goal has been pursued by other authors in the past. Our procedure starts from a list of atomic coordinates, from which we build a primary hydrodynamic model by replacing nonhydrogen atoms with spherical elements of some fixed radius. The resulting particle, consisting of overlapping spheres, is in turn represented by a shell model treated as described in our previous work. We have applied this procedure to a set of 13 proteins. For each protein, the atomic element radius is adjusted, to fit all of the hydrodynamic properties, taking values close to 3 A, with deviations that fall within the error of experimental data. Some differences are found in the atomic element radius found for each protein, which can be explained in terms of protein hydration. A computational shortcut makes the procedure feasible, even in personal computers. All of the model-building and calculations are carried out with a HYDROPRO public-domain computer program.

The Conformation of Serum Albumin in Solution: A Combined Phosphorescence Depolarization-Hydrodynamic Modeling Study

There is a striking disparity between the heart-shaped structure of human serum albumin (HSA) observed in single crystals and the elongated ellipsoid model used for decades to interpret the protein solution hydrodynamics at neutral pH. These two contrasting views could be reconciled if the protein were flexible enough to change its conformation in solution from that found in the crystal. To investigate this possibility we recorded the rotational motions in real time of an erythrosin-bovine serum albumin complex (Er-BSA) over an extended time range, using phosphorescence depolarization techniques. These measurements are consistent with the absence of independent motions of large protein segments in solution, in the time range from nanoseconds to fractions of milliseconds, and give a single rotational correlation time phi(BSA, 1 cP, 20 degrees C) = 40 +/- 2 ns. In addition, we report a detailed analysis of the protein hydrodynamics based on two bead-modeling methods. In the first, BSA was modeled as a triangular prismatic shell with optimized dimensions of 84 x 84 x 84 x 31.5 A, whereas in the second, the atomic-level structure of HSA obtained from crystallographic data was used to build a much more refined rough-shell model. In both cases, the predicted and experimental rotational diffusion rate and other hydrodynamic parameters were in good agreement. Therefore, the overall conformation in neutral solution of BSA, as of HSA, should be rigid, in the sense indicated above, and very similar to the heart-shaped structure observed in HSA crystals.

Hydrodynamic Properties of Rigid Particles: Comparison of Different Modeling and Computational Procedures

The hydrodynamic properties of rigid particles are calculated from models composed of spherical elements (beads) using theories developed by Kirkwood, Bloomfield, and their coworkers. Bead models have usually been built in such a way that the beads fill the volume occupied by the particles. Sometimes the beads are few and of varying sizes (bead models in the strict sense), and other times there are many small beads (filling models). Because hydrodynamic friction takes place at the molecular surface, another possibility is to use shell models, as originally proposed by Bloomfield. In this work, we have developed procedures to build models of the various kinds, and we describe the theory and methods for calculating their hydrodynamic properties, including approximate methods that may be needed to treat models with a very large number of elements. By combining the various possibilities of model building and hydrodynamic calculation, several strategies can be designed. We have made a quantitative comparison of the performance of the various strategies by applying them to some test cases, for which the properties are known a priori. We provide guidelines and computational tools for bead modeling.

SOLPRO: theory and computer program for the prediction of SOLution PROperties of rigid macromolecules and bioparticles

Abstract Single-valued hydrodynamic coefficients of a rigid particle can be calculated from existing theories and computer programs for either bead models or ellipsoids. Starting from these coefficients, we review the procedures for the calculation of complex solution properties depending on rotational diffusion, such as the decays of electric birefringence and fluorescence anisotropy. We also describe the calculation of the scattering form factor of bead models. The hydrodynamic coefficients and solution properties can be combined to give universal, shape-dependent functions, which were initially intended for ellipsoidal particles, and are extended here for the most general case. We have implemented all these developments in a new computer program, SOLPRO, for calculation of SOLution PROperties, which can be linked to existing software for bead models or ellipsoids.

Improved hydrodynamic interaction in macromolecular bead models

The calculation of hydrodynamic properties of macromolecules in terms of bead models requires an adequate description of the hydrodynamic interaction between the spherical elements. For this purpose, the original or modified Oseen tensor are customarily used, although it has been shown that this simple description may lead to erroneous results, particularly for rotational coefficients. In this paper we study several more elaborate theories for multisphere systems. We apply those treatments to our problem of rigid bead models, implementing them in computer programs, and making calculations for various test structures. The comparison of the results from the various theories, and from other, presumably very accurate procedures, allow us to give some guidelines to improve the treatment of hydrodynamic interactions in macromolecular bead models. These advances are introduced in new versions of our public-domain computer software.

Calculation of NMR relaxation, covolume, and scattering-related properties of bead models using the SOLPRO computer program

Abstract The hydrodynamic properties of macromolecules and bioparticles, represented by bead models, can be calculated using methods implemented in the computer routine HYDRO. Recently, a new computer routine, SOLPRO, has been presented for the calculation of various SOLution PROperties. These include (1) time-dependent electro-optic and spectroscopic properties related to rotational diffusion, (2) non-dynamic properties like scattering curves, and (3) dimensionless quantities that combine two or more solution properties in a form which depends on the shape of the macromolecule but not on its size. In the present work we describe the inclusion of more of those types of properties in a new version of SOLPRO. Particularly, we describe the calculation of relaxation rates in nuclear magnetic resonance (NMR). For dipolar coupling, given the direction of the dipole the program calculates values of the spectral density, from which the NMR relaxation times can be obtained. We also consider scattering-related properties, namely the distribution of distances for the bead model, which is directly related to the angular dependence of scattered intensity, and the particles longest distance. We have devised and programmed a procedure to calculate the covolume of the bead model, related to the second virial coefficient and, in general, to the concentration dependence of solution properties. Various shape-dependent dimensionless quantities involving the covolume are calculated. In this paper we also discuss some aspects, namely bead overlapping and hydration, that are not explicitely included in SOLPRO, but should be considered by the user.

Calculation of hydrodynamic properties of macromolecular bead models with overlapping spheres.

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

Intrinsic viscosity and rotational diffusion of bead models for rigid macromolecules and bioparticles

Abstract The conventional Kirkwood-Riseman (K-R) treatment of the intrinsic viscosity of macromolecular bead models shows a deficiency when it is applied to models with few beads, whose sizes are not much smaller than of the modelled particle. We present a complete derivation of the intrinsic viscosity up to first order in interbead distances (Oseen-type hydrodynamic interaction), finding that a term that belongs to the zeroth-order contribution is missing in the usual description. This term is simply proportional to the total volume of the bead model. The nature of this correction for viscosity is similar to a previously described correction for rotational coefficients. We discuss the performance of these corrections for various simple models, including ellipsoids as well as oligomeric structures in rodlike, chainlike and polyhedral conformations.

Universal size-independent quantities for the conformational characterization of rigid and flexible macromolecules

Solution properties of macromolecules can be combined to form dimensionless quantities that are universal in the sense of depending on the shape or conformation of the macromolecular solute, being independent of its size. A number of such quantities have been formulated at different times by different authors, and as a consequence they differ widely from one to another, not only in their notation and the order of magnitude of their values, but also in the way in which the primary solution properties enter in them. In this work we propose a new set of universal size-independent quantities in a systematic and consistent way. First, solution properties are expressed in the form of radii of an equivalent sphere. Then the equivalent radii are combined to give ratios of radii. We propose the use of the ratios of radii as indicators of macromolecular conformation. Examples of their values are given for three macromolecular structures: (1) ellipsoids, with application to globular proteins, (2) oligomeric arrays of subunits, with seed globulins as examples and (3) flexible-chain macromolecules, illustrated by polystyrene in two solvents.

Bead modeling using HYDRO and SOLPRO of the conformation of multisubunit proteins: sunflower and rape-seed 11S globulins

Oil seed globulins from sunflower and rape seed are multi-subunit, oligomeric proteins whose native 11S form is a hexamer. In this work we try to determine the spatial structure in which the six subunits of 11S globulin are arranged. Experimental values of solution properties, including radius of gyration, sedimentation and diffusion coefficients and intrinsic viscosity, are compared with theoretical predictions for hexamers of various geometries. Bead model calculations of solution properties are carried out using the HYDRO and SOLPRO computer programs. A most compact shape, the regular octahedron, is the hexameric structure that fits best the experimental values.