Hydration from hydrodynamics. General considerations and applications of bead modelling to globular proteins.

The effect of hydration on hydrodynamic properties of globular proteins can be expressed in terms of two quantities: the delta (g/g) parameter and the thickness of the hydration layer. The two paradigms on hydration that originate these alternative measures are described and compared. For the numerical calculation of hydrodynamic properties, from which estimates of hydration can be made, we employ the bead modelling with atomic resolution implemented in programs HYDROPRO and HYDRONMR. As typical, average values, we find 0.3 g/g and a thickness of only approximately 1.2 A. However, noticeable differences in this parameter are found from one protein to another. We have made a numerical analysis, which leaves apart marginal influences of modelling imperfections by simulating properties of a spherical protein. This analysis confirms that the errors that one can attribute to the experimental quantities suffice to explain the observed fluctuations in the hydration parameters. However, for the main purpose of predicting protein solution properties, the above mentioned typical values may be safely used. Particularly for atomic bead modelling, a hydrodynamic radius of approximately 3.2 A yields predictions in very good agreement with experiments.     

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.     

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.     

Low-resolution structures of proteins in solution retrieved from X-ray scattering with a genetic algorithm.

Small-angle x-ray solution scattering (SAXS) is analyzed with a new method to retrieve convergent model structures that fit the scattering profiles. An arbitrary hexagonal packing of several hundred beads containing the problem object is defined. Instead of attempting to compute the Debye formula for all of the possible mass distributions, a genetic algorithm is employed that efficiently searches the configurational space and evolves best-fit bead models. Models from different runs of the algorithm have similar or identical structures. The modeling resolution is increased by reducing the bead radius together with the search space in successive cycles of refinement. The method has been tested with protein SAXS (0.001 < S < 0.06 A(-1)) calculated from x-ray crystal structures, adding noise to the profiles. The models obtained closely approach the volumes and radii of gyration of the known structures, and faithfully reproduce the dimensions and shape of each of them. This includes finding the active site cavity of lysozyme, the bilobed structure of gamma-crystallin, two domains connected by a stalk in betab2-crystallin, and the horseshoe shape of pancreatic ribonuclease inhibitor. The low-resolution solution structure of lysozyme has been directly modeled from its experimental SAXS profile (0.003 < S < 0.03 A(-1)). The model describes lysozyme size and shape to the resolution of the measurement. The method may be applied to other proteins, to the analysis of domain movements, to the comparison of solution and crystal structures, as well as to large macromolecular assemblies.     

Determination of a molecular shape for netrin-4 from hydrodynamic and small angle X-ray scattering measurements

As part of a continuing investigation of netrins, an emerging class of extracellular matrix proteins that are involved in axon guidance activity, we have used dynamic light scattering (DLS) and small angle X-ray scattering to investigate the solution conformation of a truncated version of netrin-4 (Δnetrin-4) that lacks the C-terminal portion. The protein is characterized by a hydrodynamic (Stokes) radius (r(H)) of 4.60 (±0.20) nm, a radius of gyration (r(G)) of 4.42 (±0.20) nm and a maximum particle dimension (D(max)) of 16nm. More detailed ab initio modeling of the SAXS data indicates an extended rod like conformation for Δnetrin-4 in solution-a concept supported by the excellent agreement observed between experimental parameter estimates and those calculated for the ab initio models for Δnetrin-4 by the HYDROPRO program.     

Hydrodynamic properties of a double-helical model for DNA.

The translational and rotational diffusion coefficients of very short DNA fragments have been calculated using a double-helical bead model in which each nucleotide is represented by one bead. The radius of the helix is regarded as an adjustable parameter. The translational coefficient and the perpendicular rotation coefficient agree very well with experimental values for oligonuclotides with 8, 12, and 20 base pairs, for a single value of the helical radius of about 10 A. We have also calculated a nuclear magnetic resonance relaxation time in which the coefficient for rotation about the main axis is involved. As found previously with cylindrical models, the results deviate from experimental values, indicating that the internal motion of the bases has a remarkable amplitude. An attempt to quantify the extent of internal motions is presented.     

SOMO (SOlution MOdeler) differences between X-Ray- and NMR-derived bead models suggest a role for side chain flexibility in protein hydrodynamics

Reduced numbers of frictional/scattering centers are essential for tractable hydrodynamic and small-angle scattering data modeling. We present a method for generating medium-resolution models from the atomic coordinates of proteins, basically by using two nonoverlapping spheres of differing radii per residue. The computed rigid-body hydrodynamic parameters of BPTI, RNase A, and lysozyme models were compared with a large database of critically assessed experimental values. Overall, very good results were obtained, but significant discrepancies between X-ray- and NMR-derived models were found. Interestingly, they could be accounted for by properly considering the extent to which highly mobile surface side chains differently affect translational/rotational properties. Models of larger structures, such as fibrinogen fragment D and citrate synthase, also produced consistent results. Foremost among this method's potential applications is the overall conformation and dynamics of modular/multidomain proteins and of supramolecular complexes. The possibility of merging data from high- and low-resolution structures greatly expands its scope.     

Intrinsic viscosity of bead models for macromolecules and bioparticles

A new method based on the fractal dimension dependence of the hydrodynamic radius is proposed for calculation of the intrinsic viscosity of bead models. The method describes properly the viscosity increment except for elongated structures such as linear aggregates and ellipsoids. It is expected to be useful for very compact structures, for which the volume correction does not improve the results calculated by the modified Oseen tensor. The results obtained for the viscosity increment lie between the volume corrected ones and those determined by the cubic substitution procedure. They are close to the values recalculated from the falling velocities of the models analyzed.     

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

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. Accepted: 1 November 1996     

Modeling the hydration of proteins: prediction of structural and hydrodynamic parameters from X-ray diffraction and scattering data

The implications of protein-water interactions are of importance for understanding the solution behavior of proteins and for analyzing the fine structure of proteins in aqueous solution. Starting from the atomic coordinates, by bead modeling the scattering and hydrodynamic properties of proteins can be predicted reliably (Debye modeling, program HYDRO). By advanced modeling techniques the hydration can be taken into account appropriately: by some kind of rescaling procedures, by modeling a water shell, by iterative comparisons to experimental scattering curves (ab initio modeling) or by special hydration algorithms. In the latter case, the surface topography of proteins is visualized in terms of dot surface points, and the normal vectors to these points are used to construct starting points for placing water molecules in definite positions on the protein envelope. Bead modeling may then be used for shaping the individual atomic or amino acid residues and also for individual water molecules. Among the tuning parameters, the choice of the scaling factor for amino acid hydration and of the molecular volume of bound water turned out to be crucial. The number and position of bound water molecules created by our hydration modeling program HYDCRYST were compared with those derived from X-ray crystallography, and the capability to predict hydration, structural and hydrodynamic parameters (hydrated volume, radius of gyration, translational diffusion and sedimentation coefficients) was compared with the findings generated by the water-shell approach CRYSOL. If the atomic coordinates are unknown, ab initio modeling approaches based on experimental scattering curves can provide model structures for hydrodynamic predictions.     

Estimation of the shape and size of fribrinogen in solution from its hydrodynamic properties using theories for bead models and cylinders

The translational and rotational diffusion coefficients and the intrinsic viscosity of fibrinogen in solution are used to estimate its size, shape and hydration. Experimental data of the three hydrodynamic properties taken from the literature are compared with theoretical predictions for several molecular geometries that have been observed by electron microscopy. Modern theories for the hydrodynamics of bead models and cylindrical particles are employed in the calculations. The discrepancy between experimental results and theoretical predictions for spherical particles rules out the dodecahedral model and indicates that fibrinogen is elongated. The Hall-Slayter nodular model and its refinements perform better but still underestimate the size of the hydrated molecule. The best agreement between theoretical and experimental values is found for a cylindrical particle with length and diameter of about 48 and 6.8 nm, respectively. The hydration is calculated to be 3 g water/g protein. We speculate that, to accommodate such a large amount of water, fibrinogen in solution should be appreciably hydrated.     

Modeling the dynamics of the solvated SL1 domain of HIV-1 genomic RNA

A mode-coupling solution of the Smoluchowski diffusion equation (MCD theory), designed to describe the dynamics of wobbling macromolecules in water, is applied to a macromolecular bead model including water beads in the nearest layers. The necessary statistical averages are evaluated by time averaging along a molecular dynamics (MD) trajectory where both solute and water are introduced as atomistic models. The cross peaks in (1)H nuclear Overhauser effect spectroscopy (NOESY) NMR spectra that are routinely measured to determine biological structures are here calculated for the mutated 23 nucleotides stem-loop fragment of the SL1 domain in the HIV-1(Lai) genomic RNA. The calculations are in acceptable agreement with experiments without requiring any screening of the hydrodynamic interactions. The screening of hydrodynamics was necessary in previous MCD calculations obtained by using the same full atomistic MD trajectory, but a nonsolvated frictional model.     

On the hydrodynamic analysis of macromolecular conformation

Hydrodynamics provides a powerful complementary role to the traditional "high resolution" techniques for the investigation of macromolecular conformation, especially in dilute solution, conditions which are generally inaccessible to other structural techniques. This paper describes the state of art of hydrodynamic representations for macromolecular conformation, in terms of (1) simple but straightforward ellipsoid of revolution modelling; (2) general triaxial ellipsoid modelling; (3) hydrodynamic bead modelling; (4) the ability, especially for polydisperse macromolecular systems, to distinguish between various conformation types; (5) analysis of macromolecular flexibility.     

Analytical ultracentrifugation combined with X-ray and neutron scattering: Experiment and modelling

Analytical ultracentrifugation and solution scattering provide different multi-parameter structural and compositional information on proteins. The joint application of the two methods supplements high resolution structural studies by crystallography and NMR. We summarise the procedures required to obtain equivalent ultracentrifugation and X-ray and neutron scattering data. The constrained modelling of ultracentrifugation and scattering data is important to confirm the experimental data analysis and yields families of best-fit molecular models for comparison with crystallography and NMR structures. This modelling of ultracentrifugation and scattering data is described in terms of starting models, their conformational randomisation in trial-and-error fits, and the identification of the final best-fit models. Seven applications of these methods are described to illustrate the current state-of-the-art. These include the determination of antibody solution structures (the human IgG4 subclass, and oligomeric forms of human IgA and its secretory component), the solution structures of the complement proteins of innate immunity (Factor H and C3/C3u) and their interactions with macromolecular ligands (C-reactive protein), and anionic polysaccharides (heparin). Complementary features of joint ultracentrifugation and scattering experiments facilitate an improved understanding of crystal structures (illustrated for C3/C3u, C-reactive protein and heparin). If a large protein or its complex cannot be crystallised, the joint ultracentrifugation-scattering approach provides a means to obtain an overall macromolecular structure.     

The implementation of SOMO (SOlution MOdeller) in the UltraScan analytical ultracentrifugation data analysis suite: enhanced capabilities allow the reliable hydrodynamic modeling of virtually any kind of biomacromolecule

The interpretation of solution hydrodynamic data in terms of macromolecular structural parameters is not a straightforward task. Over the years, several approaches have been developed to cope with this problem, the most widely used being bead modeling in various flavors. We report here the implementation of the SOMO (SOlution MOdeller; Rai et al. in Structure 13:723–734, 2005) bead modeling suite within one of the most widely used analytical ultracentrifugation data analysis software packages, UltraScan (Demeler in Modern analytical ultracentrifugation: techniques and methods, Royal Society of Chemistry, UK, 2005). The US-SOMO version is now under complete graphical interface control, and has been freed from several constraints present in the original implementation. In the direct beads-per-atoms method, virtually any kind of residue as defined in the Protein Data Bank (e.g., proteins, nucleic acids, carbohydrates, prosthetic groups, detergents, etc.) can be now represented with beads whose number, size and position are all defined in user-editable tables. For large structures, a cubic grid method based on the original AtoB program (Byron in Biophys J 72:408–415, 1997) can be applied either directly on the atomic structure, or on a previously generated bead model. The hydrodynamic parameters are then computed in the rigid-body approximation. An extensive set of tests was conducted to further validate the method, and the results are presented here. Owing to its accuracy, speed, and versatility, US-SOMO should allow to fully take advantage of the potential of solution hydrodynamics as a complement to higher resolution techniques in biomacromolecular modeling.     

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.     

Calculation of hydrodynamic properties of small nucleic acids from their atomic structure

Hydrodynamic properties (translational diffusion, sedimentation coefficients and correlation times) of short B-DNA oligonucleotides are calculated from the atomic-level structure using a bead modeling procedure in which each non-hydrogen atom is represented by a bead. Using available experimental data of hydrodynamic properties for several oligonucleotides, the best fit for the hydrodynamic radius of the atoms is found to be ~2.8 Å. Using this value, the predictions for the properties corresponding to translational motion and end-over-end rotation are accurate to within a few percent error. Analysis of NMR correlation times requires accounting for the internal flexibility of the double helix, and allows an estimation of ~0.85 for the Lipari–Szabo generalized order parameter. Also, the degree of hydration can be determined from hydrodynamics, with a result of ~0.3 g (water)/g (DNA). These numerical results are quite similar to those found for globular proteins. If the hydrodynamic model for the short DNA is simply a cylindrical rod, the predictions for overall translation and rotation are slightly worse, but the NMR correlation times and the degree of hydration, which depend more on the cross-sectional structure, are more severely affected.     

Exploration of the structural features defining the conduction properties of a synthetic ion channel.

The finite-difference Poisson-Boltzmann methodology was applied to a series of parallel, alpha-helical bundle models of the designed ion channel peptide Ac-(LSSLLSL)3-CONH2. This method is able to fully describe the current-voltage curves for this channel and quantitatively explains their cation selectivity and rectification. We examined a series of energy-minimized models representing different aggregation states, side-chain rotamers, and helical rotations, as well as an ensemble of structures from a molecular dynamics trajectory. Potential energies were computed for single, permeating K+ and Cl- ions at a series of positions along a central pathway through the models. A variable-electric-field Nernst-Planck electrodiffusion model was used, with two adjustable parameters representing the diffusion coefficients of K+ and Cl- to scale the individual ion current magnitudes. The ability of a given DelPhi potential profile to fit the experimental data depended strongly on the magnitude of the desolvation of the permeating ion. Below a pore radius of 3.8 A, the predicted profiles showed large energy barriers, and the experimental data could be fit only with unrealistically high values for the K+ and Cl- diffusion coefficients. For pore radii above 3.8 A, the desolvation energies were 2kT or less. The electrostatic calculations were sensitive to positioning of the Ser side chains, with the best fits associated with maximum exposure of the Ser side-chain hydroxyls to the pore. The backbone component was shown to be the major source of asymmetry in the DelPhi potential profiles. Only two of the energy-minimized structures were able to explain the experimental data, whereas an average of the dynamics structures gave excellent agreement with experimental results. Thus this method provides a promising approach to prediction of current-voltage curves from three-dimensional structures of ion channel proteins.     

Computational reconstruction of multidomain proteins using atomic force microscopy data

Classical structural biology techniques face a great challenge to determine the structure at the atomic level of large and flexible macromolecules. We present a novel methodology that combines high-resolution AFM topographic images with atomic coordinates of proteins to assemble very large macromolecules or particles. Our method uses a two-step protocol: atomic coordinates of individual domains are docked beneath the molecular surface of the large macromolecule, and then each domain is assembled using a combinatorial search. The protocol was validated on three test cases: a simulated system of antibody structures; and two experimentally based test cases: Tobacco mosaic virus, a rod-shaped virus; and Aquaporin Z, a bacterial membrane protein. We have shown that AFM-intermediate resolution topography and partial surface data are useful constraints for building macromolecular assemblies. The protocol is applicable to multicomponent structures connected in the polypeptide chain or as disjoint molecules. The approach effectively increases the resolution of AFM beyond topographical information down to atomic-detail structures.