Hydrodynamic bead modeling of biological macromolecules

GRPY: An Accurate Bead Method for Calculation of Hydrodynamic Properties of Rigid Biomacromolecules

Two main problems that arise in the context of hydrodynamic bead modeling are an inaccurate treatment of bead overlaps and the necessity of using volume corrections when calculating intrinsic viscosity. We present a formalism based on the generalized Rotne-Prager-Yamakawa approximation that successfully addresses both of these issues. The generalized Rotne-Prager-Yamakawa method is shown to be highly effective for the calculation of transport properties of rigid biomolecules represented as assemblies of spherical beads of different sizes, both overlapping and nonoverlapping. We test the method on simple molecular shapes as well as real protein structures and compare its performance with other computational approaches.

Fibrinogen species as resolved by HPLC-SAXS data processing within the UltraScan Solution Modeler (US-SOMO) enhanced SAS module

Fibrinogen is a large heterogeneous aggregation/degradation-prone protein playing a central role in blood coagulation and associated pathologies, whose structure is not completely resolved. When a high-molecular-weight fraction was analyzed by size-exclusion high-performance liquid chromatography/small-angle X-ray scattering (HPLC-SAXS), several composite peaks were apparent and because of the stickiness of fibrinogen the analysis was complicated by severe capillary fouling. Novel SAS analysis tools developed as a part of the UltraScan Solution Modeler (US-SOMO; http://somo.uthscsa.edu/), an open-source suite of utilities with advanced graphical user interfaces whose initial goal was the hydrodynamic modeling of biomacromolecules, were implemented and applied to this problem. They include the correction of baseline drift due to the accumulation of material on the SAXS capillary walls, and the Gaussian decomposition of non-baseline-resolved HPLC-SAXS elution peaks. It was thus possible to resolve at least two species co-eluting under the fibrinogen main monomer peak, probably resulting from in-column degradation, and two others under an oligomers peak. The overall and cross-sectional radii of gyration, molecular mass and mass/length ratio of all species were determined using the manual or semi-automated procedures available within the US-SOMO SAS module. Differences between monomeric species and linear and sideways oligomers were thus identified and rationalized. This new US-SOMO version additionally contains several computational and graphical tools, implementing functionalities such as the mapping of residues contributing to particular regions of P(r), and an advanced module for the comparison of primary I(q) versus q data with model curves computed from atomic level structures or bead models. It should be of great help in multi-resolution studies involving hydrodynamics, solution scattering and crystallographic/NMR data.

Prediction of hydrodynamic and other solution properties of rigid proteins from atomic- and residue-level models

Here we extend the ability to predict hydrodynamic coefficients and other solution properties of rigid macromolecular structures from atomic-level structures, implemented in the computer program HYDROPRO, to models with lower, residue-level resolution. Whereas in the former case there is one bead per nonhydrogen atom, the latter contains one bead per amino acid (or nucleotide) residue, thus allowing calculations when atomic resolution is not available or coarse-grained models are preferred. We parameterized the effective hydrodynamic radius of the elements in the atomic- and residue-level models using a very large set of experimental data for translational and rotational coefficients (intrinsic viscosity and radius of gyration) for >50 proteins. We also extended the calculations to very large proteins and macromolecular complexes, such as the whole 70S ribosome. We show that with proper parameterization, the two levels of resolution yield similar and rather good agreement with experimental data. The new version of HYDROPRO, in addition to considering various computational and modeling schemes, is far more efficient computationally and can be handled with the use of a graphical interface.

Developments in the US-SOMO bead modeling suite: new features in the direct residue-to-bead method, improved grid routines, and influence of accessible surface area screening

The US-SOMO suite provides a flexible interface for accurately computing solution parameters from 3D structures of biomacromolecules through bead-modeling approaches. We present an extended analysis of the influence of accessible surface area screening, overlap reduction routines, and approximations for non-coded residues and missing atoms on the computed parameters for models built by the residue-to-bead direct correspondence and the cubic grid methods. Importantly, by taking the theoretical hydration into account at the atomic level, the performance of the grid-type models becomes comparable or exceeds that of the corresponding hydrated residue-to-bead models.

Sedimentation velocity ultracentrifugation analysis for hydrodynamic characterization of G-quadruplex structures

Analytical ultracentrifugation (AUC) is a powerful technique for the characterization of hydrodynamic and thermodynamic properties. The intent of this article is to demonstrate the utility of sedimentation velocity (SV) studies to obtain hydrodynamic information for G-quadruplex (GQ) systems and to provide insights into one part of this process, namely, data analysis of existing SV data. An array of data analysis software is available, mostly written and continually developed by established researchers in the AUC field, with particularly rapid advances in the analysis of SV data. Each program has its own learning curve, and this article is intended as a resource in the data analysis process for beginning researchers in the field. We discuss the application of three of the most commonly used data analysis programs, DCDT+, Sedfit, and SedAnal, to the interpretation of SV data obtained in our laboratory on two GQ systems.

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.

Analytical ultracentrifugation: sedimentation velocity and sedimentation equilibrium

Analytical ultracentrifugation (AUC) is a versatile and powerful method for the quantitative analysis of macromolecules in solution. AUC has broad applications for the study of biomacromolecules in a wide range of solvents and over a wide range of solute concentrations. Three optical systems are available for the analytical ultracentrifuge (absorbance, interference, and fluorescence) that permit precise and selective observation of sedimentation in real time. In particular, the fluorescence system provides a new way to extend the scope of AUC to probe the behavior of biological molecules in complex mixtures and at high solute concentrations. In sedimentation velocity (SV), the movement of solutes in high centrifugal fields is interpreted using hydrodynamic theory to define the size, shape, and interactions of macromolecules. Sedimentation equilibrium (SE) is a thermodynamic method where equilibrium concentration gradients at lower centrifugal fields are analyzed to define molecule mass, assembly stoichiometry, association constants, and solution nonideality. Using specialized sample cells and modern analysis software, researchers can use SV to determine the homogeneity of a sample and define whether it undergoes concentration-dependent association reactions. Subsequently, more thorough model-dependent analysis of velocity and equilibrium experiments can provide a detailed picture of the nature of the species present in solution and their interactions.

Hydrodynamic modeling: the solution conformation of macromolecules and their complexes

Hydrodynamic bead modeling (HBM) is the representation of a macromolecule by an assembly of spheres (or beads) for which measurable hydrodynamic (and related) parameters are then computed in order to understand better the macromolecular solution conformation. An example-based account is given of the main stages in HBM of rigid macromolecules, namely: model construction, model visualization, accounting for hydration, and hydrodynamic calculations. Different types of models are appropriate for different macromolecules, according to their composition, to what is known about the molecule or according to the types of experimental data that the model should reproduce. Accordingly, the construction of models based on atomic coordinates as well as much lower resolution data (e.g., electron microscopy images) is described. Similarly, several programs for hydrodynamic calculations are summarized, some generating the most basic set of solution parameters (e.g., sedimentation and translational diffusion coefficients, intrinsic viscosity, radius of gyration, and Stokes radius) while others extend to data determined by nuclear magnetic resonance, fluorescence anisotropy, and electric birefringence methods. An insight into the topic of hydrodynamic hydration is given, together with some practical suggestions for its satisfactory treatment in the modeling context. All programs reviewed are freely available.

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.

The disintegrin echistatin stabilizes integrin alphaIIbbeta3's open conformation and promotes its oligomerization

We have employed echistatin, a 5.4 kDa snake venom disintegrin, as a model protein to investigate the paradox that small ligand-mimetics can bind to the resting alphaIIbbeta3 integrin while adhesive macromolecules cannot. We characterized the interactions between purified human alphaIIbbeta3 and two recombinant echistatin variants: rEch (1-49) M28L, chosen for its selectivity toward beta3-integrins, and rEch (1-40) M28L, a carboxy-terminal truncation mutant. While both contain an RGD integrin targeting sequence, only rEch (1-49) M28L was an effective inhibitor of alphaIIbbeta3 function. Electron microscopy of rotary shadowed specimens yielded a variety of alphaIIbbeta3 conformers ranging from compact, spherical particles (maximum dimension 22 nm) to the classical "head with two tails" forms (32 nm). The population of larger particles (42-56 nm) increased from 17% to 28% in the presence of rEch (1-49) M28L, indicative of ligand-induced oligomerization. Sedimentation velocity measurements demonstrated that both full length and truncated echistatin perturbed alphaIIbbeta3's solution structure, yielding slower-sedimenting open conformers. Dynamic light scattering showed that rEch (1-49) M28L protected alphaIIbbeta3 from thermal aggregation, raising its transition mid-point from 46 degrees C to 69 degrees C; a smaller shift resulted with rEch (1-40) M28L. Sedimentation equilibrium demonstrated that both echistatin ligands induced substantial alphaIIbbeta3 dimerization. van't Hoff analysis revealed a pattern of entropy/enthalpy compensation similar to tirofiban, a small RGD ligand-mimetic that binds tightly to alphaIIbbeta3, but yields smaller conformational perturbations than echistatin. We propose that echistatin may serve as a paradigm for understanding multidomain adhesive macromolecules because its ability to modulate alphaIIbbeta3's structure resides on an RGD loop, while full disintegrin activity requires an auxiliary site that includes the carboxy-terminal nine amino acid residues.

The solution structure and oligomerization behavior of two bacterial toxins: pneumolysin and perfringolysin O

Pneumolysin (PLY), an important protein virulence factor of the human bacterial pathogen Streptococcus pneumoniae, could be a candidate for inclusion in a new anti-streptococcal vaccine. PLY solution species from monomer via multimeric intermediates to ring-shaped oligomers were studied with time-dependent sedimentation velocity in the analytical ultracentrifuge (AUC). Hydrodynamic bead modeling was used to interpret the data obtained. PLY remained mostly monomeric in solution; intermediate PLY multimers were detected in small quantities. Current understanding of PLY molecular mechanism is guided by a model built on the basis of its homology with perfringolysin O (PFO) for which there is an atomic structure. PFO, a virulence factor of the organism Clostridium perfringens, has almost the same molecular mass as PLY and shares 48% sequence identity and 60% sequence similarity with PLY. We report a comparative low-resolution structural study of PLY and PFO using AUC and small-angle x-ray scattering (SAXS). AUC data demonstrate that both proteins in solution are mostly monodisperse but PLY is a monomer whereas PFO is mostly dimeric. Ab initio dummy atom and dummy residue models for PFO and PLY were restored from the distance distribution function derived from experimental small-angle x-ray scattering curves. In solution, PLY is elongated, consistent with the shape predicted by its high-resolution homology model. The PFO dimer is also an elongated particle whose shape and volume are consistent with a staggered antiparallel dimer.

Hydrodynamic bead modelling of the 2:1 p50–IκBγ complex

NFkappaB is an important and ubiquitous transcription factor formed by various homo- and heterodimers of the NFkappaB family. The active transcription factor regulates genes involved in immune, inflammatory and survival responses. Specificity in gene regulation is achieved, at least in part, by the distinct DNA binding preferences of the various homo- and heterodimers and by the complex pathways that lead to signal-induced degradation of the IkappaB inhibitors. Analytical ultracentrifugation and hydrodynamic bead modelling were used to model the solution structures of the NFkappaB family member p50, its inhibitor IkappaBgamma and their complex. Sedimentation equilibrium (SE) and sedimentation velocity (SV) data show that p50 is a dimer in solution with a sedimentation coefficient consistent with a conformation intermediate between the closed conformation observed in the crystal structure of the p50 (N-terminal domain)-p65 heterodimer complexed with IkappaBalpha and the open conformation adopted by p50 when bound to DNA. SE and SV data show that IkappaBgamma is a monomer in solution and is prone to aggregation over time. p50 forms a 2:1 stoichiometric complex with IkappaBgamma in solution with a sedimentation coefficient consistent with a closed conformation for the p50 dimer.

A model for sedimentation in inhomogeneous media. II. Compressibility of aqueous and organic solvents

The effects of solvent compressibility on the sedimentation behavior of macromolecules as observed in analytical ultracentrifugation are examined. Expressions for the density and pressure distributions in the solution column are derived and combined with the finite element solution of the Lamm equation in inhomogeneous media to predict the macromolecular concentration distributions under different conditions. Independently, analytical expressions are derived for the sedimentation of non-diffusing particles in the limit of low compressibility. Both models are quantitatively consistent and predict solvent compressibility to result in a reduction of the sedimentation rate along the solution column and a continuous accumulation of solutes in the plateau region. For both organic and aqueous solvents, the calculated deviations from the sedimentation in incompressible media can be very large and substantially above the measurement error. Assuming conventional configurations used for sedimentation velocity experiments in analytical ultracentrifugation, neglect of the compressibility of water leads to systematic errors underestimating sedimentation coefficients by approximately 1% at a rotor speeds of 45000 rpm, but increasing to 2-5% with increasing rotor speeds and decreasing macromolecular size. The proposed finite element solution of the Lamm equation can be used to take solvent compressibility quantitatively into account in direct boundary models for discrete species, sedimentation coefficient distributions or molar mass distributions. Using the analytical expressions for the sedimentation of non-diffusing particles, the ls-g*(s) distribution of apparent sedimentation coefficients is extended to the analysis of sedimentation in compressible solvents. The consideration of solvent compressibility is highly relevant not only when using organic solvents, but also in aqueous solvents when precise sedimentation coefficients are needed, for example, for hydrodynamic modeling.

Numerical path integration technique for the calculation of transport properties of proteins

We present a new technique for the computation of both the translational diffusivity and the intrinsic viscosity of macromolecules, and apply it here to proteins. Traditional techniques employ finite element representations of the surface of the macromolecule, taking the surface to be a union of spheres or of polygons, and have computation times that are O(m(3)) where m is the number of finite elements. The new technique, a numerical path integration method, has computation times that are only O(m). We have applied the technique to approximately 1000 different protein structures. The computed translational diffusivities and intrinsic viscosities are, to lowest order, proportional respectively to N(-1/3)(R) and N(0)(R), where N(R) is the number of amino acid residues in the protein. Our calculations also show some correlation with the shape of the molecule, as represented by the ratio m(2)/m(3), where m(2) and m(3) are, respectively, the middle and the smallest of the three principal moments of inertia. Comparisons with a number of experimental results are also performed, with results generally consistent to within experimental error.

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.

Building hydrodynamic bead-shell models for rigid bioparticles of arbitrary shape

The calculation of hydrodynamic and other solution properties of rigid macromolecules, using bead-shell model methodologies, requires the specification of the macromolecular shape in a format that can be interfaced with existing programs for hydrodynamic computations. Here, a procedure is presented for such a structural specification that is applicable to arbitrarily shaped particles. A computer program (MAKEPIXB), in which the user inserts the code needed to determine the structure, produces an structural file that is interpreted by another program (HYDROPIX) which is in charge of the computation of properties. As simple and yet illustrative examples we consider two cases: (1) dimeric structures composed of ellipsoidal subunits; and (2) toroidal structures, presenting simple equations that predict the properties of toroids with varying radial ratios.

Comparative investigations of biopolymer hydration by physicochemical and modeling techniques

The comparative investigation of biopolymer hydration by physicochemical techniques, particularly by small-angle X-ray scattering, has shown that the values obtained differ over a wide range, depending on the nature of the polymer and the environmental conditions. In the case of simple proteins, a large number of available data allow the derivation of a realistic average value for the hydration (0.35 g of water per gram of protein). As long as the average properties of proteins are considered, the use of such a default value is sufficient. Modeling approaches may be used advantageously, in order to differentiate between different assumptions and hydration contributions, and to correctly predict hydrodynamic properties of biopolymers on the basis of their three-dimensional structure. Problems of major concern are the positioning and the properties of the water molecules on the biopolymer surface. In this context, different approaches for calculating the molecular volume and surface of biopolymers have been applied, in addition to the development of appropriate hydration algorithms.

Biophysical studies by ultracentrifugation

Advances in data analysis are broadening the applicability of ultracentrifugation to the characterization of macromolecular behavior in complex solution. The direct fitting of sedimentation velocity data to the Lamm equation is emerging as a very powerful means to characterize size distributions, improve the precision of data analysis and increase experimental throughput. With improvements in data acquisition and analysis, ultracentrifugation is poised to make significant contributions to our understanding of how macromolecules behave in vivo.

Binding of a fibrinogen mimetic stabilizes integrin alphaIIbbeta3's open conformation

The platelet integrin alphaIIbbeta3 is representative of a class of heterodimeric receptors that upon activation bind extracellular macromolecular ligands and form signaling clusters. This study examined how occupancy of alphaIIbbeta3's fibrinogen binding site affected the receptor's solution structure and stability. Eptifibatide, an integrin antagonist developed to treat cardiovascular disease, served as a high-affinity, monovalent model ligand with fibrinogen-like selectivity for alphaIIbbeta3. Eptifibatide binding promptly and reversibly perturbed the conformation of the alphaIIbbeta3 complex. Ligand-specific decreases in its diffusion and sedimentation coefficient were observed at near-stoichiometric eptifibatide concentrations, in contrast to the receptor-perturbing effects of RGD ligands that we previously observed only at a 70-fold molar excess. Eptifibatide promoted alphaIIbbeta3 dimerization 10-fold more effectively than less selective RGD ligands, as determined by sedimentation equilibrium. Eptifibatide-bound integrin receptors displayed an ectodomain separation and enhanced assembly of dimers and larger oligomers linked through their stalk regions, as seen by transmission electron microscopy. Ligation with eptifibatide protected alphaIIbbeta3 from SDS-induced subunit dissociation, an effect on electrophoretic mobility not seen with RGD ligands. Despite its distinct cleft, the open conformer resisted guanidine unfolding as effectively as the ligand-free integrin. Thus, we provide the first demonstration that binding a monovalent ligand to alphaIIbbeta3's extracellular fibrinogen-recognition site stabilizes the receptor's open conformation and enhances self-association through its distant transmembrane and/or cytoplasmic domains. By showing how eptifibatide and RGD peptides, ligands with distinct binding sites, each affects alphaIIbbeta3's conformation, our findings provide new mechanistic insights into ligand-linked integrin activation, clustering and signaling.

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.