Macromolecular size-and-shape distributions by sedimentation velocity analytical ultracentrifugation

Sedimentation velocity analytical ultracentrifugation is an important tool in the characterization of macromolecules and nanoparticles in solution. The sedimentation coefficient distribution c(s) of Lamm equation solutions is based on the approximation of a single, weight-average frictional coefficient of all particles, determined from the experimental data, which scales the diffusion coefficient to the sedimentation coefficient consistent with the traditional s approximately M(2/3) power law. It provides a high hydrodynamic resolution, where diffusional broadening of the sedimentation boundaries is deconvoluted from the sedimentation coefficient distribution. The approximation of a single weight-average frictional ratio is favored by several experimental factors, and usually gives good results for chemically not too dissimilar macromolecules, such as mixtures of folded proteins. In this communication, we examine an extension to a two-dimensional distribution of sedimentation coefficient and frictional ratio, c(s,f(r)), which is representative of a more general set of size-and-shape distributions, including mass-Stokes radius distributions, c(M,R(S)), and sedimentation coefficient-molar mass distributions c(s,M). We show that this can be used to determine average molar masses of macromolecules and characterize macromolecular distributions, without the approximation of any scaling relationship between hydrodynamic and thermodynamic parameters.     

Size-distribution analysis of macromolecules by sedimentation velocity ultracentrifugation and lamm equation modeling

A new method for the size-distribution analysis of polymers by sedimentation velocity analytical ultracentrifugation is described. It exploits the ability of Lamm equation modeling to discriminate between the spreading of the sedimentation boundary arising from sample heterogeneity and from diffusion. Finite element solutions of the Lamm equation for a large number of discrete noninteracting species are combined with maximum entropy regularization to represent a continuous size-distribution. As in the program CONTIN, the parameter governing the regularization constraint is adjusted by variance analysis to a predefined confidence level. Estimates of the partial specific volume and the frictional ratio of the macromolecules are used to calculate the diffusion coefficients, resulting in relatively high-resolution sedimentation coefficient distributions c(s) or molar mass distributions c(M). It can be applied to interference optical data that exhibit systematic noise components, and it does not require solution or solvent plateaus to be established. More details on the size-distribution can be obtained than from van Holde-Weischet analysis. The sensitivity to the values of the regularization parameter and to the shape parameters is explored with the help of simulated sedimentation data of discrete and continuous model size distributions, and by applications to experimental data of continuous and discrete protein mixtures.     

Modern analytical ultracentrifugation in protein science: a tutorial review

Analytical ultracentrifugation (AU) is reemerging as a versatile tool for the study of proteins. Monitoring the sedimentation of macromolecules in the centrifugal field allows their hydrodynamic and thermodynamic characterization in solution, without interaction with any matrix or surface. The combination of new instrumentation and powerful computational software for data analysis has led to major advances in the characterization of proteins and protein complexes. The pace of new advancements makes it difficult for protein scientists to gain sufficient expertise to apply modern AU to their research problems. To address this problem, this review builds from the basic concepts to advanced approaches for the characterization of protein systems, and key computational and internet resources are provided. We will first explore the characterization of proteins by sedimentation velocity (SV). Determination of sedimentation coefficients allows for the modeling of the hydrodynamic shape of proteins and protein complexes. The computational treatment of SV data to resolve sedimenting components has been achieved. Hence, SV can be very useful in the identification of the oligomeric state and the stoichiometry of heterogeneous interactions. The second major part of the review covers sedimentation equilibrium (SE) of proteins, including membrane proteins and glycoproteins. This is the method of choice for molar mass determinations and the study of self-association and heterogeneous interactions, such as protein-protein, protein-nucleic acid, and protein-small molecule binding.     

Characterization of Phospholipid Mixed Micelles by Translational Diffusion

The concentration dependence of the translational self diffusion rate, D (s), has been measured for a range of micelle and mixed micelle systems. Use of bipolar gradient pulse pairs in the longitudinal eddy current delay experiment minimizes NOE attenuation and is found critical for optimizing sensitivity of the translational diffusion measurement of macromolecules and aggregates. For low volume fractions Phi (Phi\\ le 15% v/v) of the micelles, experimental measurement of the concentration dependence, combined with use of the D (s)= D (o)(1-3.2lambdaPhi) relationship, yields the hydrodynamic volume. For proteins, the hydrodynamic volume, derived from D (s) at infinitely dilute concentration, is found to be about 2.6 times the unhydrated molecular volume. Using the data collected for hen egg white lysozyme as a reference, diffusion data for dihexanoyl phosphatidylcholine (DHPC) micelles indicate approximately 27 molecules per micelle, and a critical micelle concentration of 14 mM. Differences in translational diffusion rates for detergent and long chain phospholipids in mixed micelles are attributed to rapid exchange between free and micelle-bound detergent. This difference permits determination of the free detergent concentration, which, for a high detergent to long chain phospholipid molar ratio, is found to depend strongly on this ratio. The hydrodynamic volume of DHPC/POPC bicelles, loaded with an M2 channel peptide homolog, derived from translational diffusion, predicts a rotational correlation time that slightly exceeds the value obtained from peptide (15)N relaxation data.     

Hydrodynamics of concentrated proteoglycan solutions

The dynamics of water transport in proteoglycan compartments has been studied in relation to osmotic flow (proteoglycan diffusion) and hydraulic permeability (proteoglycan sedimentation) in concentrated solutions of proteoglycan subunit and native proteoglycan aggregate isolated from Swarm rat chondrosarcoma. A central parameter that describes the kinetics of both types of water movement is the hydrodynamic frictional coefficient of water with proteoglycan. The frictional coefficient is markedly concentration dependent, increasing with increasing concentration, and highlights important structural features and types of organization of the proteoglycans in concentrated solutions. These include the requirements that proteoglycans in the extracellular matrix not to be immobilized but to have translational diffusive mobility and concentration gradients to be osmotically active, that chondroitin sulfate segmental mobility describing translational motion largely determines osmotic flow and hydraulic permeability of the proteoglycans, and that the proteoglycans exhibit an enhanced ability to resist flow as compared to other macromolecules. Additional dynamic studies suggest the formation of transient super-aggregate structures may occur at high concentrations which endows the proteoglycan subunit hydrodynamic properties similar to proteoglycan aggregate.     

Diffusion of the Reaction Boundary of Rapidly Interacting Macromolecules in Sedimentation Velocity

Sedimentation velocity analytical ultracentrifugation combines relatively high hydrodynamic resolution of macromolecular species with the ability to study macromolecular interactions, which has great potential for studying dynamically assembled multiprotein complexes. Complicated sedimentation boundary shapes appear in multicomponent mixtures when the timescale of the chemical reaction is short relative to the timescale of sedimentation. Although the Lamm partial differential equation rigorously predicts the evolution of concentration profiles for given reaction schemes and parameter sets, this approach is often not directly applicable to data analysis due to experimental and sample imperfections, and/or due to unknown reaction pathways. Recently, we have introduced the effective particle theory, which explains quantitatively and in a simple physical picture the sedimentation boundary patterns arising in the sedimentation of rapidly interacting systems. However, it does not address the diffusional spread of the reaction boundary from the cosedimentation of interacting macromolecules, which also has been of long-standing interest in the theory of sedimentation velocity analytical ultracentrifugation. Here, effective particle theory is exploited to approximate the concentration gradients during the sedimentation process, and to predict the overall, gradient-average diffusion coefficient of the reaction boundary. The analysis of the heterogeneity of the sedimentation and diffusion coefficients across the reaction boundary shows that both are relatively uniform. These results support the application of diffusion-deconvoluting sedimentation coefficient distributions c(s) to the analysis of rapidly interacting systems, and provide a framework for the quantitative interpretation of the diffusional broadening and the apparent molar mass values of the effective sedimenting particle in dynamically associating systems.     

Size-distribution analysis of proteins by analytical ultracentrifugation: strategies and application to model systems.

Strategies for the deconvolution of diffusion in the determination of size-distributions from sedimentation velocity experiments were examined and developed. On the basis of four different model systems, we studied the differential apparent sedimentation coefficient distributions by the time-derivative method, g(s*), and by least-squares direct boundary modeling, ls-g*(s), the integral sedimentation coefficient distribution by the van Holde-Weischet method, G(s), and the previously introduced differential distribution of Lamm equation solutions, c(s). It is shown that the least-squares approach ls-g*(s) can be extrapolated to infinite time by considering area divisions analogous to boundary divisions in the van Holde-Weischet method, thus allowing the transformation of interference optical data into an integral sedimentation coefficient distribution G(s). However, despite the model-free approach of G(s), for the systems considered, the direct boundary modeling with a distribution of Lamm equation solutions c(s) exhibited the highest resolution and sensitivity. The c(s) approach requires an estimate for the size-dependent diffusion coefficients D(s), which is usually incorporated in the form of a weight-average frictional ratio of all species, or in the form of prior knowledge of the molar mass of the main species. We studied the influence of the weight-average frictional ratio on the quality of the fit, and found that it is well-determined by the data. As a direct boundary model, the calculated c(s) distribution can be combined with a nonlinear regression to optimize distribution parameters, such as the exact meniscus position, and the weight-average frictional ratio. Although c(s) is computationally the most complex, it has the potential for the highest resolution and sensitivity of the methods described.     

Improved measurement of the rotor temperature in analytical ultracentrifugation

Sedimentation velocity is a classical method for measuring the hydrodynamic, translational friction coefficient of biological macromolecules. In a recent study comparing various analytical ultracentrifuges, we showed that external calibration of the scan time, radial magnification, and temperature is critically important for accurate measurements (Anal. Biochem. 440 (2013) 81–95). To achieve accurate temperature calibration, we introduced the use of an autonomous miniature temperature logging integrated circuit (Maxim Thermochron iButton) that can be inserted into an ultracentrifugation cell assembly and spun at low rotor speeds. In the current work, we developed an improved holder for the temperature sensor located in the rotor handle. This has the advantage of not reducing the rotor capacity and allowing for a direct temperature measurement of the spinning rotor during high-speed sedimentation velocity experiments up to 60,000 rpm. We demonstrated the sensitivity of this approach by monitoring the adiabatic cooling due to rotor stretching during rotor acceleration and the reverse process on rotor deceleration. Based on this, we developed a procedure to approximate isothermal rotor acceleration for better temperature control.     

Maillard induced complexes of bovine serum albumin - a dilute solution study

Association of bovine serum albumin (BSA) on heating in the presence and absence of 2% xylose has been studied using dynamic light scattering and sedimentation velocity. When 3% solutions of the protein alone are heated at 95°C association products are formed with molar masses of 2 × 10⁶g/mol, a value which is independent of the time of heating. These aggregates can be dissociated in solvents that disrupt non-covalent bonds. When the reducing sugar xylose is present there is a continuous change in the hydrodynamic properties with time. After 80 min a molar mass in excess of 7 × 10⁶g/mol is obtained. This increase in molar mass is attributed to additional non-disulphide linkages resulting from the Maillard reaction. Information about the gross conformation of the Maillard induced association products has been obtained from MHKS (Mark-Houwink-Kuhn-Sakarada) double logarithmic plots of D_20,w and s_20,w against molar mass. The values of the MHKS coefficients obtained are most consistent with a linear rod: i.e. the association is of an end-to-end type     

Changes of the oligomeric structure of legumin from pea (Pisum sativum L.) after succinylation

The influence of various levels of succinylation on the structure of the legumin from pea seed has been studied by the techniques of sedimentation velocity, viscometry, fluorescence and circular dichroism spectroscopy, as well as dynamic light scattering. The protein dissociates gradually into the 3S subunit forming a 7S intermediate. At a level of 75-80% succinylation, sudden unfolding of the protein occurs characterized by drastic changes in viscometric and spectroscopic properties. The fluorescence spectra point to the formation of a novel organized structure at a moderate degree of modification before the molecular unfolding takes place. The succinylated subunit was shown to have a sedimentation coefficient of 3.2S, a diffusion coefficient of 5.03 x 10(-7) cm2 . s-1 a Stokes' radius of 4.24 nm, a partial specific volume of 0.703 ml/g, an intrinsic viscosity of 0.13 dl/g, a molar mass of 52.2 kDa and a frictional ratio of 1.74.     

Some measurements of the shape and hydrodynamic properties of yeast phosphoglycerate kinase (E.C.2.7.2.3).

Using values obtained for sedimentation and diffusion constants the relative mass of phosphoglycerate kinase was calculated to be 45 800 +/- 1700. This value is higher than was previously estimated and the difference is thought to be caused by contamination of earlier crystalline preparations. Using the coordinates from X-ray crystallography it was found possible to calculate a frictional ratio for a linear dumb-bell (1.115) which compared well with the ratio calculated from diffusion (1.114 +/- 0.033). Since the calculated ratio for a bent molecule was 1.020 the natural state of the molecule in solution is essentially linear. From the concentration dependence of sedimentation and diffusion was calculated the effective interactive radius which resembles haemoglobin in its relationship to the molecular radius.     

Molecular mass determination by sedimentation velocity experiments and direct fitting of the concentration profiles.

A new method for the direct molecular mass determination from sedimentation velocity experiments is presented. It is based on a nonlinear least squares fitting procedure of the concentration profiles and simultaneous estimation of the sedimentation and diffusion coefficients using approximate solutions of the Lamm equation. A computer program, LAMM, was written by using five different model functions derived by Fujita (1962, 1975) to describe the sedimentation of macromolecules during centrifugation. To compare the usefulness of these equations for the analysis of hydrodynamic results, the approach was tested on data sets of Claverie simulations as well as experimental curves of some proteins. A modification for one of the model functions is suggested, leading to more reliable sedimentation and diffusion coefficients estimated by the fitting procedure. The method seems useful for the rapid molecular mass determination of proteins larger than 10 kDa. One of the equations of the Archibald type is also suitable for compounds of low molecular mass, probably less than 10 kDa, because this model function requires neither the plateau region nor a meniscus free of solute.     

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.     

Interpretation of the stokes radius of macromolecules determined by gel filtration chromatography

From a comparison of the gel chromatographic properties of large randomly-coiled polypeptides in 6 M guanidine hydrochloride and of large globular proteins, we found that the distribution coefficient was more closely correlated with the intrinsic viscosity-based Stokes radius than with the translational frictional coefficient-based Stokes radius. This means that the effect of the hydrodynamic flow of dissolved molecules during gel chromatography should be considered. The ratio of transport of solute by bulk flow as compared with that by net diffusion (i.e., Brownian motion) is large under some conditions. On the other hand, we consider that the distribution coefficient obtained in static equilibrium experiments should be determined by the translational frictional coefficient-based Stokes radius, since the solvent does not flow. On this basis, we discuss the meaning of the Stokes radius and the separation mechanism of macromolecules by gel filtration.     

Adair was right in his time

The subunit molar mass of hemoglobin was established in the 19th century by chemical analysis, the tetramer structure by osmotic pressure determination in 1924 and by the newly developed analytical ultracentrifuge in 1926, which became a powerful tool for biological macromolecule molar mass determinations. The Svedberg equation was derived by eliminating the translational friction coefficient relating to sedimentation and diffusion in the ultracentrifuge in a strictly solute/solvent vanishing concentration two-component system analysis. A differential equation describing the radial equilibrium concentration distribution in the ultracentrifuge was also derived, both yielding the buoyant molar mass (1-nu2rho)M2 term. Many years later it was realized that solutions of biological macromolecules are multicomponent systems and the two-component analysis leads to minor or major erroneous results. Thermodynamic derivation of an equation for multicomponent systems redefines the buoyant molar mass terms by (deltarho/deltac2)muM2, leading to correct molar mass (g/mol) values following determination of the density increment at constant chemical potentials of diffusible solutes, and powerfully connects the analytical sedimentation equation to the osmotic pressure concentration derivative and, in a broad complementary sense, to light, X-ray and neutron scattering experiments. Macromolecular interactions can be studied with high precision and solute-solvent interactions yield powerful information relating to "thermodynamic" hydration, closely related to hydration derived from X-ray diffraction, as well as solute-cosolute interactions. A series of examples is given to demonstrate the correctness and usefulness of the thermodynamic multicomponent system approach. It is a strange fact that in current analytical ultracentrifugation analysis the elegant and powerful multicomponent solution technology is almost totally disregarded and the classical limited validity Svedberg approach is used uniquely.     

Analytical ultracentrifugation as a tool for studying protein interactions

The last two decades have led to significant progress in the field of analytical ultracentrifugation driven by instrumental, theoretical, and computational methods. This review will highlight key developments in sedimentation equilibrium (SE) and sedimentation velocity (SV) analysis. For SE, this includes the analysis of tracer sedimentation equilibrium at high concentrations with strong thermodynamic non-ideality, and for ideally interacting systems, the development of strategies for the analysis of heterogeneous interactions towards global multi-signal and multi-speed SE analysis with implicit mass conservation. For SV, this includes the development and applications of numerical solutions of the Lamm equation, noise decomposition techniques enabling direct boundary fitting, diffusion deconvoluted sedimentation coefficient distributions, and multi-signal sedimentation coefficient distributions. Recently, effective particle theory has uncovered simple physical rules for the co-migration of rapidly exchanging systems of interacting components in SV. This has opened new possibilities for the robust interpretation of the boundary patterns of heterogeneous interacting systems. Together, these SE and SV techniques have led to new approaches to study macromolecular interactions across the entire spectrum of affinities, including both attractive and repulsive interactions, in both dilute and highly concentrated solutions, which can be applied to single-component solutions of self-associating proteins as well as the study of multi-protein complex formation in multi-component solutions.     

Hydrodynamic and molecular characteristics of polyelectrolyte complexes between sodium dextransulfate and chitosan hydrochloride

Hydrodynamic and molecular characteristics of particles of polyelectrolyte complexes (PEC) between sodium dextransulfate and chitosan hydrochloride were studied by various physicochemical methods (high-rate sedimentation, viscosimetry, turbidimetry, and diffusion). As was shown, the complex formation is accompanied by increase in the average sizes with simultaneous changes in the shape of the particles of the investigated PEC. According to the proposed polycomplexes model, side by side aggregation of the taken macromolecules could cause disordering of adjacent helical parts of polyanion-matrix that facilitates the formation of sphere-like polycomplexes particles.     

Prediction of the rotational diffusion behavior of biopolymers on the basis of their solution or crystal structure

Two low structure-resolution methods are proposed for prediction of rotational diffusion parameters. The indirect procedure is based on the structure of a molecule in solution or in crystal, and uses the structure parameters of radius of gyration, and low-resolution molecular surface and volume, determined from measured or theoretically calculated small-angle x-ray scattering intensities, to estimate a frictional equivalent ellipsoid of revolution. The direct method starts mainly from the crystallographic structure of a molecule and calculates the triaxial inertia equivalent ellipsoid, experimentally calibrated by translation diffusion data, to simulate the frictional behavior. The predicted harmonic mean of the rotational correlation times of compact globular macromolecules with molar masses of 14,000-65,000 g/mol agree with experimental results within the error limits. The prediction method is recommended for expert systems in structure research and for detection of internal protein flexibility or marker mobility by nmr and electron paramagnetic resonance experiments.     

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.