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.     

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.     

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.     

Analytical ultracentrifugation for the study of protein association and assembly

Analytical ultracentrifugation remains pre-eminent among the methods used to study the interactions of macromolecules under physiological conditions. Recent developments in analytical procedures allow the high resolving power of sedimentation velocity methods to be coupled to sedimentation equilibrium approaches and applied to both static and dynamic associations. Improvements in global modeling based on numerical solutions of the Lamm equation have generated new sedimentation velocity applications with an emphasis on data interpretation using sedimentation coefficient or molar mass distributions. Procedures based on the use of multiple optical signals from absorption and interference optics for the analysis of the sedimentation velocity and equilibrium behavior of more complex interactions have now been developed. New applications of tracer sedimentation equilibrium experiments and the development of a fluorescence optical system for the analytical ultracentrifuge extend the accessible concentration range over several orders of magnitude and, coupled with the new analytical procedures, provide powerful new tools for studies of both weak and strong macromolecular interactions in solution.     

Non-ideality by sedimentation velocity of halophilic malate dehydrogenase in complex solvents

We have investigated the potential of sedimentation velocity analytical ultracentrifugation for the measurement of the second virial coefficients of proteins, with the goal of developing a method that allows efficient screening of different solvent conditions. This may be useful for the study of protein crystallization. Macromolecular concentration distributions were modeled using the Lamm equation with the approximation of linear concentration dependencies of the diffusion constant, D = D(o) (1 + k(D)c), and the reciprocal sedimentation coefficient s = s(o)/(1 + k(s)c). We have studied model distributions for their information content with respect to the particle and its non-ideal behavior, developed a strategy for their analysis by direct boundary modeling, and applied it to data from sedimentation velocity experiments on halophilic malate dehydrogenase in complex aqueous solvents containing sodium chloride and 2-methyl-2,4-pentanediol, including conditions near phase separation. Using global modeling for three sets of data obtained at three different protein concentrations, very good estimates for k(s) and s degrees and also for D degrees and the buoyant molar mass were obtained. It was also possible to obtain good estimates for k(D) and the second virial coefficients. Modeling of sedimentation velocity profiles with the non-ideal Lamm equation appears as a good technique to investigate weak inter-particle interactions in complex solvents and also to extrapolate the ideal behavior of the particle.     

Variable-Field Analytical Ultracentrifugation: I. Time-Optimized Sedimentation Equilibrium

Sedimentation equilibrium (SE) analytical ultracentrifugation (AUC) is a gold standard for the rigorous determination of macromolecular buoyant molar masses and the thermodynamic study of reversible interactions in solution. A significant experimental drawback is the long time required to attain SE, which is usually on the order of days. We have developed a method for time-optimized SE (toSE) with defined time-varying centrifugal fields that allow SE to be attained in a significantly (up to 10-fold) shorter time than is usually required. To achieve this, numerical Lamm equation solutions for sedimentation in time-varying fields are computed based on initial estimates of macromolecular transport properties. A parameterized rotor-speed schedule is optimized with the goal of achieving a minimal time to equilibrium while limiting transient sample preconcentration at the base of the solution column. The resulting rotor-speed schedule may include multiple over- and underspeeding phases, balancing the formation of gradients from strong sedimentation fluxes with periods of high diffusional transport. The computation is carried out in a new software program called TOSE, which also facilitates convenient experimental implementation. Further, we extend AUC data analysis to sedimentation processes in such time-varying centrifugal fields. Due to the initially high centrifugal fields in toSE and the resulting strong migration, it is possible to extract sedimentation coefficient distributions from the early data. This can provide better estimates of the size of macromolecular complexes and report on sample homogeneity early on, which may be used to further refine the prediction of the rotor-speed schedule. In this manner, the toSE experiment can be adapted in real time to the system under study, maximizing both the information content and the time efficiency of SE experiments.     

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.     

Sliding clamp of the bacteriophage T4 polymerase has open and closed subunit interfaces in solution.

The sliding clamps of bacteriophage T4 (gp45), Escherichia coli (beta clamp), and yeast (PCNA) are required for processive DNA synthesis by their cognate DNA polymerases. The X-ray crystal structures of all three of these clamps have been shown to be closed, circular complexes. This paper reports investigations of the solution structure of bacteriophage T4 gp45 by analytical ultracentrifugation, fluorescence, and hydrodynamic modeling. Mutants of gp45 with inter- and intrasubunit disulfide bonds were created to alter the solution structure of gp45, with additional mutagenesis used to investigate the importance of the proline-rich loop region found between the two domains of each gp45 monomer. The wild-type gp45 trimer assembles from monomers cooperatively with a dissociation constant of 0.21 microM2 and values between 0.088 and 0. 32 microM2 for the mutants. Velocity ultracentrifugation experiments showed that wild-type gp45 possesses a sedimentation coefficient strongly dependent on concentration, typical of asymmetric or elongated molecules, that when extrapolated to zero concentration yields a sedimentation coefficient of 4.0 S. The loop and the disulfide mutants exhibited sedimentation coefficients with little concentration dependence, typical of symmetric or spherical molecules, that when extrapolated to zero concentration yielded sedimentation coefficients of 4.4-4.8 S. The lower sedimentation coefficient in the former case is consistent with wild-type gp45 being more asymmetric or elongated than the mutant forms. Fluorescence-resonance energy-transfer experiments were used to measure the distance between two amino acids (W91 and V162C-coumarin) on opposite sides of the gp45 subunit interface. For an intrasubunit disulfide mutant, the distance between these two amino acids was determined to be 19 A (14 A in the X-ray crystal structure), consistent with a closed complex. For the mutants without intrasubunit disulfides, the efficiency of fluorescence-resonance energy transfer was in accord with a model of gp45 being an open complex composed of two closed subunit interfaces and a third open interface separated by a distance of 35-38 A. The collective data supplemented with hydrodynamic modeling were consistent with gp45 subunit separation achieved within the plane of the gp45 ring.     

Sedimentation velocity analysis of highly heterogeneous systems

This article discusses several improvements to the van Holde-Weischet (vHW) method [Biopolymers 17 (1978) 1387] that address its capability to deal with sedimentation coefficient distributions spanning a large range of s values. The method presented here allows the inclusion of scans early and late in the experiment that ordinarily would need to be excluded from the analysis due to ultracentrifuge cell end effects. Scans late in the experiment are compromised by the loss of a defined plateau region and by back-diffusion from the bottom of the cell. Early scans involve partial boundaries that have not fully cleared the meniscus. In addition, a major refinement of the algorithm for determining the boundary fractions is introduced, taking into account different degrees of radial dilution for different species in the system. The method retains its desirable model-independent properties (the analysis of sedimentation data does not require prior knowledge of a user-imposed model or range of sedimentation coefficients) and reports diffusion-corrected s value distributions, which can be presented either in a histogram format or the traditional integral distribution format. Data analyzed with the traditional vHW method are compared with those of the improved method to demonstrate the benefit from the added information in the analysis.     

Enhanced resolution of sedimentation coefficient distribution profiles by extrapolation to infinite time

Sedimentation velocity experiments can be used to identify two or more independent non-interacting macromolecules, which differ in their size by only a few percent. The procedure requires the extrapolation of differential apparent sedimentation coefficient distributions obtained at different running time to t → ∞ and works because it eliminates or greatly reduces diffusion effects. Here, we present an improved time extrapolation function of sedimentation distribution profiles originally presented by Stafford (In: Harding, Rowe, Horton (eds.) Analytical ultracentrifugation in biochemistry and polymer science, 1992). We describe a computing procedure with the program lamm to analyze concentration profiles obtained by absorbance or interference optics that utilizes suitable smoothing methods for noisy data sets and present examples which include time invariant noises.     

Accounting for thermodynamic non-ideality in the Guinier region of small-angle scattering data of proteins

Hydrodynamic studies of the solution properties of proteins and other biological macromolecules are often hard to interpret when the sample is present at a reasonably concentrated solution. The reason for this is that solutions exhibit deviations from ideal behaviour which is manifested as thermodynamic non-ideality. The range of concentrations at which this behaviour typically is exhibited is as low as 1–2 mg/ml, well within the range of concentrations used for their analysis by techniques such as small-angle scattering. Here we discuss thermodynamic non-ideality used previously used in the context of light scattering and sedimentation equilibrium analytical ultracentrifugation and apply it to the Guinier region of small-angle scattering data. The results show that there is a complementarity between the radially averaged structure factor derived from small-angle X-ray scattering/small-angle neutron scattering studies and the second virial coefficient derived from sedimentation equilibrium analytical ultracentrifugation experiments.     

Evaluating the stoichiometry of macromolecular complexes using multisignal sedimentation velocity

Gleaning information regarding the molecular physiology of macromolecular complexes requires knowledge of their component stoichiometries. In this work, a relatively new means of analyzing sedimentation velocity (SV) data from the analytical ultracentrifuge is examined in detail. The method depends on collecting concentration profile data simultaneously using multiple signals, like Rayleigh interferometry and UV spectrophotometry. If the cosedimenting components of a complex are spectrally distinguishable, continuous sedimentation-coefficient distributions specific for each component can be calculated to reveal the molar ratio of the complex's components. When combined with the hydrodynamic information available from the SV data, a stoichiometry can be derived. Herein, the spectral properties of sedimenting species are systematically explored to arrive at a predictive test for whether a set of macromolecules can be spectrally resolved in a multisignal SV (MSSV) experiment. Also, a graphical means of experimental design and criteria to judge the success of the spectral discrimination in MSSV are introduced. A detailed example of the analysis of MSSV experiments is offered, and the possibility of deriving equilibrium association constants from MSSV analyses is explored. Finally, successful implementations of MSSV are reviewed.     

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.     

Density contrast sedimentation velocity for the determination of protein partial-specific volumes

The partial-specific volume of proteins is an important thermodynamic parameter required for the interpretation of data in several biophysical disciplines. Building on recent advances in the use of density variation sedimentation velocity analytical ultracentrifugation for the determination of macromolecular partial-specific volumes, we have explored a direct global modeling approach describing the sedimentation boundaries in different solvents with a joint differential sedimentation coefficient distribution. This takes full advantage of the influence of different macromolecular buoyancy on both the spread and the velocity of the sedimentation boundary. It should lend itself well to the study of interacting macromolecules and/or heterogeneous samples in microgram quantities. Model applications to three protein samples studied in either H(2)O, or isotopically enriched H(2) (18)O mixtures, indicate that partial-specific volumes can be determined with a statistical precision of better than 0.5%, provided signal/noise ratios of 50-100 can be achieved in the measurement of the macromolecular sedimentation velocity profiles. The approach is implemented in the global modeling software SEDPHAT.     

Sedimentation velocity of intrinsically disordered proteins: what information can we actually obtain?

Intrinsically disordered proteins are a challenge to characterise structurally because of their innate flexibility. Hydrodynamic methods such as sedimentation velocity analytical ultracentrifugation have been proposed as methods for their characterisation. By examining in detail this assumption we show that although velocity measurements do yield information on gross conformation, the information is restricted to only the weight averaged sedimentation and diffusion coefficients of the conformational ensemble.     

Turbidimetric ultracentrifugation. Application to the study of human serum very low density lipoprotein distributions.

In this communication it is shown that the sedimentation coefficient distribution may be accurately measured for very large particles using turbidimetric techniques and the ultraviolet-scanning analytical ultracentrifuge. A principal advantage is that turbidity is a function of the product of concentration and molecular weight; thus, large particles may be observed even when present in very small amounts. We propose to call this method of analysis "turbidimetric ultracentrifugation." We have used turbidimetric ultracentrifugation ot determine the sedimentation coefficient distribution for a sample of human serum very low density lipoproteins. This distribution is compared to that found with conventional schlieren techniques with good agreement.     

Investigation of β-carotene–gelatin composite particles with a multiwavelength UV/vis detector for the analytical ultracentrifuge

A multiwavelength UV/vis detector for the analytical ultracentrifuge (MWL-AUC) has been developed recently. In this work, β-carotene–gelatin composite particles are investigated with MWL-AUC. Band centrifugation with a Vinograd cell is used to ensure maximum sample separation. Spectral changes of the system are observed in dependence of the sedimentation coefficient and are attributed to a previously unknown inhomogeneity of the β-carotene chemical composition with both H- and J-aggregates coexisting in a mixture. In addition, our data suggest that pure H- and J-aggregates exist in a particle while their relative concentrations in a mixture determine the color characteristics of the sample. The unique abilities and properties of MWL-AUC include sedimentation coefficient distributions for all possible wavelengths, full UV/vis spectra of each different species in the mixture and 3D movies of the sedimentation process. These properties significantly extend the scope of the analytical ultracentrifuge technique and show that complex biopolymer multicomponent mixtures can be resolved into their individual species.     

Hybrid colloid analysis combining analytical ultracentrifugation and flow-field flow fractionation

Ferritin, the iron storage protein, is an organic-inorganic hybrid colloid consisting of a hollow protein capsule, which is filled with ferrihydride with up to 4500 iron atoms. Owing to the varying iron content and the resulting density differences, as well as the protein oligomerization, a particle size distribution is superimposed with a density distribution, making a precise analysis of ferritin by analytical ultracentrifugation difficult. This study describes how the information of the sedimentation coefficient distribution can be combined with the diffusion coefficient distribution obtained from flow-field flow fractionation to yield the buoyant molar mass of the oligomers in the mixture, extending the information content of each individual analytical method. In addition, the sedimentation and diffusion coefficients are compatible with a simple hard-sphere aggregation model, suggesting that the ferritin oligomers up to the pentamer have a globular solution structure.Presented at the conference for Advances in Analytical Ultracentrifugation and Hydrodynamics, 8–11 June 2002, Grenoble, France     

A model for sedimentation in inhomogeneous media. I. Dynamic density gradients from sedimenting co-solutes

Macromolecular sedimentation in inhomogeneous media is of great practical importance. Dynamic density gradients have a long tradition in analytical ultracentrifugation, and are frequently used in preparative ultracentrifugation. In this paper, a new theoretical model for sedimentation in inhomogeneous media is presented, based on finite element solutions of the Lamm equation with spatial and temporal variation of the local solvent density and viscosity. It is applied to macromolecular sedimentation in the presence of a dynamic density gradient formed by the sedimentation of a co-solute at high concentration. It is implemented in the software SEDFIT for the analysis of experimental macromolecular concentration distributions. The model agrees well with the measured sedimentation profiles of a protein in a dynamic cesium chloride gradient, and may provide a measure for the effects of hydration or preferential solvation parameters. General features of protein sedimentation in dynamic density gradients are described.     

Assessing sedimentation equilibrium profiles in analytical ultracentrifugation experiments on macromolecules: from simple average molecular weight analysis to molecular weight distribution and interaction analysis

Molecular weights (molar masses), molecular weight distributions, dissociation constants and other interaction parameters are fundamental characteristics of proteins, nucleic acids, polysaccharides and glycoconjugates in solution. Sedimentation equilibrium analytical ultracentrifugation provides a powerful method with no supplementary immobilization, columns or membranes required. It is a particularly powerful tool when used in conjunction with its sister technique, namely sedimentation velocity. Here, we describe key approaches now available and their application to the characterization of antibodies, polysaccharides and glycoconjugates. We indicate how major complications, such as thermodynamic non-ideality, can now be routinely dealt with, thanks to a great extent to the extensive contribution of Professor Don Winzor over several decades of research.