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.     

Determination of the sedimentation coefficient distribution by least-squares boundary modeling

A new method is presented for the calculation of apparent sedimentation coefficient distributions g*(s) for the size-distribution analysis of polymers in sedimentation velocity experiments. Direct linear least-squares boundary modeling by a superposition of sedimentation profiles of ideal nondiffusing particles is employed. It can be combined with algebraic noise decomposition techniques for the application to interference optical ultracentrifuge data at low loading concentrations with significant systematic noise components. Because of the use of direct boundary modeling, residuals are available for assessment of the quality of the fits and the consistency of the g*(s) distribution with the experimental data. The method can be combined with regularization techniques based on F statistics, such as used in the program CONTIN, or alternatively, the increment of s values can be adjusted empirically. The method is simple, has advantageous statistical properties, and reveals precise sedimentation coefficients. The new least-squares ls-g*(s) exhibits a very high robustness and resolution if data acquired over a large time interval are analyzed. This can result in a high resolution for large particles, and for samples with a high degree of heterogeneity. Because the method does not require a high frequency of scans, it can also be easily used in experiments with the absorbance optical scanning system. Published 2000 John Wiley & Sons, Inc.     

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.     

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.     

On the analysis of protein self-association by sedimentation velocity analytical ultracentrifugation

Analytical ultracentrifugation is one of the classical techniques for the study of protein interactions and protein self-association. Recent instrumental and computational developments have significantly enhanced this methodology. In this paper, new tools for the analysis of protein self-association by sedimentation velocity are developed, their statistical properties are examined, and considerations for optimal experimental design are discussed. A traditional strategy is the analysis of the isotherm of weight-average sedimentation coefficients s(w) as a function of protein concentration. From theoretical considerations, it is shown that integration of any differential sedimentation coefficient distribution c(s), ls-g(*)(s), or g(s(*)) can give a thermodynamically well-defined isotherm, as long as it provides a good model for the sedimentation profiles. To test this condition for the g(s(*)) distribution, a back-transform into the original data space is proposed. Deconvoluting diffusion in the sedimentation coefficient distribution c(s) can be advantageous to identify species that do not participate in the association. Because of the large number of scans that can be analyzed in the c(s) approach, its s(w) values are very precise and allow extension of the isotherm to very low concentrations. For all differential sedimentation coefficients, corrections are derived for the slowing of the sedimentation boundaries caused by radial dilution. As an alternative to the interpretation of the isotherm of the weight-average s value, direct global modeling of several sedimentation experiments with Lamm equation solutions was studied. For this purpose, a new software SEDPHAT is introduced, allowing the global analysis of several sedimentation velocity and equilibrium experiments. In this approach, information from the shape of the sedimentation profiles is exploited, which permits the identification of the association scheme and requires fewer experiments to precisely characterize the association. Further, under suitable conditions, fractions of incompetent material that are not part of the reversible equilibrium can be detected.     

A method for directly fitting the time derivative of sedimentation velocity data and an alternative algorithm for calculating sedimentation coefficient distribution functions

The time-derivative method for deriving the sedimentation coefficient distribution, g(s*), from sedimentation velocity data that was developed by Walter Stafford has many advantages and is now widely used. By fitting Gaussian functions to the g(s*) distribution both sedimentation and diffusion coefficients (and therefore molecular masses) for individual species can be obtained. However, some of the approximations used in these procedures limit the accuracy of the results. An alternative approach is proposed in which the dc/dt data are fitted rather than g(s*). This new approach gives improved accuracy, extends the range to sedimentation coefficients below 1 S, and enhances resolution of multiple species. For both approaches the peaks from individual species are broadened when the data cover too wide a time span, and this effect is explored and quantified. An alternative algorithm for calculating ?(s*) from the dc/dt curves is presented and discussed. Rather than first averaging the dc/dt data for individual scan pairs and then calculating ?(s*) from that average, the ?(s*) distributions are calculated for every scan pair and then subsequently averaged. This alternative procedure yields smaller error bars for g(s*) and somewhat greater accuracy for fitted hydrodynamic properties when the time span becomes large.     

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.     

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.     

Characterization of weak protein dimerization by direct analysis of sedimentation equilibrium distributions: the INVEQ approach

Closer scrutiny has been accorded a recently reported procedure for characterizing weak protein dimerization by sedimentation equilibrium (INVEQ) in which the equilibrium distribution is analyzed as a dependence of radial distance on solute concentration rather than of solute concentration on radial distance. By demonstrating theoretically that the fundamental parameter derived from the analysis is simply the difference between the dimerization constant and the osmotic second virial coefficient for monomer-monomer interaction, this investigation refutes the original claim that independent estimates of these two parameters can be obtained by nonlinear curve fitting of the sedimentation equilibrium distribution. This criticism also applies to conventional analyses of sedimentation distributions by the commonly employed Beckman Origin and NONLIN software. Numerically simulated distributions are then analyzed to demonstrate limitations of the procedure and also to indicate a means of improving the reliability of the returned estimate of the dimerization constant. These features are illustrated by applying the original and revised analytical procedures to a sedimentation equilibrium distribution for alpha-chymotrypsin (pH 4.0, I 0.05 M).     

An extrapolation method for reducing equilibration times in sedimentation equilibrium experiments.

We present a detailed investigation of the use of an extrapolation technique to decrease running times of sedimentation equilibrium experiments. If concentration profiles are available at time delta tau, 2delta tau, 3delta tau,...., cn(r) = c(r, n delta tau), then the Aitken transformation replaces the cn(r) + ĉn(r) = [cn + 1(r) cn - 1(r) - c2n(r)]/[cn + 1(r) + cn - 1(r) - 2cn(r)]. We show that the ĉn(r) converge to the equilibrium values c infinity (r) much more quickly than the cn(r). Savings in time are shown to range from a factor of approximately 2 for meniscus depletion experiments to factors of between 4 and 8 for lower speeds or smaller molecular weights. It is also shown that the technique is quite sensitive to noise, so that an accurate optical system is required to allow its optimal use.     

Sedimentation velocity analysis of heterogeneous protein-protein interactions: sedimentation coefficient distributions c(s) and asymptotic boundary profiles from Gilbert-Jenkins theory

Interacting proteins in rapid association equilibrium exhibit coupled migration under the influence of an external force. In sedimentation, two-component systems can exhibit bimodal boundaries, consisting of the undisturbed sedimentation of a fraction of the population of one component, and the coupled sedimentation of a mixture of both free and complex species in the reaction boundary. For the theoretical limit of diffusion-free sedimentation after infinite time, the shapes of the reaction boundaries and the sedimentation velocity gradients have been predicted by Gilbert and Jenkins. We compare these asymptotic gradients with sedimentation coefficient distributions, c(s), extracted from experimental sedimentation profiles by direct modeling with superpositions of Lamm equation solutions. The overall shapes are qualitatively consistent and the amplitudes and weight-average s-values of the different boundary components are quantitatively in good agreement. We propose that the concentration dependence of the area and weight-average s-value of the c(s) peaks can be modeled by isotherms based on Gilbert-Jenkins theory, providing a robust approach to exploit the bimodal structure of the reaction boundary for the analysis of experimental data. This can significantly improve the estimates for the determination of binding constants and hydrodynamic parameters of the complexes.     

Nonequivalence of second virial coefficients from sedimentation equilibrium and static light scattering studies of protein solutions

Experimental data for ovalbumin and lysozyme are presented to highlight the nonequivalence of second virial coefficients obtained for proteins by sedimentation equilibrium and light scattering. Theoretical considerations confirm that the quantity deduced from sedimentation equilibrium distributions is B(22), the osmotic second virial coefficient describing thermodynamic nonideality arising solely from protein self-interaction. On the other hand, the virial coefficient determined by light scattering is shown to reflect the combined contributions of protein-protein and protein-buffer interactions to thermodynamic nonideality of the protein solution. Misidentification of the light scattering parameter as B(22) accounts for published reports of negative osmotic second virial coefficients as indicators of conditions conducive to protein crystal growth. Finally, textbook assertions about the equivalence of second virial coefficients obtained by sedimentation equilibrium and light scattering reflect the restriction of consideration to single-solute systems. Although sedimentation equilibrium distributions for buffered protein solutions are, indeed, amenable to interpretation in such terms, the same situation does not apply to light scattering measurements because buffer constituents cannot be regarded as part of the solvent: instead they must be treated as non-scattering cosolutes.     

Anomalies in the sedimentation behavior of high molecular weight DNA isolated from Proteus mirabilis

The sedimentation properties of a P. mirabilis DNA sample have been investigated at different concentrations and rotor speeds. Pronounced speed effects occurred at high angular velocities. The s value evaluated from low-speed experiments amounts to 61 S., indicating a mean molecular weight of 105 million. Anomalous concentration distributions in the ultracentrifuge cell have been observed at low speeds. At the boundary, a pile-up of concentration occurs which exceeds the total plateau concentration. The concentration elevation decreases with increasing time due to convection which is caused by the existence of a negative density gradient. Despite this convection, accurate mean sedimentation coefficients could be obtained even at extremely low concentrations. A careful analysis of sedimentation coefficient distributions shows, however, that the lending and trailing tails of the boundaries are disturbed by convection. Thus it may be expected that the effect produces difficulties in determining true sedimentation coefficient distributions of polydisperse DNA samples of very high molecular weight.     

Determination os sedimentation coefficient distributions for cartilage proteoglycans

Sedimentation coefficient distributions of widely polydisperse proteoglycan preparations were made using a previously developed transport sedimentation methodology. Boundary stability was improved by centrifuging samples in a preformed CsCl density gradient (0.016 g/cm⁴). The results were compared with the distributions obtained with an interferometric analytical centrifugation method. When these two techniques were applied to analyze A1 and A1–D1 proteoglycan preparations, results were in substantial agreement with respect to the mean sedimentation coefficients of the peaks, average S value, sedimentation coefficient distribution, skewness, proportion of monomer and aggregates, and linearity of the plot ln(s) versus C extrapolations to zero concentration. The lower solute concentration compatible with the transport (velocity gradient) method makes this technique particularly suitable for studying the details of proteoglycan distribution of molecular sizes, especially for aggregates.     

The effective time of centrifugation for the analysis of boundary spreading in sedimentation velocity experiments

This investigation establishes a likely order of magnitude for the zero-time correction factor governing the effective time of centrifugation that is pertinent in the analysis of boundary spreading in sedimentation velocity experiments. This correction is shown to be too small to unduly affect the magnitudes of sedimentation and diffusion coefficients deduced from the application of computer software incorporating the printout value of ω2t and an effective position of the air-solution meniscus that is obtained as an additional parameter in the analysis involving nonlinear least-squares curve-fitting of sedimentation velocity distributions to the Lamm equation. Although this procedure slightly underestimates the actual meniscus position (r(a)), uncertainty about its exact location precludes adoption of the alternative approach with r(a) fixed and the correction to ω2t regarded as the additional curve-fitting parameter.     

The 15 SCR flexible extracellular domains of human complement receptor type 2 can mediate multiple ligand and antigen interactions

Complement receptor type 2 (CR2, CD21) is a cell surface protein that links the innate and adaptive immune response during the activation of B cells. The extracellular portion of CR2 comprises 15 or 16 short complement regulator (SCR) domains, for which the overall arrangement in solution is unknown. This was determined by constrained scattering and ultracentrifugation modelling. The radius of gyration of CR2 SCR 1-15 was determined to be 11.5 nm by both X-ray and neutron scattering, and that of its cross-section was 1.8 nm. The distance distribution function P(r) showed that the overall length of CR2 SCR 1-15 was 38 nm. Sedimentation equilibrium curve fits gave a mean molecular weight of 135,000 (+/- 13,000) Da, in agreement with a fully glycosylated structure. Velocity experiments using the g*(s) derivative method gave a sedimentation coefficient of 4.2 (+/- 0.1) S. In order to construct a model of CR2 SCR 1-15 for constrained fitting, homology models for the 15 SCR domains were combined with randomised linker peptides generated by molecular dynamics simulations. Using an automated procedure, the analysis of 15,000 possible CR2 SCR 1-15 models showed that only those models in which the 15 SCR domains were flexible but partially folded back accounted for the scattering and sedimentation data. The best-fit CR2 models provided a visual explanation for the versatile interaction of CR2 with four ligands C3d, CD23, gp350 and IFN-alpha. The flexible location of CR2 SCR 1-2 is likely to facilitate interactions of C3d-antigen complexes with the B cell receptor.     

Analytical sedimentation studies of turkey gizzard myosin light chain kinase and telokin.

Sedimentation equilibrium and velocity studies were performed with turkey gizzard myosin light chain kinase (MLCK) and telokin, a small protein apparently corresponding to the sequence of the COOH-terminal end of MLCK. The measurements carried out with MLCK give values for the monomer molecular weight (M(r)), sedimentation coefficient (S20 degrees,w), and virial coefficient (A2) of 108,000, 3.74 S, and -1.95 x 10(-4) mol.ml.g-2, respectively. In the case of telokin, M(r) = 18,500; S20 degrees, w = 1.63 S; and A2 = 5.81 x 10(-4)mol.ml.g-2. Combination of the results of the two kinds of experiment shows that MLCK is a rod-shaped molecule (a/b = 18.9) with a Stoke's radius of 69 A. Telokin is also elongated (a/b = 8.3) with a Stoke's radius of 29 A. MLCK apparently exhibits self-association, with 15% of the protein sedimenting as a dimer in the experiments.     

Interpretation of thermodynamic non-ideality in sedimentation equilibrium experiments on proteins

This investigation re-examines theoretical aspects of the allowance for effects of thermodynamic non-ideality on the sedimentation equilibrium distribution for a single macromolecular solute, and thereby resolves the question of the constraints that pertain to the definition of the activity coefficient term in the basic sedimentation equilibrium expression. Sedimentation equilibrium results for ovalbumin are then presented to illustrate a simple procedure for evaluating the net charge (valence) of a protein from the magnitude of the second virial coefficient in situations where the effective radius of the protein can be assigned. Finally, published sedimentation equilibrium results on lysozyme are reanalysed to demonstrate the feasibility of employing the dependence of the second virial coefficient upon ionic strength to evaluate both the valence and the effective radius of the non-interacting solute.     

Improved methods for fitting sedimentation coefficient distributions derived by time-derivative techniques

Time-derivative approaches to analyzing sedimentation velocity data have proven to be highly successful and have now been used routinely for more than a decade. For samples containing a small number of noninteracting species, the sedimentation coefficient distribution function, g(s *), traditionally has been fitted by Gaussian functions to derive the concentration, sedimentation coefficient, and diffusion coefficient of each species. However, the accuracy obtained by that approach is limited, even for noise-free data, and becomes even more compromised as more scans are included in the analysis to improve the signal/noise ratio (because the time span of the data becomes too large). Two new methods are described to correct for the effects of long time spans: one approach that uses a Taylor series expansion to correct the theoretical function and a second approach that creates theoretical g(s *) curves from Lamm equation models of the boundaries. With this second approach, the accuracy of the fitted parameters is approximately 0.1% and becomes essentially independent of the time span; therefore, it is possible to obtain much higher signal/noise when needed. This second approach is also compared with other current methods of analyzing sedimentation velocity data.