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.     

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.     

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.     

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.     

Determination of molecular parameters by fitting sedimentation data to finite-element solutions of the Lamm equation.

A method for fitting experimental sedimentation velocity data to finite-element solutions of various models based on the Lamm equation is presented. The method provides initial parameter estimates and guides the user in choosing an appropriate model for the analysis by preprocessing the data with the G(s) method by van Holde and Weischet. For a mixture of multiple solutes in a sample, the method returns the concentrations, the sedimentation (s) and diffusion coefficients (D), and thus the molecular weights (MW) for all solutes, provided the partial specific volumes (v) are known. For nonideal samples displaying concentration-dependent solution behavior, concentration dependency parameters for s(sigma) and D(delta) can be determined. The finite-element solution of the Lamm equation used for this study provides a numerical solution to the differential equation, and does not require empirically adjusted correction terms or any assumptions such as infinitely long cells. Consequently, experimental data from samples that neither clear the meniscus nor exhibit clearly defined plateau absorbances, as well as data from approach-to-equilibrium experiments, can be analyzed with this method with enhanced accuracy when compared to other available methods. The nonlinear least-squares fitting process was accomplished by the use of an adapted version of the "Doesn't Use Derivatives" nonlinear least-squares fitting routine. The effectiveness of the approach is illustrated with experimental data obtained from protein and DNA samples. Where applicable, results are compared to methods utilizing analytical solutions of approximated Lamm equations.     

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.     

Band centrifugation of macromolecules in self-generating density gradients. II. Sedimentation and diffusion of macromolecules in bands

Three approaches to the simultaneous sedimentation and diffusion of hands or zones of noninteracting homogeneous macromolecules are examined: (1) The authors' method of moments: (2) the transport me of Sehumaker and Rosenbloom; and (3) the stochastic solution of the Lamm equation due to Gehatia and Katehalski. All three methods indicate that the motion of the maximum of the hand may be used to evaluate the sedimentation coefficient. The moment, method provides relations which appear to be useful for measuring diffusion coefficients. Relations are given for the analysis of resolved components. The problem of measuring sedimentation coefficients of macromolecules with concentration-dependent sedimentation coefficients is examined. Methods are described for evaluating the sedimentation coefficient in these systems and for obtaining the sedimentation coefficient at infinite dilution. Methods are described for determining the weight-average sedimentation coefficient in Multi-component systems, and the differential and integral distribution of sedimentation coefficients of macromolecules with low-diffusion coefficients.     

Boundary analysis in sedimentation transport experiments: a procedure for obtaining sedimentation coefficient distributions using the time derivative of the concentration profile.

A procedure is described for computing sedimentation coefficient distributions from the time derivative of the sedimentation velocity concentration profile. Use of the time derivative, (delta c/delta t)r, instead of the radial derivative, (delta c/delta r)t, is desirable because it is independent of time-invariant contributions to the optical baseline. Slowly varying baseline changes also are significantly reduced. An apparent sedimentation coefficient distribution (i.e., uncorrected for the effects of diffusion), g*(s), can be calculated from (delta c/delta t)r as [formula: see text] where s is the sedimentation coefficient, omega is the angular velocity of the rotor, c0 is the initial concentration, r is the radius, rm is the radius of the meniscus, and t is time. An iterative procedure is presented for computing g*(s)t by taking into account the contribution to (delta c/delta t)r from the plateau region to give (delta c/delta t)corr. Values of g*(s)t obtained this way are identical to those of g*(s) calculated from the radial derivative to within the roundoff error of the computations. Use of (delta c/delta t)r, instead of (delta c/delta r)t, results in a significant increase (greater than 10-fold) in the signal-to-noise ratio of data obtained from both the uv photoelectric scanner and Rayleigh optical systems of the analytical ultracentrifuge. The use of (delta c/delta t)r to compute apparent sedimentation coefficient distributions for purposes of boundary analysis is exemplified with an antigen-antibody system.     

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.     

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.     

Sedimentation velocity analysis of heterogeneous protein-protein interactions: Lamm equation modeling and sedimentation coefficient distributions c(s)

We describe algorithms for solving the Lamm equations for the reaction-diffusion-sedimentation process in analytical ultracentrifugation, and examine the potential and limitations for fitting experimental data. The theoretical limiting case of a small, uniformly distributed ligand rapidly reacting with a larger protein in a "constant bath" of the ligand is recapitulated, which predicts the reaction boundary to sediment with a single sedimentation and diffusion coefficient. As a consequence, it is possible to express the sedimentation profiles of reacting systems as c(s) distribution of noninteracting Lamm equation solutions, deconvoluting the effects of diffusion. For rapid reactions, the results are quantitatively consistent with the "constant bath" approximation, showing c(s) peaks at concentration-dependent positions. For slower reactions, the deconvolution of diffusion is still partially successful, with c(s) resolving peaks that reflect the populations of sedimenting species. The transition between c(s) peaks describing reaction boundaries of moderately strong interactions (K(D) approximately 10(-6) M) or resolving sedimenting species was found to occur in a narrow range of dissociation rate constant between 10(-3) and 10(-4) s(-1). The integration of the c(s) peaks can lead to isotherms of species populations or s-value of the reaction boundary, respectively, which can be used for the determination of the equilibrium binding constant.     

Conformation parameters of linear macromolecules from velocity sedimentation and other hydrodynamic methods

Linear macromolecules constitute a broad class of synthetic and natural polymers which are highly useful in various technologies and represent the key molecular systems in living nature. The study of the molecular characteristics of these polymers represents an important problem in fundamental and applied science. The methods of molecular hydrodynamics have been and remain an important way of studying the molar mass, molar mass distribution, size and conformation of linear polymers. This paper discusses the approaches to the problems of hydrodynamic methods, in particular analytical velocity ultracentrifugation, in the study of various types of linear macromolecule. The velocity sedimentation data were processed with three different methods: Sedanal and Sedfit software, and the classical approach of evaluating the rate at which the sedimentation boundary moves. The Sedfit program also allows an evaluation of the frictional ratio values, i.e., the coefficient of translational diffusion. It will be discussed for which systems the estimation of the frictional ratio obtained by Sedfit is adequate and for which it is not. The applications of other hydrodynamic methods (intrinsic viscosity, translational diffusion) are also discussed with a view to obtaining the conformational characteristics of linear macromolecules.     

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.     

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.     

Activated ERK2 is a monomer in vitro with or without divalent cations and when complexed to the cytoplasmic scaffold PEA-15

The extracellular signal-regulated protein kinase, ERK2, fully activated by phosphorylation and without a His(6) tag, shows little tendency to dimerize with or without either calcium or magnesium ions when analyzed by light scattering or analytical ultracentrifugation. Light scattering shows that ~90% of ERK2 is monomeric. Sedimentation equilibrium data (obtained at 4.8-11.2 μM ERK2) with or without magnesium (10 mM) are well described by an ideal one-component model with a fitted molar mass of 40180 ± 240 Da (without Mg(2+) ions) or 41290 ± 330 Da (with Mg(2+) ions). These values, close to the sequence-derived mass of 41711 Da, indicate that no significant dimerization of ERK2 occurs in solution. Analysis of sedimentation velocity data for a 15 μM solution of ERK2 with an enhanced van Holde-Weischet method determined the sedimentation coefficient (s) to be ~3.22 S for activated ERK2 with or without 10 mM MgCl(2). The frictional coefficient ratio (f/f(0)) of 1.28 calculated from the sedimentation velocity and equilibrium data is close to that expected for an ~42 kDa globular protein. The translational diffusion coefficient of ~8.3 × 10(-7) cm(2) s(-1) calculated from the experimentally determined molar mass and sedimentation coefficient agrees with the value determined by dynamic light scattering in the absence and presence of calcium or magnesium ions and a value determined by NMR spectrometry. ERK2 has been proposed to homodimerize and bind only to cytoplasmic but not nuclear proteins [Casar, B., et al. (2008) Mol. Cell 31, 708-721]. Our light scattering data show, however, that ERK2 forms a strong 1:1 complex of ~57 kDa with the cytoplasmic scaffold protein PEA-15. Thus, ERK2 binds PEA-15 as a monomer. Our data provide strong evidence that ERK2 is monomeric under physiological conditions. Analysis of the same ERK2 construct with the nonphysiological His(6) tag shows substantial dimerization under the same ionic conditions.     

Determination of sedimentation coefficients for small peptides.

Direct fitting of sedimentation velocity data with numerical solutions of the Lamm equations has been exploited to obtain sedimentation coefficients for single solutes under conditions where solvent and solution plateaus are either not available or are transient. The calculated evolution was initialized with the first experimental scan and nonlinear regression was employed to obtain best-fit values for the sedimentation and diffusion coefficients. General properties of the Lamm equations as data analysis tools were examined. This method was applied to study a set of small peptides containing amphipathic heptad repeats with the general structure Ac-YS-(AKEAAKE)nGAR-NH2, n = 2, 3, or 4. Sedimentation velocity analysis indicated single sedimenting species with sedimentation coefficients (s(20,w) values) of 0.37, 0.45, and 0.52 S, respectively, in good agreement with sedimentation coefficients predicted by hydrodynamic theory. The described approach can be applied to synthetic boundary and conventional loading experiments, and can be extended to analyze sedimentation data for both large and small macromolecules in order to define shape, heterogeneity, and state of association.     

An improved function for fitting sedimentation velocity data for low-molecular-weight solutes.

Many traditional approaches to the analysis of sedimentation velocity data work poorly with data for low-molecular-weight solutes, which have sedimentation boundaries that are severely broadened by diffusion. An approach that has previously had some success is to directly fit these broad boundaries to approximate solutions of the Lamm equation that directly account for the high diffusion. However, none of the available approximate solutions work well at times both early and late in the run, or give boundary shapes that are highly accurate, especially for species of molecular weight < 10,000. An improved fitting function has been developed to overcome some of these limitations. The new function adds two correction terms to the Fujita-MacCosham solution. The optimum coefficients for these new correction terms were determined by a least-squares approach. The accuracy and limitations of fitting with this new function were tested against synthetic data sets obtained by finite-element methods, for analysis of samples containing either single species or several noninteracting species. We also compare the strengths and weaknesses of this method of analysis, and its ability to work with noisy data, relative to recently developed time-derivative methodologies.     

Self-association mode of a flavoenzyme D-amino acid oxidase from hog kidney. I. Analysis of apparent weight-average molecular weight data for the apoenzyme in terms of models

The self-association of D-amino acid oxidase apoenzyme in 0.1 M sodium pyrophosphate, pH 8.3, at 25 degrees C was studied by low-angle laser light scattering. The concentration (c) dependence of the apparent weight-average molecular weight (Mwapp) was determined over a wide concentration range of 0.04 to 6.1 mg/ml. The extrapolated Mwapp value, to zero enzyme concentration, corresponded to the Mr value of the monomer. The self-association mode of the apoenzyme was systematically explored with nonlinear least-squares analysis of the Mwapp versus c data. The simplest model that fitted the data well was a model of isodesmic indefinite self-association of the monomer with the isodesmic association constant of 0.467 +/- 0.034 liter/g. The monomer-dimer model proposed previously, but only in a low enzyme concentration range of less than 0.9 mg/ml at 5-20 degrees C (Henn, S. W., and Ackers, G. K. (1969) Biochemistry 8, 3829-3838), did not fit the Mwapp versus c data either in the limited low concentration range or in the whole concentration range examined at 25 degrees C. To test the validity of the chosen model, the observed sedimentation boundary profiles were compared with the idealized boundary profiles calculated for the better-fit models. The profile calculated with the model of the isodesmic indefinite self-association mechanism was qualitatively consistent with the observed ones. The utility of the nonlinear least-squares procedure for analyzing self-associating systems was demonstrated.     

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.