A comparison of weight average and direct boundary fitting of sedimentation velocity data for indefinite polymerizing systems

Analysis of sedimentation velocity data for indefinite self-associating systems is often achieved by fitting of weight average sedimentation coefficients (s(20,w)) However, this method discriminates poorly between alternative models of association and is biased by the presence of inactive monomers and irreversible aggregates. Therefore, a more robust method for extracting the binding constants for indefinite self-associating systems has been developed. This approach utilizes a set of fitting routines (SedAnal) that perform global non-linear least squares fits of up to 10 sedimentation velocity experiments, corresponding to different loading concentrations, by a combination of finite element simulations and a fitting algorithm that uses a simplex convergence routine to search parameter space. Indefinite self-association is analyzed with the software program isodesfitter, which incorporates user provided functions for sedimentation coefficients as a function of the degree of polymerization for spherical, linear and helical polymer models. The computer program hydro was used to generate the sedimentation coefficient values for the linear and helical polymer assembly mechanisms. Since this curve fitting method directly fits the shape of the sedimenting boundary, it is in principle very sensitive to alternative models and the presence of species not participating in the reaction. This approach is compared with traditional fitting of weight average data and applied to the initial stages of Mg(2+)-induced tubulin self-associating into small curved polymers, and vinblastine-induced tubulin spiral formation. The appropriate use and limitations of the methods are discussed.     

Using Lamm-Equation modeling of sedimentation velocity data to determine the kinetic and thermodynamic properties of macromolecular interactions

The interaction of macromolecules with themselves and with other macromolecules is fundamental to the functioning of living systems. Recent advances in the analysis of sedimentation velocity (SV) data obtained by analytical ultracentrifugation allow the experimenter to determine important features of such interactions, including the equilibrium association constant and information about the kinetic off-rate of the interaction. The determination of these parameters is made possible by the ability of modern software to fit numerical solutions of the Lamm Equation with kinetic considerations directly to SV data. Herein, the SV analytical advances implemented in the software package SEDPHAT are summarized. Detailed analyses of SV data using these strategies are presented. Finally, a few highlights of recent literature reports that feature this type of SV data analysis are surveyed.     

The use of analytical sedimentation velocity to extract thermodynamic linkage

For 25 years, the Gibbs Conference on Biothermodynamics has focused on the use of thermodynamics to extract information about the mechanism and regulation of biological processes. This includes the determination of equilibrium constants for macromolecular interactions by high precision physical measurements. These approaches further reveal thermodynamic linkages to ligand binding events. Analytical ultracentrifugation has been a fundamental technique in the determination of macromolecular reaction stoichiometry and energetics for 85 years. This approach is highly amenable to the extraction of thermodynamic couplings to small molecule binding in the overall reaction pathway. In the 1980s this approach was extended to the use of sedimentation velocity techniques, primarily by the analysis of tubulin-drug interactions by Na and Timasheff. This transport method necessarily incorporates the complexity of both hydrodynamic and thermodynamic nonideality. The advent of modern computational methods in the last 20 years has subsequently made the analysis of sedimentation velocity data for interacting systems more robust and rigorous. Here we review three examples where sedimentation velocity has been useful at extracting thermodynamic information about reaction stoichiometry and energetics. Approaches to extract linkage to small molecule binding and the influence of hydrodynamic nonideality are emphasized. These methods are shown to also apply to the collection of fluorescence data with the new Aviv FDS.     

The analysis of macromolecular interactions by sedimentation equilibrium

The study of macromolecular interactions by sedimentation equilibrium is a highly technical method that requires great care in both the experimental design and data analysis. The complexity of the interacting system that can be analyzed is only limited by the ability to deconvolute the exponential contributions of each of the species to the overall concentration gradient. This is achieved in part through the use of multi-signal data collection and the implementation of soft mass conservation. We illustrate the use of these constraints in SEDPHAT through the study of an A+B+B?AB+B?ABB system and highlight some of the technical challenges that arise. We show that both the multi-signal analysis and mass conservation result in a precise and robust data analysis and discuss improvements that can be obtained through the inclusion of data from other methods such as sedimentation velocity and isothermal titration calorimetry.     

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 strategy for efficient characterization of macromolecular heteroassociations via measurement of sedimentation equilibrium.

A method is proposed for the selection of experimental conditions for sedimentation equilibrium experiments that will provide maximal information about the values of equilibrium association constants within a given scheme for heteroassociation of two solute components. A discriminator function is proposed that indicates the sensitivity of the experimentally observed gradient or gradients to alterations in the underlying association constants. The value of this function is plotted or tabulated as a function of the concentrations of the two components, over a broad range of solution compositions. It is suggested that experiments performed with loading compositions corresponding to large absolute values of the discriminator function will yield the most information with respect to determination of the underlying association constants. This method was tested by predicting optimal conditions for three different types of sedimentation equilibrium experiments: (i) measurement of total (natural) solute absorbance; (ii) measurement of individual component gradients via measurement of tracer absorbance; and (iii) global analysis of multiple experiments. Experimental data resulting from sedimentation equilibrium experiments carried out under the specified conditions were simulated by addition of realistic levels of random error to calculated equilibrium gradients. The simulated data were then analyzed exactly as real experimental data, i.e., without prior knowledge of the underlying association constants. It was found that the highest accuracy and precision in determination of heteroassociation constants are obtained by global analysis of multiple experiments performed using significantly different loading compositions, each of which is selected from 'sensitive' regions of the discriminator map.     

Characterization of macromolecular heterogeneity by equilibrium sedimentation techniques

New graphical procedures have been developed to investigate the heterogeneity of protein preparations using sedimentation equilibrium. The heterogeneous systems that can be studied include self-associating systems contaminated by incompetent monomer, self-associating systems contaminated by non-dissociating oligomer and simple non-interacting monomer-oligmer disperse systems. The new procedures are based on the concentration dependence of the apparent association constants estimated by a non-linear least square fitting program (NONLIN), on the assumption of conservation of mass during sedimentation and on the applications of several standard techniques for statistical inferences of NONLIN estimations. The procedures outlined here can detect various types of heterogeneity, discriminate amongst different types of heterogeneity, estimate the amount of contaminant causing heterogeneity and determine the true equilibrium constant of the self-associating components. The procedures appear to be sensitive, accurate and easily applicable when tested using both protein samples and computer simulated data.     

Use of difference boundary sedimentation velocity to investigate nonspecific protein-nucleic acid interactions

The difference boundary sedimentation velocity technique of Schachman and co-workers is demonstrated to be applicalbe to the measurement of binding constants (Kobsd) in the range 10(2)-10(5) M(-1) for the nonspecific interactions of proteins with DNA. The difference technique can reproducibly detect a 2% change in the sedimentation coefficient of the DNA upon binding ligands, corresponding to average extents of association as low as 10 molecules of protein (in the cases of Escherichia coli lac repressor and E. coli RNA polymerase) per molecule of bacteriophage T7 DNA. At these low binding densities, it is plausible to assume that the primary effect of ligand binding is on the buoyant mass of the complex and not on the frictional coefficient of the flexible DNA coil. Binding constants calculated by using this assumption agree well with literature values for the nonspecific interactions of RNase and lac repressor proteins with double-stranded DNA. Advantages of the method are that it is relatively rapid, requires the optical detection of the DNA only, and can be performed on small amounts of sample. The method appears useful for surveying (to an accuracy of +/-50% in Kobsd or +/-10% in log Kobsd) the effects of solution variables on Kobsd of protein-DNA interactions. Applications of the method to the nonspecific interactions of RNA polymerase core and holoenzymes with T7 DNA are discussed.     

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.     

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.     

Analyses of sedimentation equilibrium data

A numerical procedure is presented which can quite adequately compute the molecular weight averages as a function of solute concentration from sedimentation equilibrium data for homogeneous systems and for monomer-dimer associating systems with a possible extension to heterogeneous systems where monotonic variation in the weight average molecular weight is observed such as in weakly associating or dissociating systems. The procedure utilizes the method of orthogonal polynomials for curve fitting which allows for a rapid determination of best fit with minimal round off error. The procedure is particularly applicable in cases where the concentration of solute at the meniscus can be considered to be neither appreciable and reasonably well determined as in low speed sedimentation equilibrium experiments, nor essentially zero as in high speed sedimentation equilibrium experiments where the calculations become somewhat more simplified. The use of moderate speed sedimentation equilibrium has the advantage of providing a more broad concentration distribution in the centrifuge cell which yields more extensive information concerning dissociating systems yet still provides results at low solute concentrations where most solutes can be considered to be behaving ideally.     

Tools for the Quantitative Analysis of Sedimentation Boundaries Detected by Fluorescence Optical Analytical Ultracentrifugation

Fluorescence optical detection in sedimentation velocity analytical ultracentrifugation allows the study of macromolecules at nanomolar concentrations and below. This has significant promise, for example, for the study of systems of high-affinity protein interactions. Here we describe adaptations of the direct boundary modeling analysis approach implemented in the software SEDFIT that were developed to accommodate unique characteristics of the confocal fluorescence detection system. These include spatial gradients of signal intensity due to scanner movements out of the plane of rotation, temporal intensity drifts due to instability of the laser and fluorophores, and masking of the finite excitation and detection cone by the sample holder. In an extensive series of experiments with enhanced green fluorescent protein ranging from low nanomolar to low micromolar concentrations, we show that the experimental data provide sufficient information to determine the parameters required for first-order approximation of the impact of these effects on the recorded data. Systematic deviations of fluorescence optical sedimentation velocity data analyzed using conventional sedimentation models developed for absorbance and interference optics are largely removed after these adaptations, resulting in excellent fits that highlight the high precision of fluorescence sedimentation velocity data, thus allowing a more detailed quantitative interpretation of the signal boundaries that is otherwise not possible for this system.     

Average quantities accessible to the analysis of sedimentation equilibrium data for self-association systems

This paper shows that analysis of sedimentation equilibrium data, searching for average molecular weights, gives quantities which are dependent on the total protein concentration of the samples, while knowledge of the molecular weight of the monomeric subunits allows a more meaningful search for the concentrations of the individual polymeric components of the system. From these the various average molecular weights can be construed, and the various dissociation equilibrium constants evaluated. Also, in this paper considerations are proposed on the meaning of nonideality terms in associating systems and possible ways for estimating them. As an example the proposed procedures have been applied to measurements of sedimentation equilibrium in carbonmonoxyhemoglobin.     

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.     

Characterization of heterologous protein-protein interactions using analytical ultracentrifugation.

Methods for quantitative characterization of heterologous protein-protein interactions by means of analytical ultracentrifugation (AUC) include sedimentation equilibrium, tracer sedimentation equilibrium, sedimentation velocity, and analytical band sedimentation. Fundamental principles governing the behavior of macromolecules in a centrifugal field are summarized, and the application of these principles to the interpretation of data obtained from each type of experiment is reviewed. Instrumentation and software for the acquisition and analysis of data obtained from different types of AUC experiments are described.     

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.     

Applications of analytical ultracentrifugation to protein size-and-shape distribution and structure-and-function analyses

The rebirth of modern analytical ultracentrifugation (AUC) began in 1990s. Since then many advanced AUC detectors have been developed that provide a vast range of versatile choices when characterizing the physical and chemical features of macromolecules. In addition, there have been remarkable advances in software that allow the analysis of AUC data using more sophisticated models, including quaternary structures, conformational changes, and biomolecular interactions. Here we report the application of AUC to protein size-and-shape distribution analysis and structure-and-function analysis in the presence of ligands or lipids. Using band-sedimentation velocity, quaternary structural changes and an enzyme's catalytic activity can be observed simultaneously. This provides direct insights into the correlation between quaternary structure and catalytic activity of the enzyme. On the other hand, also in this study, we have applied size-and-shape distribution analysis to a lipid-binding protein in either an aqueous or lipid environment. The sedimentation velocity data for the protein with or without lipid were evaluated using the c(s,f(r)) two-dimensional distribution model, which provides a precise and quantitative means of analyzing the protein's conformational changes.