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.     

The molecular weight and oligomerization of rabbit thrombomodulin as assessed by sedimentation equilibrium

Lubrol-solubilized rabbit thrombomodulin has been examined by equilibrium sedimentation in buffers that include sufficient D2O to make the detergent neutrally buoyant. Data were acquired at rotor speeds from 12,000 to 28,000 rpm from two thrombomodulin preparations, at protein concentrations from 0.01 to 0.07%, and in buffer containing 0.01 to 0.23% Lubrol. Examination of the data from different rotor speeds shows that the thrombomodulin exists as a heterogeneous mixture containing monomer (Mr 65,000), trimer, and higher oligomers. The oligomers do not equilibrate over the time scale of the experiment. The weight fraction as monomer varies from preparation to preparation, and appears to be independent of detergent concentration. Thus, experimenters should be cautious when interpreting binding or kinetic results obtained under similar buffer conditions.     

Sedimentation analysis of noninteracting and self-associating solutes using numerical solutions to the Lamm equation.

The potential of using the Lamm equation in the analysis of hydrodynamic shape and gross conformation of proteins and reversibly formed protein complexes from analytical ultracentrifugation data was investigated. An efficient numerical solution of the Lamm equation for noninteracting and rapidly self-associating proteins by using combined finite-element and moving grid techniques is described. It has been implemented for noninteracting solutes and monomer-dimer and monomer-trimer equilibria. To predict its utility, the error surface of a nonlinear regression of simulated sedimentation profiles was explored. Error contour maps were calculated for conventional independent and global analyses of experiments with noninteracting solutes and with monomer-dimer systems at different solution column heights, loading concentrations, and centrifugal fields. It was found that the rotor speed is the major determinant for the shape of the error surface, and that global analysis of different experiments can allow substantially improved characterization of the solutes. We suggest that the global analysis of the approach to equilibrium in a short-column sedimentation equilibrium experiment followed by a high-speed short-column sedimentation velocity experiment can result in sedimentation and diffusion coefficients of very high statistical accuracy. In addition, in the case of a protein in rapid monomer-dimer equilibrium, this configuration was found to reveal the most precise estimate of the association constant.     

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.     

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.     

Analysis of binding in macromolecular complexes: a generalized numerical approach

In this work, we introduce a generalized, global numerical methodology for analysis of binding phenomena in complex macromolecular assemblies. On the basis of a numerical algorithm (EQS) to solve systems of simultaneous free energy equations, binding profiles of simple to highly complex interacting systems can be analyzed over any concentration region without any need to generate an analytical form to describe the data. The output of the numerical algorithm is the concentration of each individual species in solution, allowing the generation of all possible binding profiles of the system (e.g., protein saturation by ligand). We present here the application of this approach to the DNA-protein subunit-ligand interactions of the trp repressor system as a typical example. From a practical point of view, the analysis program is capable of the rapid and simultaneous analysis of multiple binding profiles in terms of internally consistent sets of free energies. Given both the enormous complexity, as well as the underlying subtlety, involved in the regulation of biological function, the present generalized approach to analyzing macromolecular binding should find wide applications.     

Mass Balance Model with Donnan Equilibrium Accurately Describes Unusual pH and Excipient Profiles during Diafiltration of Monoclonal Antibodies

There is extensive experimental data showing that the final pH and buffer composition after protein diafiltration (DF), particularly with monoclonal antibodies, can be considerably different than that in the DF buffer due to electrostatic interactions between the charged protein and the charged ions. Previous models for this behavior have focused on the final (equilibrium) partitioning and are unable to explain the complex pH and concentration profiles during the DF process. The objective of this study is to develop a new model for antibody DF based on solution of the transient mass balance equations, with the permeate concentrations of the charged species evaluated assuming Donnan equilibrium across the semipermeable membrane in combination with electroneutrality constraints. Model predictions are in excellent agreement with experimental data obtained during DF of both acidic and basic monoclonal antibodies, with the protein charge determined from independent electrophoretic mobility measurements. The model is able to predict the entire pH/histidine concentration profiles during DF, providing a framework for the development of DF processes that yield the desired antibody formulation.     

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.     

Separating distinct structures of multiple macromolecular assemblies from cryo-EM projections

Single particle analysis for structure determination in cryo-electron microscopy is traditionally applied to samples purified to near homogeneity as current reconstruction algorithms are not designed to handle heterogeneous mixtures of structures from many distinct macromolecular complexes. We extend on long established methods and demonstrate that relating two-dimensional projection images by their common lines in a graphical framework is sufficient for partitioning distinct protein and multiprotein complexes within the same data set. The feasibility of this approach is first demonstrated on a large set of synthetic reprojections from 35 unique macromolecular structures spanning a mass range of hundreds to thousands of kilodaltons. We then apply our algorithm on cryo-EM data collected from a mixture of five protein complexes and use existing methods to solve multiple three-dimensional structures ab initio. Incorporating methods to sort single particle cryo-EM data from extremely heterogeneous mixtures will alleviate the need for stringent purification and pave the way toward investigation of samples containing many unique structures.     

Direct sedimentation analysis of interference optical data in analytical ultracentrifugation

Sedimentation data acquired with the interference optical scanning system of the Optima XL-I analytical ultracentrifuge can exhibit time-invariant noise components, as well as small radial-invariant baseline offsets, both superimposed onto the radial fringe shift data resulting from the macromolecular solute distribution. A well-established method for the interpretation of such ultracentrifugation data is based on the analysis of time-differences of the measured fringe profiles, such as employed in the g(s*) method. We demonstrate how the technique of separation of linear and nonlinear parameters can be used in the modeling of interference data by unraveling the time-invariant and radial-invariant noise components. This allows the direct application of the recently developed approximate analytical and numerical solutions of the Lamm equation to the analysis of interference optical fringe profiles. The presented method is statistically advantageous since it does not require the differentiation of the data and the model functions. The method is demonstrated on experimental data and compared with the results of a g(s*) analysis. It is also demonstrated that the calculation of time-invariant noise components can be useful in the analysis of absorbance optical data. They can be extracted from data acquired during the approach to equilibrium, and can be used to increase the reliability of the results obtained from a sedimentation equilibrium analysis.     

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.     

Systematic interpolation method predicts protein chromatographic elution from batch isotherm data without a detailed mechanistic isotherm model

Predicting protein elution for overloaded ion exchange columns requires models capable of describing protein binding over broad ranges of protein and salt concentrations. Although approximate mechanistic models are available, they do not always have the accuracy needed for precise predictions. The aim of this work is to develop a method to predict protein chromatographic behavior from batch isotherm data without relying on a mechanistic model. The method uses a systematic empirical interpolation (EI) scheme coupled with a lumped kinetic model with rate parameters determined from HETP measurements for non‐binding conditions, to numerically predict the column behavior. For two experimental systems considered in this work, predictions based on the EI scheme are in excellent agreement with experimental elution profiles under highly overloaded conditions without using any adjustable parameters. A qualitative study of the sensitivity of predicting protein elution profiles to the precision, granularity, and extent of the batch adsorption data shows that the EI scheme is relatively insensitive to the properties of the dataset used, requiring only that the experimental ranges of protein and salt concentrations overlap those under which the protein actually elutes from the column and possess a ±10% measurement precision.     

Quaternary structure of the HSC70 cochaperone HIP

HSC70 interacting protein (HIP) is an essential cytoplasmic cochaperone involved in the regulation of HSC70 chaperone activity and the maturation of progesterone receptor. To determine the quaternary structure and the gross conformation of the protein in solution, a wide array of biochemical and biophysical techniques has been used. Size-exclusion chromatography and sedimentation velocity indicate the presence of a single species with a Stokes radius, R(s), of 55 A and a sedimentation coefficient, s degrees (20,w), of 4.34 S. The combination of these data gives a molecular mass of 101 000 Da, a value close to that of the theoretical molecular mass of a dimer (87 090 Da). Sedimentation equilibrium, performed at various protein concentrations and rotor speeds, gives a molecular mass of 88 284 Da, almost in exact agreement with the molecular mass of a dimer. On the basis of these data, a frictional ratio f/f(0) of 1.6 is obtained, suggesting an elongated shape for the HIP dimer. Secondary structure predictions, supported by circular dichroism experiments, indicate that HIP is an almost all alpha-protein, able to form extended coiled coils. Using threading and comparative model building methods, a structural model of a segment of HIP involved in HSC70 binding has been constructed and potential sites of interaction between HIP and HSC70 are proposed on the basis of electrostatic as well as shape complementarity. Altogether, these results indicate that HIP is an elongated dimer, able to bind two HSC70 molecules through its TPR regions, and suggest the existence of a versatile binding site on HSC70 that may be involved in the interaction of the chaperone with the cochaperones or other interacting proteins.     

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.     

Conformational spectra--probing protein conformational changes

Stafford [Biophys. J. 17 (1996) MP452] has shown that it is possible, using the analytical ultracentrifuge in sedimentation velocity mode, to calculate the molecular weights of proteins with a precision of approximately 5%, by fitting Gaussian distributions to g(s*) profiles so long as partial specific volume and the radial position of the meniscus are known. This makes possible the analysis of systems containing several components by the fitting of multiple distributions to the total g(s*) profile. We have found the Stafford relationship to hold for a range of protein solutes, particularly good agreement being found when the g(s*) profiles are computed from Schlieren (dc/dr vs. r) data using the Bridgman equation [J. Am. Chem. Soc. 64 (1942) 2349] . On this basis, we have developed a new approach to the analysis of systems where two or more distinguishable conformations of a single species are present, either in the same sample cell or in different cells in the same rotor. In the former case, this allows us to analyse a given solution of pure protein (i.e. monodisperse with respect to M) to reveal the presence in that solution of two or more conformers under identical solvent conditions. In the latter case, we can detect with high sensitivity any conformational change occurring in the transition from one set of solvent conditions to another. Alternatively, in this case, we can analyse slightly different proteins (e.g. deletion mutants) for conformational changes under identical solvent conditions. Examples of these procedures using well-defined protein systems are given.     

Soft interactions and crowding

The intracellular milieu is complex, heterogeneous and crowded—an environment vastly different from dilute solutions in which most biophysical studies are performed. The crowded cytoplasm excludes about a third of the volume available to macromolecules in dilute solution. This excluded volume is the sum of two parts: steric repulsions and chemical interactions, also called soft interactions. Until recently, most efforts to understand crowding have focused on steric repulsions. Here, we summarize the results and conclusions from recent studies on macromolecular crowding, emphasizing the contribution of soft interactions to the equilibrium thermodynamics of protein stability. Despite their non-specific and weak nature, the large number of soft interactions present under many crowded conditions can sometimes overcome the stabilizing steric, excluded volume effect.     

Analysis of heterologous interacting systems by sedimentation velocity: curve fitting algorithms for estimation of sedimentation coefficients, equilibrium and kinetic constants

Analytical ultracentrifugation (AUC) has played and will continue to play an important role in the investigation of protein-protein, protein-DNA and protein-ligand interactions. A major advantage of AUC over other methods is that it allows the analysis of systems free in solution in nearly any buffer without worry about spurious interactions with a supporting matrix. Large amounts of high-quality data can be acquired in relatively short times. Advances in software for the treatment of AUC data over the last decade have eliminated many of the tedious aspects of AUC data analysis, allowing relatively rapid analysis of complicated systems that were previously unapproachable. A software package called sedanal is described that can perform global fits to AUC sedimentation velocity data obtained for both interacting and non-interacting, macromolecular multi-species, multi-component systems, by combining data from multiple runs over a range of sample concentrations and component ratios. Interaction parameters include both forward and reverse rate constants, or equilibrium constants, for each reaction, as well as concentration dependence of both sedimentation and diffusion coefficients. sedanal fits to time-difference data to eliminate time-independent systematic errors inherent in AUC data. The sedanal software package is based on the use of finite-element numerical solutions of the Lamm equation.     

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.     

Modern analytical ultracentrifugation in protein science: a tutorial review

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