Some statistical properties of differencing schemes for baseline correction of sedimentation velocity data

An automated interface for sedimentation velocity analysis in SEDFIT

Sedimentation velocity analytical ultracentrifugation (SV-AUC) is an indispensable tool for the study of particle size distributions in biopharmaceutical industry, for example, to characterize protein therapeutics and vaccine products. In particular, the diffusion-deconvoluted sedimentation coefficient distribution analysis, in the software SEDFIT, has found widespread applications due to its relatively high resolution and sensitivity. However, a lack of suitable software compatible with Good Manufacturing Practices (GMP) has hampered the use of SV-AUC in this regulatory environment. To address this, we have created an interface for SEDFIT so that it can serve as an automatically spawned module with controlled data input through command line parameters and output of key results in files. The interface can be integrated in custom GMP compatible software, and in scripts that provide documentation and meta-analyses for replicate or related samples, for example, to streamline analysis of large families of experimental data, such as binding isotherm analyses in the study of protein interactions. To test and demonstrate this approach we provide a MATLAB script mlSEDFIT.

An automated interface for sedimentation velocity analysis in SEDFIT

Sedimentation velocity analytical ultracentrifugation (SV-AUC) is an indispensable tool for the study of particle size distributions in biopharmaceutical industry, for example, to characterize protein therapeutics and vaccine products. In particular, the diffusion-deconvoluted sedimentation coefficient distribution analysis, in the software SEDFIT, has found widespread applications due to its relatively high resolution and sensitivity. However, a lack of available software compatible with Good Manufacturing Practices (GMP) has hampered the use of SV-AUC in this regulatory environment. To address this, we have created an interface for SEDFIT so that it can serve as an automatically spawned module with controlled data input through command line parameters and output of key results in files. The interface can be integrated in custom GMP compatible software, and in scripts that provide documentation and meta-analyses for replicate or related samples, for example, to streamline analysis of large families of experimental data, such as binding isotherm analyses in the study of protein interactions. To test and demonstrate this approach we provide a MATLAB script mlSEDFIT.

Efficient data acquisition with three-channel centerpieces in sedimentation velocity

Three-channel 3D printed centerpieces with two sample sectors next to a joint solvent reference sector were recently described as a strategy to double the throughput of sedimentation velocity analytical ultracentrifugation experiments [Anal. Chem. 91 (2019) 5866-5873]. They are compatible with Rayleigh interference optical detection in commercial analytical ultracentrifuges, but require the rotor angles of data acquisition to be repeatedly adjusted during the experiment to record data from the two sample sectors. Here we present an approach to automate this data acquisition mode through the use of a secondary, general-purpose automation software, and an accompanying data pre-processing software for scan sorting.

Variable Field Analytical Ultracentrifugation: II. Gravitational Sweep Sedimentation Velocity

Sedimentation velocity (SV) analytical ultracentrifugation is a classical biophysical technique for the determination of the size-distribution of macromolecules, macromolecular complexes, and nanoparticles. SV has traditionally been carried out at a constant rotor speed, which limits the range of sedimentation coefficients that can be detected in a single experiment. Recently we have introduced methods to implement experiments with variable rotor speeds, in combination with variable field solutions to the Lamm equation, with the application to expedite the approach to sedimentation equilibrium. Here, we describe the use of variable-field sedimentation analysis to increase the size-range covered in SV experiments by ∼100-fold with a quasi-continuous increase of rotor speed during the experiment. Such a gravitational-sweep sedimentation approach has previously been shown to be very effective in the study of nanoparticles with large size ranges. In the past, diffusion processes were not accounted for, thereby posing a lower limit of particle sizes and limiting the accuracy of the size distribution. In this work, we combine variable field solutions to the Lamm equation with diffusion-deconvoluted sedimentation coefficient distributions c(s), which further extend the macromolecular size range that can be observed in a single SV experiment while maintaining accuracy and resolution. In this way, approximately five orders of magnitude of sedimentation coefficients, or eight orders of magnitude of particle mass, can be probed in a single experiment. This can be useful, for example, in the study of proteins forming large assemblies, as in fibrillation process or capsid self-assembly, in studies of the interaction between very dissimilar-sized macromolecular species, or in the study of broadly distributed nanoparticles.

Sedimentation coefficient distributions of large particles

The spatial and temporal evolution of concentration boundaries in sedimentation velocity analytical ultracentrifugation reports on the size distribution of particles with high hydrodynamic resolution. For large particles such as large protein complexes, fibrils, viral particles, or nanoparticles, sedimentation conditions usually allow migration from diffusion to be neglected relative to sedimentation. In this case, the shape of the sedimentation boundaries of polydisperse mixtures relates directly to the underlying size-distributions. Integral and derivative methods for calculating sedimentation coefficient distributions g*(s) of large particles from experimental boundary profiles have been developed previously, and are recapitulated here in a common theoretical framework. This leads to a previously unrecognized relationship between g*(s) and the time-derivative of concentration profiles. Of closed analytical form, it is analogous to the well-known Bridgman relationship for the radial derivative. It provides a quantitative description of the effect of substituting the time-derivative by scan differences with finite time intervals, which appears as a skewed box average of the true distribution. This helps to theoretically clarify the differences between results from time-derivative method and the approach of directly fitting the integral definition of g*(s) to the entirety of experimental boundary data.

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.

A histogram approach to the quality of fit in sedimentation velocity analyses

The quality of fit of sedimentation velocity data is critical to judge the veracity of the sedimentation model and accuracy of the derived macromolecular parameters. Absolute statistical measures are usually complicated by the presence of characteristic systematic errors and run-to-run variation in the stochastic noise of data acquisition. We present a new graphical approach to visualize systematic deviations between data and model in the form of a histogram of residuals. In comparison with the ideally expected Gaussian distribution, it can provide a robust measure of fit quality and be used to flag poor models.

Accounting for photophysical processes and specific signal intensity changes in fluorescence-detected sedimentation velocity

Fluorescence detected sedimentation velocity (FDS-SV) has emerged as a powerful technique for the study of high-affinity protein interactions, with hydrodynamic resolution exceeding that of diffusion-based techniques, and with sufficient sensitivity for binding studies at low picomolar concentrations. For the detailed quantitative analysis of the observed sedimentation boundaries, it is necessary to adjust the conventional sedimentation models to the FDS data structure. A key consideration is the change in the macromolecular fluorescence intensity during the course of the experiment, caused by slow drifts of the excitation laser power, and/or by photophysical processes. In the present work, we demonstrate that FDS-SV data have inherently a reference for the time-dependent macromolecular signal intensity, resting on a geometric link between radial boundary migration and plateau signal. We show how this new time-domain can be exploited to study molecules exhibiting photobleaching and photoactivation. This expands the application of FDS-SV to proteins tagged with photoswitchable fluorescent proteins, organic dyes, or nanoparticles, such as those recently introduced for subdiffraction microscopy and enables FDS-SV studies of their interactions and size distributions. At the same time, we find that conventional fluorophores undergo minimal photobleaching under standard illumination in the FDS. These findings support the application of a high laser power density for the detection, which we demonstrate can further increase the signal quality.

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.

Multi-signal sedimentation velocity analysis with mass conservation for determining the stoichiometry of protein complexes

Multi-signal sedimentation velocity analytical ultracentrifugation (MSSV) is a powerful tool for the determination of the number, stoichiometry, and hydrodynamic shape of reversible protein complexes in two- and three-component systems. In this method, the evolution of sedimentation profiles of macromolecular mixtures is recorded simultaneously using multiple absorbance and refractive index signals and globally transformed into both spectrally and diffusion-deconvoluted component sedimentation coefficient distributions. For reactions with complex lifetimes comparable to the time-scale of sedimentation, MSSV reveals the number and stoichiometry of co-existing complexes. For systems with short complex lifetimes, MSSV reveals the composition of the reaction boundary of the coupled reaction/migration process, which we show here may be used to directly determine an association constant. A prerequisite for MSSV is that the interacting components are spectrally distinguishable, which may be a result, for example, of extrinsic chromophores or of different abundances of aromatic amino acids contributing to the UV absorbance. For interacting components that are spectrally poorly resolved, here we introduce a method for additional regularization of the spectral deconvolution by exploiting approximate knowledge of the total loading concentrations. While this novel mass conservation principle does not discriminate contributions to different species, it can be effectively combined with constraints in the sedimentation coefficient range of uncomplexed species. We show in theory, computer simulations, and experiment, how mass conservation MSSV as implemented in SEDPHAT can enhance or even substitute for the spectral discrimination of components. This should broaden the applicability of MSSV to the analysis of the composition of reversible macromolecular complexes.

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 the 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.

Overview of current methods in sedimentation velocity and sedimentation equilibrium analytical ultracentrifugation

Modern computational strategies have allowed for the direct modeling of the sedimentation process of heterogeneous mixtures, resulting in sedimentation velocity (SV) size-distribution analyses with significantly improved detection limits and strongly enhanced resolution. These advances have transformed the practice of SV, rendering it the primary method of choice for most existing applications of analytical ultracentrifugation (AUC), such as the study of protein self- and hetero-association, the study of membrane proteins, and applications in biotechnology. New global multisignal modeling and mass conservation approaches in SV and sedimentation equilibrium (SE), in conjunction with the effective-particle framework for interpreting the sedimentation boundary structure of interacting systems, as well as tools for explicit modeling of the reaction/diffusion/sedimentation equations to experimental data, have led to more robust and more powerful strategies for the study of reversible protein interactions and multiprotein complexes. Furthermore, modern mathematical modeling capabilities have allowed for a detailed description of many experimental aspects of the acquired data, thus enabling novel experimental opportunities, with important implications for both sample preparation and data acquisition. The goal of the current unit is to describe the current tools for the study of soluble proteins, detergent-solubilized membrane proteins and their interactions by SV and SE.

High-precision isothermal titration calorimetry with automated peak-shape analysis

Isothermal titration calorimetry (ITC) is a powerful classical method that enables researchers in many fields to study the thermodynamics of molecular interactions. Primary ITC data comprise the temporal evolution of differential power reporting the heat of reaction during a series of injections of aliquots of a reactant into a sample cell. By integration of each injection peak, an isotherm can be constructed of total changes in enthalpy as a function of changes in solution composition, which is rich in thermodynamic information on the reaction. However, the signals from the injection peaks are superimposed by the stochastically varying time-course of the instrumental baseline power, limiting the precision of ITC isotherms. Here, we describe a method for automated peak assignment based on peak-shape analysis via singular value decomposition in combination with detailed least-squares modeling of local pre- and postinjection baselines. This approach can effectively filter out contributions of short-term noise and adventitious events in the power trace. This method also provides, for the first time, statistical error estimates for the individual isotherm data points. In turn, this results in improved detection limits for high-affinity or low-enthalpy binding reactions and significantly higher precision of the derived thermodynamic parameters.

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.

Analysis of PKR-RNA interactions by sedimentation velocity

PKR is an interferon-induced kinase that plays a pivotal role in the innate immunity pathway for defense against viral infection. PKR is activated to undergo autophosphorylation upon binding to RNAs that contain duplex regions. Some highly structured viral RNAs do not activate and function as PKR inhibitors. In order to define the mechanisms of activation and inhibition of PKR by RNA, it is necessary to characterize the stoichiometries, affinities, and free energy couplings governing the assembly of the relevant complexes. We have found sedimentation velocity analytical ultracentrifugation to be particularly useful in the study of PKR-RNA interactions. Here, we describe protocols for designing and analyzing sedimentation velocity experiments that are generally applicable to studies of protein-nucleic acid interactions. Initially, velocity data obtained at multiple protein:RNA ratios are analyzed using the dc/dt method's to define the association model and to test whether the system is kinetically limited. The sedimentation velocity data obtained at multiple loading concentrations are then globally fitted to this model to determine the relevant association constants. The frictional ratios of the complexes are calculated using the fitted sedimentation coefficients to determine whether the hydrodynamic properties are physically reasonable. We demonstrate the utility of this approach using examples from our studies of PKR interactions with simple dsRNAs, the HIV TAR RNA, and the VAI RNA from adenovirus.

Extended Fujita approach to the molecular weight distribution of polysaccharides and other polymeric systems

In 1962 H. Fujita (H. Fujita, Mathematical Theory of Sedimentation Analysis, Academic Press, New York, 1962) examined the possibility of transforming a quasi-continuous distribution g(s) of sedimentation coefficient s into a distribution f(M) of molecular weight M for linear polymers using the relation f(M)=g(s)·(ds/dM) and showed that this could be done if information about the relation between s and M is available from other sources. Fujita provided the transformation based on the scaling relation s=κ(s)M(0.5), where κ(s) is taken as a constant for that particular polymer and the exponent 0.5 essentially corresponds to a randomly coiled polymer under ideal conditions. This method has been successfully applied to mucus glycoproteins (S.E. Harding, Adv. Carbohyd. Chem. Biochem. 47 (1989) 345-381). We now describe an extension of the method to general conformation types via the scaling relation s=κM(b), where b=0.4-0.5 for a coil, ∼0.15-0.2 for a rod and ∼0.67 for a sphere. We give examples of distributions f(M) versus M obtained for polysaccharides from SEDFIT derived least squares g(s) versus s profiles (P. Schuck, Biophys. J. 78 (2000) 1606-1619) and the analytical derivative for ds/dM performed with Microcal ORIGIN. We also describe a more direct route from a direct numerical solution of the integral equation describing the molecular weight distribution problem. Both routes give identical distributions although the latter offers the advantage of being incorporated completely within SEDFIT. The method currently assumes that solutions behave ideally: sedimentation velocity has the major advantage over sedimentation equilibrium in that concentrations less than 0.2mg/ml can be employed, and for many systems non-ideality effects can be reasonably ignored. For large, non-globular polymer systems, diffusive contributions are also likely to be small.

The boundary structure in the analysis of reversibly interacting systems by sedimentation velocity

Sedimentation velocity (SV) experiments of heterogeneous interacting systems exhibit characteristic boundary structures that can usually be very easily recognized and quantified. For slowly interacting systems, the boundaries represent concentrations of macromolecular species sedimenting at different rates, and they can be interpreted directly with population models based solely on the mass action law. For fast reactions, migration and chemical reactions are coupled, and different, but equally easily discernable boundary structures appear. However, these features have not been commonly utilized for data analysis, for the lack of an intuitive and computationally simple model. The recently introduced effective particle theory (EPT) provides a suitable framework. Here, we review the motivation and theoretical basis of EPT, and explore practical aspects for its application. We introduce an EPT-based design tool for SV experiments of heterogeneous interactions in the software SEDPHAT. As a practical tool for the first step of data analysis, we describe how the boundary resolution of the sedimentation coefficient distribution c(s) can be further improved with a Bayesian adjustment of maximum entropy regularization to the case of heterogeneous interactions between molecules that have been previously studied separately. This can facilitate extracting the characteristic boundary features by integration of c(s). In a second step, these are assembled into isotherms as a function of total loading concentrations and fitted with EPT. Methods for addressing concentration errors in isotherms are discussed. Finally, in an experimental model system of alpha-chymotrypsin interacting with soybean trypsin inhibitor, we show that EPT provides an excellent description of the experimental sedimentation boundary structure of fast interacting systems.

Limiting the sedimentation coefficient for sedimentation velocity data analysis: partial boundary modeling and g(s) approaches revisited

Brown and coworkers (Eur. Biophys. J. 38 (2009) 1079-1099) introduced partial boundary modeling (PBM) to simplify sedimentation velocity data analysis by excluding species outside the range of interest (e.g., aggregates, impurities) via restricting the sedimentation coefficient range being fitted. They strongly criticized the alternate approach of fitting g(s) distributions using similar range limits, arguing that (i) it produces "nonoptimal fits in the original data space" and (ii) the g(s) data transformations lead to gross underestimates of the parameter confidence intervals. It is shown here that neither of those criticisms is valid. These two approaches are not truly fitting the same data or in equivalent ways; thus, they should not actually give the same best-fit parameters. The confidence limits for g(s) fits derived using F statistics, bootstrap, or a new Monte Carlo algorithm are in good agreement and show no evidence for significant statistical distortion. Here 15 g(s) measurements on monoclonal antibody samples gave monomer mass estimates with experimental standard deviations of less than 1%, close to the confidence limit estimates. Tests on both real and simulated data help to clarify the strengths and drawbacks of both approaches. New algorithms for computing g(s) and a scan-differencing approach for PBM are introduced.