Calculations and Publication-Quality Illustrations for Analytical Ultracentrifugation Data

CHAPTER FIVE

Calculations and PublicationQuality Illustrations for Analytical Ultracentrifugation Data Chad A. Brautigam1 Department of Biophysics, The University of Texas Southwestern Medical Center, Dallas, Texas, USA 1 Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. General Features of GUSSI 2.1 The svdfr Module 2.2 The cofs Module 2.3 The General dfr Module 3. Utility Functions of GUSSI 3.1 Assembling SV-Based Isotherms 3.2 Sorting SE Data 3.3 Determining the Oligomeric State of Glycoproteins and Membrane Proteins 4. Summary Acknowledgments References

110 111 113 113 115 116 116 118 118 130 130 130

Abstract The analysis of analytical ultracentrifugation (AUC) data has been greatly facilitated by the advances accumulated in recent years. These improvements include refinements in AUC-based binding isotherms, advances in the fitting of both sedimentation velocity (SV) and sedimentation equilibrium (SE) data, and innovations in calculations related to posttranslationally modified proteins and to proteins with a large amount of associated cosolute, e.g., detergents. To capitalize on these advances, the experimenter often must prepare and collate multiple data sets and parameters for subsequent analyses; these tasks can be cumbersome and unclear, especially for new users. Examples are the sorting of concentration-profile scans for SE data, the integration of sedimentation velocity distributions (c(s)) to arrive at weighted-average binding isotherms, and the calculations to determine the oligomeric state of glycoproteins and membrane proteins. The significant organizational and logistical hurdles presented by these approaches are streamlined by the software described herein, called GUSSI. GUSSI also creates publication-quality graphics for documenting and illustrating AUC and other

Methods in Enzymology, Volume 562 ISSN 0076-6879 http://dx.doi.org/10.1016/bs.mie.2015.05.001

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2015 Elsevier Inc. All rights reserved.

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biophysical experiments with minimal effort on the user's part. The program contains three main modules, allowing for plotting and calculations on c(s) distributions, SV signal versus radius data, and general data/fit/residual plots.

1. INTRODUCTION Analytical ultracentrifugation (AUC) is a first-principles technique that is increasingly utilized by researchers in the biological sciences. The method can be used to hydrodynamically characterize macromolecules in solution and to study the energetics of their interactions. Currently, two experimental modes comprise the majority of AUC experiments. The first, called “sedimentation velocity” (SV), examines the transport of macromolecules in a high centrifugal field (ca. 100,000–200,000
g), providing highresolution hydrodynamic information (Schuck, 2000). Moreover, SV can be used to study self- and hetero-interactions, providing information such as stoichiometry (Balbo et al., 2005) and association constants (Brautigam, 2011; Brown, Balbo, & Schuck, 2008; Stafford & Sherwood, 2004). The data obtained from SV, i.e., radial concentration profiles of the sedimenting species captured at various points in time, are described by the Lamm equation (Lamm, 1929), a partial differential equation with no known exact analytical solutions; it is approachable with numerical solutions (Claverie, Dreux, & Cohen, 1975). The other AUC mode commonly used is called “sedimentation equilibrium” (SE), in which a lower centrifugal field (ca. 3000–30,000
g) is applied to the solution, resulting in a final equilibrium concentration gradient. These data may be rigorously analyzed to yield properties such as molar mass and virial coefficients (Casassa & Eisenberg, 1964). SE is also widely used to characterize interacting systems (Ghirlando, 2011). In the following, familiarity with AUC and the attendant data formats is assumed. Those inexperienced in AUC theory and practice are referred to the abovementioned citations, to earlier chapters in this volume (Correia & Stafford, 2015), and to other general treatments (Laue, 1999; Stafford, 2003; Zhao, Brautigam, Ghirlando, & Schuck, 2013) for introductions to SV and SE. In recent years, significant improvements have been made in the AUC field. For example, advances in the calculation of numerical solutions to the Lamm equation (Brown & Schuck, 2008; Cao & Demeler, 2008; Stafford & Sherwood, 2004) facilitated the extraction of hydrodynamic parameters

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directly from SV data and fitting diffusion-deconvoluted size distributions (Schuck, 2000) to these data. In addition, deriving thermodynamic quantities for interacting species was eased by advancements in (1) calculating Lamm-equation solutions coupled to thermodynamic/kinetic parameters (Cao & Demeler, 2008; Correia & Stafford, 2009; Dam, Velikovsky, Mariuzza, Urbanke, & Schuck, 2005), (2) isotherm analysis of SV data (Dam & Schuck, 2005; Schuck, 2010), (3) the modeling of SE data (Gillis et al., 2013; Vistica et al., 2004), among others. Whereas these approaches are powerful, they can increase the demands on the experimenter. For example, isotherm analysis of SV data requires the researcher to integrate several differential (c(s), ls-g*(s), or g(s*)) distributions, record the signal population and the weighted-average sedimentation coefficients, assemble these values into an electronic file, and load this file into the analysis program. Additionally, taking advantage of SE analytical advances can require the user to sort through many data scans according to sample identity, rotor speed, and wavelength of acquisition, then assembling the scans into the proper file formats. Furthermore, once preliminary analysis of AUC data is complete, other calculations can be performed. Examples of such calculations include the determination of the oligomeric states of glycoproteins and membrane proteins; these calculations can require assembly and collation of a large number of data sets, parameters, and equations. Also, AUC and other biophysical data sets can be large and/or difficult to illustrate. Commercial graphing packages are often either ill-suited to these data types or cumbersome to use, requiring the user to perform actions repetitively or to learn specialized scripting protocols. In this chapter, a software program called GUSSI (Grapher that Understands data from SV, SE, and Isotherm analyses) is introduced. This program contains utilities that drastically reduce the data-preparation procedures needed to apply advanced analytical techniques to the user’s data. GUSSI also solves the aforementioned challenges to plotting several biophysical data types through its ability to automatically parse and present the attendant data formats. The program is designed to be flexible enough for advanced users, but to have an intuitive interface that is inviting to novices.

2. GENERAL FEATURES OF GUSSI The analytical software programs SEDFIT and SEDPHAT (Zhao, Brautigam, et al., 2013) have plotting functions that conform to GUSSI’s formatting requirements. For this reason, this chapter will often reference

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the interaction of SEDFIT, SEDPHAT, and GUSSI. However, data from any program may be imported into GUSSI as long as they adhere to GUSSI’s formatting requirements, which are fully documented in its accompanying manual. Indeed, it has been observed that the power and flexibility of GUSSI’s user interface and the quality of the output encourage researchers to use it for purposes afield from the biophysical data formats described herein. The program operates only under the Windows operating system. GUSSI may be invoked from menu commands in SEDFIT/SEDPHAT, or as a stand-alone executable. In the former case, the software provides a data file to GUSSI; in the latter, the user employs GUSSI menu commands to open a properly formatted file. Next, GUSSI automatically parses and presents the data (Fig. 1). Its graphical user interface shows the graph, and the user can easily manipulate the plot’s features, such as line color and width, marker existence/appearance, labeling text, and axes appearances. Changes are automatically updated on the graph, giving the user instantaneous feedback. Menu functions allow writing figures in one of four popular file formats, as well as saving the program’s “state,” which allows the

Figure 1 The organization and appearance of GUSSI. (A) The overall organization of the program. The hierarchical relationships between the main modules (“cofs” for c(s) plots, “svdfr” for SV data/fit/residual plots, and “gen. dfr” for general data/fit/residual plots) are shown. The abbreviations are: g.p., glycoprotein; m.p., membrane proteins; SE dfr, SE data/fit/residuals; pop., population; ITC, isothermal titration calorimetry; Fluo., fluorescence; SPR, surface plasmon resonance; DLS, dynamic light scattering, field autocorrelation mode; iDLS, dynamic light scattering, intensity autocorrelation mode; and MST, microscale thermophoresis. (B) A screenshot of the program's user interface. On the left is the subject graph, and the panel on the right is the “Control Panel,” which houses various oft-used controls affecting the appearance of the graph. Calculations performed by GUSSI are accessed from the top menu bar.

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figure to be recalled later in GUSSI for examination or revision. This state file is particularly useful because it is completely encapsulated, i.e., it contains all of the information, even the data, needed to reproduce the graph. This represents a convenient means to document analytical results in a publication-ready format. Also, GUSSI state files may be easily shared among collaborators wishing to perfect a graph or illustrate points to one another. GUSSI features three different “modules”: (1) a data/fit/residual (dfr) module for sedimentation velocity (svdfr), (2) a c(s) distribution module (cofs), and (3) a dfr module for all other data structures (general dfr) (Fig. 1A).

2.1 The svdfr Module Following the analysis of SV data, the data points are often presented as markers along with a line representing the fit of the model to the data. It is also good practice to include a plot of the residuals, i.e., the deviations between the data points and the fit lines. There are usually 30–400 “scans” of data that represent the radial concentration profiles of the sedimenting species at a given point in time. When svdfr data are presented to GUSSI, several defaults are automatically applied. For example, because of the high information density of SV data, GUSSI automatically shows only every third scan/fit pair, as well as only every third data point. Importantly, the program does not discard the hidden data/fits. On the contrary, scan and data point sampling are user-adjustable parameters. Second, GUSSI colors the scans in progressive rainbow colors. The colors of the data points are inextricably tied to those of the fits and the residuals. The color scheme can be adjusted; several formats are available, including black-and-white and some that respond to user input (Fig. 2). The residuals can be shown as lines or as markers.

2.2 The cofs Module The cofs module of GUSSI is by far the most feature-rich, reflecting the popularity of the c(s) analysis as well as the many variations on the analysis that have been introduced over the years. Thus, this module features many integration tools and plot types; those that present the most unique and timesaving features are presented here. Others are also presented in the documentation file that accompanies the program. A powerful feature of GUSSI is the ease with which multiple distributions may be rendered simultaneously in the same plot (Fig. 2B). After

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Figure 2 The GUSSI svdfr and c(s) plots. (A) The default svdfr plot. In the upper panel, the individual data points are circles, and the fits to those data are shown as lines. The lower panel displays the residuals. Time-invariant noise features are subtracted from the data and fit lines. (B) A c(s) plot. The c(s) analysis for the data shown in (A) is displayed with the usual regularization (P-level ¼ 0.683) along with the distribution after the application of prior knowledge (Brown, Balbo, & Schuck, 2007), i.e., that the species sedimenting at 2.7 S is a single, discrete species (“with prior knowledge”).

invocation, additional distributions can be copied (e.g., from SEDFIT) and pasted into GUSSI. A legend can be shown, and the legend’s location is useradjustable. The properties of individual lines can be manipulated by making them “active,” which can be accomplished in multiple ways; the easiest is to point and click on the desired distribution on the graph. GUSSI can present the output from discrete/continuous analyses and multisignal SV (Padrick et al., 2010) analyses as well, and the sedimentation coefficients of c(s) distributions can be converted to s-values (s20,w) that are corrected to standard conditions, i.e., water at 20 °C. Further, in a single mouse-click, the displayed distributions can be normalized by the integrated area or by the maximum c(s) value in a user-provided s-range. This feature is extremely useful, allowing the simultaneous display of distributions with widely varying total signals. Arguably, the most important information that can come from a c(s) analysis is obtained by integrating the individual peaks. GUSSI features a tool for the integration of the distributions that gives the user the weighted-average s-values and the area under the distribution (in signal units) within the integrated range for a single distribution or for all distributions simultaneously. An integral form of the c(s) distribution can be superscribed over the normalized distributions, allowing the user to easily identify the overall and peak signal magnitudes in any given distribution.

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2.3 The General dfr Module GUSSI is capable of displaying the results of analyses of many other biophysical experiments (Fig. 1A). For example, the results of a single-speed or multispeed SE analysis can be graphed. Also, many different kinds of isotherms can be plotted, including those derived from SV. The “general dfr” module of GUSSI is responsible for handling such results (Figs. 1 and 3). Although these disparate biophysical results are plotted under a single computational rubric, GUSSI recognizes that the respective data structures require special features. For example, for long-column SE data, there can be a large number of data points, and thus data downsampling is enabled (Fig. 3A). In SV isotherms, the abscissa is scaled logarithmically (Fig. 3B). GUSSI is well adapted to displaying other biophysical data types, such as those from dynamic light scattering, fluorescence spectroscopy, and microscale thermophoresis. The program is particularly useful in displaying isothermal titration calorimetry data and fits that result from the powerful analytical combination of the programs NITPIC (Keller et al., 2012; Scheuermann & Brautigam, 2015) and SEDPHAT (Zhao, Piszczek, & Schuck, 2015). In all general dfr plots, legends are available. Also, the residual portion of the plot may be toggled off and on. Error bars can also be added to the data points.

Figure 3 General dfr plots from GUSSI. (A) SE data. As in Fig. 2A, the markers represent data points, and the lines fits to them. The residuals are shown as respective markers. A legend has been activated, indicating the rotor speeds of the respective concentration profiles. Only every third data point is shown. (B) SV isotherm. Globally analyzed sw (stars) and sfast (i.e., the sedimentation coefficient of the fast-sedimenting species; squares) isotherms are shown, along with fits to those data using effective-particle theory (Schuck, 2010).

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3. UTILITY FUNCTIONS OF GUSSI GUSSI contains three utility functions that significantly simplify dataprocessing steps required for SV isotherm analysis, SE analysis, and SV analysis of glycoproteins and membrane proteins. The first two prepare the data for subsequent analyses, and the third provides a means to determine the oligomeric state of the subject protein.

3.1 Assembling SV-Based Isotherms Isotherms derived from c(s) distributions have recently proved very useful to evaluate the association constants of both self- and hetero-association of macromolecules (Ayaz et al., 2014; Dam & Schuck, 2005; Zhao et al., 2012). Ordinarily, the experimenter fits the SV data from several (usually at least five) separate experiments using the c(s) model. The resulting distributions are integrated, and the experimenter records the signal population of the observed peaks and their signal-weighted sedimentation coefficients (i.e., the “weighted-average” s-value, or “sw”). Once this is accomplished, these values, along with the concentrations of the sedimenting species, are assembled into an electronic file for subsequent analysis. GUSSI streamlines and improves this process considerably. In the GUSSI workflow, the user loads all of the distributions into the cofs module. Then, the “Isotherm Maker” is initiated. After integration limits are chosen, a table appears bearing all of the information that was integrated and prompting the user for the species’ concentrations. Optionally, parameters destined for use in SEDPHAT can be entered in this table. Once the user completes the table, the isotherm and any associated files are saved for subsequent analysis. This workflow offers several improvements over the previous one. Importantly, the integration of all isotherms is accomplished at the same time with identical integration limits, thus strictly enforcing consistency across all distributions. Also, the tedious task of assembling the isotherm file is now accomplished automatically and without typographical errors. Isotherm construction based on the mass transport of all species, on effective-particle theory, and on the species’ populations are supported (Dam & Schuck, 2005; Schuck, 1998, 2010). Further, a framework for excluding contributions from contaminating species is available. This latter point is here illustrated in a brief example. Consider a simulated system (Fig. 4A) in which protein A (2.0 S) interacts with protein B (8.0 S) to form an 8.65-S complex with a dissociation constant (KD) of

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Figure 4 Correction of sw isotherms in GUSSI. The simulation assumes that species A associates with species B to form a 1:1 AB complex with a dissociation constant of 1 μM. Also, the sedimentation coefficient of A (sA) is 2.0 S, sB is 8.0 S, and sAB is 8.65 S. The preparation of A is contaminated with a 4.0 S, inert species. The [A] was held constant at 1 μM in the titration, while [B] was varied as described in the inset to part (A). (A) The c(s) distributions for the simulation. The simulated data were analyzed in SEDFIT. The distributions are normalized, and thus the contaminant at 4 S appears to have varying signal amplitudes. (B) Isotherm fitted to the data with (squares) or without (triangles) the correction applied as described in the text. Without the correction, KD refined to 4.1 μM, whereas the corrected data yielded the simulated value when evaluated, i.e., 1.0 μM.

1 μM. However, the protein A preparation is contaminated with a 4.0-S species that does not participate in the interaction. A titration is performed in which [A] is held constant at 1 μM (thus, the signal of the inert contaminant is constant), but [B] is varied from 0.5 to 10 μM. Obviously, simply integrating the distributions over an appropriate range (1–10 S) and analyzing the sw isotherm will result in an inaccurate KD (Fig. 4B, triangles; the fitted KD was 4.1 μM). However, in GUSSI, the user can define an “Exclusion Zone,” i.e., an s-range that is excluded from the calculation. The corrected sw values, here termed sw,corr, are given by sw, corr ¼

sw ctot  sw, e ctot, e ctot  ctot, e

(1)

where ctot is the total integrated signal over the entire integration region, and sw,e and ctot,e are, respectively, the weighted-average s-value and the total integrated signal derived from the Exclusion Zone (Zhao, Lomash, et al., 2013). This simple correction to the simulated system described above results in the isotherm shown in Fig. 4B (squares; the Exclusion Zone was from 3 to 5 S), and fitting this isotherm results in the correct, simulated KD of 1.0 μM.

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3.2 Sorting SE Data Recent innovations in the analysis of SE data are best utilized when the data are acquired at multiple rotor speeds and wavelengths (Cole, 2004; Vistica et al., 2004). However, these strategies present several bookkeeping hurdles. Hundreds of data files can be acquired, but their respective rotor speeds and wavelengths of acquisition are encoded not in the filenames but in a terse file header. Furthermore, the achievement of thermodynamic and mechanical equilibrium, which is necessary for the validity of the analysis, may be open to question. Finally, the meniscus and bottom of the solution column must be identified and sensible fitting limits chosen. Thus, the experimenter may face a formidable postexperimental sorting and evaluation chore. The GUSSI SE Data Sorter efficiently handles all of these tasks. The Sorter is invoked from the general dfr module of GUSSI (Fig. 1A). The user is prompted to identify the directory(ies) in which the SE data reside. Next, the program reads the all of the data files, parsing them according to cell number, wavelength, and rotor speed. It then performs a series of analyses. First, the meniscus and bottom of each scan are located. Next, the program evaluates the achievement of equilibrium using an algorithm similar to that employed by the program WinMATCH (similar algorithms are used for this purpose in SEDFIT and HeteroAnalysis (Cole, 2004)). In short, the scans are examined as a function of time and compared to the final scan; when only small differences are consistently calculated, equilibrium is judged to have been achieved. Next, a table is presented that displays the results of the finding and matching routines. Graphs detailing these choices are available, and unsatisfactory scans may be excluded from further consideration. The bottom part of the table prompts the user for information that is optional but useful in subsequent analyses in SEDPHAT. After the user has made all necessary exclusions and adjustments, the program “sorts” the data. The final scan in each cell/wavelength/rotor-speed category is written to a user-selected directory with succinct but informative filenames (e.g., “Eq9k_250.RA3” for absorbance data from Cell 3 acquired at 250 nm at a rotor speed of 9000 rpm). Experimental files needed for analysis of the data in SEDPHAT are also (optionally) written at this time, as is a comprehensive log file.

3.3 Determining the Oligomeric State of Glycoproteins and Membrane Proteins Finally, routines are present in the GUSSI cofs module that can help the user to establish the oligomeric state of a glycoprotein or membrane protein

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(Fig. 1A). Whereas the molar mass (and therefore the oligomeric state) of a protein can be simply derived from an SV experiment (Schuck, Perugini, Gonzales, Howlett, & Schubert, 2002), this task is more difficult for glycoproteins and membrane proteins because the partial specific volume ð
vÞ of the sedimenting species can be indeterminate. For glycoproteins, this complication arises from the possibility that the extent and composition of the modifying carbohydrates may not be known. In the case of membrane proteins, this indeterminacy can be due to unknown detergent content in the sedimenting, protein-containing micelle. GUSSI’s approach to these problems is graphical. It calculates the molar mass (or f/f0) of the sedimenting species as a function of a hypothesized value, e.g., oligomeric state and extent of carbohydrate modification. It then displays graphs depicting these calculations. In many cases, this allows the user to make an unambiguous determination of the oligomeric state of the protein. In the following, contributions from nondetergent cosolutes such as ions or cosmotropes are considered to be negligible; also, it is assumed that a single association state is populated at all concentrations under investigation. 3.3.1 Protein, Solution, and Chemical Information In performing these calculations, the user must know some information regarding the protein. These values include the monomeric molar mass, MP,mono, the partial specific volume, v
P , and the mass-based extinction coefficient (at a given wavelength), εP. For unmodified proteins, these parameters can conveniently be estimated from the amino acid sequence. SEDFIT (Zhao, Brautigam, et al., 2013), SEDNTERP (Laue, Shah, Ridgeway, & Pelletier, 1992), and UltraScan (Demeler, 2005) are among the platforms that can be used for this purpose; the needed values are obtained after pasting the one-letter amino acid sequence into the respective program’s user interface (and providing the experimental temperature). SEDNTERP is also very widely used to calculate the density (ρ) and viscosity (η) of the solution. For this calculation, the user provides the concentrations of solution constituents that are chosen from a table of chemicals with known physical properties. If the solution components are not tabulated in SEDNTERP, ρ and η can be measured using densimetry and viscometry, respectively. Often, dn/dcP, the protein’s refractive-index increment, is estimated at 0.187 cm3/g, as most proteins have values close to that. However, there are notable excursions from this value (Zhao, Brown, & Schuck, 2011), and a calculator available in SEDFIT supplies an estimate of the value based on actual amino acid

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composition. The properties of the cosedimenting chemicals, i.e., carbohydrates and detergents in glycoproteins and membrane proteins, respectively, must be known also as well as possible. With glycoproteins, the identities of the carbohydrate (noted with the subscript C) adducts are often not certain, and thus an approach to determining their oligomerization states taking this uncertainty into account is warranted. With membrane proteins, the identity of the cosedimenting detergent (noted with subscript D) is known, and thus v
D and dn/dcD can usually be obtained from databases or the manufacturer. Therefore, AUC experiments in which the amount of cosedimenting detergent can be firmly established are possible, enabling several powerful and distinct approaches to gleaning oligomeric state. These differences between glycoproteins and membrane proteins inform the distinct strategies outlined below. 3.3.2 Glycoproteins When studying a glycoprotein, it is possible that the identity of the carbohydrate moieties, extent of glycosylation, and oligomeric state are unknown. SV studies can be carried out to estimate the latter two quantities by calculating the molar mass of the sedimenting species (MGP,s) via the Svedberg equation: MGP, s ¼

s RT
GP ; DGP 1 
vGP, h ρ

(2)

where sGP is the sedimentation coefficient, R is the gas constant, T is the absolute temperature, DGP is the translational diffusion coefficient, and vGP, h is the hypothetical partial specific volume (“GP” refers to the glyco
protein). The
vGP, h must be hypothetical because the identity and extent of glycosylation is unknown in this scenario. Conveniently, the partial specific volumes of most protein-associating carbohydrate moieties ð
vC Þ fall in a narrow range: 0.58–0.68 cm3/g (Lewis & Junghans, 2000; Perkins, 1986), allowing the user to make an educated guess regarding the range of
vGP, h to be investigated. The approach to calculating v
GP, h is similar to that employed before in the context of SE (Ghirlando et al., 1995; Lewis & Junghans, 2000; Shire, 1992). MP,mono is known from the amino acid sequence. Together with a user-supplied guess regarding the oligomeric state (n, an integer) and an assumed probable mass percentage of the carbohydrate conjugated to the species (100
q), a hypothetical mass of the glycoprotein can be obtained:

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MGP, h ¼

nMP, mono ð1  qÞ

(3)

An assumption is made that v
GP, h can be calculated as the weight-average of
vP and a hypothesized partial specific volume of the carbohydrate,
vC, h : v
GP, h ¼

nMP, mono
vP + MC, h
vC, h ; MGP, h

(4)

or, cast in terms of q, this reduces to vC, h +
vP ð1  qÞ: vGP, h ¼ q



(5)

With these quantities determined, the only two obstacles to using Eq. (2) are determining sGP and DGP. The former can be arrived at by integrating c(s) distributions calculated by SEDFIT, defining sGP as the sw of the relevant peak. DGP can be calculated from the SEDFIT-refined frictional ratio, (f/f0)r, using that program’s scaling law (Brown, Balbo, & Schuck, 2007): pffiffiffi  1=2
3=2 2
vu 1=2 DGP ¼ kTsGP ηðf =f0 Þr ; (6) 18π 1 
vu ρ where k is the Boltzmann constant and v
u is the user-supplied
v employed in the SEDFIT analysis. Thus, any range of MGP,h and MGP,s may be calculated for any combination of hypothesized q and
vC, h . To utilize this feature of GUSSI, the user analyzes the SV data from one or more concentrations of a glycoprotein, noting the parameters used (uniformly for all data sets) and the (f/f0)r’s obtained. Then, the resulting differential distributions (usually c(s) from SEDFIT) are loaded into GUSSI, and the glycoprotein routine is engaged. The user is prompted to select the s-range of interest, after which GUSSI determines the sGP - values together with the standard error of these values. Next, the user inputs the necessary solution, protein, (f/f0)r, and carbohydrate parameters. Upon actuation, the program calculates and plots MGP,h and MGP,s for 1000 q-values from 0.05 to 0.8 (the latter represents a glycoprotein in which the carbohydrate has four times the mass of the protein, a very unusual situation). The plot of MGP,h versus q yields a single line, while that of MGP,s results in a swath of probable values taking into consideration the errors in sGP, DGP, and the user-inputted range of
vC, h (Fig. 5A, Table 1). The upper and lower bounds are calculated by inputting the limits of these values that result in the highest and lowest (respectively) MGP,s, resulting in conservative error estimates. This

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Figure 5 Glycoprotein calculations. (A) The GUSSI calculation on NPC1-NTD with the hypothesis of a monomer. The black, positively trending line represents the hypothetical mass based on a monomer of the protein and carbohydrate composition, and the other lines indicate the mass and limits based on sGP and DGP. The shaded area represents the most likely molar mass and carbohydrate content, and qint (Eq. 7) is indicated, along with the corresponding molar mass. (B) The same calculation, assuming that the protein is a dimer. The value of qint does not appear on this graph because it is negative.

Table 1 Parameters for the Carbohydrate Analysis Analysis Parameters

vu (cm3/g)


0.7212

3

ρ (g/cm )

1.00058

η (Poise)

0.01009

T (K)

293

Frictional ratios from analyses

Exp. 1

1.339

Exp. 2

1.354

Exp. 3

1.400

Protein parameters

vP (cm3/g)


0.7212

εP (L/g cm)

1.109

MP,mono (Da)

25,722

Carbohydrate parameters

vC range (cm3/g)


0.602–0.64

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bounding strategy is recapitulated in all calculations described in this chapter. MGP,h and MGP,s have opposite trends with increasing q; if the two intersect at positive values of q, this indicates that the hypothesized oligomerization state is consistent with the data and that the probable extent of glycosylation may be read from the abscissa of the intersection, while the molar mass of the sedimenting species is derived from the ordinate. Indeed, it can be shown that there is an exact analytical expression for the q-value of intersection, qint: qint ¼

1  ½ψ ð1 
vP ρÞ ; 1 + ½ψ ðv
P 
vC Þ

(7)

nMP, mono DGP : sGP RT

(8)

with ψ¼

Thus, by inputting the best values of sGP, DGP, and v
C (taken, respectively, in some cases as the mean experimental s and D, and the mean of the user-inputted range of v
C, h ), then these values may be inserted into Eqs. (7) and (8), allowing the exact calculation of qint. GUSSI calculates qint using these assumptions and displays it in the resulting plot (Fig. 5A). Alternative oligomerization states may be hypothesized and graphed (Fig. 5B). Importantly, there are cases in which two or more hypothesized oligomerization states will lead to intersections of the MGP,h and MGP,s lines; i.e., qint has more than one solution that leads to positive values of q. In such cases, GUSSI detects this issue and warns the user, and all experimental knowledge must be weighed to ascertain which value of qint is the most rational. Two critical assumptions are made in the above calculations. First, there must be no microheterogeneity in the sedimenting glycoprotein. This is important because this flaw would cause an underestimate of (f/f0)r (thus an overestimate of DGP), invalidating the subsequent calculations. However, this assumption can be conveniently checked in SEDFIT through the use of prior information in the c(s) distribution (Brown et al., 2007). In essence, this approach tests the hypothesis that the glycoprotein can be treated as a single species, suggesting no detectable microheterogeneity. The second assumption is that the signal from the glycoprotein comprises the vast majority of the total signal in the experiment. If this condition is not met, the refinement of the (f/f0)r may be skewed. There are advanced approaches that may relax this requirement, such as bimodal or size-and-shape distributions (Brown & Schuck, 2006), but they are beyond the scope of this chapter.

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As an illustration of the power of this approach, the data (exemplified in Fig. 2A) for a glycosylation mutant of the amino-terminal domain of the Niemann–Pick type C protein 1 (hereafter called NPC1-NTD) (Kwon et al., 2009) were reevaluated. Kwon et al. used a strategy similar to that delineated above, but did a separate SEDFIT analysis for each hypothesized q; only one such analysis is required by GUSSI. Three c(s) distributions for a concentration range of NPC1-NTD were obtained by analysis in SEDFIT, yielding very similar (f/f0)r’s (Table 1). When all the experimental information was input into GUSSI, the program produced Fig. 5A, showing an intersection of MGP,h and MGP,s at about 16% carbohydrate (qint ¼ 0.161) and a total molar mass of about 31 kDa, consistent with a monomer of NPC1-NTD. If the hypothesized oligomer was changed to a dimer, the two mass lines did not intersect at positive values of q (qint ¼ 0.373; Fig. 5B), indicating that a dimer of NPC1-NTD is very unlikely. Importantly, the hypothesis of little or no microheterogeneity was tested using a prior assumption (Brown et al., 2007) that the main species could be represented by a delta function (i.e., a homogeneous single species) at the s-value of the majority species. Violation of the prior would result in shoulder peaks outside the main peak, but none were observed (Fig. 2B). Thus, the data were consistent with the prior assumption, and the (f/f0)r values obtained from the analysis were used without hesitation. 3.3.3 Membrane Proteins The goal of the membrane-protein routines in GUSSI is essentially the same as that in the glycoprotein routine, i.e., to use SV to determine the oligomeric state of the protein. In this case, the sedimenting species is a protein-containing micelle of uncertain composition. GUSSI’s routines follow closely the protocols elaborated by M. le Maire, C. Ebel, and coworkers (Le Maire et al., 2008; Le Roy et al., 2015; Salvay, Santamaria, le Maire, & Ebel, 2008). In all of the following, an assumption is made that the detergent has no absorbance at the protein-detection wavelength. Further, contributions from lipids are neglected below, but they can be included in GUSSI. There are three strategies to address the oligomeric state of a membrane protein using the tools incorporated into GUSSI. All of them require some knowledge about the protein, the detergent, the protein-containing micelle, and the experimental setting: MP,mono,
vP ,
vD , δD (the amount of detergent bound to the protein in units of g/g), ρ, and η, and T. The value of δD is probably not known by the user, but may be measured in an SV experiment in which the evolution of the concentration profiles is monitored by both

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laser interferometry and absorbance optics. The magnitude of the protein’s signal in the interference data can be predicted by knowing its signal magnitude in the absorbance data along with εP, dn/dcP, and the wavelength of the laser used in the interference optics (λ). Any interferometric signal in excess of that expected from the protein is attributed to the detergent. If we denote the species’ signal magnitude from the absorbance optics as aa and that from the interference optics (using the same centrifugation cell) as ai, then δD ¼

ai εP λ dn=dcP  : aa  dn=dcD dn=dcD

(9)

With δD in hand, the first calculation that can be accomplished is that of a hypothetical f/f0 for the sedimenting species, given a hypothetical molar mass for the protein oligomer, MH ¼ nMP,mono. This is undertaken by realizing that f/f0 ¼ RS/Rmin, where RS is the Stokes radius of the species and Rmin is the minimum possible radius of the particle given its mass. In this context, 

 3MH ð
vP + δD
vD Þ 1=3 Rmin ¼ ; 4πNA

(10)

where NA is Avogadro’s number, and RS ¼

MH ð1  ϕ0 ρÞ ; 6πηsNA

(11)

where vD  ϕ0 ¼ v
P + δD


δD : ρ

(12)

The appearance of δD in Eq. (12) explicitly makes ϕ0 dependent on the relative absorbance and interference signals (see Eq. 9). The buoyant molar mass of the protein equals the product of MH and (1  ϕ0 ρ), as formulated in the trailblazing work of Casassa and Eisenberg (1964) and Tanford and Reynolds (1976). If the f/f0 derived from this analysis is not in the typical range of 1.1–1.5, the hypothesis of MH is likely incorrect, and a new calculation with a different hypothesized oligomeric state (n) is warranted. The second two strategies for testing the likelihood of putative protein oligomers use a modification of the Svedberg equation to calculate the molar

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mass of the protein component of the protein/detergent complex, MP, and then to compare it to MH. This modification makes use of ϕ0 : MP ¼

sRT : Dð1  ϕ0 ρÞ

(13)

These two methods differ only in how they obtain information on D. In the first, D is calculated from the SV result by using Eq. (6), with the caveats pointed out above (Section 3.3.2) for using (f/f0)r holding here as well. The second method relies on external information regarding RS obtained from another experiment, e.g., size-exclusion chromatography. In this case, the Stokes–Einstein equation is used: D¼

kT : 6πηRS

(14)

In practice, the user would conduct one or more SV experiments, collecting both interference and absorbance data (Le Maire et al., 2008). These data sets would be loaded into SEDFIT and analyzed using the c(s) model, with the user inputting the same analysis parameters for all sets and noting the (f/f0)r’s for each set. All of the distributions can then be imported into the cofs module of GUSSI, and the membrane-protein routine invoked. After the user selects the peaks of interest, GUSSI determines the sw values and signal magnitudes associated with the peaks (δD is calculated from the latter; see Eq. 9). Then, the user is prompted to input the needed information. Once this is accomplished, the user may toggle between the three types of calculations, supplying hypothetical oligomers and observing how well they conform to the SV data. To illustrate these functions, SV data for the Ca2+-ATPase SERCA1a (Salvay et al., 2008) were reevaluated.1 This membrane protein (termed “Ca2+-ATPase” hereafter), solubilized in dodecyl-β-D-maltoside (DM), has a monomeric molar mass of approximately 110 kDa based on its amino acid sequence. Both interference and absorbance data were acquired; because the reference buffer contained no detergent, free micelles of DM were detected in the interference data (Fig. 6A). In sum, six data sets were evaluated (absorbance and interference for each sample): (1) 0.8 mg/mL Ca2+-ATPase in buffer with 1 mg/mL DM, (2) 0.3 mg/mL Ca2+-ATPase 1

Data originally used in the Journal of Biological Physics, “Analytical ultracentrifugation sedimentation velocity for the chracterization of detergent-solubilized membrane proteins Ca++-ATPase and ExbB.”, vol. 33, 2007, pages 399–419, Salvay, A.Gl, Santamaria, M. le Maire, M., Ebel, C. Copyright Springer Science + Business B.V. 2008.

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Figure 6 The Ca2+-ATPase distributions and the membrane-protein f/f0 calculation. (A) Exemplary c(s) distributions for the Ca2+-ATPase/DM data. Shown are the distributions resulting from the analysis of the interference and absorbance data for sample (1) in the text, i.e., 0.8 mg/mL Ca2+-ATPase in a solution containing 1 mg/mL DM. “M” denotes the free detergent micelle, and “PD” the protein/detergent complex. In panels (B) and (C), the lines represent the high, optimal, and low limits for f/f0 for the protein/ micelle species given the parameters. In panel (B), the user has hypothesized a monomer, and the projection from the middle line to the X-axis shows this molar mass. The f/f0 is shown with an error interval. Panel (C) shows the same calculation with hypothesized dimer.

in buffer with 1 mg/mL DM, and (3) 0.3 mg/mL Ca2+-ATPase in buffer with 0.3 mg/mL DM. The data were analyzed using the c(s) model in SEDFIT (Table 2, Fig. 6A). After loading all of the c(s) distributions into GUSSI and initiating the membrane-protein routine, the program first calculated that the f/f0 for a hypothetical monomer was 1.21 (Fig. 6B), well within the range expected for a protein-containing micelle. GUSSI also takes into account the measurement errors in s and δD to arrive at an estimate of the standard error in this quantity (Fig. 6B). If the hypothesized oligomer was changed to a dimer, the calculated f/f0 was 1.92, a value that is physically unlikely (Fig. 6C). This test pointed to a monomeric protein. The second test was to use s from SV and RS from chromatography to calculate MP (Eqs. 13 and 14). This quantity is compared with MH to assess how well the data fit the hypothesis. The experimentally determined RS (5.5 nm (Salvay et al., 2008)) was used for this calculation (Table 2, Fig. 7). This resulted in MP ¼ 135 kDa, close to the hypothetical monomer, and well away from the hypothetical dimer. Therefore, this second test was also most consistent with the hypothesis of monomeric Ca2+-ATPase. The final test was to calculate MP based on s and D from the SV experiment. Calculating D using Eq. (6) and then inputting that into Eq. (13) yielded MP ¼ 111 kDa (Fig. 8; a range of likely MP values given the errors

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Table 2 Parameters for the Membrane-Protein Analysis Analysis/Experimental Parameters

vu (cm3/g)


0.7425

ρ (g/cm )

1.004

η (Poise)

0.0100

T (K)

293

λ (nm)

675

3

Frictional ratios from analyses (absorbance scans only)

Exp. 1

1.246

Exp. 2

1.324

Protein parameters

vP (cm3/g)


0.7425

εP (L/g cm)

0.966

MP,mono (Da)

109,490

3

dn/dcP (cm /g)

0.187

RS (nm)

5.5

Detergent parameters

vD (cm3/g)


0.82 3

dn/dcP (cm /g)

0.143

Calculated by GUSSI

δD (g/g)

0.47

in s and D is displayed, 101–121 kDa in this example). The hypothesis of the monomer is again closest to the calculated mass of the protein. Critically, the (f/f0)r values from the absorbance data only were used in this calculation because these values from the interference data were probably skewed by the strong signals from the empty detergent micelles (Fig. 6A, Table 2). Also, Cell 3 was excluded from this test because the homogeneity criterion did not appear to hold for this sample. All three tests conclusively pointed to the monomer of this protein being the entity present in the sedimenting protein–detergent complex; hypotheses of n > 1 may be safely rejected. On a final note, the δD calculated for Ca2+-ATPase (0.5 g/g) in this work was substantially lower than those reported earlier (0.8–0.9 g/g) using the same data (Le Maire et al., 2008; Salvay et al., 2008). However, these other

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Figure 7 The RS membrane-protein calculation. The bars represent the MH given the supplied molar mass of the protein and the hypothesized oligomeric state (the highlighted bar; neighboring bars are shown for reference). A shaded band with an optimal value shows the result of the molar-mass calculation when D is calculated from a supplied RS. GUSSI notes the exact MP with an arrow, and terms this mass “MP(ai, aa, s, RS)” to emphasize that the mass is a function of the measured signal amplitudes, the sedimentation coefficient, and the Stokes radius.

Figure 8 The “fitted f/f0” membrane-protein calculation. This figure has the same format as Fig. 7. The band represents the molar-mass calculation with D calculated from s and (f/f0)r (the latter are taken from the absorbance data only). In this case, the mass is noted as “MP(ai, aa, s, D)” to emphasize the contributions of the signal amplitudes, and the sedimentation and diffusion coefficients to the calculation.

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authors also came to the conclusion that Ca2+-ATPase is a monomer. It is likely that the variation in the reported δD resulted from different integration limits in the c(s) distributions derived from the absorbance data (i.e., variation in ai in Eq. 9). These results demonstrate that the determination of oligomeric state can be somewhat insensitive to the exact value of δD. Also, the value δD calculated herein is similar to that gleaned from a densityvariation SV experiment on Ca2+-ATPase if a frictional ratio of 1.25 is assumed (Le Roy et al., 2015), buttressing the validity of the value presented above.

4. SUMMARY The new software presented above, GUSSI, enables the straightforward graphing of analyses from SEDFIT and SEDPHAT in a publishable form. The graphs can be obtained in just a few mouse clicks. These graphs are highly user-customizable, and the program offers innovative plot formats. The program also provides a convenient platform for simple operations on AUC data. GUSSI is undergoing active development, and it currently encompasses many more functions than detailed in this chapter. These are fully documented in a manual that accompanies the program. The software may be obtained at http://biophysics.swmed.edu/MBR/software.html.

ACKNOWLEDGMENTS The author gratefully acknowledges the advice, suggestions, and testing efforts of Drs. Patrick Brown, Christine Ebel, Rodolfo Ghirlando, Shae Padrick, Grzegorz Piszczek, Peter Schuck, and Huaying Zhao. The author also thanks Dr. Schuck for adding GUSSI plotting menu items to SEDFIT and SEDPHAT. Drs. Hyock Joo Kwon and Johann Deisenhofer are thanked for providing the NPC1-NTD data, as are Drs. Aline Le Roy and Ebel for graciously supplying the Ca2+-ATPase data. Dr. Thomas Scheuermann and Drs. Padrick, Zhao, Ghirlando, and Ebel are thanked for providing comments on a draft of this chapter. Comprehensive citation of the expansive AUC literature was not possible due to format and space requirements; the author apologizes for any oversights.

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