INTRODUCTION

TOP ABSTRACT INTRODUCTION EXPERIMENTAL DATA ANALYSIS RESULTS DISCUSSION REFERENCES

Analyzing the size-distribution of biological or synthetic macromolecules in solution, for example, the study of the oligomeric state of proteins, is a very important application of analytical ultracentrifugation with a long history (Signer and Gross, 1934; Svedberg and Pedersen, 1940; Bridgman, 1942; Baldwin and Williams, 1950; Vinograd and Bruner, 1966; Scholte, 1968; van Holde and Weischet, 1978; Stafford, 1992a). Because of the relatively large size dependence of macromolecular migration in a gravitational field, sedimentation velocity studies have the potential for high resolution. In the last decades, two approaches for the determination of sedimentation coefficient distributions have become most popular: the integral sedimentation coefficient distributions G(s) by van Holde and Weischet (vHW) (1978) and the differential apparent sedimentation coefficient distribution g*(s) obtained as a transform of the time-derivative of the signal, dc/dt (Stafford, 1992a). Exploiting the increased computational capabilities now available, we have recently proposed new methods for obtaining the apparent differential sedimentation coefficient distribution g*(s), termed ls-g*(s) (Schuck and Rossmanith, 2000), and for calculating a differential sedimentation coefficient distribution c(s) in which corrections for diffusion are made (Schuck, 2000). Both methods are based on direct least-squares modeling of the sedimentation boundary, using linear combinations of sedimentation profiles for nondiffusing species or linear combinations of Lamm equation solutions, respectively. They can be applied to larger data sets and, by virtue of regularization, exhibit substantially less noise in the calculated distributions than previous methods. First applications of the methods indicate a high versatility and significant advantages in sensitivity and resolution over the classical methods (Perugini et al., 2000; Schuck et al., 2000; Schuck and Rossmanith, 2000; Cole and Garsky, 2001; Sedlák and Cölfen, 2001; Hatters et al., 2001). However, a systematic exploration of differences and relationships among the different approaches, the statistical and experimental prior knowledge needed, as well as the implicit assumptions and their practical relevance are not available at present.

One of the classical problems in the theory of ultracentrifugal sedimentation is the treatment of diffusion in size-distribution analysis. The deconvolution of boundary diffusion can significantly increase the amount of detail that can be learned from a sedimentation experiment, and how diffusion is described constitutes a central difference among the existing approaches. One difficulty is that a set of two parameters (such as sedimentation and diffusion coefficient) is required to describe the sedimentation of each species in the distribution. Although the apparent sedimentation coefficient distributions g(s*) and ls-g*(s) do not allow consideration of diffusion, the vHW method for calculating G(s) is designed to take diffusion into account in a model-independent way by extrapolation of boundary fractions to infinite time. In the present paper, we describe a similar model-free method for calculating G(s) by extrapolation of the ls-g*(s) distributions to infinite time. Although complete model independence can be appealing, the extrapolation process can limit the resolution. Further, we will show that this extrapolation strategy for the deconvolution of diffusion fails for heterogeneous mixtures of species with overlapping sedimentation boundaries. The Lamm equation method c(s) uses an intermediate strategy, by estimating size-dependent diffusion coefficient via the Stokes-Einstein and Svedberg relationships and by utilizing prior knowledge, such as the weight-average frictional ratio, the molar mass of a main component, or similar information relating size and shape of the distribution. It also uses maximum entropy regularization, a Bayesian strategy to achieve numerical stability and optimal resolution, which is routinely used in many other fields of physics and biophysics for problems of similar mathematical structure. In contrast to the more classical data transformations, direct boundary models such as c(s) provide a criterion for the goodness-of-fit, which has potential use in nonlinear regression of distribution parameters. So far, however, this has remained unexplored.

In the present communication, we apply the different methods [g(s*), ls-g*(s), G(s) by vHW and by extrapolation of ls-g*(s), and c(s)] to four different data sets with different broadness of size distribution and different extent of diffusion. First, we examine a previously proposed theoretical model system with four species of closely spaced sedimentation coefficients (Stafford, 1992b). We then compare the information obtained from the analysis of experimental data from a predominantly single species with a trace impurity, a self-associating protein with several discrete oligomeric states, and a continuous distribution of large lipid emulsion particles. Besides questions of sensitivity and of resolution of species that do not exhibit clearly distinguishable sedimentation boundaries, we also examine the stability of the c(s) approach with respect to the prior assumption needed. In particular, we show how the knowledge of the weight-average frictional ratio can be extracted from the experimental data by combination with nonlinear regression.

	EXPERIMENTAL

TOP ABSTRACT INTRODUCTION EXPERIMENTAL DATA ANALYSIS RESULTS DISCUSSION REFERENCES

Analytical ultracentrifugation

For sedimentation velocity experiments, a Optima XL-I analytical ultracentrifuge (Beckman Coulter, Fullerton, CA) with absorbance and interference optical detection system was used. Epon double-sector centerpieces were filled with 400 µl of sample solution and PBS, respectively, and centrifuged at a rotor speed of 40,000 or 55,000 rpm and at rotor temperatures of 5 or 20°C, respectively. Absorbance data were acquired at a wavelength of 280 or 230 nm, respectively, and in time intervals of 2 min, with the radial increment set to 0.002 cm and taking two averages per scan; interference scans were taken in time intervals of 1 min. Buffer viscosity, protein partial specific volumes and frictional ratios were calculated using the software Sednterp (Laue et al., 1992).

Sedimentation equilibrium studies were conducted in a Beckman Optima XL-A equipped with absorbance optics. Double-sector or six-channel charcoal-filled epon centerpieces were filled with 140 µl of sample at loading concentration between 0.1 and 0.6 mg/ml, respectively. Sedimentation equilibrium was attained at a rotor temperature of 4°C at rotor speeds of 10,000 and 13,000 rpm, respectively, and absorbance profiles were acquired at wavelengths of 230 and 280 nm. Extinction coefficient ratios at different wavelengths were estimated spectrophotometrically.

	DATA ANALYSIS

TOP ABSTRACT INTRODUCTION EXPERIMENTAL DATA ANALYSIS RESULTS DISCUSSION REFERENCES

Lamm equation modeling

Sedimentation velocity data analysis was performed with the program Sedfit (which can be obtained from http://www.AnalyticalUltracentrifugation.com). For direct boundary modeling with distributions of Lamm equation solutions (Schuck, 2000), the measured absorbance or interference profiles a(r, t) were modeled as an integral over the differential concentration distribution c(s)

(1)

with  denoting noise components, and (s, D, r, t) denoting the solution of the Lamm equation for a single species (Lamm, 1929)

(2)

(where r denotes the distance from the center of rotation, and  the rotor angular velocity), which was solved by finite element methods on a static or moving frame of reference as described in (Claverie et al., 1975; Schuck, 1998; Schuck et al., 1998). For each species, the diffusion coefficient D(s) was estimated as a function of the sedimentation coefficient s based on the known partial specific volume of the protein, and on an estimated anhydrous frictional ratio f/f₀ (Schuck, 2000). In some cases, this can be combined with a predetermined interval of s-values for which diffusion coefficients D(s) are calculated from the Svedberg equation, using prior knowledge of the buoyant molar mass (Schuck et al., 2000).

The integral Eq. 1 was solved numerically by discretization into a grid of 100-200 sedimentation coefficients and calculating the best-fit concentrations for each species in a linear least squares fit. Systematic noise components of the data were estimated by using an algebraic method (Schuck and Demeler, 1999) (see below). Numerical stability was achieved by the maximum entropy method (Amato and Hughes, 1991),

(3)

which is a regularization procedure that minimizes not only the deviation between model and data at each radius value r_i and each time t_j, but simultaneously maximizes the information entropy  c ln c ds. The maximum entropy constraint  is adjusted such that the increase in the ² of the constrained fit, as compared to the unconstrained fit ( = 0), corresponds to a confidence level of one or two standard deviations (p = 0.68 or 0.95, respectively) as calculated by F-statistics (Provencher, 1982a; Bevington and Robinson, 1992; Johnson and Straume, 1994; Schuck, 2000). This method has the virtue of providing the simplest distribution of all possible distributions that are consistent with the data, while allowing for sharp peaks in the distribution if necessary for modeling the data. Alternatively, when prior knowledge is available about the smoothness of the distribution, the maximum entropy constraint can be replaced by a Tikhonov-Phillips term  (d²c/ds²)² ds that minimizes the second derivative of the distribution. Because of its linearity, this term can be implemented in the computationally simpler matrix form. However, it cannot describe isolated peaks as well as the maximum entropy constraint, and tends to produce smoother distributions, which can make it more robust.

For critical inspection of the quality of fits to sedimentation velocity data, we have developed a two-dimensional picture representation of the residuals to avoid the usual loss of information on systematic deviations in the common overlay presentations. The bitmap representation of the residuals was calculated in the following way: The residual values of all points in all scans R(r, t) were transformed to a gray value n(r, t) between 0 and 255, with n = 0 for R(r, t) < 0.05, n = 255 for R(r, t) > 0.05, and with a linear transformation for the residuals 0.05 < R(r, t) < 0.05. This transformation results in neutral gray (n = 128) for a perfect fit with R(r, t) = 0, brighter pixels for positive and dark pixels for negative residuals. In the bitmap, the pixels were ordered in rows that correspond to the scan number, and columns that correspond to the radius values. This representation of the residuals results in a uniformly gray picture without any structure for a good fit with randomly distributed residuals. If a structure is visible, this corresponds to systematic residuals. In this way, systematic residuals from vibrations of the camera, which produce vertical patterns, can be distinguished from those of an imperfect boundary model resulting in diagonal structures. Also, the presence of isolated bad scans can be diagnosed from horizontal lines, and they can be identified from a separate output file of Sedfit of the local rms error for each file.

Systematic noise analysis

Components of systematic time-invariant and radial-invariant noise were calculated using the algebraic approach described previously (Schuck and Demeler, 1999). In brief, the time-invariant baseline signal b_i at each radius r_i was minimized by least-squares according to

(4)

In this equation, S denotes any model for the sedimentation boundary, such as the solution of the Lamm equation (Eq. 2), or the sedimentation of a size-distribution (Eq. 3), which might, in general, depend on a set of parameters {p}. It can be easily shown that, for any given value of {p} (or if S has no further parameters), the best-fit time-invariant noise is given by

(5)

where the quantities

(6)

(with the total number of scans N_s) represent an average scan, and an average sedimentation model, respectively. This leads to a least-squares problem for the calculation of the remaining parameters {p}

(7)

An analogous procedure can be used for the radial-invariant signal offsets (Schuck and Demeler, 1999). For the distribution analysis, it is straightforward to solve Eq. 7 directly by linear least squares methods (Schuck, 2000; Schuck and Rossmanith, 2000).

Inspection of Eq. 7 shows that the only information that can be extracted from the data is that of a time-difference (here with an average scan as a reference). It should be noted that no explicit estimate of the time-invariant noise is used in Eq. 7. Nevertheless, for fundamental reasons, modeling the time-difference introduces new degrees of freedom into the data analysis, which is a consequence of the unknown radial-dependent baseline offsets. This can lead to slight correlation with parameters of the boundary model, in particular with those describing very slow sedimentation processes (Schuck and Demeler, 1999; Kar et al., 2000). Such correlation can be minimized by using a large data set that includes large boundary displacement (Kar et al., 2000). The calculation of an explicit estimate of the baseline parameters with Eq. 5 follows after the nonlinear regression in Eq. 7 and allows comparing of the sedimentation model with the data in the original data space (direct boundary modeling). It follows from Eq. 5 that the best estimate of the time-invariant signal is an average over all scans of the residuals profiles. Therefore, it is dependent on the parameters of the sedimentation model, and realistic estimates of the time-invariant baseline signal are obtained only if the sedimentation model fits the data well. However, this step does not introduce any additional correlation in the estimates of the sedimentation parameters {p}.

Calculation of the sedimentation coefficient distributions g*(s) and G(s)

For obtaining the apparent sedimentation coefficient distribution g*(s), the direct boundary model for a distribution of nondiffusing particles ls-g*(s) (Schuck and Rossmanith, 2000) was used, as implemented in the software Sedfit. In this method, ls-g*(s) is calculated using the same concepts and numerical framework as the distribution c(s) described above, but by replacing the Lamm equation solution (s, D(s), r, t) in Eq. 1 with the theoretical sedimentation profiles of nondiffusing species, i.e., step-functions U(s, r, t),

(8)

(9)

U(s, r, t) describes the ideal behavior of initially uniformly distributed particles with sedimentation coefficient s, but without any diffusion, during sedimentation and radial dilution in the sector-shaped solution column in the centrifugal field (with the meniscus position of the solution column r_m, and a boundary position r*(t)) (Fujita, 1962; Stafford, 1992a; Schuck and Rossmanith, 2000). Regularization was applied using the Tikhonov-Phillips method, at a confidence level of p = 0.95, unless stated otherwise.

Eq. 8 can be combined with Eqs. 7 and 5 for systematic noise analysis. As described above, as a consequence of modeling the time difference in Eq. 7, some additional correlation and increased error of the distribution at small s-values can occur. Both can be minimized best in the ls-g*(s) analysis by using data sets with large boundary displacement and scans where the boundary has cleared the meniscus. It should be noted that, if used for the analysis of small molecules where diffusion is not negligible, Eq. 9 is not a good approximation, and, dependent on the size of the data set, large residuals and relatively poor estimates of the time-invariant signal may be obtained.

The differential sedimentation coefficient distribution defined by Eqs. 8 and 9 is termed ls-g*(s) to indicate its basis on the least-squares data modeling (ls), and the neglect of diffusion (g*). A similar differential sedimentation coefficient distribution can be calculated using the time-derivative dc/dt, as approximated by the time difference c/t (Philo, 2000; Stafford, 1992a). This is termed g(s*) to indicate the use of a transformation of the radial variable r into an apparent sedimentation coefficient s* in the course of its calculation. In this method, the pairwise time difference between scans is used to eliminate systematic time-invariant noise. The final apparent sedimentation coefficient distribution g(s*) from dc/dt can be transformed back into a boundary model (with rhs of Eq. 8 and 9) for comparison of model and data, and explicit estimates of the time-invariant noise can be calculated via Eq. 5 (data not shown). Because both methods g(s*) and ls-g*(s) are based on equivalent definitions of the apparent sedimentation coefficient distribution, when applied to the same data sets, this leads to equivalent results for both the g*(s) distribution (Schuck and Rossmanith, 2000) and the time-invariant noise estimates (data not shown). However, the absence of a differentiation step in ls-g*(s) allows larger boundary displacements between the scans, and avoids artificial broadening effects that can be introduced by the approximation of dc/dt by c/t (Schuck and Rossmanith, 2000; Philo, 2000). For comparison, where possible, g(s*) analysis of time-difference sedimentation data was performed with the program dcdt+ (J. S. Philo, 3329 Heatherglow Ct., Thousand Oaks, CA 91360).

The integral sedimentation coefficient distribution G(s) was calculated as described by van Holde and Weischet (1978). This method is based on the Faxén-type approximate solution of the Lamm equation, which can be written as

(10)

with the boundary position of a nondiffusing species r*(t) as defined in Eq. 9, and the error function  (van Holde and Weischet, 1978; Demeler et al., 1997). Following the derivation of vHW, the radial positions R_i of fractional plateau concentrations c_i (c_i = ic_p/N, with i denoting the fraction number, and N denoting the total number of divisions of the plateau concentration c_p) can be transformed in apparent sedimentation coefficients s^*_app,i = ln(R_i/r_m)/²t, so that

(11)

With the inverse error function ¹ applied to both sides of Eq. 11, we arrive at

(12)

(van Holde and Weischet, 1978), which shows that a linear extrapolation of s^*_app,i on a t^0.5 scale to infinite time allows for determination of s, and deconvolution of diffusion effects on the sedimentation boundary. The resulting s-values from the different boundary fractions form the integral sedimentation coefficient distribution G(s) (van Holde and Weischet, 1978; Demeler et al., 1997).

We have used the implementation of the vHW approach outlined earlier (Schuck, 2000). In brief, after determination of the plateau signals for each curve, the boundary was divided in N (usually 20 to 50) fractions of equal concentration increments dh. The radial position of the boundary fraction is calculated as R_i = mean {r, with dh × (i  0.5) < c(r) < dh × (i + 0.5)}, i.e., as the average of the radial values of all data points that have signal values as defined by the limits of the boundary fraction. This method is designed for a high number of fractions, where the boundary increment dh for each fraction are comparable in size to the noise of the data, and it extracts the boundary positions in a least-square sense, not requiring smoothing of the data. In the algorithm implemented in Sedfit, it is ensured that all boundary fractions in all scans have at least one data point, otherwise the number of boundary fractions N is automatically reduced.

As an alternative strategy for the calculation of G(s), we have implemented the following extrapolation of ls-g*(s) to infinite time: The total set of scans used for analysis was subdivided in sequential sets of scans, each taken at a time interval centered at t_i. (For example, sets of 10 scans were used for the analysis of interference optical data.) For each set, a differential sedimentation coefficient distribution ls-g*(s)_i was calculated and divided into N equal area fractions A_j. Because the area under the ls-g*(s) curves corresponds to the loading concentration (Eq. 8), these fractional areas are equivalent to boundary fractions, and the average sedimentation coefficient s_ij(A_j) in a given area fraction A_j at time t_i directly corresponds to the s-values s^*_app,i calculated for each boundary fractions in the vHW method. As a consequence, the same extrapolation procedure, Eq. 12, can be applied to generate a distribution G(s). (It should be noted that this method, like vHW, requires the existence of solution plateaus to define consistent area fractions, and that it requires some depletion at the meniscus to avoid correlation of ls-g*(s) with baseline and systematic noise parameters.) Each of the ls-g*(s)_i curves can be calculated taking into account time-invariant and radial-invariant systematic noise (Eq. 7), but best results were obtained if only time-invariant noise was considered (vertical alignment of the scans, e.g., close to the meniscus, may be achieved separately). After the linear regression with Eq. 12, the best-fit values of s_ij(A_j) can be transformed back into equivalent boundary positions, and a step-function model of the boundary in the original data space can be generated (Eq. 9). This allows for calculating overall best-fit estimates for the systematic noise contributions via Eq. 5 (see above).

There is a subtle difference between the two procedures. The division of the boundary into equal fractions of the plateau signal introduces small errors in the linear approximation of s_app,i versus t^0.5. However, by substituting Eq. 10 with an improved approximate solution of the Lamm equation, a more complex relationship s_app,i versus t^0.5 can be derived (Eq. 17 of van Holde and Weischet, 1978). In the division of the ls-g*(s) area, the division is made in units of equivalent loading concentrations, and the fractions are propagated in time according to Eq. 9, such that they generate boundary fractions that have experienced different radial dilutions. This can be expected to slightly alter the precision of the linear approximation. However, in studies with synthetic data, we found this to produce similar accuracy of the linear extrapolation s_app,i versus t^0.5 (both have errors < 1%, data not shown). An independent theoretical justification of the ls-g*(s) approach can be derived from the observation that the apparent sedimentation coefficient distributions exhibit a sharpness increasing with time, and by interpreting Eq. 10 as describing diffusional spread in a space of "apparent sedimentation coefficients." In this view, the division of ls-g*(s) in area fractions and linear extrapolation on a t^0.5 scale can be understood solely as a rational method for extrapolating ls-g*(s) to infinite time (with the transformation to the integral G(s) distribution providing increased numerical stability of the extrapolation). However, because of the close relationship of the methods, we interpret the extrapolation of area fractions of the ls-g*(s) distributions as an extension of the vHW method, in a sense that ls-g*(s) can be calculated easily in the presence of systematic time-invariant noise of interference optical data by applying algebraic noise decomposition (Schuck and Demeler, 1999).

Sedimentation equilibrium analysis

Sedimentation equilibrium data were analyzed by global modeling of 3-6 data sets obtained at different loading concentrations and rotor speeds using the commercial mathematical modeling software Mlab (Civilized Software, Silver Spring, MD). Least-squares fits of the measured absorbance profiles a(r) were calculated using models based on the exponential equilibrium distribution of ideally sediment-ing oligomeric species (Svedberg and Pedersen, 1940)

(13)

where r₀ is an arbitrary reference radius, and, with the monomer molar mass and partial specific volume M and , respectively, the solvent density , the absolute temperature T, the gas constant R, the molar extinction coefficient at wavelength , and the thickness of the centerpiece d (1.2 cm). Dependent on the particular model used, the molar concentrations of the i-mer c_i(r₀) were coupled by mass action law. In all models, the absence of partial specific volume changes upon oligomerization was assumed.

A transformation of the absorbance distribution (of a single scan) into a continuous molar mass distribution c(M) combined with maximum entropy regularization was achieved by replacing the kernel in Eq. 3 by sedimentation equilibrium exponentials (Eq. 13) for a single species. To allow for a rational comparison of the concentration of the different species, average loading concentrations were used as concentration units, obtained by integration of each species from meniscus to bottom. This method is similar to the Laplace transform with regularization described by Wiff and Gehatia (1976). Because of the high sensitivity of the shape of c(M) on the location of the bottom of the solution column, for this analysis, the meniscus and bottom were predetermined using intensity scans.

Dynamic light scattering

Dynamic light scattering experiments were conducted using a Protein Solutions DynaPro 99 instrument with a DynaPro-MSTC200 microsampler (Protein Solutions, Charlottesville, VA). Protein samples were centrifuged for 5 min in a microcentrifuge to remove dust particles, and a 20-µl sample was inserted in the cuvette with the temperature control set to 20°C. The light-scattering signal was collected at 90°, and autocorrelation coefficients were exported for analysis with the software Sedfit, adapted for dynamic light-scattering analysis by replacing the Lamm equation solutions with the following models for the field autocorrelation function:

(14)

where  is the decay time and q = (4n/)sin(/2), with the solvent refractive index n, the wavelength of the incident light , and the scattering angle  (Murphy, 1997). Continuous size distributions were calculated from the autocorrelation data analogs to the analysis of the sedimentation coefficient distribution, but using the correlation functions Eq. 14 as kernel in the integral Eq. 3. This results in a distribution analysis that is similar to the maximum entropy method described by Livesey et al. (1986) and similar to that implemented by Provencher (1979, 1982b) in CONTIN (which uses a Tikhonov-Phillips regularization). The regularization was adjusted to a confidence level between 0.45 and 0.55 (Provencher, 1992).

	RESULTS

TOP ABSTRACT INTRODUCTION EXPERIMENTAL DATA ANALYSIS RESULTS DISCUSSION REFERENCES

Comparison of the resolution using synthetic data

To examine the potential of Lamm equation modeling for resolving species in sedimentation velocity experiments when no clear sedimentation boundary is visible, we have simulated data from the model system that was proposed earlier by Stafford (1992b) (Fig. 1). It consists of four species with sedimentation coefficients 6, 7, 8, and 9 S. For each, theoretical sedimentation profiles were calculated with a starting concentration of 0.25 (arbitrary signal units), and to the sum of their signals, normally distributed noise at a magnitude of 0.01 was added. From inspection of the resulting broad sedimentation profiles, the presence of a heterogeneous mixture is obvious, but no distinct boundaries can be identified (Fig. 1 A).

View larger version (38K): [in this window] [in a new window]   FIGURE 1   Study of the resolution of c(s), ls-g*(s), and dc/dt-based g(s*) distributions for a four-component model system of elongated molecules as suggested in Stafford (1992b). (A) Sedimentation profiles of species with relative molar masses 341,000, 398,000, 441,000, and 490,000 and sedimentation coefficients of 6, 7, 8, and 9 S, respectively (corresponding to anhydrous frictional ratios of 2.93, 2.79, 2.61, and 2.49). The simulated rotor speed was 50,000 rpm, rotor temperature 20°C, partial specific volume 0.73 cm3/g, and the loading concentration for each species 0.25. The simulated data (thin lines, only every second data set shown) was generated by adding the distributions of all species and 0.01 normally distributed noise. c(s) analysis was performed with 150 species with sedimentation coefficients between 4.5 and 10.5 S, with regularization at a confidence level of 0.68, and a floating frictional ratio and floating baseline (similar results were obtained with and without consideration of time-invariant noise). The frictional ratio converged at a value of 2.9, close to the weight-average frictional ratio of 2.7 of the simulated species. The best-fit distributions are shown as bold dashed lines. (B) The calculated c(s) distribution (bold solid line) is shown, and, for comparison, the results of the ls-g*(s) analysis of scans 10-30 (bold dashed line), dc/dt analysis of scans 18-21 (thin solid line), vHW analysis of scans 1-25 (circles), and G(s) by extrapolation of ls-g*(s) to infinite time (diamonds). Also shown is the distribution c(s) obtained with a value of f/f0 of 2.0 (dotted line, reduced in scale by factor 0.2).

The distributions obtained greatly differ for the different methods (Fig. 1 B). As can be expected, the apparent sedimentation coefficient distributions (both ls-g*(s) and g(s*)) only reveal the range of s-values, without finer structure in the size distribution. The results obtained with the vHW method for the integral sedimentation coefficient distribution G(s) (van Holde and Weischet, 1978), and for G(s) by extrapolation of ls-g*(s) to infinite time are only slightly better as they represent the range of s-values more accurately. Although they can, in principle, unravel the effects of diffusion, the close spacing of the sedimentation coefficients clearly exceeds the resolution. However, the four species can be discriminated with the c(s) method, if combined with the optimization of the weight-average frictional ratio (see below).

The deconvolution of diffusion by G(s) and c(s) merits more detailed consideration. The extrapolation of the boundary fractions of both G(s) methods is shown in Fig. 2. In comparison, the extrapolated ls-g*(s) distribution has fewer time points for the extrapolation (due to the need for a whole set of curves for calculating a single ls-g*(s) distribution), but it can be subdivided into more boundary fractions (Fig. 2 B). However, as expected, the results are very similar in both methods. Interestingly, although the linear regression of the boundary fractions appears to be of high quality, it is clear that they do not contain the information required for resolving the species underlying the model system.

View larger version (50K): [in this window] [in a new window]   FIGURE 2   Sedimentation coefficients of the boundary fractions of scans 1-25 of the data shown in Fig. 1 A, determined directly by the vHW method and by extrapolation of area fractions of ls-g*(s) to infinite time. (A) For the vHW analysis, the maximal number of boundary fractions was 20, and shown are the apparent s-values (circles) and extrapolations (lines) for fractions 1-19. (Fraction 20 near the plateau was dominated by noise, not shown). (B) The ls-g*(s) extrapolation was based on calculated ls-g*(s) distributions with a grid of 50 s-values between 1 and 15 S, with regularization at a confidence level of p = 0.9, allowing for a floating baseline. This was applied to a "sliding subset" of five sequential scans. Each ls-g*(s) distribution was divided into 30 equal area fractions, for which the weight-average sedimentation coefficient was determined (circles) and extrapolated to infinite time (solid line). The distributions G(s) are shown in Fig. 1 B.

The application of the distribution of Lamm equation solutions c(s) is based on one piece of additional information that can link s and D, which can be obtained most conveniently by estimating a frictional ratio f/f₀ of the species under study. Because, in most cases, the data will not contain enough information for defining f/f₀ as a function of sedimentation coefficient (or even a distribution of f/f₀ for all species with the same sedimentation coefficient), the c(s) method is restricted to use only a single value, corresponding to a weight-average frictional ratio of all species. An estimate may initially be based, for example, on the expected hydrodynamic behavior for the type of macromolecule under study (e.g., for globular protein, or random coil). In the application to the data shown in Fig. 1 A, we started with an initial guess of f/f₀ of 1.2, which resulted in two peaks at 6.5 and 8.5 S, but with a poor fit with an rms deviation of 0.015, and clearly systematic residuals (data not shown). The model functions calculated from c(s) were too broad, particularly in the earlier scans, indicating too large diffusion coefficients predicted by a too small prior estimate of f/f₀ = 1.2. Therefore, we increased f/f₀ to a value of 2 (resulting in the distribution with three peaks shown as dotted lines in Fig. 1 B), and then floated this parameter to be optimized in a nonlinear regression. This resulted in a c(s) distribution that exhibits peaks at the correct position of the species underlying the simulated data. Furthermore, the parameter f/f₀ converged to a value of 2.9, which is close to the weight-average frictional ratio of 2.7 for the simulated, highly elongated, species. This indicates that nonlinear regression of f/f₀ can be a useful technique to obtain good estimates for this parameter. Nonlinear regression was also found to be a good technique for determining the exact meniscus position, as well as the sedimentation and diffusion coefficients of co-sedimenting small molecules (such as buffer components that are not matched in the reference and sample side of the ultracentrifugal cell; data not shown).

When inspecting the details of the resulting best-fit c(s) distribution, it should be considered that the maximum entropy regularization causes the peaks not to be very sharp, as one could expect for an ideal measurement. By design of the Bayesian procedure, it restricts the structure in the c(s) distribution to the features that are essential for describing the raw data within the limits of the noise. According to this result, any sharper distributions would not lead to significantly better fits (at a confidence limit of 0.68). Therefore, the c(s) distribution in Fig. 1 B depicts the information that can be extracted reliably from the data in Fig. 1 A, and, by design, it does not represent the best fit, which would, in most cases, be too unstable. However, it should be noted that even in the presence of noise, all four species can be unraveled with the c(s) analysis even without their boundary separation. Interestingly, the four species cannot be resolved if only the data subsets suitable for vHW analysis are taken into consideration. We have shown earlier that, for broad continuous distributions, the regularization can produce artificial oscillations (Schuck, 2000). The analysis here also highlights another well-known property of maximum entropy regularization, an inherent tendency to merge closely spaced peaks, in particular in nonoptimal fits (where the predefined F-ratio results in higher fractional increase in ², and consequently higher regularization). This is shown in the dotted line in Fig. 1 B. To achieve optimal resolution, therefore, it appears important to balance the distribution parameters (the confidence level and prior estimate of f/f₀) with the inspection of the quality of the fit. As indicated above, this can include optimization of f/f₀ by least-squares regression.

Study on the sensitivity of the methods using experimental data from an immunoglobulin G sample

Next, we compared the performance of the methods using data from a nearly homogeneous immunoglobulin G (IgG) sample. The experimental fringe-shift data are shown in Fig. 3 A. A fit with discrete solutions of the Lamm equation reveals one main species with 6.60 S, but with a (statistically highly significant) 3% contamination of a dimeric species with 9.58 (± 0.2) S. Similarly, a fit of dc/dt with the Fujita-MacCosham-Philo function (Philo, 1997, 2000) converges at a value of 5% of a faster sedimenting species (data not shown). This may serve as a test for the sensitivity in the detection of trace amounts of species.

View larger version (26K): [in this window] [in a new window]   FIGURE 3   VHW analysis by extrapolation of ls-g*(s) of interference data from an immunoglobulin sample. (A) Experimental fringe profiles at 40,000 rpm, 20°C, scans 30-170. For clarity of presentation, only every 10th data set is shown, and the integral fringe shift is removed. The best-fit time-invariant noise contribution from the G(s) analysis is shown as a dotted line. (B) ls-g*(s) distributions (solid lines) calculated for sets of 20 scans, with 50 s-values between 2 and 11 S, and regularization at p = 0.9. The circles represent the integral sedimentation coefficient distributions G(s), obtained by extrapolating 30 area fractions of ls-g*(s) to infinite time. For comparison, G(s) is shown when derived from ls-g*(s) distributions with lower resolution (30 s-values from 2 to 11 S, dashed line), or from ls-g*(s) distributions calculated on the basis of fewer scans (6, dotted line), and from conventional vHW analysis after removing best-fit systematic noise components (fractions 2-29, squares). (C) sapp values of the area fractions (circles) and linear extrapolation to infinite time on a t0.5 scale (lines).

Figure 4 shows differential sedimentation coefficient distributions that are uncorrected for the effects of diffusion, which are g(s*) by dc/dt (solid line), and the ls-g*(s) distribution (circles). If applied to the same data subset, both distributions are very similar, and both clearly reveal the presence of the larger species. Figure 3, B and C, presents the integral sedimentation coefficient distribution G(s) by extrapolation of ls-g*(s) to infinite time (circles in Fig. 3, B and C, calculated systematic noise shown as the dotted line in Fig. 3 A) and by conventional analysis after subtraction of systematic noise components (taken, for comparison only, from the best fit with the discrete Lamm equation modeling) (open squares in Fig. 3 B). It can be seen that the s-value of the main species is at ~6.7 S and that the highest fraction indicates the presence of a faster component. However, quantitation is difficult because of the relatively large noise present in the highest boundary (or area) fractions. Finally, the differential sedimentation coefficient distribution c(s), with diffusion deconvoluted assuming an average frictional ratio of 1.58, is shown as a dotted line in Fig. 4.

View larger version (26K): [in this window] [in a new window]   FIGURE 4   Differential sedimentation-coefficient distributions from the data shown in Fig. 3 A. Scans 110-139 were analyzed with the dc/dt-method (solid line), and with the ls-g*(s) method using 100 s-values between 4 and 12 S, and Tikhonov-Phillips regularization at a confidence level of 0.68 (circles). The c(s) analysis (dotted line) was calculated on the basis of scans 1-200, with an average frictional ratio of 1.58, 200 s-values between 4 and 14 S, and maximum entropy regularization at a confidence level of 0.95 (leading to an rms error of the fit of 0.00267 fringes). The inset shows the same distributions at an expanded view for visualizing the contaminating larger species.

As in the first example, the increase in resolution in the c(s) method is achieved in part because of the larger number of files that can be included in the analysis, but mainly through estimates of the extent of diffusion, which will be examined in the following. Figure 5 shows the dependence of the quality of fit obtained at different values for the frictional ratio. It can be seen that the rms error has a clearly defined minimum. With nonoptimal values, the boundary shape is not well described, as illustrated by the diagonal pattern in the residual bitmaps. It should be noted that this occurs, to a similar extent, both at too low and too high values of f/f₀, and that the assumption of the average shape to be spherical ( f/f₀ = 1) is equally poor as the limit of nondiffusing particles. (This limit of no diffusion is identical to the ls-g*(s) and g(s*) distribution). In contrast, the residual bitmap shows very little systematic patterns at the optimal value of 1.58. As a consequence, like in the first example, we can extract an average frictional ratio from the data itself by virtue of the criterion of the quality of fit. How the different values of f/f₀ affect the calculated c(s) distribution is shown in Fig. 6. Consistent with previous observations (Schuck, 2000), the position of the main peak remains essentially constant, whereas smaller values of f/f₀ lead to sharper peaks. However, the location of the peak of the trace component was found to be correlated with f/f₀. If the diffusion is over corrected, information on the contaminating faster-sedimenting species is lost, and the smaller peak appears reduced in area and at higher s-values (inset in Fig. 6). Alternatively, this is accompanied by a sharp decrease in the quality of the fit (Fig. 5), which helps in determining the s-value of the faster species. In the inset of Fig. 6, the two solid lines indicate two distributions that are indistinguishable on a confidence level of 0.9, suggesting an error estimate in the order of 0.3 S (best fit at 9.38 S).

View larger version (37K): [in this window] [in a new window]   FIGURE 5   Dependency of the quality of fit with the c(s) method on the assumed average frictional ratio. Scans 1-200 of the immunoglobulin experiment shown in Fig. 3 A were analyzed with 200 s-values from 4 to 14 S, p = 0.9, and using different values of f/f0. (The resulting distributions are shown in Fig. 6.) The circles represent the rms error of the fits obtained with different values of r × f/f0 (r = 1.02). A linear regression converges at a value of r × f/f0 = 1.58. For comparison, the rms error of the discrete two-component model is shown as a horizontal dotted line. The corresponding residual bitmaps at representative values of r × f/f0 are shown as insets, with residual values at each data point depicted as bright (positive) or dark (negative) pixel. Different radius values are represented by pixel columns (meniscus left, bottom right), and scans are represented as pixel rows (first scans top, last scans bottom). The diagonal structures visible indicate large residuals propagating in time with the sedimentation boundary.

View larger version (42K): [in this window] [in a new window]   FIGURE 6   Effect of the deconvolution of diffusion on the calculated c(s) distributions for the immunoglobulin experiment. Distributions are calculated for scans 1-200, at p = 0.9, and normalized at the main peak. The differences of the quality of the fit are shown in Fig. 5. The limiting case of no consideration of diffusion is illustrated by the apparent sedimentation coefficient distribution ls-g*(s) (dashed line). This corresponds to an infinite value of r × f/f0. With finite values, the main peak of the c(s) distribution decreases in width with decreasing r × f/f0 at constant peak position (solid lines, values indicated in graph). The best-fit value of r × f/f0 = 1.58 is shown as a bold line. c(s) curves at values smaller than 1.3 virtually superimpose the 1.3 curve (data not shown). The inset shows the dependence of the location of the minor peak of the c(s) curves on the value of r × f/f0: 1.70 (dotted line), 1.58 (bold solid line), 1.55 (solid line), 1.30 (dotted line). With a number of data points ~300,000, the curves for 1.58 and 1.55 are statistically indistinguishable on a 90% confidence limit, suggesting an error of the peak position of ~0.3 S. The different heights of the peaks in the inset are a result of the normalization of the distribution at the main peak.

Application to the study of preparations of the herpes simplex capsid protein VP5

A more complex problem is the analysis of a protein with extended, slow oligomerization. This is illustrated by experiments with preparations of the herpes simplex capsid protein VP5 (the biological implications will be discussed elsewhere). Previous reports of sucrose gradient centrifugation suggested that the 149 kDa protein is monomeric (Newcomb et al., 1999). However, Fig. 7 A shows typical sedimentation profiles with a sloping plateau region, indicating the existence of large aggregates, together with two separate major boundaries from discrete smaller species. Experiments at different loading concentrations and rotor speeds led to similar distributions, but with slightly different peak areas, consistent with a slow and at least partially reversible oligomerization.

View larger version (33K): [in this window] [in a new window]   FIGURE 7   Sedimentation velocity experiment with preparations of the herpes simplex virus capsid protein VP5. The protein was prepared as described in Newcomb et al. (1999) followed by dialysis against PBS. The absorbance profiles were acquired at 230 nm, at a protein concentration of 0.16 mg/ml, a rotor temperature of 8°C and a rotor speed 55,000 rpm. (A) Measured absorbance distributions (thin lines) and best-fit distribution from the Lamm equation model c(s) (dashed bold lines). For clarity, only every third scan is shown. (B) Residuals of the fit, which has an rms error of 0.0123 OD230. (C) Apparent sedimentation coefficient distributions g(s*) calculated by dc/dt (solid line) and distributions ls-g*(s) (dashed line). (D) Best-fit c(s) sedimentation coefficient distribution (solid line), allowing for systematic time-invariant noise. The 68% confidence band from Monte Carlo simulations (1000 iterations, calculated at slightly lower resolution in s) is shown as dashed and dotted lines.

With the data of Fig. 7 A, both versions of G(s) are not applicable because of the absence of a solution plateau. No consistent boundary fractions (or area fractions, respectively) can be defined. Therefore, only the differential sedimentation coefficient distributions can be compared (Fig. 7, C and D). All distributions clearly show two maxima corresponding to the two visible separating boundaries, with the peaks in c(s) clearly being the best resolved. To avoid broadening from large time-intervals, only a small subset of absorbance scans can be used in the calculation of g(s*) by dc/dt (solid line in Fig. 7 C), limiting the range of g*(s) under conditions where the peaks are well resolved. Because the ls-g*(s) method does not require the approximation of dc/dt by c/t (Schuck and Rossmanith, 2000), a much larger number of scans can be incorporated in the analysis, which shows the presence of a high number of larger aggregates with s-values up to 25 S, consistent with the results from the c(s) analysis. However, although the c(s) distribution may suggest separate peaks for the larger species, a Monte Carlo statistical analysis reveals that the apparent peaks at s-values larger than ~15 S may be induced by oscillations from the regularization procedure and not significant within the given level of noise in the data. However, this only refers to the exact position of the c(s) peaks, but not the existence of material at large s-values, which is highly significant.

Because of the formation of distinct boundaries, at least for the two slower sedimenting species, the oligomerization is slow on the time scale of the sedimentation, and it seems possible to assign oligomeric states to the individual peaks. This, however, requires additional information that we have sought in sedimentation equilibrium and dynamic light-scattering experiments. Global modeling of sedimentation equilibrium data at multiple rotor speeds and concentrations show that the majority of the protein is monomeric, but with significant contributions of small oligomers (data not shown; an example of sedimentation equilibrium profile modeled with monomer, dimer, and tetramer is shown in the inset of Fig. 8 A). Although the self-association scheme could not be identified, the data were consistent with an isodesmic association with contaminations of incompetent monomer. These results from sedimentation equilibrium show that the main peak of the c(s) corresponds to the monomer, and suggest that the second peak is a dimer (or possibly a trimer). With a monomer sedimentation coefficient of 6.8 S, we can calculate a Stokes radius (R_S) of 5.2 nm, and a frictional ratio of 1.5 (equivalent to a prolate ellipsoid with 2a = 25.4 and 2b = 4.5 nm). This value was applied in the c(s) analysis of Fig. 7 D for diffusional deconvolution. Nonlinear regression of the weight-average frictional f/f₀ (with a starting value of 1.0) converged to a best-fit value of 1.25, but with a final rms error of the fit that was not statistically different from that obtained with f/f₀ = 1.5. Further, the c(s) curves calculated using both values virtually superimpose (data not shown). This confirms that a nonlinear regression of f/f₀ leads to values sufficient in precision for the deconvolution of diffusion in the sedimentation coefficient distributions, although the obtained average frictional ratio itself is not suitable for the transformation of c(s) into precise molar mass distributions c(M).

View larger version (23K): [in this window] [in a new window]   FIGURE 8   (A) Sedimentation-equilibrium experiment with preparations of VP5 at a loading concentration of 0.26 mg/ml, a rotor speed of 10,000 rpm and a temperature of 4°C, with data acquired at a wavelength of 230 nm. The calculated c(M) distribution obtained with maximum entropy regularization (p = 0.9) is shown. The inset shows the raw data (circles) modeled (as part of a global fit) with monomer, dimer, and tetramer (solid line), and the calculated distributions for monomer (dotted line), dimer (dashed line), and tetramer (dash-dotted line). (B) Distribution of Stokes radii as obtained upon analysis of the autocorrelation function from dynamic light-scattering experiments. The distribution of scattering intensity versus radius is shown as a solid line and the estimated relative weight concentrations (based on a constant frictional ratio) are shown as a dashed line.

Interesting from the methodological point of view is a transformation of the sedimentation equilibrium data into a "model-free" molar mass distribution c(M), as suggested earlier by Wiff and Gehatia (1976) (Fig. 8 A). This c(M) transform does not take advantage of our knowledge of the molar mass of the different oligomers, and it is mathematically equivalent to the continuous size-distribution analysis of the sedimentation velocity data. It shows a main peak at a molar mass ~200 kDa, distinctly higher than the molar mass of the monomer (149 kDa), clearly indicating the presence of oligomeric species. Unfortunately, the data at molar mass >600 kDa are not reliable in this transformation because they are mainly governed by assumptions of sedimentation in the region of optical artifacts close to the bottom of the cell. In contrast to sedimentation velocity, the c(M) transform does not have sufficient information to resolve the different species. A similar situation is encountered in the interpretation of the dynamic light-scattering data, which are commonly transformed to distributions of Stokes radii, R_S (Fig. 8 B). The scattering intensity has a peak at ~5 nm, but also extends to larger species. To better compare the relative abundance of the different species, the distribution was rescaled into relative weight concentrations as shown by the dotted line in Fig. 8 B. From these data, it appears that particles with R_S > 8 nm are in very low abundance, despite their significant contribution to the scattered intensity. Like in the c(M) transform of the sedimentation equilibrium data, no resolution of the oligomers is achieved. Nevertheless, the virtual absence of species with R_S > 8 nm indicates that the 10.3-S peak seen in sedimentation velocity may be a dimer (R_S = 6.9 nm), and less likely a trimer (R_S = 10.3 nm) or even larger oligomers. Similarly, the 13.2-S peak may be a trimer (R_S = 8.3 nm) but less likely a tetramer (R_S = 10.6 nm) or even a pentamer or a hexamer.

This example illustrates the current potential and limitations of the sedimentation coefficient distributions from complex oligomeric mixtures. It demonstrates that complementary information from sedimentation equilibrium and dynamic light scattering can be used (and is required) for the detailed interpretation. This is despite the much lower resolution of these methods due to the significantly more ill-conditioned analysis of exponentials as compared to the Lamm equation solutions. Correspondingly, the additional information from the c(s) distribution on the number and approximate size of species can be very important for the correct interpretation of the sedimentation equilibrium data.

Analysis of continuous size distributions of emulsions

As a last example, we analyzed a truly continuous size distribution of lipid emulsion particles. General physical characteristics of such particles and their use for the study of apolipoproteins have been described by MacPhee et al. (1977) and (M. A. Perugini, P. Schuck, G. J. Howlett, submitted). In the current context, for illustrating the behavior of the size distribution, we considered the data from a mixture of two different elution fractions after sucrose-gradient ultracentrifugation. Figure 9 A shows the experimental flotation data exhibiting a bimodal boundary. For both of the fractions, we have measured the average diffusion coefficient by dynamic light scattering (with hydrodynamic radii of 34 and 62 nm, respectively). The dashed lines in Fig. 9 A are the calculated best-fit distributions based on two discrete species with the predetermined diffusion coefficients. The comparison with the boundary spread of the experimental data shows that the distributions of the fractions are broad, and that boundary broadening by diffusion is relatively small, but cannot be neglected.

View larger version (38K): [in this window] [in a new window]   FIGURE 9   Flotation experiments with mixtures of fractionated lipid emulsions. (A) Experimental absorbance distributions of a mixture of two sucrose-gradient ultracentrifugation fractions, measured in intervals of 360 s (circles, every second scan and every second data point of sets 1-25 analyzed are shown). For experimental details, see M. A. Perugini, P. Schuck, G. J. Howlett (submitted). To illustrate the extent of diffusion of the particles, the dashed lines show the best-fit models with separate species for each of the two fractions, using the diffusion coefficients as predetermined by dynamic light scattering for each fraction. (B) Differential sedimentation coefficient distributions, calculated with a frictional ratio of 1.0, a partial-specific volume of 1.055 ml/g, and with regularization at p = 0.9. Shown are c(s) with Tikhonov-Phillips regularization (solid line) and maximum entropy regularization (dotted line), and ls-g*(s) (dashed line). The integral sedimentation coefficient distribution G(s) by the vHW method, based on scans 7-14, is shown with solid circles, scaled to 0.003.

Here, the analysis with the c(s) method can be based on the knowledge that the emulsion particles are spherical, i.e., that f/f₀ = 1.0. (For simplicity, we have used the mean partial specific volume of 1.055 ml/g of the components of the emulsion mixture; a refinement taking into account the full size-dependence of the partial-specific volume of the particles is included in [M. A. Perugini, P. Schuck, G. J. Howlett, submitted]). The resulting size distributions are shown in Fig. 9 B. When using maximum entropy regularization, we obtained very noisy c(s) distributions with several artificial spikes (dotted line). This is consistent with previous findings that use of the maximum entropy method for broad continuous distributions can cause artificial oscillations (Provencher, 1992). However, this difficulty can be circumvented effectively by the use of Tikhonov-Phillips regularization (solid line). Through the second derivative minimization of this procedure, one can make use of the broadness and smoothness of the distributions as prior knowledge.

The ls-g*(s) method leads to a similar distribution, which is only slightly broader because of the limited extent of diffusion (Fig. 9 B, dashed line). Further artificial broadening would be expected from the approximation of dc/dt by c/t in the g(s*) analysis, due to the large boundary displacement between the absorbance scans (Schuck and Rossmanith, 2000; Philo, 2000). (This results in a convolution of the g*(s) distribution with a hyperbola segment of width s = st/t [Schuck and Rossmanith, 2000]; when restricting the analysis to scans 8-13, the broadening for a single nondiffusing species would be ~100 S for the first peak, and ~200 S for the second peak.) The vHW analysis applied to a suitable data subset results in similar information as ls-g*(s) or c(s) (Fig. 9 B, circles). However, less information on the faster floating particles is obtained, and the two peaks from the two lipid emulsion fractions appear not as well resolved as in the c(s) analysis.

	DISCUSSION