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PLoS One. 2013; 8(5): e62694.

Published online 2013 May 16. doi: 10.1371/journal.pone.0062694

PMCID: PMC3656001

Hendrik W. van Veen, Editor^{}

University of Cambridge, United Kingdom

* E-mail: peter.schuck/at/nih.gov

Conceived and designed the experiments: CAB SBP PS. Performed the experiments: CAB SBP PS. Analyzed the data: CAB SBP PS. Contributed reagents/materials/analysis tools: CAB PS. Wrote the paper: CAB PS. Designed the software used in the analysis: PS.

Received 2012 December 18; Accepted 2013 March 25.

This is an open-access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.

This article has been cited by other articles in PMC.

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

The study of protein interactions in multi-component systems is key to improve our understanding of signaling pathways, which ubiquitously possess dynamically assembled multi-protein complexes as critical nodes for integrating different information flows and regulating downstream events. Hallmarks of such complexes are multi-valent interactions and cooperativity, which are notoriously difficult to characterize. We have recently developed a global multi-method analysis for interacting systems with multiple binding sites [1] that is useful for determining thermodynamic parameters of the interactions, including association constants, enthalpy changes, and cooperativity constants. However, often one of the most difficult steps is to identify the thermodynamic states, i.e. to ascertain which complexes exist in solution. This goal can be far from trivial to achieve for two-component interactions and be very difficult for three-component or higher-order systems.

Several years ago, the multi-signal sedimentation velocity (MSSV) approach was introduced [2] as a new tool to address this problem. It takes advantage of the strongly size-dependent migration in the centrifugal field in a configuration that leaves complexes always in a bath of their components, such as to maintain populated complexes in solution during the experiment despite their differential sedimentation velocities. MSSV exploits the relatively high resolution that can be achieved in modern diffusion-deconvoluted sedimentation coefficient distributions [3] and synergistically combines this with spectral deconvolution of absorbance and/or refractive index optical signals [2], [4], [5]. Among the virtues of this method are the relatively fast experimental time, the ability to detect multiple co-existing complexes, the orthogonal observations of composition and complex size and hydrodynamic shape often allowing for an internal test for consistency of the derived complex stoichiometry, and the relative independence of estimates of sample concentrations. Dependent on the particular molecules under study, these often make the MSSV more attractive than other solution methods such as isothermal titration calorimetry, sedimentation equilibrium, or single-signal sedimentation velocity. Many applications of MSSV to two- and three-component systems have demonstrated the power of this approach [6]–[16].

One potential drawback of MSSV is that interacting systems with rapid chemical interconversion on the time-scale of sedimentation will not hydrodynamically resolve the different chemical species during the sedimentation process, but exhibit coupled migration and produce so-called reaction boundaries. This previously limited the application of MSSV to systems that either have slow reaction kinetics (i.e. complex lifetimes on the order of hours) or to conditions where complex formation can be substantially saturated. However, since the original development of MSSV, significant progress has been made in the theory and conceptual understanding of reaction boundaries [17]–[19]. In particular, the effective particle theory (EPT) establishes simple rules for the composition of reaction boundaries [17], opening these for quantitative interpretation with regard to the binding affinity and/or stoichiometry. One objective of the present work was to illustrate how EPT can be applied in the context of MSSV analyses of rapidly interacting systems.

A very useful feature of MSSV is that, due to the high statistical precision of data acquisition, components may be distinguished in the context of significant spectral overlap. In many cases, intrinsic differences in UV absorbance from the content of aromatic amino acids may suffice to resolve components, and extrinsic chromophoric labels may not be necessary [2]. Recently, two of us (C.A.B. and S.B.P.) have developed criteria for the reliability of the MSSV analysis [5], and, based on the component extinction coefficient matrix at the different signals, introduced a quantitative predictive measure for the feasibility of spectral discrimination in MSSV. When spectral discrimination is insufficient, misassignment of signals can occur, with the result of incorrect identification of species' compositions. One of the tell-tales of spectral misassignment is that the total integral over the components sedimentation coefficient distribution does not approximate the known loading concentrations. This observation motivated the question of how the approximate knowledge of total loading concentrations of each component could be used as a constraint to stabilize the MSSV data analysis.

Mass conservation constraints have long been successfully used in the analysis of sedimentation equilibrium of interacting systems [20]–[23], but previously not been used in sedimentation coefficient distribution analyses. In the present work, we describe a novel approach, termed mass conservation constrained multi-signal sedimentation velocity (MC-MSSV), that allows one to introduce approximate total loading concentration either as a strict constraint or as a ‘soft’ regularization parameter. Specifically, we will first show theoretically how mass conservation constraints can achieve a well-conditioned analysis with unambiguous solutions even where the extinction coefficient matrix is singular. Next, we will demonstrate the behavior of MC-MSSV with simulated data, and introduce the combination with constraints in the sedimentation coefficient range of one of the components. Finally, we illustrate the practical application of MC-MSSV on an experimental model system from the interaction of bovine lactoferrin and Tp34 from *Treponema pallidum*.

Briefly, MSSV is a generalization of the *c(s)* method [3] for the global analysis of SV data for macromolecular mixtures acquired at multiple signals [2]. Let us assume we have *m* macromolecular components (1…*M*) with signal coefficients *ε _{m}^{λ}* at the different signals

(1)

which is approximated by the discretization into a grid of *s*-values, *c _{k,s}*, and computed by least-squares,

(2)

i.e. the minimization of the squared difference between the measured signals and the theoretical contributions of all classes of components to each signal. In Eq. 2, for simplicity the contributions to the different signals from radial-dependent baselines *b ^{λ}(r)* and time-dependent baselines

When solving Eq. 2 in the standard MSSV method, the apparent meniscus position of the solution column (separate for the absorbance and interference system due to unavoidable inconsistencies in the radial calibration) and the frictional ratio *f _{r}* can be included as unknowns in a non-linear regression. SEDPHAT provides the flexibility to divide the s-range into different segments in which the sedimenting components can be defined separately with respect to their stoichiometry

A precondition for Eq. 1 and 2 to have a unique solution is that the spectral contributions of the macromolecular components are distinguishable, i.e., that the extinction coefficient matrix has a non-vanishing determinant, >0. While this condition ensures a mathematical solution, it is not sufficient for the MSSV data analysis in practice. As shown by Padrick & Brautigam [5], a better metric for predicting whether a set of extinction coefficients will be sufficiently different to be distinguishable in the MSSV analysis is the quantity *D _{norm}*, defined as

(3)

By representing the fractional volume of a parallelepiped whose edges are the vectors of the component extinction coefficients, , relative to the maximum volume of a hyperrectangle from edges of the same length, it is a measure the spectral ‘orthogonality’ [5]. *D _{norm}* values above 0.065 for two-component analyses with dual-signal experiments are desirable for a promising MSSV study. (For for three-component analysis with three-signal experiments, more limited data suggest values >0.01 to be promising.) A

For the practical implementation of MSSV, it is very important to note that sufficiently distinguishable signals with large *D _{norm}* values may be obtained for many pairs of macromolecules without requiring extrinsic labels. For example, in some cases, differences in the content of aromatic amino acids and/or in the fraction of carbohydrate moieties can be sufficient for two or three proteins to be distinguishable on the basis of UV absorbance (e.g., at 280 nm and/or 250 nm) and/or refractometric signal increment in the interference optics [2]. In other cases, extrinsic chromophores have been attached to proteins to increase the spectral discrimination.

A statistical problem in the global analysis of signals from different sources, especially when using different optical systems, is that the number of data points as well as the overall signal amplitudes can be very dissimilar. For example, the interference optics provides routinely a higher density of data points than the absorbance system. It can be advantageous to apply corrections to the statistical weights of the different data sets that compensate for the number of data points and/or the signal amplitudes [31].

The above methodology was extended to make use of estimates of the total molar concentration of each macromolecular component in solution, *C _{m}^{tot}*. Usually this quantity is not known with complete accuracy but may be estimated from stock concentrations and the pipetting schedule, or better from

(4)

where Eq. 3 is extended by a penalty term that describes the sum of the squared mass deficit for all components. The scaling parameter α can be iteratively adjusted in two different ways: (1) In a ‘soft’ mass conservation approach, it can be adjusted such that the quality of fit to the raw data degrades by no more than a statistically indistinguishable level, pre-calculated by F-statistics. This is similar to the standard regularization [3], [29], and results in the *c _{k}(s)* distributions that, among all possible distributions that fit the data, is closest to preserving the total mass. Vice versa, any remaining mass differences result from significant features of the distribution. (2) Since very large values of α can enforce arbitrarily strict mass conservation, it may be adjusted so that the total mass loss is within a preset range, or related, so that the maximum mass defect for any component is within a preset tolerance

To study the effect of mass conservation regularization in relation to spectral discrimination of components it is of interest to follow the solution of Eq. 4. As a quadratic minimization problem we can as usual obtain a linear equation system by taking partial derivatives with respect to any particular unknown, for example, that of component *κ* at *s*-value *σ*, *c _{κσ}*, composed of spectral components as given by the stoichiometry

(5)

where we use abbreviations analogous to those introduced previously for the sedimentation related vector and matrix (in vector matrix notation and ) [3], [29] and introduce a matrix of the species' extinction coefficients with (in vector-matrix notation ). In the single-signal *c(s)* method, Eq. 5 would corresponds to a standard linear system that can be easily solved for non-negative concentrations with standard algebraic methods [3], [29] provided is non-singular. In MSSV without mass conservation constraints (*α*=0), for a unique solution we rely on the matrix being non-singular. If we simplify the problem by assuming all radial points, time-points, and menisci for the different signals are the same (which is approximately true), then will be independent of signal. For the subset of all *c _{k,s}* at the same

In the implementation in SEDPHAT, an additional refinement was introduced that allows the restriction of the summation range of *c _{k,s}* to be considered for mass conservation assessment to a user-defined interval of

For reversible binding events that produce complexes with lifetimes on the order of hours or longer, SV will result in the hydrodynamic separation of complex species. These will appear in the diffusion-deconvoluted sedimentation coefficient distributions generally as separate peaks at different *s*-values for different complex species. In MSSV they will produce co-localized peaks in the component *c _{k}*(

First, due to the ergodicity of a stable reaction boundary, co-sedimenting populations of the free smaller species must always remain in excess of those of co-sedimenting free populations of the larger species. As a consequence, if we denote with *R _{AB}* the molar ratio of total A (the slower sedimenting molecule) to B (the faster sedimenting molecule), then

(6)

(with *c _{Atot}* and

For simplicity, assuming a 11 reaction with the smaller species A providing the undisturbed boundary, the sedimentation velocity of the reaction boundary is described by

(7)

and the composition of the reaction boundary follows

(8)

[17]. It can be discerned from the latter expression that the reaction boundary composition will always be 1.0, i.e. reflecting the complex stoichiometry, if the sedimentation coefficients of the free species *s _{A}* and

Given an experimental value of the reaction boundary stoichiometry and species *s*-values, it is possible to estimate directly the binding constant as

(9)

again for the case if the slower sedimenting component constitutes the undisturbed boundary.

Recombinant Tp34 was overexpressed in *E. coli* and prepared as described [11]. The preparation of bovine lactoferrin (bLF; Sigma Chemical Corp.) was also described before [11]. Both proteins were stored in Buffer A (20 mM HEPES pH 7.5, 100 mM NaCl, 2 mM n-octyl-β-D-glucopyranoside) at 4°C.

The method of Pace [32] was used to determine ε_{280} for bLF and Tp34. Briefly, bLF was denatured in 6 M guanidinium hydrochloride, and its absorbance at 280 nm was determined. The extinction coefficient of the protein under these conditions was taken to be the weighted sum of the coefficients of the chromophoric amino acids in its primary sequence. With this knowledge, the concentration of the denatured protein could be calculated. The absorbance of an identical solution under non-denaturing conditions was then obtained; because the concentration of this sample was known, its extinction coefficient could be calculated. This coefficient was used for all experiments. The ε_{280} of Tp34 was determined in the same way.

Prior to centrifugation, bLF and Tp34 were diluted from their stock solutions into Buffer B (20 mM Tris pH 7.5 and 20 mM NaCl). To maintain a compositional balance between the sample and reference sectors, references were prepared in parallel by diluting Buffer A into Buffer B in amounts that mimicked the protein-containing samples. Three samples were prepared: one containing only 13.8 µM Tp34, another containing only 4.6 µM bLF, and the third having a mixture comprising 6.9 µM Tp34 and 2.3 µM bLF. The concentrations of the first two samples were chosen to give convenient pipetting volumes during their preparation. Although not necessary, we sometimes plan the experiment such that concentrations of the components alone are exactly the same as those in the mixture. This strategy allows the experimenter to insert the concentration values derived the analyses of the components alone directly into the mass-conservation calculations (see below). In this case, we used exactly half of the concentration of the proteins in the mixture, another computationally convenient approach. The samples and references were placed in the respective sectors of dual-sectored Epon centerpieces; each centerpiece had been sandwiched between two sapphire windows. Three assembled centrifugation cells (one containing the Tp34 alone, another containing the bLF alone, and the third containing the Tp34/bLF mixture) were inserted into an An60-Ti rotor, which was placed in a Beckman Optima XL-I analytical ultracentrifuge (Beckman-Coulter) and allowed to equilibrate to the experimental temperature (20°C) under vacuum for approximately 1.5 hours. After that, centrifugation was commenced at 50,000 rpm and continued until both proteins had completely sedimented. Data were acquired either in combination of interferometry with absorbance at 280 nm, or in combination of absorbance 250 nm and 280 nm. In all cases, the absorbance optical system was set to scan in continuous mode, with a radial resolution of 0.003 cm. These are our standard settings for all SV experiments; they offer an excellent balance of scanning speed and radial resolution. There is no need to accelerate radial scans in MSSV detection over the standard settings when using absorbance data acquisition at a single wavelength, as the frequency of measuring the boundary position is not different in MSSV from that of standard SV experiments, even though data are acquired sequentially at different signals. The use of wavelength scans in the centrifuge prior to sedimentation, although reporting on relative extinction coefficients at different wavelengths, does not provide sufficiently precise data for use in determining the mass constraints or extinction coefficients in the context of MSSV. Instead, the analytical strategy outlined in the Results section is used.

SEDPHAT version 10.31 was used for all of the analyses of the experimental (i.e. non-simulated) data. The buffer density and viscosity were estimated using SEDNTERP [33]. However, we chose to fix the partial specific volumes of the proteins at 0.73 mL/g. This action has the advantage of giving a common *s*_{20,w} grid for all data presented, with the drawback of small inaccuracies in representing the frictional ratios, *s*_{20,w} values, and masses of the proteins, although compensatory corrections could be easily applied. Unless otherwise mentioned, all analyses had two “spectra” per segment of *s*_{20,w}-space. Spectrum 1 corresponded to the *c _{k}*(

As a first test of the MC-MSSV analysis approach, sedimentation profiles were simulated for a 100 kg/mol, 6S-protein ‘B’ (ε_{IF}=275,000 M^{−1} cm^{−1} and ε_{280}=100,000 M^{−1} cm^{−1}) binding a 20 kg/mol, 2 S-protein ‘A’ (ε_{IF}=55,000 M^{−1} cm^{−1}) with different absorbance extinction coefficients ε_{280}=20,630, 23,180, or 26,500 M^{−1} cm^{−1} corresponding to the different *D _{norm}* values 0.01, 0.05 and 0.10, respectively, creating a 7 S complex with

Due to the relatively small size difference between the complex and the larger component, as well as the potential for different hydrodynamic shapes, the interpretation of the observed *s*-value in terms of complex stoichiometry would be ambiguous. However, the composition of the reaction boundary as observed in an MSSV experiment should give unequivocal information about the 11 stoichiometry.

To assess how well the molar ratio can be defined by the data, we probed the change in the quality of fit that occurred when the *c _{k}*(

To further improve the molar ratio resolution a similar steep increase can be achieved towards high molar ratio values if, in addition to mass conservation, we also enforce the condition that there can be no B in the low-*s* region. This is very plausible and known *a priori* because free B sediments faster than free A, therefore populations of B can be safely excluded in the range of the undisturbed boundary formed by free A. This can be achieved easily in SEDPHAT by using a multi-segmented *c _{k}*(

A second scenario was simulated (System 2) that is more challenging in that, besides small *D _{norm}* values, very dissimilar overall signal contributions are present. To this end, the sedimentation of 1.9 µM of a 200 kg/mol, 8.5 S protein B with ε

In order to verify this picture from the error surface projections independently, we carried out a series of simulations with this System 2, at eight *D _{norm}* values and at ten replicate simulations each with independent normally distributed noise. The distribution of best-fit complex molar ratio values as a function of

These simulations also showed that, within the statistical uncertainty of the results, Tikhonov-Phillips regularization can have a side effect of not only providing the most parsimonious *c _{k}*(

Earlier [5], a set of four criteria had been established to assess the success of an MSSV analysis. They may be summarized as mass conservation (Criterion 1), distribution rationality (Criterion 2), molar-ratio rationality (Criterion 3), and molar-ratio distinguishability (Criterion 4). We used this framework to assess the performance of MC-MSSV under realistic experimental conditions. We chose bovine lactoferrin (bLF; a ~80 kg/mol glycoprotein) and Tp34 (~20 kg/mol) from *Treponema pallidum* as a model system. The interaction of these two proteins had been characterized before using isothermal titration calorimetry [11]. These earlier experiments established that the molar ratio of the interaction is 11, and the calorimetric data were fitted with a 11 binding model; the best-fit association constant was 1.6×10^{6} M^{−1} (*K _{d}*=0.63 µM).

First, the spectral properties of the individual proteins were determined. We have generally found that determination of both extinction coefficient using a standalone UV-Vis spectrometer to result in inaccuracies that interfere with the analysis. Instead we recommend choosing one extinction coefficient and using it as a reference for calibrating the others in a preliminary *c _{k}*(

Next, an SV experiment containing a mixture of the two proteins was examined. Based on the known pipetted volumes and on integration of the entire distributions in Figures S5 and S6, the concentrations of Tp34 and bLF were 6.9 µM and 2.3 µM, respectively. From the calorimetry results and mass action law we expected that 90% of the bLF would be in a 11 complex having an *s _{20,w}* -value >5.2 S. We used standard MSSV without constraints for our first analysis of these data. (These data contained an instrumental artifact that caused a decreasing slope in the absorbance data only at late times and high radial values (not shown); the data were truncated to exclude as much of these artifactual data as possible.) The resulting fits are shown in Fig. 4 and the

In a second analysis, we used mass conservation to constrain the total concentrations of the components to be within 5% of their known values. As shown in Figure S7, an essentially identical quality of fit was achieved. However, again, this strategy failed to satisfy the above criteria: Although the mass conservation succeeded in constraining the total concentrations present over all segments, bLF alone was still detected at low *s _{20,w}* –values (Fig. 5B and Figure S7C), which is physically impossible, thus failing Criterion 2. Considering Criterion 3, the ratio of Tp34 to bLF in the complex was even more unrealistic (11.71). Thus, mass conservation constraints alone were not enough to accurately analyze these data.

In a third analysis, in addition to mass conservation, we applied our prior knowledge that only free Tp34 can sediment at low *s*-values. To this end, we divided sedimentation-coefficient space into a low-*s _{20,w}* segment (0.2 to 4 S), where only (free) Tp34 was expected, and a high-

It is interesting to note that in the *c _{k}*(

With the first three criteria met in the doubly constrained analysis, we focused on the fourth and examined the statistical significance of the molar ratio value of 0.7. Using the χ^{2} of the fit and F-statistics [35], we chose to test whether varying the molar ratio by 0.5X and 2X would result in a statistically worse fit. Constraining the composition of the high-*s* range to values of either 0.351 or 1.41, while both mass conservation and low-*s* constraints were in place, resulted in significantly (>2σ) worse fits. We therefore judge that Criterion 4 for the success of an MSSV experiment, molar-ratio distinguishability, is also met in this case when applying both mass conservation and low-*s* constraints.

There is an additional fact that supports the reliability of the 0.7 molar ratio detected in the constrained MC-MSSV experiment. To confirm the analysis shown above, we analyzed a follow-up experiment that was conducted almost identically to that described above. However, no interferometric signal was acquired in this second experiment; instead, two UV wavelengths (280 nm and 250 nm) were used to monitor the sedimentation. From the SV experiments of both proteins separately, ε_{250} of bLF and Tp34 were found to be 48,800 and 11,300 AU×M^{−1}×cm^{−1}, respectively, corresponding to a *D _{norm}* value of 0.08, which, according to our simulations (see above and [5]), should afford excellent spectral discrimination of Tp34 and bLF. The MSSV analysis without any constraints yields

Having confirmed the measured molar ratio of 0.7 of the reaction boundary, based on knowledge that bLF and Tp34 form 11 complexes, we can use effective particle theory to obtain an independent estimate of the association solely from the observed molar ratio, the known loading concentrations, and the measured or estimated *s*-values of all species. As outlined in the theory section, at a molar excess of the smaller component we are in all cases below the phase transition line such that the conditions of Eq. 9 are fulfilled. Assuming a value of *s _{AB}*=6.2 S (which is not directly measured), then the measured molar ratio of 0.7 results in an estimate of

In the last decade, sedimentation velocity analytical ultracentrifugation has re-emerged as a popular tool for the study of protein interactions, as the introduction of new theoretical and computational data analysis methods has significantly increased the resolution and precision of this approach. Initially, our focus in the development of sedimentation velocity was the spatio-temporal analysis of the evolution of a single signal. More recently we have broadened the analysis to the deconvolution of spectral dimensions, which can offer significant advantages when studying heterogeneous protein interactions.

In such systems, different approaches have been described for the quantitative determination of binding constants: direct fitting with solutions to coupled systems of Lamm equations that embed a certain reaction scheme [37]–[39], and the analysis of the isotherms of boundary patterns (their overall weighted-average *s*-value, reaction boundary *s*-value, and amplitudes of reaction and undisturbed boundaries) [19], [40]. We have explored and implemented in SEDPHAT both approaches, and found that the direct Lamm equation modeling can in some cases provide useful information on kinetic rate constants, whereas the boundary pattern analysis is significantly more tolerant for sample imperfections. In our experience, the latter is increasingly important for the study of interactions between more components involving more binding interfaces. Such multi-component and/or multi-site interactions also exacerbate a more fundamental problem, which is the question of what type of complexes can form. Even though both direct Lamm equation fitting as well as boundary pattern analysis may be applied with different interaction schemes to rule out incorrect interaction models by trial and error, this becomes increasingly impractical for more complicated systems. In this context, the strength of the MSSV analysis is the ability to define, in few experiments, the stoichiometry of the complexes formed. This is often one of the most important, and non-trivial facts that informs on possible reaction pathways and is fundamentally a prerequisite for any thermodynamic analysis. In addition to *c _{k}(s)* providing a relatively model-free analysis of co-sedimentation that can guide the formulation of the reaction scheme for a thermodynamic analysis, in conjunction with the effective particle theory [17], [19], it also highlights experimental designs that will be particularly informative on stoichiometries of complex formation, producing data that can be utilized initially for the

MSSV exploits spectral information, which, together with the boundary velocity and boundary spread, forms a set of three independent sources of information on the composition of complexes by mass, *s*-value, and composition. Within the precision of typical results of boundary analysis, molar mass estimates and considerations of hydrodynamic shape alone will often fail to deliver unambiguous estimates for the complex stoichiometry both for systems with similar-sized components as well as for very dissimilar sized components. If components are spectrally distinguishable, such ambiguity can be resolved by MSSV. Furthermore, we have shown previously that spectral deconvolution can be synergistic to the hydrodynamic separation of complexes by sedimentation coefficient [2], which for compact particles, scales with the 2/3 power of the molar mass. Interestingly, an extension of another hydrodynamic approach, fluorescence correlation spectroscopy, to the simultaneous global analysis of multiple signals, F3CS, has recently been developed and been described to be similarly applicable to define the thermodynamic states of complex systems with ternary complexes [41]. While there appear to be several analogies, there are also important differences with regard to the required sample concentrations, spectral properties, complex life-times, and hydrodynamic resolution.

One important virtue of MSSV is that, dependent on protein amino acid composition (or, more generally, macromolecular UV/VIS extinction properties), it may be applied label-free. Differences in the ratio of aromatic amino acids leading to different extinction profiles can be sufficient for spectral discrimination in sedimentation velocity using the absorbance and/or interference detection. (Unfortunately, only a single wavelength is currently available for the commercial fluorescence detector, which excludes the simultaneous operation of other detectors.) This is despite the fact that the quality of the commercial spectrophotometer in the ultracentrifuge is far lower than that of most common bench-top spectrophotometers. For example, the limited reproducibility in the monochromator control when scanning at alternating absorbance wavelengths exhibited by individual instruments imposes some restrictions in the selection of wavelengths to be near a minimum or maximum of absorbance or at a spike of lamp intensity (e.g.230 nm). Alternatively, sorting of scans according to the actual detection wavelength, which is more precisely measured than controlled, is possible and supported in a SEDFIT utility function. In practice, it should also be noted that wavelength accuracy depends on accurate wavelength calibration and may be different in different ultracentrifuges. This potential limitation can be circumvented by measuring absorbance spectra and locating peak absorbances of all components in the ultracentrifuge used for the MSSV experiment.

The excellent spectral resolution in MSSV ultimately rests on the exquisite precision of measuring sedimentation boundary heights, which is also exploited in the familiar application of sedimentation velocity to determine trace amounts of aggregates [42], where routinely boundaries with signal amplitudes less than the noise of a single data point can be detected and quantitated due to the large number of points (typically on the order of 10^{5}) determining boundaries and plateau signals. Thus, in scenarios featuring a small signal-to-noise ratio (e.g. <10), we expect the current methodology to perform well. Likewise, even in cases featuring a large difference in sedimentation coefficients between the free and complexed species, wherein the smaller species is overrepresented in the analysis (e.g., see Figure S7), there are usually ample data to spectrally resolve the underrepresented species (presumably the complex), and there is no requirement for all species to contribute evenly or to the same number of scans (e.g., see Figure S2). It should be noted that the precision of relative signal contributions obtained from global modeling of SV data far exceeds the precision of wavelength scans, which could only give a very coarse estimate of extinction coefficients and would not be suitable in conjunction with MSSV. An analytical methodology similar in principle to MSSV could be applied to data acquired from size-exclusion chromatography (SEC) when the elution profiles are acquired with different in-line detectors, e.g. a UV detector and a refractometer. However, such a strategy would rely on the complex of interest having a long lifetime on the timescale of the chromatography experiment. This limitation is not present in MSSV [2], [4], [5]. It would further require sufficient resolution between the complex and any free species; when performed properly, sedimentation velocity generally has superior resolution to SEC.

In the present work, we were concerned with cases where spectral discrimination, with or without extrinsic chromophores, is too poor for a reliable spectral deconvolution. We have shown that the introduction of additional knowledge of the approximate total concentrations of loaded concentrations can substitute for insufficient spectral discrimination. While mass conservation principles have been very successfully applied in sedimentation equilibrium analyses [20], [21], [23], and in some form are common as hard constraints in direct Lamm equation modeling [37], [39] and boundary structure analyses [19], [40], they are novel in sedimentation coefficient distribution analyses, which are conceptually more related to data transforms.

In the weakest form, mass conservation prior knowledge may be used as a form of spectral regularization, which probes the space of different possible multi-component decompositions and reveals the solution where the calculated total loading concentration of each component after integration of the distribution is closest to that known to be inserted into the experimental mixture. When used in conjunction with Tikhonov-Phillips regularization for parsimony of the sedimentation coefficient domain, the second regularization term poses the problem of how to scale it independently. In the current implementation of SEDPHAT, we decided *ad hoc* for an adjustment to produce the same relative increase in the χ^{2} of the fit as applied to the Tikhonov-Phillips term and predicted by F-statistics on a certain confidence level. Furthermore, we found that Tikhonov-Phillips regularization will invariably be correlated with spectral assignment (as it similarly biases peak areas within the statistically permitted extent [43]). Therefore, in this weak form as purely a regularization term, once mass conservation is combined with Tikhonov regularization it may not be more useful than allowing one to explore the flexibility of the spectral assignment. For a statistically better-defined result, we would recommend its use in the absence of simultaneous Tikhonov-Phillips regularization.

A stronger variant, which we envision to be the predominant use of MC-MSSV, is the requirement that mass conservation be strictly fulfilled within a preset tolerance, the achievement of which governs the scaling of the penalty term independent from Tikhonov-Phillips regularization. In SEDPHAT this is programmed such that the tolerance is determined by the user, derived, for example, from the estimated experimental reproducibility of pipetting, and comparison with single-component experiments conducted side-by-side. As long as protein concentrations are based on signal coefficients determined in these separate sedimentation velocity experiments on single-component solutions, actual loss of material in the mixture seems unlikely. Pathological processes such as co-precipitation or altered surface adsorption of complexes, as compared to individual components, are possible. Some of these may be flagged in a standard MSSV analysis by mass balances of all components to be negative, in contrast to spectral mis-assignment that over-estimates one component at the cost of the other. At present, MSSV analysis also assumes the absence of hypo- or hyper-chromicity at the detection wavelengths, which could potentially affect apparent mass conservation. The absence of significant spectral changes upon binding may be verified in a bench-top spectrophotometer, and, if present, be eliminated from consideration in MSSV by detection at the isosbestic point. Importantly, complications of mass conservation analyses historically familiar from sedimentation equilibrium analyses, including those related to baseline signals and the location of the base of the cell, are not present in sedimentation velocity.

Although mass conservation constraints can stabilize the MSSV analysis even in the limit where spectral discrimination of components is completely absent, in itself it cannot provide information on which sedimentation coefficient the mass of each component is attributed to. As an extreme example, one may even load data from the same signal twice and perform a stable MC-MSSV analysis, but this will not yield reliable information on how each component partitions into two sedimenting species (beyond the limits set by mass conservation and non-negativity of concentrations). The strongest use of MC-MSSV, therefore, is an analysis where mass conservation is combined with additional knowledge that excludes components from certain sedimentation coefficient ranges. Especially when studying components with dissimilar sized free species, this knowledge will be obvious from inspection of the *c _{k}(s)* traces of the samples from the individual components. Conceptually, the introduction of this information is akin to ‘fringe counting’ or ‘free pool’ experiments, where the complex stoichiometry is indirectly assessed based on the observed boundary amplitudes of the remaining free species of one component and the complex. As shown in the Results section, exploiting such principles in the context of MC-MSSV analyses can be very powerful. Obviously, even when spectra are distinguishable, the introduction of such robust knowledge is highly desirable, and can be expected to leverage MSSV to study more complex multi-component systems.

Besides the previously developed initial criteria for distribution rationality and molar ratio rationality [5], the success of the MC-MSSV analysis is best studied in the framework of a segmented *c _{k}(s)* distribution probing the error surface projections for different molar ratio values pre-constrained in a segment covering the sedimentation coefficient of the complex (or reaction boundary, see below). In the limit of very low spectral distinguishability, by MC-MSSV one can expect only to determine the average composition of all complexes in the relevant segment. Additional systematic errors may occur, for example, due to errors in the signal coefficients on which MSSV is based. In the present experimental work we have used the method of Pace [32]. When the absorbance extinction coefficients are predicted from amino acid composition, reported errors are typically <5%, but errors exceeding 15% may occur [32]. For proteins without post-translational modifications it may be advantageous to measure the extinction coefficients experimentally in multi-signal experiments with refractive index optics and using the refractive index increments as a fixed point instead of the extinction coefficient. The refractive index increment is less composition dependent [44], although the compositional prediction of the refractive index increment may be warranted to achieve better accuracy especially for small proteins [44]. Obviously, the assessment of the complex stoichiometry is greatly facilitated by the fact that a complex can only contain an integral number of subunits, which for small complexes (as apparent chiefly from the

However, non-integral (or non-rational) values for the boundary composition can be expected for reaction boundaries of rapidly interacting systems. In fact, as we have shown in the effective particle theory, these reaction boundaries can be understood and quantitatively well approximated as regular sedimentation and diffusion processes from ‘pseudo-particles’ or ‘effective particles’ that are composed of all interacting species transiently co-migrating in non-stoichiometric amounts [17], [18]. As such, they are equally accessible experimentally by MSSV or MC-MSSV as stable boundaries of real physical species. However, in addition to the reaction stoichiometry, the composition and velocity of the reaction boundary will depend, in a simple relationship, on the *s*-values of all species, the loading concentrations, and the affinity constants [17]. In fact, a well-known hallmark of reaction boundaries is a concentration-dependence of the peak *s*-values of *c(s)*, and this similarly holds true for *c _{k}(s)*. Most notably, for fundamental reasons, the content of the slower sedimenting component will be concentration-dependent and always be less than stoichiometric. However, when one of the components is in >5 fold molar excess over the stoichiometry, the composition will converge to represent essentially the properties of a complex saturated with the excess component [4], [17].

More quantitatively, we have shown in the present work how the measured reaction boundary stoichiometry can be used, in a back-of-the-envelope calculation, to estimate *K _{d}* (and the complex

In summary, we have developed a new analytical technique for MSSV, called MC-MSSV, which can compensate for poor spectral resolution of the components of a complex and result in excellent outcomes for the determination of complex stoichiometries. It utilizes our *a priori* knowledge of the component concentrations to constrain the MSSV analysis. Previously [5], we had recommended that the *D _{norm}* (Eq. 3) of a two-component system be greater than 0.065 in order to reliably distinguish two components using their spectral differences. Our present work demonstrates that this limit is virtually eliminated in an MC-MSSV analysis. Further, the addition of a constraint limiting a certain (low)

**A series of screenshots and graphs that illustrates how to use the MC-MSSV method in the SEDPHAT software.** The relevant SEDPHAT parameter box entries that relate to the MC-MSSV tools described in the present work are explained.

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**Simulated data of System 1, which consists of a 100 kg/mol, 6 S-protein ‘B’ (ε _{IF}=275,000 M^{−1} cm^{−1} and ε_{280}=100,000 M^{−1} cm^{−1}) binding a 20 kg/mol, 2 S-protein ‘A’ (ε_{IF}=55,000 M^{−1} cm^{−1}) with absorbance extinction coefficients ε_{280}=23,180 M^{−1} cm^{−1} corresponding to **

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**Simulated data of System 2 consisting of a 200 kg/mol, 8.5 S-protein ‘B’ (ε _{IF}=550,000 M^{−1} cm^{−1} and ε_{280}=140,850 M^{−1} cm^{−1}) binding a 10 kg/mol, 1.2 S-protein ‘A’ (ε_{IF}=27,500 M^{−1} cm^{−1}) with absorbance extinction coefficients ε_{280}=8,000 M^{−1} cm^{−1} corresponding to different **

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**Replicate simulations of System 2 (as in Fig. S2) at different low ***D _{norm}*

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**Same as Fig. S3, but calculated with Tikhonov-Phillips regularization at a confidence level of 0.68.**

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**MSSV analysis of Tp34 alone.** (A) Interference data, fits, and residuals. (B) Absorbance data at 280 nm, fits, and residuals. (C) The *c _{k}*(

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**MSSV analysis of bLF alone.** (A) Interference data, fits, and residuals. (B) Absorbance data at 280 nm, fits, and residuals. (C) The *c _{k}*(

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**Mass-constrained MSSV analysis of the Tp34/bLF mixture.** (A) Interference data, fits, and residuals. (B) Absorbance data at 280 nm, fits, and residuals. (C) The *c _{k}*(

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**MSSV analysis of the Tp34/bLF mixture with both low-***s*** and mass-conservation constraints.** (A) Interference data, fits, and residuals. (B) Absorbance data at 280 nm, fits, and residuals. (C) The *c _{k}*(

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**MSSV analysis of the Tp34/bLF mixture with low-***s*** constraint but without and mass-conservation constraints.** (A) Interference data, fits, and residuals. (B) Absorbance data at 280 nm, fits, and residuals. (C) The *c _{k}*(

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**Unconstrained MSSV analysis of the experiment with dual 280 nm/250 nm absorbance data acquisition.** (A) The data, fit, and residuals for data collected at 280 nm. (B) The data, fit, and residuals for data collected at 250 nm. (C) The *c _{k}*(

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The authors thank Drs. Ranjit K. Deka and Michael V. Norgard for supplying stocks of bLF and Tp34.

This work was supported by the intramural research program of the National Institute of Biomedical Imaging and Bioengineering, National Institutes of Health, DHHS. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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