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Current Methods in Sedimentation Velocity and Sedimentation

Equilibrium Analytical Ultracentrifugation

Huaying Zhao*, Chad A. Brautigam#, Rodolfo Ghirlando$, and Peter Schuck*

*Dynamics of Macromolecular Assembly Section, Laboratory of Cellular Imaging and

Macromolecular Biophysics, National Institute of Biomedical Imaging and Bioengineering,

National Institutes of Health, Bethesda, Maryland 20892

#Department of Biochemistry, The University of Texas Southwestern Medical Center, Dallas,

Texas 75390

$Laboratory of Molecular Biology, National Institute of Diabetes and Digestive and Kidney

Diseases, National Institutes of Health, Bethesda, Maryland 20892

Abstract

Significant progress in the interpretation of analytical ultracentrifugation (AUC) data in the last

decade has led to profound changes in the practice of AUC, both for sedimentation velocity (SV)

and sedimentation equilibrium (SE). Modern computational strategies have allowed for the direct

modeling of the sedimentation process of heterogeneous mixtures, resulting in SV size-

distribution analyses with significantly improved detection limits and strongly enhanced

resolution. These advances have transformed the practice of SV, rendering it the primary method

of choice for most existing applications of AUC, such as the study of protein self- and hetero-

association, the study of membrane proteins, and applications in biotechnology. New global multi-

signal modeling and mass conservation approaches in SV and SE, in conjunction with the

effective-particle framework for interpreting the sedimentation boundary structure of interacting

systems, as well as tools for explicit modeling of the reaction/diffusion/sedimentation equations to

experimental data, have led to more robust and more powerful strategies for the study of reversible

Internet Resources

There are abundant resources available on the internet covering software, detailed experimental step-by-step protocols, video

instructions, and discussion forums. The following is only a small subset facilitating the application of the methods described in the

present Unit.

General practical tools to help designing AUC experiments can be found at https://sedfitsedphat.nibib.nih.gov/tools/Protocols/Forms/

AllItems.aspx, including detailed step-by-step protocols for conducting SE and SV experiments, a table with criteria for buffer

selection, a grid for SV run conditions with different optical systems, instructions for radial calibration of the interference system, and

a tutorial for the test for equilibrium. A video showing the assembly of cells can be found at http://www.jove.com/video/1530/

assembly-loading-and-alignment-of-an-analytical-ultracentrifuge-sample-cell (Balbo et al., 2009).

All methods discussed above are implemented in SEDFIT and SEDPHAT, unless otherwise stated. They can be freely downloaded

from https://sedfitsedphat.nibib.nih.gov/software/default.aspx, and an extensive online help system and documentation can be found at

http://www.analyticalultracentrifugation.com/default.htm, including a tutorial for ‘getting started’ in SEDFIT, and step-by-step

examples for the data analysis with c(s) http://www.analyticalultracentrifugation.com/examples.htm. ‘Getting started’ tools for

SEDPHAT are available at http://www.analyticalultracentrifugation.com/sedphat/concepts_for_getting_started.htm.

Tutorials on special topics of AUC and/or SEDFIT or SEDPHAT analyses can be accessed at http://

www.analyticalultracentrifugation.com/tutorials.htm. Movies illustrating the mechanism of migration o f there action boundary can be

found at https://sedfitsedphat.nibib.nih.gov/tools/Reaction%20Boundary%20Movies/Forms/AllItems.aspx.

GUSSI is are available for download at http://biophysics.swmed.edu/MBR/software.

SEDNTERP for the calculation of buffer densities and viscosities, as well as protein extinction coefficients and partial specific

volumes can be found at John Philo’s website http://www.jphilo.mailway.com/download.htm

Active discussion forums are SEDFIT-L (https://list.nih.gov/cgi-bin/wa.exe?SUBED1=SEDFIT-L&A=1), SEDPHAT-L (https://

list.nih.gov/cgi-bin/wa.exe?A0=SEDPHAT-L), and RASMB (http://rasmb.bbri.org/cgi-bin/mailman/listinfo/rasmb ).

The websites https://sedfitsedphat.nibib.nih.gov/workshop/default.aspx and http://www.analyticalultracentrifugation.com/default.htm

will have current information on upcoming workshops.

NIH Public Access

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Published in final edited form as:

Curr Protoc Protein Sci. 2013 February ; 0 20: . doi:10.1002/0471140864.ps2012s71.

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protein interactions and multi-protein complexes. Furthermore, modern mathematical modeling

capabilities have allowed for a detailed description of many experimental aspects of the acquired

data, thus enabling novel experimental opportunities, with important implications for both sample

preparation and data acquisition. The goal of the current commentary is to supplement previous

AUC protocols, Current Protocols in Protein Science 20.3 (1999) and 20.7 (2003), and 7.12

(2008), and provide an update describing the current tools for the study of soluble proteins,

detergent-solubilized membrane proteins and their interactions by SV and SE.

Keywords

sedimentation equilibrium; sedimentation velocity; chemical equilibria; reversible interaction;

size-distribution; multi-protein complex; analytical ultracentrifugation; protein hydrodynamics

INTRODUCTION

Analytical ultracentrifugation (AUC) is a classical first-principle method of physical

biochemistry, firmly rooted in solution thermodynamics. It allows for the real-time

observation of the redistribution of proteins in dilute solution following the application of a

strong centrifugal field. In sedimentation velocity (SV), the evolution of the sedimentation

process is studied, whereas in sedimentation equilibrium (SE), the final equilibrium

distribution is examined. SV and SE are complementary and together provide an

information-rich characterization of many aspects of protein behavior in solution, including

the protein mass and size, density, hydrodynamic shape, size-distribution and purity, weak

protein interactions, specific reversible interactions, and the formation of multi-protein

complexes. Among the virtues of AUC are the absence of surfaces, the absence of label

requirements, and the powerful size resolution for protein-sized macromolecules and larger.

By adjusting the rotor speed, peptides and proteins with molar masses ranging from 100 g/

mol to 108 g/mol can be studied; furthermore the broad dynamic range of the detection

systems in combination with current data analysis strategies allows for the characterization

of interacting systems having equilibrium dissociation (Kd) between 10 nM and 10 mM

(Zhao et al., 2012; Rowe, 2011).

Current major applications in protein science include the study of peptide and protein self-

association and the characterization of oligomeric state, the characterization of

heterogeneous protein-protein interactions and multi-protein complexes with regard to the

number and size of co-existing complexes, as well as their affinity, the study of membrane

proteins in detergent solutions, and the study of protein solution conformation and ligand-

induced conformational changes.

Compared to the classical approaches, a profound change in the application of AUC has

taken place approximately a decade ago, mainly due to the confluence of three new

developments for SV: (1) Sophisticated and efficient algorithms for the precise numerical

solution of the differential equation of the sedimentation/diffusion process, which can be

applied routinely for the non-linear regression of experimental data using modern personal

computer hardware. Historically, this has been a major limitation in SV methods for most of

the previous century. (2) The introduction of mathematical tools, borrowed from modern

image analysis, for direct data fitting based on integral equations for determining an

unknown size distribution. In this manner, the level of detail of the mathematical model is

adjusted to the experimental sensitivity of the method for polydispersity and trace

impurities. (3) The adaptation of the model to accommodate the specific noise structure

displayed by SV data, which is a combination of time-invariant and radius-invariant baseline

offsets. These can now be directly determined from the least-squares fit of the data as part of

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the model. This advance avoids ‘manual’ alignment, ambiguous data subset selection, and

noise-amplifying data differencing. Consequently, the experimenter can now take full

advantage of the large number of experimental data points, leading to lower detection limits,

higher dynamic range, and an improved precision of the parameter estimates. These,

together with other developments, have spawned many new approaches for studying

mixtures of non-interacting and interacting proteins. The generalization of these concepts to

multi-signal and density-contrast global analysis has led to new tools particularly suitable

for heterogeneous protein interactions, multi-protein complexes, and membrane proteins in

detergent solution. Complementarily, a new, more physically intuitive theoretical framework

for the interpretation of the boundary structure of rapidly interacting systems was developed,

which can be used for data analysis as well as experimental planning. Furthermore, more

precise predictions of hydrodynamic friction from structure-based models are available.

New software capabilities embed all these tools.

As a consequence of these developments, the majority of AUC applications, including the

study of detergent-solubilized membrane proteins, have shifted towards the use of the SV

mode. However, fresh developments have also taken place in SE. In particular, global-

modeling capabilities have been extended, permitting a multi-signal, multi-speed global

analysis and a global density-contrast analysis. In combination with new implicit mass-

conservation constraints, these extend both the range of interacting systems and affinities

that can be measured by SE, and the size range of binding partners in heterogeneous protein

interactions.

Because the basic principles of both the instrumentation and the technique have not

changed, the present commentary is aimed at providing additional and updated information

to the previous more detailed Units for SE and SV of soluble proteins (Current Protocols in

Protein Science Units 20.3 (1999), 20.7 (2003), 7.13 (2010)), as well as SE for detergent-

solubilized membrane proteins (Current Protocols in Protein Science 7.12 (2008)).

Elements of the newer techniques were presented previously in Current Protocols in

Immunology 18.8 (2007) and 18.15 (2008) and will be referenced. A more recent detailed

introduction to SV can be found in (Schuck et al., 2010), and a review for SE can be found

in (Ghirlando, 2011). For the data analysis, we recommend our software SEDFIT and

SEDPHAT, in which are implemented all tools discussed below. It can be downloaded from

https://sedfitsedphat.nibib.nih.gov and an extensive online help system can be found at

http://www.analyticalultracentrifugation.com. Workshops on current AUC methodology are

held regularly in our laboratory at the National Institutes of Health, Bethesda, Maryland.

BASIC PRINCIPLES

For a description of the basic setup of AUC, we refer the reader to the previous Units cited

above, as well as recent reviews (Rowe, 2010; Schuck, 2012; Zhao et al., 2012).

Sedimentation Velocity Analysis of Non-Interacting Systems

Basic Theory—The basic theory is also described in Unit 20.7, but briefly recapitulated

here to provide the context and to introduce the symbols. We can define the sedimentation

coefficient s as the linear velocity u of sedimentation a protein exhibits per unit gravitational

field ω2r (with rotor angular velocity ω and distance from the center of rotation r):

Equation 1

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It is measured in units of Svedberg, with 1S = 10−13 sec. Because it is dependent on buffer

viscosity and density, this experimental s-value (sexp) is often normalized to standard

solution conditions of water at 20° C (s20,w) according to

Equation 2

with ρ and η denoting the buffer density and viscosity, respectively, and v̄ the partial-

specific volume of the protein. The sedimentation coefficient depends on the translational

friction coefficient f and molar mass M

Equation 3

(with Avogadro’s number NA, and can be related to the Stokes radius RS). It is useful to

relate the frictional coefficient to that of an ideal, smooth, compact sphere of the same

density and mass, which leads to the frictional ratio f/f0 and to the ratio of s-values of the

observed particle and the corresponding ideal, smooth, compact sphere:

Equation 4

We note that sedimentation velocities in excess of any particle’s equivalent sphere,

ssphere, 20w, are impossible. Typically, due to frictional contribution from bound hydration

water, f/f0 -values between 1.1 – 1.2 are obtained for highly globular particles, 1.3 – 1.5 for

moderately asymmetric particles, and in excess of 1.5 for highly asymmetric particles. From

Eq. 3 we can also derive the Svedberg equation

Equation 5

, which is of key importance because it relates the sedimentation coefficient, s, and diffusion

coefficient, D, with the protein molar mass, M. It is based on the insight that, for dilute

solutions, the frictional coefficient for sedimentation is the same as that governing diffusion

(Svedberg and Pedersen, 1940).

Direct Boundary Modeling with Lamm Equation Solutions—In SV, we are not

observing single particles, but rather an ensemble of particles with a radial- and time-

dependent concentration χ(r,t). Generally, sedimentation starts with a well-mixed sample at

a uniform loading concentration χ(r,t=0)=c0. For a single class of proteins, the evolution of

the concentration for a sector-shaped cell in the centrifugal field is governed by the Lamm

equation (Lamm, 1929)

Equation 6

The lack of analytical solutions to this equation has presented a significant obstacle for fully

exploiting SV during most of the 20th century, even though numerical solutions have been

sparsely used to simulate and analyze data (Claverie et al., 1975; Dishon et al., 1967; Cox,

1965). Abundant computational resources and the introduction of highly efficient numerical

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solutions have now enabled a transformation of SV: rather than measuring the velocity of

the boundary midpoint and calculating s via Eq. 1, we can fit directly solutions of Eq. 6 to

experimental data a(r,t) in a direct, least-squares boundary model (see below). Different

strategies for solving Eq. 6 are implemented in different software packages: SEDFIT and

SEDPHAT are based on the solutions described in (Brown et al., 2008) with adaptive

adjustment to pre-determined precision. A different approach has been proposed by (Cao

and Demeler, 2005) and implemented in ULTRASCAN, but if applied as suggested by Cao

& Demeler it lacks the precision required for modeling SV boundaries of medium and large

proteins (Schuck, 2009). Examples for the evolution of concentration profiles of different

size particles are shown in Figure 1.

When experimental data are acquired using the absorbance optical (ABS) system, the radial

profiles are not always well described in terms of a temporal snapshot due to the finite

velocity of the scanner reaching higher radii with a time delay. This can produce significant

errors in the sedimentation coefficients of large proteins and protein complexes (Brown et

al., 2009). However, by evaluating the sedimentation profiles that would be measured in

scan i as χ′(r,ti) = χ(r,ti+(r−r1)/vscan) and assigning later times to larger radii than the initial

scan time stamp ti, SEDFIT and SEDPHAT are able to account for the finite scanning

velocity vscan (typically 40 μm/sec).

The use of Eq. 6 in modeling data requires a definition of the initial conditions, including the

geometric limits of the solution column, specifically the radial positions for the meniscus, m,

and the cell bottom, b. Traditionally, the assignment of m was separate from, and had to

precede, the evaluation of the scans. This was true in the pre-computer approach of plotting

the movement of the boundary midpoint log(r/m) as a function of time, which has a slope of

ω2s based on an integral form of Eq. 1. Similarly, this was true for the g(s*) transformation

(Stafford, 1992), as implemented in SEDANAL and DCDT+. However, in the modern least-

squares modeling of SV data, the meniscus is a fitted parameter and thus can be subject to

the same optimization based on the quality of fit, much like other fitting parameters.

Different groups have established that such a treatment of the meniscus parameter leads to a

well-defined estimate, which is more precise than that discerned from visual inspection of

the experimental scans (Brown et al., 2009; Besong et al., 2012; Gabrielson et al., 2007),

mainly due to the simple facts that the location of the true meniscus location is obscured by

optical imaging artifacts, and that the radial resolution of experimental scans is not

sufficiently high. Correspondingly, refining the meniscus parameter usually improves the fit

significantly and also leads to well-determined macromolecular sedimentation parameters.

Correlation of the meniscus parameter with protein sedimentation parameters of interest can

occur for very broad boundaries either of small peptide sedimentation, of data at low rotor

speeds, of SV configurations with short columns, and/or data with highly polydisperse

material or extended association schemes. For this reason, it is recommended to use

graphically determined bounds as upper and lower limits for the non-linear regression of the

meniscus parameter.

If normal high-speed SV experiments with well-formed sedimentation boundaries between

clear solvent and solution plateaus lead to optimized meniscus values at or outside these

bounds, this is usually a strong indication for the presence of convection. In this case, the

best practice is to repeat the experiment, aiming for a more thorough temperature

equilibration. However, some information can be rescued from such data by excluding some

initial scans prior to temperature stabilization and either (1) letting the meniscus position

refine to its best-fit position as determined by the progression of the boundaries after

convection has ceased (Brown et al., 2009); or (2) using a model where sedimentation is

initialized at some point after start of the sedimentation with the concentration distribution

extracted from an experimental scan at that time (Cox, 1966; Schuck et al., 1998).

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In analogy, the bottom position of the solution column can also be refined from the non-

linear regression of the data. However, this step is only necessary when the experimental

data exhibit back-diffusion in the radial range to be analyzed. Unless there are small solute

components in the sample, this is usually not the case, since the back-diffusion of proteins in

typical high-speed SV is too steep to optically image reliably, and produces concentrations

that are too high for Eq. 6 to be sensible. Thus, the steep back-diffusion sections of the data

are customarily discarded, and the bottom position may become irrelevant for the model.

Another important ingredient for modeling experimental SV data is the ability to describe

the experimental imperfections that produce signals offsetting the macromolecular

sedimentation signal of interest. For example, Figure 2 shows raw SV data acquired at a

series of time points with the interference optical (IF) detection system during the

sedimentation of a protein. Two types of offsets can clearly be discerned. (1) Time-

dependent vertical offsets of the whole signal pattern caused partly by inconsistent offsets in

the zero fringe shift assignment (which is intrinsically arbitrary due to the periodicity of the

fringe pattern), and partly by vibrations or other adventitious time-dependent changes in the

optical pathlengths on the nm scale. These offsets are termed RI noise, for ‘radius-invariant’

noise. (2) A constant radial signal pattern that is superimposed equally on all scans. This

phenomenon is caused by radius-dependent imperfections in the optical system, and can

have both elements of short and long spatial correlations. It is referred to as TI noise, for

‘time-invariant’ noise. These noise components are also present when other SV data

acquisition systems are used, even though their magnitude is smaller. For example, data

from the ABS system often have negligible RI noise, but invariably show significant TI

noise components that consist of a background profile with short spatial correlation.

Previous approaches to address these noise components have required ad hoc alignment of

scans to eliminate RI noise, e.g., from operator-selected data subsets in regions that are

thought not to change with time. To eliminate the TI noise, schemes for pair-wise

subtraction of scans were devised to eliminate the TI noise from difference data. This

approach will usually introduce some bias and invariably lead to noise amplification. This

strategy is implemented, for example, in SEDANAL and DCDT+. Recently, a more

straightforward method was introduced that does not pose any of these drawbacks, by

simply incorporating these TI and RI signal terms directly into the model as terms b(r) and

β(t), respectively (Schuck and Demeler, 1999). The latter approach rigorously honors all

degrees of freedom presented by the analysis problem, and was shown to be statistically

optimal (Schuck, 2010c). The additional linear baseline parameters can be easily solved for

when using appropriate modern optimization techniques.

The practical significance of modern noise decomposition is an increased precision,

resulting, for example, in the ability to extract reliable information from SV data at very low

signal/noise ratio and to use lower protein concentrations (such as 10 nM of a 50 kg/mol

protein (Zhao et al., 2012)). After the fit, the noise estimates can be subtracted from the raw

data to allow the visualization of the remaining macromolecular sedimentation data in the

original data space for critical visual inspection of the model and of the information content

of the data (Dam and Schuck, 2004).

To extend the description of the baselines, we can add to the SV model expressions for the

signal arising from a mismatch in the geometry of the sample and reference solution column

(solution column height) and/or of the buffer composition in the sample and reference sector

(Zhao et al., 2010). Buffer salts exhibit significant sedimentation, which can be described

extremely well with Eq. 6 (typically, with D ~ 1.4 ×10−5 cm2/s and s ~ 0.14 S for NaCl in

water at 20° C). If the resulting signal contributions from sedimentation in both sectors do

not cancel, a contribution bbuff(r,t) is superimposed on the data, usually taking the form of a

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slowly varying characteristic tilt of the baseline. The ability to include this term into the

sedimentation model, where necessary, relieves experimental constraints in the perfectness

of sample preparation and filling of the centrifugal cells (see below). When such buffer

signal offsets are present, their inclusion in the model prevents errors in the protein

sedimentation parameters, and often results in better fits with tighter confidence limits,

despite the introduction of additional sedimentation parameters for the co-solvents. Despite

this newfound ability to computationally compensate for meniscus and buffer mismatches,

the best experimental practice is still to match them as rigorously as possible.

At very high co-solvent concentrations, whether or not they contribute to the signal directly,

the sedimentation of the co-solvents will generate significant local changes in solvent

density and viscosity that will affect, in turn, the sedimentation process of the protein of

interest. For example, these concentration gradients can result in characteristic retardation of

macromolecular sedimentation process with time, and even produce isopycnic bands of the

protein in the density gradients after long times. It is possible to extend Eq. 6 to conditions

of locally varying and time-dependent density and viscosity, and we have described a

method how the linked sedimentation of co-solvent and macromolecule can be numerically

simulated and fitted to experimental data (Schuck, 2004). Applications of this approach have

been described in the characterization of protein pharmaceuticals (Gabrielson et al., 2009).

In cases where the protein itself is at a concentration exceeding the limit where

hydrodynamic non-ideality is small (e.g., 1–2 mg/ml for globular proteins, lower limits for

more elongated structures), terms can be added to the Lamm equation (Eq. 6) that describe

the concentration-dependence of sedimentation, s(w), and diffusion coefficients, D(w),

arising from the long-range hydrodynamic interactions (Dishon et al., 1967; Solovyova et

al., 2001). SEDFIT, SEDPHAT, and SEDANAL provide for such models, which use linear

approximation of the sedimentation coefficient with a non-ideality coefficient ks. An

example for the application of direct boundary modeling with a non-ideality model can be

found in the study of malate dehydrogenase in complex solvents (Solovyova et al., 2001).

These models can be extended to multiple species, where Johnston-Ogston effects (i.e. the

expulsion of slower sedimenting species from the region of the faster-sedimenting boundary

at higher total concentration) are displayed. However, even though modeling of such

systems is possible, SV data cannot be expected to provide sufficient information for the

reliable determination of different non-ideality coefficients and mutual interaction

coefficients necessary in systems of multi-species and multi-component sedimentation with

hydrodynamic interactions. In this regard, direct boundary modeling of non-ideal solutions

is currently still limited. Furthermore, the linear approximation of the concentration-

dependent sedimentation coefficient breaks down at higher than 5% total macromolecular

volume exclusion (e.g. ~ 70 mg/ml protein). This fact renders impossible the full,

quantitative interpretation of the sedimentation boundaries measured under crowded

conditions.

Whichever model is used for the macromolecular sedimentation concentration profiles c(r,t)

of the proteins of interest, in the direct-boundary model, the experimental data are fitted by

least-squares with terms of the form

Equation 7

, where the brackets emphasize the non-linear parameters that depend on the precise model

of the sedimenting species, as well as non-linear parameters that depend on the solution

geometry. In these optimizations it is of critical importance to monitor the residuals of the

fit. As illustrated in (Dam and Schuck, 2004), fits with incorrect models may sometimes

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roughly describe the boundary data, yet provide qualitatively inaccurate answers if the

boundary broadening from unrecognized heterogeneity is modeled as though originating

from diffusion. To inspect the residuals, in addition to an overlay of the residuals to all

scans, we have introduced a residuals bitmap that displays the radius- and time-dependent

residuals in the form of a greyscale picture (Dam and Schuck, 2004). An example for an

unacceptable fit to the data of Figure 4A using a single-species model and an acceptable fit

with a distribution model is shown in Figure 4B and 4C, respectively.

Usually, fits within the noise of data acquisition and without systematicity should be

expected from SV analyses. It has proven useful to compare models using criteria from F-

statistics (Johnson and Straume, 1994; Johnson, 1992).

Direct Boundary Modeling With Size Distributions—A key development for the

direct-boundary modeling of SV was the explicit, constructive description of polydisperse

systems as a size distribution. SV data are highly sensitive to polydispersity and even trace

impurities due to the strongly size-dependent hydrodynamic separation. Without equipping

the model with appropriate terms to describe the effect of polydispersity, in most cases no

satisfactory fit will be achieved. The basic approach comprises the description of the

ensemble of sedimenting molecules as a differential distribution c(s) that describes the

concentration of species with sedimentation coefficients between s and s+ds (Schuck, 2000)

Equation 8

; for greater clarity, from here onwards we omit the baseline, noise, and buffer terms of Eq.

7.

Diffusion takes place with a

occurs with a ew2st time dependence. Thus, the description of the diffusion process and the

sedimentation coefficient distribution usually do not strongly correlate. The solution to D(s)

exploits some knowledge of the sedimenting macromolecules, often in the form of a

hydrodynamic scaling law for certain classes of particles. Sometimes, different rules for

calculating D(s) are applied in different segments of the distribution, dependent on s (see

below). This generality and adaptability is possible due to the numerical evaluation of the

Fredholm integral equation Eq. 8, which exploits techniques borrowed from modern image

analysis, with details outlined elsewhere (Schuck, 2000, 2009).

dependence, but separation from differential sedimentation

Data fitting to a size-distribution of diffusionally broadened sedimentation profiles

deconvolutes diffusion effects from the sedimentation boundary, much like removing

blurring from point-spread functions in image processing. An example of the gain in

resolution of c(s) when compared to a ls-g*(s) ‘data transformation’ that reflects largely the

boundary shape is shown in Figure 3. In addition to the gain in resolution, because the c(s)

model can usually describe the complete set of experimental data, it can report with

exquisite sensitivity the presence of trace aggregates. With detection limits of 1% or better

for oligomers of antibodies, this approach was found to be highly useful in biotechnology

applications (Berkowitz, 2006; Gabrielson and Arthur, 2011). Another feature of the fit of

Eq. 8 is the ability to include scans from the entire time-range of sedimentation from all

species, without requirement that sedimentation boundaries of all species present be visible

at the same time. This advantage results in a very wide (often 100–1000-fold) range of s-

values that can be covered in the c(s) distribution from a single experiment.

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For folded proteins, typically the most productive scaling law is that for compact particles,

where D(s) is governed by an average frictional ratio f/f0,w for all particles (in a given

segment) (Schuck, 2000; Schuck et al., 2002). The value of f/f0,w is usually refined as part of

the non-linear regression. This model exploits the fact that actual shapes must vary quite

significantly in order to produce a different frictional ratio, and many different shapes can

have the same frictional ratio. Usually considered a downside of hydrodynamics for the

assessment of protein shapes, the low shape resolution turns out to be advantageous in the

present case. Furthermore, as shown in (Schuck et al., 2002), deviations of the best-fit

frictional ratio from the actual frictional ratio will mainly result in broadening of the c(s)

peaks. Due to the high resolution, robustness, and sensitivity of this approach, the vast

majority of the > 1000 applications of c(s) reported in the literature for folded proteins have

used this c(s) method with a refined average frictional ratio as scaling law D(s). In this

manner, this analysis has become the first tool used for the understanding of experimental

SV data of unknown samples, providing a stepping stone towards the appropriate data

analysis and improved experimental design.

Once sedimentation and best-fit frictional ratios have been assigned, each pair of s, and D

can be inserted in the Svedberg equation Eq. 5 to produce a molar mass distribution c(M).

Provided that correct parameter values for solvent density and viscosity and for the protein

partial specific volume are available, the precision of molar masses reported from single

c(M) peaks is ~ 5–10%.

If there are multiple boundaries formed, then multiple c(s) segments can be constructed with

a different frictional ratio attributed to each boundary. Usually, not more than one piece of

diffusion information can be extracted in a well-conditioned manner from a single

sedimentation boundary (see below). For example, for data showing two sedimentation

boundaries, a model with a bimodal frictional ratio f/f0,w can be used, and molar masses of

isolated peaks corresponding to each sedimentation boundary may be obtained.

Other scaling-law c(s) variants available in SEDFIT or SEDPHAT are for worm-like chains

(applicable, for example, to amyloid fibrils (MacRaild et al., 2003)), arbitrarily user-defined

scaling laws (applicable to some unfolded biological macromolecules (Harding et al., 2011),

and proteins with ligand-induced conformational changes. The latter is a variation of the

segmented distribution: for a certain s-range, the diffusion coefficient D(s) is calculated by

the Svedberg equation (Eq. 5) based on a constant molar mass. Thus, this strategy reflects

different possible hydrodynamic friction coefficients exhibited by the same protein (Schuck

et al., 2000). For protein complexes that are known to have a certain frictional ratio, for

example, after imaging of globular or spherical particles by electron microscopy, it is also

possible to phrase the unknown extent of diffusion D(s) in terms of a known frictional ratio

but unknown partial-specific volume, which can be refined in the SV boundary analysis. In

this way, partial-specific volumes can be evaluated, for example, for samples of detergent-

solubilized membrane proteins (Ebel, 2011), or for nano-particles.

Another noteworthy special case is the limit of f/f0,w being infinite, appropriate for non-

diffusing species (Schuck and Rossmanith, 2000). In this case, the resulting c(s) distribution

is referred to as ls-g*(s) due to its relationship with the previously derived apparent

sedimentation coefficient distribution g(s*) from the time-derivative approach (Stafford,

1992), which was popular in the 1990s. In principle, the latter is theoretically also based on

– though not practically applicable to – non-diffusing particles. Many nano-particles, large

proteins, or other macromolecular complexes can be considered ‘non-diffusing’ species for

the purpose of SV. This approximation is valid because, during the short experimental time

provided by their rapid sedimentation, the extent of diffusional boundary broadening is

negligible (which may be the case even though the boundary shows broadening from

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differential sedimentation of polydisperse mixtures). The application of ls-g*(s) allows the

sedimentation coefficient distribution of such “non-diffusing” species to be derived. When

applied to diffusing species, a misfit of ls-g*(s) arises, which can be removed by restricting

the data to a subset of the scans, similar to the g(s*) method. In this case, due to the lack of

diffusional deconvolution, a diffusion-broadened sedimentation coefficient distribution is

produced, reflecting the raw boundary shapes in the dimension of apparent s-values (hence

the attribute ‘g*’). Like the previous g(s*) method, one could model the spread of apparent

sedimentation coefficients to estimate the molar mass of a discrete species. However, much

more accurate and robust estimates can be achieved using the c(s)/c(M) method to evaluate

diffusion information directly, accounting at the same time for polydispersity (Brown et al.,

2009). In contrast to the previous g(s*) from the time-derivative method (Stafford, 1992),

however, ls-g*(s) does not suffer from the same artificial broadening and peak shifts as

reported for g(s*) caused by the problematic approximation of the time-derivative by finite

time differences (Schuck and Rossmanith, 2000). Thus, ls-g*(s) has been used often when

diffusional deconvolution was not desired. For example, ls-g*(s) has been applied to large,

essentially ‘non-diffusing’ particles and/or to distributions of particles with a very wide

range of sedimentation coefficients.

Importantly, like c(s), the ls-g*(s) method has generally no limitation in the shape, time-

interval or total time-range of scans and can incorporate boundaries that are not

simultaneously visible in the observed radial range. In comparison with the time-derivative

method common in the 1990s, the ls-g*(s) approach eliminates practical constraints that

impose the selection of relatively low rotor speeds in SV, while maintaining a 1000-fold

dynamic range of sedimentation coefficients in a single run. Regarding experimental design,

this possibility enables the use of the highest possible rotor speed (50,000 or 60,000 rpm as

limited by the rotor) to achieve the best possible hydrodynamic separation in SV.

Information Content and Regularization—One practically important aspect of the

direct boundary modeling with size distributions, well-known in all fields employing

Fredholm integral equations of the type Eq. 8, is that the solution is ill-conditioned and

requires regularization. When Eq. 8 is solved strictly as stated, typically the absolute best-fit

distributions consist of series of sharp, baseline-separated spikes, the number and location of

which is dependent on very small features in the data, such as details of the noise. Although

it may seem tempting to take this best-fit distribution as if reflecting a true size distribution,

these spikes are highly unreliable and do not reflect the true information content of the data.

In fact, a broad set of such spiky distributions usually fit the data almost equally well. From

these properties of the inverse problem arises the requirement for regularization.

Regularization is the selection, from all the distributions that fit the data statistically

indistinguishably, of one particular solution that has the least information content. As

described in detail by Provencher (Provencher, 1982), by minimizing the information

content of the distributions and adjusting the detail of the distribution to what is reliably

extracted from the experimental data, misleading details of the peak structure are avoided

and the risk for over-interpretation is minimized. This is typically accomplished by a

secondary optimization that quantifies the level of detail in the distribution. The exact form

can be chosen dependent on the system under study. To achieve that, the error surface of the

fit is skewed by a regularization penalty term and minimized, while the magnitude of the

penalty term is adjusted such that the difference in the quality of fit to the raw data of the

penalized solution does not diminish relative to the un-penalized overall best-fit by more

than a permissible factor governed by F-statistics on a given, user-determined confidence

level (e.g., P = 0.683). This produces a statistically acceptable fit that conforms as much as

possible to the parsimony expressed in the regularization terms. (Conversely, it will be

appreciated that, by this design of regularization, the original spiky best-fit solution cannot

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provide a statistically better fit, and that therefore any information from a naive,

unregularized c(s) solution will present details that are misleading.)

There is a large body of literature on regularization, as it is an important and well-studied

problem in data analysis in many fields of physical science. Regularization methods often

employed in biophysics are Tikhonov regularization and maximum-entropy regularization.

As implemented in SEDFIT and SEDPHAT, Tikhonov regularization uses a measure of the

total curvature -∫(c″(s))2 ds to favor distribution curves that have as few peaks as possible,

and as broad peaks as consistent with the experimental data. This is appropriate, for

example, for macromolecules with a broad intrinsic size distribution, for example, polymeric

products, or heavily glycosylated proteins with an extensive and polydisperse degree of

glycosylation. Maximum-entropy regularization, also implemented in SEDFIT, minimizes

the information content by maximizing the Shannon-Jaynes entropy -Òc(s)ln (c(s)/p(s))ds,

where p(s) is a default distribution usually taken to be constant (see below). This model is

particularly attractive for systems that likely have a few intrinsically discrete species, such

as purified protein samples. This is the default choice in SEDFIT. Because maximum

entropy regularization has a tendency to produce oscillations for broader distributions

(Schuck et al., 2002), in such cases the user should switch to the Tikhonov method. No

regularization option is currently available for the c(s) implementation in ULTRASCAN.

In practice, for typical SV data with a high signal/noise ratio, both Tikhonov and maximum

entropy regularization will result in very similar distributions. For data with a low signal/

noise ratio, some distinct differences in the peak shapes and widths might be observed. The

type and extent of regularization can be a dominating factor for the peak widths in the c(s)

method. When considering fine details of the distribution, it is generally useful to probe the

family of distributions that fit the data equally well and ask whether or not a certain feature

is imposed by the data or by the regularization. This can be achieved by alternating the

regularization method and the regularization level (P-value).

An additional, highly useful tool is the customization of the regularization to what is already

known or to what can be hypothesized about the sample. This modification of the

regularization can be achieved by imposing Bayesian prior distributions. For example, a

non-constant prior distribution (p(s), above) can be employed (Brown et al., 2007b).

SEDFIT and SEDPHAT allow these priors to be constructed in variety of ways. It is often

useful to use a c(s) distribution obtained from one sample as a prior hypothesis for the

expected distribution of another sample. For example, this can be applied to a dilution series

to determine whether the sedimentation coefficient distribution of a more dilute sample is

consistent with that of a higher concentration. In this manner, one can address the problem

observed at the lower concentrations, namely that the decreased signal/noise level results in

broader peaks with the default regularization. For instance, this approach can better address

the question of whether an oligomer dissociates upon dilution. Also, one can hypothesize

that a certain peak may be described by a single species by using the numerical equivalent of

a Dirac δ-function as a prior hypothesis (Figure 4F). If microheterogeneity contributes to a

c(s) peak, then the resulting distribution, termed c(pδ)(s), will not conform to the sharpness

of the δ-function. (Due to its wide utility when studying purified proteins, this analysis can

be conducted by pressing the keyboard shortcut control-X after obtaining the c(s)

distributions.) The δ-function approach can help to assess the purity of a preparation, and/or

to adjust the regularization of the c(s) distribution to utilize prior knowledge, as may be

obtained from mass spectrometry. Alternatively, when obtaining slightly bimodal, not fully

separated c(s) distributions in studies of species with similar s-values, we may use Gaussians

placed at the centroids of each partial peak to try to baseline-separate the signals from both

species. This can help in the quantification of subpopulations. Vice versa, we may test

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whether a single, broader peak would also be consistent with the data, or if the data have

statistically unambiguous information that the peaks can be sub-divided.

The way in which the Bayesian approach utilizes this additional knowledge is fundamentally

different from hard-wired constraints. Due to the imperfections of the experimental data

alone (originating from both data acquisition and sample), a hard-wired constraint that

strictly imposes a certain model to the analysis will in most cases lead to a statistically

significantly worse fits, which often then requires further judgment regarding whether it

should be rejected or is still acceptable. In contrast, the Bayesian approach always fully

honors the complete information of the data and, by design, ensures the same quality of fit,

by allowing the result to refute the constraint if it is wrong, or by subtly adding on missing

pieces in the interpretation. Notably, the Bayesian approach described above does not

provide a quantitative measure of the agreement with the prior; rather, that is assessed

graphically by the user.

In summary, SV analysis has progressed from a simple ‘data transform’ to a size-

distribution analysis that fully utilizes all the data and invites the utilization of our certain

knowledge of the system, such as the validity of the laws of diffusion and hydrodynamic

scaling laws, to arrive at higher-resolution results. The Bayesian analysis represents a highly

useful further step that now lets us actively probe the consistency of different interpretations

with the experimental data.

Multi-dimensional generalizations of c(s)—Several useful generalizations of the c(s)

are available that make use of additional data or distribution dimensions.

First, it is possible to generalize the sedimentation coefficient distribution to a size-and-

shape distribution c(s,f/f0) (Brown and Schuck, 2006) that allows for a distribution of

hydrodynamic friction values for each single s-value, thereby abandoning the need for a

hydrodynamic scaling law:

Equation 9

An example is shown in Figure 4. The size-and-shape distribution may equivalently be

presented in s-D or s-RS coordinates, although it is easier to create an efficiently discretized

mesh in s-f/f0 coordinates. Computationally, this problem can be solved exactly, in the same

rigorous way as the standard c(s) distribution available in SEDFIT. A fundamentally

different approach, called ‘2DSA’, with a rather unique computational strategy was

proposed by Demeler & colleagues (Brookes et al., 2009) to achieve a size-and-shape

distribution in ULTRASCAN, as described in CPPS Unit 7.13 (2010). However, we are

unaware of attempts to prove the correctness of the algorithm, and 2DSA remains empirical.

In fact, upon detailed examination (Schuck, 2009), several key points of this method were

found to be in conflict with mathematics and incompatible with general scientific principles

of data analysis, and the method is therefore unlikely to give correct results.

Ordinarily, SV data do not contain sufficient information to create well-defined peaks in the

diffusion dimension except for the major peaks (Brown and Schuck, 2006), thus providing

very similar information as c(s). In fact, by integration of the diffusion dimension, c(s,f/f0)

can be collapsed back to a sedimentation coefficient distribution, called ‘general c(s,*)’, that

is independent of scaling laws but typically is virtually identical to the standard c(s).

However, in applications where scaling laws cannot be easily phrased or where species with

very different molar masses co-sediment at the same s-value, the c(s,f/f0) and its related

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distributions may be very useful. An example of the former case occurs in ensembles of

particles with density distributions (Carney et al., 2011).

Another extension of c(s) is for the global modeling of SV data sets acquired in experiments

at different solvent densities (Brown et al., 2011)

Equation 10

In units of sedimentation and diffusion parameters at standard conditions (20,w), all

experiments share the same sedimentation coefficient distribution, which will be mapped to

the measured experimental conditions based on the experimental solvent density and

viscosity. A scaling factor αρ accounts for the experimental fact that not all samples may be

precisely at the same concentration, for example, due to slight dilution errors. This allows us

to treat the protein partial-specific volume as an unknown parameter to be refined globally

in the non-linear regression, and thereby provides a method for the determination of the

partial-specific volumes, for example, of proteins or protein-detergent complexes.

This density-contrast SV approach improves on the precision of the classical Edelstein-

Schachman SE approach due to the significantly higher precision of sedimentation

coefficients, and due to the allowance for protein heterogeneity and impurities (Brown et al.,

2011).

Undoubtedly the most powerful extension of the c(s) method for multi-component mixtures

is the global analysis of SV data acquired in the same experiment simultaneously with

different optical signals λ (Balbo et al., 2005):

Equation 11

The signals can be absorbance data at different wavelengths and/or a combination of

interference and absorbance optical signals. If a total number of Λ different signals are

available, then up to the same number of different macromolecular components can be

spectrally discriminated based on their molar extinction (or signal) coefficient ελk, and

separate sedimentation coefficient distributions ck(s) can be calculated for each component k

(Figure 5). (The remaining required quantity in Eq. 11 is the optical pathlength, d, which

will be 1.2 cm for standard double sector centerpieces.) As shown in (Balbo et al., 2005), the

spectral and hydrodynamic resolution can exhibit synergy, and the information content of

the set of component ck(s) distributions can be far beyond that of separate standard c(s)

analyses of the individual signals.

This multi-signal SV (MSSV) technique is naturally of great utility in the study of protein

interactions, where the stoichiometry of protein complexes can be deduced from the area

ratio of co-localized ck(s) peaks. MSSV has found many applications, as recently reviewed

in (Padrick and Brautigam, 2011; Padrick et al., 2010). In the current version of SEDPHAT,

up to three different signals may be included in the analysis, thus enabling the study of co-

existing binary and ternary complexes formed in mixtures of three different proteins.

Another important application is the analysis of protein-detergent mixtures, where this

approach can reveal the protein/detergent ratio of the detergent-solubilized protein (Salvay

et al., 2007).

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For this multi-signal SV approach to work, sufficiently different spectral signatures of the

interacting components are required. Padrick & Brautigam (Padrick and Brautigam, 2011)

have determined a predictor, Dnorm, that is calculated based on the known extinction

coefficients and allows the prediction of whether different components are likely to be

distinguishable in a two- or three-component MSSV analysis (see below). Often, interacting

proteins have sufficiently different fractions of aromatic amino acids for the combination of

interference signals with absorbance at 280 and/or 250 nm to generate a sufficiently high

Dnorm. The same is true for many protein interactions with nucleic acids (Berke and Modis,

2012), lipids or detergents (Ebel, 2011), or strongly glycosylated proteins. Thus, MSSV can

often be conducted label-free (Balbo et al., 2005). Examples for the analysis of MSSV to

mixtures of three protein components can be found in (Houtman et al., 2006; Barda-Saad et

al., 2010). Detailed instructions for the application of MSSV can be found in the web-based

SEDPHAT help system and in (Padrick et al., 2010). Although Dnorm is based on simple

matrix manipulations, some may find it non-trivial to calculate. As a courtesy, SEDPHAT

can calculate and present this quantity to allow for experimental planning.

A current limitation is that the absence of hyper- or hypochromicity is assumed.

Furthermore, it should be noted that the integrals of ck(s) peaks of different components

foremost reflects the composition of the sedimentation boundary. For these quantities to

reflect the complex stoichiometry, the complex must be quasi stable, which is the case close

to saturation or for slow dissociation kinetics, both of which can be tested via concentration-

independence of the c(s) peaks (see below). An exception can be the reaction boundaries of

rapidly interacting molecules with very similar sedimentation coefficient in their free form

(see below).

Sedimentation Velocity Analysis of Interacting Systems

Direct boundary modeling with a set of coupled Lamm equations for reacting

systems—A key application of SV in protein science is the study of interacting systems. In

recent years, it has become possible to use solutions of coupled sets of Lamm equations of

chemically reacting systems directly for the non-linear regression of experimental SV data.

This takes the form (Fujita, 1975)

Equation 12

where i enumerates the species participating in the interaction with their signal contribution

εidci(r, t), and qi is the local chemical reaction flux between the species. For example, for a

simple bimolecular reaction forming a 1:1 complex, qi becomes q1 = q2 = −q3 = −q with q =

konχ1χ2−koffχ3, and the equilibrium dissociation constant Kd = koff/kon. Because the

sedimentation behavior and fractional saturation strongly depend on the molar loading

concentrations, it is generally highly advisable to run multiple experiments at concentrations

that bracket Kd by a factor 10, if possible, on either side. After a reaction scheme has been

determined, both SEDPHAT and SEDANAL are capable of globally modeling all

experimental data with Eq. 12, SEDANAL in the data differencing mode (Correia and

Stafford, 2009) and SEDPHAT in the direct boundary modeling mode (Dam et al., 2005).

Applications and strategies have been reviewed recently by Brautigam (Brautigam, 2011).

In principle, the Lamm-equation modeling for an interacting system allows kinetic rate

constants to be extracted, as this parameter modulates the shape of the boundaries and their

separation. However, as shown in (Dam et al., 2005), and not surprisingly, the sensitivity of

the data for different kinetic rate constants is limited roughly to the time-scale on which the

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SV experiment takes place, which is dependent on protein size, but typically restricts the

range of koff to around 10−4/sec – 10−3/sec.

Furthermore, Eq. 12 is intrinsically a discrete model, and may not always yield a good fit

and/or accurate results if sample imperfections such as microheterogeneity and trace

impurities are present that will affect the boundary broadening. These deviations from

single-species behavior are most critical for the kinetic rate constants, as the information on

their parameter values also resides in the boundary broadening. For heterogeneous

associations between proteins that do not self-associate, sufficient purity can be tested by the

applicability of single discrete-species models to the individual components. The impact of

aggregate impurities may be reduced by restricting the radial range of the analysis, such as

in partial boundary modeling (Brown et al., 2009). Unfortunately, the influence of

breakdown products or impurities sedimenting slower than the species of interest cannot be

excluded in the same way, as their diffusionally broadened boundary often makes significant

contributions to the sedimentation data across the whole available radial range and typically

overlaps the boundary of interest, especially at early times.

Although very powerful, the direct-boundary modeling with Lamm equations of reacting

systems is usually not the first method of choice, and more robust methods that are more

consistent with the sensitivity of SV for sample imperfections will be highly desirable,

especially if kinetic rate constants are not of primary importance. One could consider it a

virtue of the Lamm-equation modeling of reacting systems that all peculiarities of the

coupled reaction/diffusion/sedimentation process are delegated to the Lamm equation solver

and modeling. However, for successfully designing and interpreting SV studies of

heterogeneous interacting systems, it is essential to understand the salient features of

reaction boundaries.

Effective particle model for the sedimentation of heterogeneous interacting

systems—For rapidly interacting systems (on the time-scale of sedimentation, i.e. usually

koff > 10−2/sec), not all sedimentation boundaries reflect real sedimenting species. To

examine this phenomenon, let us consider at first a simple 1:1 interaction of the type A + B

forming an AB complex in a rapid reaction, with the nomenclature such that sA < sB < sAB.

It has been long known that sedimentation boundaries of rapidly interacting proteins exhibit

several seemingly non-intuitive properties (Fujita, 1975): (1) The data show not three but at

most two boundaries, one always sedimenting at the s-value of either free A or free B,

termed undisturbed boundary, and the other at a composition-dependent s-value between sB

and sAB, termed reaction boundary (2) In a titration series, the s-value of the undisturbed

boundary could switch from sA to sB, but at a transition point where the solution composition

does not correspond to the complex stoichiometry. (3) At a certain solution composition,

also not equal to the complex stoichiometry, there can be only a single boundary. It is

crucially important not to mistake the reaction boundary for that of a stable, independently

sedimenting species.

These properties can be intuitively better understood in the framework of the recently

developed effective particle theory (EPT) for the coupled sedimentation/reaction process

(Schuck, 2010b), which was derived directly from Eq. 8 in the approximation of non-

diffusing particles and rectangular geometry. Due to the instability of the complex in rapid

equilibrium, obviously no separate boundary can stably exist for the complex. Instead, the

whole system with components of free A, free B, and complex AB, as determined by mass

action law, sediments jointly. Thus, in the reaction boundary, there is always a mixture of

free A, free B, and complex. There can be only one undisturbed boundary comprised of the

excess material of one of the binding partners (since the presence of the other would lead to

complex formation, hence a reaction boundary). From the ergodicity of the sedimentation

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