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Using Lamm-Equation Modeling of Sedimentation Velocity Data

to Determine the Kinetic and Thermodynamic Properties of

Macromolecular Interactions

Chad A. Brautigam

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

5323 Harry Hines Blvd., Dallas, TX 75390-8816

Abstract

The interaction of macromolecules with themselves and with other macromolecules is

fundamental to the functioning of living systems. Recent advances in the analysis of sedimentation

velocity (SV) data obtained by analytical ultracentrifugation allow the experimenter to determine

important features of such interactions, including the equilibrium association constant and

information about the kinetic off-rate of the interaction. The determination of these parameters is

made possible by the ability of modern software to fit numerical solutions of the Lamm Equation

with kinetic considerations directly to SV data. Herein, the SV analytical advances implemented in

the software package SEDPHAT are summarized. Detailed analyses of SV data using these

strategies are presented. Finally, a few highlights of recent literature reports that feature this type

of SV data analysis are surveyed.

Keywords

Analytical ultracentrifugation; sedimentation velocity; Lamm equation modeling

1 Introduction

Macromolecular interactions lie at the heart of modern molecular biology. Proteins can

interact with themselves (a homo-association) or with other molecules (a hetero-

association), like small metabolites, other proteins, nucleic acids, and carbohydrates. In

studying these interactions in vitro, one of the most relevant quantities for biochemists to

determine is strength of the interaction, expressed as the equilibrium association constant

(KA). Commonly, this quantity is reported as the dissociation constant, or Kd (Kd = 1/KA).

Numerous means have been devised to measure this quantity, including isothermal titration

calorimetry, fluorescence quenching, fluorescence anisotropy, electrophoretic mobility,

equilibrium dialysis, liquid chromatography, and others (e.g. [1,2]).

Analytical ultracentrifugation (AUC) is emerging as a potent means to study the interactions

of macromolecules [1,3–7]. Sedimentation equilibrium (SE) has long been used to determine

the values of KA for homo- and hetero-associations [1,8]. In this method, the samples are

© 2010 Elsevier Inc. All rights reserved.

Correspondence: chad.brautigam@utsouthwestern.edu, phone: +1-214-645-6384, fax: +1-214-645-5383.

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Author Manuscript

Methods. Author manuscript; available in PMC 2012 May 1.

Published in final edited form as:

Methods. 2011 May ; 54(1): 4–15. doi:10.1016/j.ymeth.2010.12.029.

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centrifuged at speeds that generate shallow concentration gradients. These gradients are

stable once all species have achieved physical and thermodynamic equilibrium. The shape of

the gradients contains information regarding the masses of all species present, and

experiments performed at several component concentrations and rotor speeds may be

analyzed to yield complex masses and dissociation constants. Although several

disadvantages to SE are known, recent advances in data analysis have overcome some of

them [9,10]. The technique remains a valuable tool for the study of macromolecular

interactions, but its main disadvantage lies in the time necessary to complete the experiment.

The most data-rich SE method, called “long-column” SE, may take as long as a week to

complete, with the concomitant demand that the sample remains stable during this time.

“Short-column” SE may take only hours to complete [11], but also has a significantly

smaller data basis.

By contrast, the sedimentation velocity (SV) configuration of analytical ultracentrifugation

takes only hours and is comparatively data rich. In the work that follows, new advances in

the direct modeling of SV data with numerical solutions to the Lamm Equation coupled to

reaction fluxes are summarized. To introduce new practitioners to how these approaches can

be used in SEDPHAT, detailed analyses are presented: one for a homo-association, the other

characterizing a hetero-association. Finally, some recent results using this approach are

discussed.

2 Theory

2.1 Background

SV is very useful for studying non-interacting solutes, and it may also be used for the study

of macromolecular interactions. This methodology features higher rotor speeds than SE. The

macromolecular solutes thus migrate through the solution column and become localized

very close to the centrifugal portion (“bottom”) of the centrifugation cell. The concentration

gradients formed are monitored during the entire course of the experiment (Fig. 1A). A

sedimenting macromolecule will give rise to a sigmoid concentration profile; roughly

speaking, the inflection point of this feature is called a “boundary.” The shape and velocity

of the boundaries contains information regarding the size and shape of the sedimenting

particle. As centrifugal force moves the boundary centrifugally, diffusion acts to make the

boundaries progressively shallow during the course of the experiment (i.e. they become

“diffusionally broadened”). The presence of two particles of sufficiently divergent size

results in two boundaries (Fig. 1B), and so on.

The partial differential equation that describes the evolution of the boundaries was first

formulated by Lamm in 1929 [12]. It is thus called the “Lamm Equation” (LE), and may be

formulated thus for an ideal particle sedimenting in a sector-shaped centrifugal cell:

(Eq. 1)

where χ, D, and s are the concentration, diffusion coefficient, and sedimentation coefficient,

respectively, of the particle, t is time, r is the radius from the center of rotation, and ω is the

angular velocity of the rotor. The LE describes the ideal transport processes occurring

during the course of the SV experiment, including sedimentation, diffusion, and even

floatation. There are no known exact analytical solutions to the LE, but there are many

approximate analytical solutions that consider the cases of no diffusion, rectangular cell

geometry, etc. [13]. However, with the advent of inexpensive, powerful computers,

numerical solutions to the LE are now readily available and routinely used [8].

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The implications of the accessibility of quickly calculated numerical solutions to the LE are

manifold. The solutions allow SV data obtained from ideal, non-interacting macromolecules

to be directly modeled. Several software programs are available for this purpose, including

LAMM, SEDANAL, ULTRASCAN, SEDFIT, and SEDPHAT [6,14–17]. In addition, the

ease of this computation facilitates the direct description of the boundaries as a continuous

distribution that scales a large number (≥ 50) of LE solutions. This approach was first

described by Schuck [18], and the mathematical formalism is:

(Eq. 2)

where a(r,t) is the signal measured by the centrifuge, χ is a concentration profile that

represents an LE solution of a non-interacting species as a function of the parameters listed,

and D(s) is the diffusion coefficient calculated as a function of s and with the assumption of

all species having the same frictional ratio (fr).

Although the scheme of directly fitting LE solutions to SV data of proteins and other

biological macromolecules can work well, the LE as formulated above (Eq. 1) does not

account for chemical reactions, i.e. the interaction of multiple species present in the

centrifugal cell. Thus, adjustments must be made to the LE so that it may be used to model

SV data in which such interactions are known to occur. Examples of macromolecular

interactions are abundant in biology. Proteins interact with themselves to form oligomers (a

homo-association) and with other proteins, forming complexes (hetero-associations). In

order to use the LE to analyze such interactions in the SV setting, the LE must be combined

with reaction fluxes and information on the equilibrium association constant (KA). For an

instantaneously equilibrating system, the data may be treated using a weighted average for s

and a gradient average for D [19]. For hetero-associations, explicit reaction fluxes, qk, may

be considered in an equation system:

(Eq. 3)

where χk is the concentration of the component k, Dk and sk are its diffusion and

sedimentation coefficient, respectively, qk is the local reaction rate, and Jk,tr is component’s

transport flux [13]. The fluxes are dependent on the component and complex concentrations,

which are dependent on KA. Both of these approaches are implemented in the freeware

program SEDPHAT [5,19]. Hereafter, this type of analysis is referred to as “LEq,” for

“Lamm Equation coupled to reaction fluxes q.”

2.2 Recent Advances

Recently, Brown and Schuck have described a new algorithm for the fast numerical

calculation of LE solutions. A full mathematical description of this algorithm is beyond the

scope and intention of the present paper, and it is detailed elsewhere [20]. A brief

summation is presented here. The overall methodology is a modification of one adopted 35

years ago by Claverie [21]. That author modeled the concentration profiles obtained in an

SV experiment as a weighted sum of overlapping triangular “hat” functions. These functions

were equally spaced on a radial grid ranging from the meniscus to the bottom of the

centrifugal cell. Values for the individual weighting terms for the hat functions can be

obtained by multiplying the LE by the hat functions and integrating from the meniscus to the

cell bottom. Brown and Schuck [20] introduced several innovations that improved the

accuracy of this approach. First, they changed the grid spacings; the Brown and Schuck

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spacing scheme has a small increment (Δr) near to the “left fitting limit”

generally located a few hundredths of a centimeter to the centrifugal (“right”) side of the

meniscus. Moving centrifugally from , the Δr’s increase as the bottom is approached. This

strategy is justified by the fact that a population of species will exhibit the steepest gradients

near to (and the meniscus), but will become diffusionally broadened (i.e. shallower) as it

moves centrifugally. Thus, it is advantageous to sample radial space finely near to , and

increasingly more coarsely with higher radius. The initial value of Δr (Δr0) is inversely

proportional to the square root of the buoyant molar mass of the particle; thus, for large

species, a finer sampling near to the left fitting limit is achieved compared to the sampling

calculated for a small species. The concentration gradients near to the left fitting limit are

steeper for larger species, thus justifying the mass dependence of Δr0.

, which is

In a second change to the Claverie algorithm, Brown and Schuck take advantage of the fact

that simple analytical solutions are available for specific sedimentation conditions, obviating

the computationally intensive numerical solution when these conditions are met. For

example, at infinite time, the Lamm Equation reduces to a Boltzmann exponential

distribution:

(Eq. 4)

where c0 is the initial concentration of the solute, Mb is the buoyant molar mass of the

particle, rb and rm are the radial positions of the bottom and meniscus, and ξ(r) = ω2r2/2RT,

where R is the universal gas constant, and T is the temperature in Kelvin. The value of ceq(r)

places an upper limit on the amount of back-diffusion that can be observed at radial values

above the hinge point of the distribution represented as Eq. 4.

Another analytical solution that is utilized is the case for non-diffusing species. In this case,

the concentration profiles reduce to the equation system:

(Eq. 5)

Although this equation system does not represent the boundary well, it is accurate for

regions of the concentration profile at the solvent portion and solute plateau (Fig. 1). The

algorithm of Brown and Schuck calculates which areas of the concentration profile may be

safely considered to be outside the boundary, and uses Eq. 5 to calculate the concentration in

these radial domains. Thus, the algorithm dynamically determines whether a grid point is

eligible for numerical calculation (“active”) or analytical calculation (“inactive”). This

strategy results in a significant computational savings [20].

Finally, the new algorithm implements a refinement to the LE solutions that had been

introduced before [5]—a “semi-infinite” solution column. Stated simply, the evolution of the

concentration profiles is treated as if the bottom of the solution column were missing. This

simplification speeds the numerical calculation of the LE solutions because back-diffusion

from the bottom of the centrifugal cell may be neglected. In most cases, it is justified,

because the area of significant back-diffusion is eliminated from the analysis. The algorithm

can calculate whether the application of the semi-infinite column is appropriate, and switch

to a finite column if necessary for cases wherein back-diffusion is significant.

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In the above, the species for which the LE is solved are considered to be ideal and

noninteracting. However, the focus of this work is macromolecular interactions; thus,

reaction fluxes of the individual components must be calculated along with their

sedimentation and diffusion. A detailed description of this process is found elsewhere [5]. In

essence, the reaction flux is multiplied by the Δt step taken in the numerical simulation of

the experiment, and component concentrations are adjusted accordingly. A correction factor

that accounts for the fact that the reaction was occurring between time t and time t + Δt is

applied. For instantaneous reactions, the local concentrations of components are allowed to

relax to their equilibrium values at all radial positions. For slower (“finite”) reactions, a

linear approximation of the rate equations is used to calculate the concentration changes at a

given time interval. The time step is limited such that the concentration change due to

reaction fluxes is not greater than that due to sedimentation.

An analysis of SV data of an interacting system using LE solutions coupled with reaction

fluxes ordinarily attempts to account for the entire concentration profile (e.g. Fig. 1).

However, the presence of significant contaminants, including aggregated material and

degradation products, will deteriorate the quality of the analysis. A method to ameliorate the

problem of contaminated SV data was recently introduced by Brown et al. [22]; they term it

partial boundary modeling (PBM). Instead of analyzing the entire concentration profile, the

experimenter may constrain the analysis to consider only a certain s-range. This constraint

has the effect of limiting the radial range of each concentration profile (“scan”) included in

the analysis. Thus, contributions of slowly sedimenting and/or quickly sedimenting species

may be eliminated from the analysis, allowing for more accurate parameter determination in

the presence of contaminant species. In SEDPHAT, this is implemented by defining an s-

range (slow to shigh), which leads to the radial constraint rmexp[ω2slowt] ≤ r ≤

rmexp[ω2shight] for each scan.

Although this radial constraint is straightforwardly applied to LEq analysis, it complicates

the calculation of time-invariant (TI) and radially invariant (RI) noise [20,23]. Ordinarily,

the entire concentration profile is analyzed, allowing for the facile treatment of these noise

elements as linear parameters [23]. However, in PBM, only a limited radial range of the

concentration profile is analyzed, undercutting the mathematical basis of the noise analysis.

The RI and TI noise may still be calculated as long as there is sufficient overlap in scans (i.e.

all radii must be represented at least twice in the scans analyzed). The noise elements (which

are still linear parameters) are determined in an iterative fashion instead of the single-step

matrix operation used before [23].

Another innovation introduced by Brown et al. is LE solutions that account for t in Eqs. 1–3

and Eq. 5 (termed TODA here) from the absorbance optical system of the Beckman Optima

XL-A or XL-I ultracentrifuge. The concentration profiles are obtained from this system by

moving a slit along a rail that is positioned over a photomultiplier tube. Light shines from

above the rotor, through the centrifugation cell, and onto the slit, which moves in one

dimension (radially). The photomultiplier tube records the light intensity as a function of the

radial position of the slit. The apparatus takes about 30 seconds to scan the 1.2-cm solution

column of a typical SV experiment. During this time, sedimentation is occurring, but the

time stamp included in the data file (which is used by SEDFIT and SEDPHAT as t in Eqs.

1–3 and Eq. 5) notes only the beginning of the scan. Thus, species appear to sediment

erroneously quickly. The effect is negligible for small species (< 10 S at 50,000 rpm), but

becomes increasingly significant for large species and at high rotor speeds. Another

consequence of sedimentation occurring during the scan is that the boundary is artificially

broadened, leading to erroneous determinations of the diffusion coefficient (and the related

quantities, molar mass and frictional ratio) of the species. SEDPHAT (and SEDFIT) can

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compensate for these problems by calculating the apparent concentration profile instead of

the ideal one. This is accomplished using the equation

(Eq. 6)

where tscan is the time recorded in the header of the data file, r0 is the radial position where

scanning begins, and vscan is the velocity of the slit in cm/s. Whether TODA correction is

used or not is under user control in SEDPHAT and SEDFIT. The introduction of TODA

correction slows the calculation of the LE solutions significantly. It is therefore

recommended only to use this option when necessary, i.e. when accurate information on

quickly sedimenting species is essential to the analysis. Further, the new LE solution

algorithm depends on the time stamp for the simplifying case of calculating the solution and

solute plateaus (Eq. 5). The time lag effects described above affect the radial position of the

step function. It also introduces a slight slope to the solute plateau region due to the radial

dilution that all solutes undergo in the standard sector-shaped centrifugal cells. These effects

are also compensated for when the user activates the TODA compensation.

3 Results

3.1 Protein Methods

The protein solutions analyzed below were purified as described [24,25]. KinA PAS-A was

in a buffer comprising 50 mM Tris pH 7.5 and 100 mM NaCl. GST-VCA and Arp2/3 were

dialyzed against a buffer containing 50 mM KCl, 10 mM imidazole, 0.5 mM EGTA

(Ethylene glycol-bis(2-aminoethylether)-N,N,N',N'-tetraacetic acid), and 0.5 mM MgCl2.

The partial specific volume of the proteins and the density and viscosity of the buffers were

estimated using SEDNTERP [26]. For the calculation of the buffer values of the GST-VCA

buffer, ethylenediaminetetraacetic acid was used in place of EGTA, and imidazole was

omitted in the calculation.

3.2 Centrifugation Methods

Proteins (sample side) and buffers (reference side) were placed in a 1.2-cm Epon dual-

sectored centrifugation cell that was sandwiched between two sapphire windows. The

assembled cells were inserted into an An50-Ti rotor, equilibrated at 20° C for several hours,

then subjected to centrifugation in an Optima XL-I ultracentrifuge (Beckman-Coulter, Brea,

CA) at 50,000 rpm for the KinA PAS-A domain, and 42,000 rpm for the GST-VCA/Arp2/3

system. Data were acquired using the absorbance optics of the centrifuge tuned to 280 nm.

3.3 Designing an LEq experiment

It is important to consider many aspects of the experiment before performing a study that is

to be analyzed using LEq. Some knowledge of the samples is useful at this preparative step.

The quality of the subsequent data analysis is enhanced by the quality of the sample

preparation. The samples should be as pure as achievable, and free of aggregates. Although

there are computational means (e.g. PBM) to mitigate the effects that these sample flaws

have on the data analysis, it is always advantageous to have pure, monodisperse samples to

study.

The molar signal increments (ε) of each component being studied should be determined.

SEDPHAT depends on these quantities for the calculation of component and complex

concentrations. For a protein, ε can be calculated for the absorption optical system (ελ) of the

Beckman XL-I centrifuge by taking the weighted sum of the extinction coefficients of the

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chromophoric amino acids present in the polypeptide. Several web-based calculators are

available for this calculation. Alternatively, ελ may be determined experimentally [27]. For

the Raleigh interferometer, the quantity to be determined is referred to in this work as εIF.

An excellent estimate for the εIF of a protein is obtained by multiplying the molar mass of

the protein by 2.75 [28,29].

An initial estimate of KA is also very helpful. This knowledge guides the choices of

experimental macromolecular concentrations to be used. Most often, several parallel SV

experiments are carried out in a titration series. As the concentrations of the components

change, their populations will also change according to mass action law. Ideally, the span of

concentrations studied in an LEq analysis should be chosen such that each posited species

has a detectable population at some point in the titration series. This situation may not be

achievable in cases of extreme cooperativity.

Two important constraints on the concentrations available to the experimenter in the titration

series exist. First, if using the absorbance optical system, the total optical density (OD) of

the experimental sample should not exceed 1.0–1.2. Beyond this limit, the absorbance signal

becomes nonlinear, which is a severe hindrance to this type of analysis. Second, for proteins,

the concentration of the sample should not significantly exceed 1 mg/mL. This constraint

comes about because effects due to hydrodynamic and thermodynamic nonideality become

more pronounced at high concentrations. The concentration at which nonideality becomes a

problem is dependent on the individual physical characteristics of the macromolecule under

study. SEDPHAT has means of compensating for hydrodynamic nonideality in LEq

analyses [30], but it is best to avoid the necessity of introducing this additional complication

to the analysis.

In LEq analyses, SEDPHAT imposes an association model on the data analysis. Thus, the

experimenter must have some idea of the stoichiometry of the association. A preliminary

multisignal SV experiment [29,31,32] can be used to garner information about

stoichiometry, and, of course, other means are available. Many models for both hetero- and

self-associations are currently implemented in the software. Although the analysis itself

could be used to test the goodness-of-fit of various models, this strategy is not recommended

for LEq analysis because of the large cost in computational time needed to perform the

calculations.

In practice, almost all of the experimental considerations introduced above can be addressed

with preliminary experiments analyzed with the c(s) distribution. As mentioned above, SV

data are very well treated by considering them as a sum (formally, an integral; see Eq. 2) of

LE solutions with different s-values scaled by a continuous distribution [16,18]. The purity

and monodispersity of the components may be assessed by performing SV experiments on

them and checking for aggregates or other contaminants. Analyzing such experiments with

the c(s) distribution very sensitively detects these defects in the sample [33]. Further, a trial

concentration series can be performed and analyzed with c(s) distributions to assess whether

all species are represented in the titration, the stoichiometry of interaction, an initial estimate

for KA and scomplex, and the koff regime. This last point is best illustrated with an example.

In Fig. 2, we show three simulated titrations for a simple A + B ↔ AB system. In all of the

simulations, the sedimentation coefficient of component A (sA) is 4 S, sB is 8 S, and sAB is

9.7 S. The KA is also the same (106 M−1) in all of the systems. In System I (Fig. 2A), the

value of koff is 10−6 s−1 (very slow on the timescale of the sedimentation experiment), while

that value for System III (Fig. 2C) is 10−1 s−1 (essentially instantaneous on this timescale).

The koff for System II is intermediate between I and III: 10−4 s−1. In both systems, the total

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concentration of A ([A]tot) is changed while [B]tot is held constant. The solution system was

water and the speed of centrifugation was 50,000 rpm.

The c(s) distributions for System I show three discernible peaks, one for each sedimenting

species (Fig. 2A). As more A is added to solution, the relative concentrations of the species

change, but the s-values of the peaks do not. However, the situation is very different in

System III. Under a given concentration, there is evidence for free A, but the complex peak

exhibits a concentration-dependent movement, approaching sAB only when [A]tot ≫ [B]tot.

This phenomenon is a consequence of the fast kinetics of System III, and has been described

in the seminal work of Gilbert and Jenkins [34]. A mathematically intuitive and quantitative

approach to the analysis of this behavior was recently reported by Schuck [35]. System II,

with its intermediate koff, shows intermediate behavior. Thus, as has been pointed out before

[4], the c(s) analysis can be used to diagnose the kinetic regime of the interaction.

Estimates for scomplex and KA are also available from the c(s) analysis. In System I, scomplex

is directly available because the complex does not significantly dissociate on the timescale

of the SV experiment. In Systems II and III, scomplex is approached at high [A]tot, and a

reasonable estimate of the value may be made. For KA, some information may be estimated

intuitively. For example, in System I, there could be a point in titrating in increasing [A]tot

that displays equal concentrations for free A, free B, and the AB complex; thus, KA is easily

estimated for this situation. Similar arguments apply to Systems II and III, but the

populations are more difficult to estimate by inspection alone; a focus on the free [A] may

be necessary. Stoichiometry may be hypothesized by examining scomplex and the frictional

ratio of the complex in order to estimate its molar mass.

SV isotherm analysis [4,6] could also be employed with the data displayed in Fig. 2. In

particular, an “sw isotherm” could be useful for a quick estimate of scomplex, KA, and

stoichiometry. In such an analysis with the systems described above, the entire SV

experiment at a given [A]tot is reduced to a single number, a weighted s-value (sw). This

single value contains information on the bulk transport properties of the solutes. It is thus

dependent on the relative populations of free A, free B and the AB complex. Of course,

these populations are dependent on the KA, and this quantity can be fitted in such an

isotherm. Further, there is information about scomplex in the isotherm, and this may be fitted

as well (N.B., this isotherm has less information on scomplex than a “moving boundary”

isotherm, which can be employed when fast kinetics are observed [4]). Finally, because the

isotherm can be fitted in a fraction of a second, the experimenter is free to explore various

hypothetical stoichiometries and to assess the respective goodnesses of fit to the data.

The complex stoichiometry may be difficult to estimate from an uncharacterized hetero-

association. Recently, multi-signal SV (MSSV) has been used with excellent results

[29,31,36]. In this type of analysis, the differing spectral properties of the associating species

are utilized to decompose a c(s) distribution into ck(s) distributions representing the

contributions from individual components k. In parts of the distribution that evince

cosedimentation, the relative areas beneath the peaks represent the molar ratio of the

cosedimenting species. Considering this ratio and the estimated mass of the complex allows

the derivation of the complex stoichiometry.

Finally, scomplex may be estimated by hydrodynamic considerations. For example, given the

mass of the postulated complex, a hypothesized frictional ratio, and information on the

solution properties, a theoretical scomplex may be calculated. A convenient calculator for this

purpose is included in SEDFIT. Also, if a crystal structure of the complex is available,

scomplex may be calculated from that information. Programs available for such calculations

are HYDROPRO [37], SOMO [38], and BEST [39].

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3.4 An LEq analysis for a self-associating protein

Below is presented an LEq analysis of data obtained on the I95E mutant of the PAS-A

domain of the Bacillus subtilis protein KinA (hereafter, this protein is referred to as I95E).

The wild-type form of this protein was known to be a tightly associated dimer from

crystallographic and SV studies [24]. The I95E mutation was introduced by Lee et al. in an

attempt to test predictions based on the crystal structure, i.e. to disrupt the KinA PAS-A

dimer. SV data sets at three concentrations were obtained in order to study the self-

association of the protein. These concentrations were approximately 9 µM, 36 µM, and 81

µM (the concentrations are given on a monomer basis, i.e. these are the concentrations of the

protein if it were all monomeric). The 36 µM sample exhibited a contaminant, and was thus

excluded from the LEq analysis (not shown). The 9 and 81 µM samples were analyzed using

the c(s) distribution, and the results are shown in Fig. 3. It was clear that a c(s) distribution

used to analyze these data displayed a single prominent peak at 1.7 S at low concentration,

but was resolved into two peaks (1.4 S and 2.0 S) at the higher concentration1,2. Thus, the

signal-average s-value of these distributions increased from 1.76 S to 1.90 S. It was

hypothesized that the I95E mutation had substantially lowered the KA of the dimerization.

For various reasons, no other SV data were collected on the system. Thus, the question at

hand is: can the two SV data sets provide enough information to reliably determine the KA

of dimerization? With only two data points, it was suspected that SV isotherm analysis

would yield inferior results. Thus, LEq was attempted, as detailed below. A step-by-step

protocol detailing this analysis is provided as “Supplemental Protocol 1.”

3.4.1 The analysis—First, the data for the global analysis were loaded into SEDPHAT.

For this analysis, absorbance data collected at 280 nm were considered. For both the 9 µM

and 81 µM samples, every third scan from scans from 4–151 (inclusive) was loaded. This

span of scans represents approximately 10 hours in the course of the sedimentation. The

experimental parameters that were used in both cases are displayed in Table 1. Importantly,

the calculated extinction coefficient of I95E at 280 nm

at this step, and was held constant for both data sets. Good initial estimates for the meniscus

and the bottom were chosen graphically, and the data-fitting limits were chosen to avoid

back-diffusion and the optical artifacts near to the meniscus.

was input

The SEDPHAT model “Monomer-Dimer Self Association” was chosen for this system. The

global parameters, i.e. those that are common to both data sets, were entered as shown in

Table 2. The refined parameters were: sedimentation coefficient of the monomer (S1),

sedimentation coefficient of the dimer (S2), the log of the association constant of the

interaction (log(Ka)), and the log of the koff of the interaction (log(k−)).

Because a successful and efficient LEq analysis is dependent on good initial guesses for

these parameters, it is worthwhile to examine the bases of the choices made here. The molar

mass of the protein (13,001 g/mol) was known, and it was therefore not necessary to refine

it. All of the values mentioned below were allowed to freely refine. For the sedimentation

coefficient of the monomer, 1.4 S was chosen, which is the signal-weighted average of the

peak of the slower (minority) species shown in Fig. 3. Because the s-value of the dimeric

1These s-values are slightly different that those reported by Lee et al. [23]. This is likely a consequence of more accurate buffer

parameters being used here.

2The figure shows that there is a small amount of contaminant in the low-concentration sample (at 2.9 S). This contaminant accounts

for about 0.001 absorbance units, or about 1% of the total sedimenting material. It was unclear to this author whether this signal was

an artifact of using c(s) to analyze an interacting system or it was really present. Thus, as described in section 3.4.1, the analysis

proceeded by ignoring it. Once the parameters for the interaction were derived, the system was simulated, and it was concluded that

this contaminant was probably actually present. Therefore, the analysis of 3.4.1 was repeated and the contaminant was explicitly taken

into account (not shown). Its presence did not alter the refined values of the parameters.

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wild-type KinA is known to be about 2.2 S, that was input as the value of the dimeric I95E

as well. If the signal populations of the monomer and dimer may be taken as those of the

two peaks shown as the dotted line in Fig. 3, then it may be guessed that there is about 2X as

much dimer as monomer present on a molar basis (the extinction coefficient of the dimer is

twice that of the monomer). Therefore, at this concentration (81 µM), the Kd of the

association would be about 8 µM, which translates to a log(KA) of about 5.1, which was

input. Finally, we note that a single peak is observed in the c(s) distribution of the low-

concentration sample, whereas two peaks can be discerned in the high-concentration

experiment. It therefore appears the koff of the interaction may be close to the slow side of

the range that is discernible by SV. Thus, the value log(k−) was set to −4 (generally, values

between −3 and −4 are able to be refined in standard SV experiments). The concentrations

of the samples, of course, must be input; they were set to 9 µM and 81 µM for the low- and

high-concentration samples, respectively.

The linear parameters (in this case, the noise elements) for such a fitting session may be

optimized using “Global Run” in SEDPHAT. It is good practice to initiate a Global Run in

order to assess whether the starting guesses are good or not. In the current case, the fits to

the high-concentration absorbance data after a Global Run are shown in Fig. 4 (light lines).

Clearly, the chosen concentration of 81 µM is not high enough, because the amplitude of the

fitted absorbance is lower than that of the raw data (circles). Consequently, the initiated

value of the concentration of the I95E in the high-concentration sample was adjusted upward

to 85 µM. The Global Run performed after this adjustment is shown in Fig. 4A (bold lines).

The amplitude of the signal from this concentration of I95E conforms much more closely to

the raw data, and it was deemed an acceptable starting point. Because the fitted line was

close to the raw data, all of the other initial guesses were thought to be close enough to their

real values to allow for the convergence of the fit.

At this point, a Global Fit was initiated, using the Marquardt-Levenberg (ML) minimization

algorithm to search parameter space. In this way, the non-linear parameters (S1, S2, log(Ka),

log(k−), and the sample menisci) were optimized. A new fit was rapidly achieved. However,

in this fit, the meniscus of the low-concentration experiment refined to its upper limit.

Further, there was significant systematicity in the residuals of the low-concentration sample

(see Supplemental Protocol 1). These flaws are indicators of a poor fit (see section 3.4.2). It

was thus deemed that the meniscus might be unduly influenced by the starting parameters

used. To alleviate this effect, the meniscus of this sample was fixed at the value that it had

refined to in a c(s) analysis (not shown), and the Marquardt-Levenberg minimization

repeated. A fit with no apparent problems was rapidly achieved. Because it searches

parameter space in a fundamentally different manner, the Simplex minimization algorithm

was then chosen, and another Global Fit begun. At this point, the meniscus of the low

concentration was allowed to refine, and it did not exhibit any further pathologies. The

minimization algorithm was alternated between ML and Simplex until no further

improvement in the fitting statistics was observed. Thus was the fitting session ended; the

session was saved, and the values of the refined parameters noted. The final refined values

of the fitted parameters are displayed in Table 2. The Kd, which was the most desired

parameter to be gleaned from this exercise, was about 110 µM. Thus, although the current

analysis used slightly different buffer parameters than those previously reported, the result

was very similar [24]. The data and the final fits to them are depicted in Figs. 4B & 4C.

3.4.2 Assessing the quality of the fit—SEDPHAT’s goodness-of-fit statistic is the

global reduced . Therefore, the fitting session (above) was continued until this statistic

remained essentially unchanged in consecutive Global Fit optimizations. In this case, the

final was 0.3647286, and the local root-mean-square deviations (r.m.s.d.’s) were 0.005

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and 0.007 for the low- and high-concentration samples, respectively. For the instrument on

which these data were collected, the r.m.s.d. values were close to instrumental noise levels.

It should be noted that values of 1 are not necessarily expected in SEDPHAT, because

these values depend on the estimate of experimental noise supplied by the user (0.01 signal

units is the default for SV data). The residuals between the data and the fits should be non-

systematic, as the residuals of this fit are (Fig. 4B). Systematicity would likely indicate that

there are aspects of the data that are not well fit by the combination of the imposed model

and the refined parameters of that model.

The final refined parameters (Table 2, Figs. 4B & 4C), along with the goodness-of-fit

statistics, should be carefully scrutinized to assess the success of the fit. The refined values

of S2 and log(k−) conform well to the initial expectations. The refined value of S1 is greater

than the initial guess that was based on a c(s) analysis (Fig. 3). However, simulations of this

system demonstrated that the s-value of the slower material is consistently underreported in

c(s) analyses (not shown). It seems likely that the combination of the reaction kinetics with

the (here incorrect) assumption of non-interacting species is the cause of this phenomenon.

The same factors probably cause the apparent errors in the species populations that led to the

erroneously high initial estimate for log(Ka). Despite these defects in the starting guesses for

these two parameters, the fitting algorithms efficiently found the global parameters that best

fit the data.

Another criterion for the acceptability of the refined parameters is that they be physically

meaningful. In other words, physically impossible values of the refined parameters should

be rejected. In the current case, this criterion mainly applies to the refined s-values S1 and

S2. Their values should conform to the expectation that the frictional ratios for the

components cannot be less than 1.0. The refined values of S1 and S2 (Table 2) represent

frictional ratios of 1.14 and 1.37, respectively, therefore meeting the physicality criterion.

Finally, in addition to the global parameters, local parameters should also be examined. The

final refined concentrations of the protein were very close to the initial estimates, and the

sample menisci did not vary far from their initiated positions; both of these facts indicate a

stable analysis. By the criteria set out above, the I95E data are well modeled by the

Monomer-Dimer Self Association model, with a dissociation constant of 110 µM.

3.4.3 Contingencies—What can be done if one of the criteria in section 3.4.2 is not met

in the analysis? The answer to this question depends on the nature of the violation. Large

and/or systematic residuals most likely indicate that the analytical model and/or its

parameters do not account well for the features observed in the data. The choice of model or

its built-in assumptions should be examined for their suitability. Parameter values that refine

to improbable or impossible values should be discarded. Often, a good approach is to restart

the fitting session with the recalcitrant parameter fixed at a realistic value. After the other

parameters have been refined, the parameter in question can sometimes be successfully

refined thereafter (see section 3.4.1). This strategy holds for both global (e.g. KA) and local

(e.g. menisci, sample concentrations) parameters. If the ill-behaved parameter still refines to

a problematic value, then the data probably do not contain enough information to arrive at

an acceptable fit for this parameter. Additional data may be required, or the applicability of

the chosen model may be questioned.

3.4.4 Error intervals—A question that naturally arises from such an analysis is “how

precise are the refined parameters?” To assess precision, error intervals are required.

Because SEDPHAT uses as its goodness-of-fit criterion, the calculation of error intervals

lends itself well to F-statistics. SEDPHAT has an F-statistic calculator that can, for a given

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confidence level, report what the resulting

For example, as reported above, the

(1 σ), the value rises to a “critical” value of 0.366693. Following notation previously

would be at that level for the current analysis.

value is 0.3647286. For a confidence level of 0.683

established [29], this higher value of

is here called . If a parameter is moved away

from its best-fit value by an amount that causes

of the parameter is outside the 68.3% confidence interval. One can therefore probe

to rise above , then this “test” value

parameter space with test values of the parameter until

confidence intervals for all relevant parameters. In practice, the test value of the parameter is

fixed, and all other fitted parameters are allowed to refine. The magnitude of

and further adjustments are made to the fixed test value as necessary. This iterative

procedure results in the desired confidence interval of the probed parameter. It is repeated

for every parameter of interest. This methodology is called the “error surface projection”

method, and it has been introduced elsewhere [40,41].

is exceeded, constructing

is observed,

The 68.3% error intervals of the refined parameters in the I95E analysis are shown in Table

2. The error intervals for the values of S1 and S2 are small; they may only vary by several

hundredths of a Svedberg unit before deleteriously affecting the fit (according to the chosen

confidence level). The interval for log(Ka) is also tightly restrained; however, small

differences in log(Ka) space may lead to large differences in Kd, the most important of the

refined parameters. Here it is found that the 68.3% confidence interval for Kd is from about

90 µM to about 130 µM. Thus, the Kd of this interaction (~110 µM) is well established, but

probably not statistically distinguishable from Kd’s 20–25% different from the best-fit value.

Finally, it is noted that the error interval for log(k−) is very large and asymmetric (Table 2).

The error interval essentially spans the entire range of values distinguishable by SV. Given

that koff is the most poorly determined value in an SV analysis [5], this result is not

surprising. Therefore, the approximate order of magnitude of koff is determined from such an

analysis, but detailed information about this constant is still obscure.

It is notable that only about one third of the data available for this characterization was used

(see section 3.4.1). Conceivably, the inclusion of all of the data could reduce the size of the

error intervals. However, no reduction in the span of the error intervals was evident when

using all of the data available for the analysis presented above in preliminary trials.

In the above analysis, a Kd of 110 µM was refined for a system in which the highest

concentration explored was 87.3 µM. Thus, at the highest concentration used, less than half

of the signal detected was due to the dimer. Intuitively, it may seem improbable that

accurate parameters for the association could be garnered by using the experimental strategy

outlined above. However, the data basis of LEq is very large; over 39,000 data points were

fitted in section 3.4.1. Also, a significant number of “imposter” parameters were explored

during the error analysis stage; none gave a fit that was better than that summarized in Table

2. As long as the assumptions built into the model are not violated (a monomer-dimer

protein interaction with no hydrodynamic or thermodynamic non-ideality), it may safely be

concluded that those parameters accurately describe the data. Ideally, of course, it would be

better to have a larger number of data sets and a wider range of concentrations to more

accurately arrive at the parameters.

3.5 An analysis of a hetero-association

The analysis presented in section 3.4.1 was for a relatively simple homo-association. Below

is analyzed a more difficult problem: a hetero-association with a small amount of

contamination in the samples. The two proteins that interact are the VCA domain from

human Wiskott-Aldrich Syndrome Protein fused to glutathione S-transferase (GST-VCA)

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and the bovine Arp2/3 complex. These proteins have been shown to interact using SV [25],

and the deduced stoichiometry was 1:1 [29]. LEq was used on a single data set in [24] to

derive information on the equilibrium association constant (KA), the off rate (koff), and the

sedimentation coefficient of the 1:1 complex (scomplex). In the present analysis, three data

sets with different concentrations of the two proteins are included in a global analysis of the

hetero-interaction.

An important point about the experiments presented here is that the concentrations (in the

micromolar range) used are all well above Kd as measured elsewhere (~26 nM) [25]. Indeed,

the experiments presented here were not designed to measure Kd. Thus, the case given

below serves as a test of whether LEq can be used to estimate the Kd of a hetero-interaction

under such imperfect conditions.

Another imperfection in this analysis is inherent in the samples: the presence of

contaminants. Experience has demonstrated that even high-quality protein preparations have

contaminants that can be detected by c(s) analysis [33]. The c(s) distributions of the proteins

alone suggest that there are some contaminants and aggregates in the samples (not shown).

The contaminants comprise ~2% of the signal, but this level of contamination can

deleteriously affect the analysis. In the following, various strategies available in SEDPHAT

are utilized to address the contamination and arrive at an informative LEq fit to the data. A

step-by-step protocol detailing these analyses is supplied as supplemental information

(“Supplemental Protocol 2”).

3.5.1 The analysis of GST-VCA alone—First, it is essential to derive the sedimentation

behavior of the individual components, so that their properties may be introduced as fixed

parameters in the LEq analysis of the mixture. The LE analysis of GST-VCA is presented

here first. GST-VCA alone was sedimented at a concentration of about 4 µM. GST-VCA is

assumed to be a dimer having a molar mass of approximately 70,000 g/mol (the calculated

molecular weight is 70,214). A previous analysis [25] had established that the sedimentation

coefficient of this protein was about 3.8 S. Also, that analysis illuminated the presence of

minor contaminating species (not shown). In order to find the correct LE parameters and to

account for the contaminants, the “Hybrid Local Continuous Distribution and Global

Discrete Species” model was chosen for the analysis; GST-VCA would be evaluated as a

single species, and continuous distributions at lower and higher portions of s-space would

model the contaminants.

It is important to note that all of the Experimental Parameters for this analysis were input

correctly except for v̄GST-VCA. This constant was purposely set to the incorrect value of 0.73

cm3/g. The reason behind this expedient was to put all components (GST-VCA, Arp2/3, and

the GST-VCA/Arp2/3 complex) on the same v̄ scale. This is necessary because SEDPHAT

cannot consider three separate values of v̄, and the correct value for the complex is not

necessarily known. Therefore, all masses should be put on the same v̄ scale—any v̄ could

have been chosen, and one close to the v̄ of most proteins was selected in this case. The

refined molar masses of GST-VCA and Arp2/3 may have diverged significantly from their

true values as a result. However, this deviation is of no consequence, because it does not

affect the values of the desired parameters from the overall analysis, namely KA, koff, and

scomplex.

The result of the hybrid discrete/continuous analysis is shown in Fig. 5. The vast majority of

the sedimenting signal was accounted for by the single discrete species, i.e. dimeric GST-

VCA. The refined values of molar mass and sGST-VCA were 71,164 g/mol and 3.79 S,

respectively. The overall quality of the fit was excellent (r.m.s.d. = 0.004817). The mass and

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sedimentation coefficient were noted for the analysis of the mixture of GST-VCA and

Arp2/3.

3.5.2 The analysis of Arp2/3 alone—An analogous analysis to that described in 3.5.1

was performed for Arp 2/3 alone. A sample containing approximately 1.5 µM Arp2/3 was

centrifuged. The same model (“Hybrid Local Continuous Distribution and Global Discrete

Species”) was used to analyze these data as that in section 3.5.1. In this case, the starting s-

value was 9.0 S and the starting molar mass was set to 224,000 g/mol, which is close to the

calculated molecular weight of Arp2/3 (223,824). Again, the v̄Arp2/3 was set to the spurious

value of 0.73 cm3/g, for the reasons enumerated above. The result of this analysis is shown

in Fig. 6. The final refined values for MArp2/3 and sArp2/3 were 206,010 g/mol and 9.0 S,

respectively. Again, the quality of the fit was excellent (r.m.s.d. = 0.004408), and the large

majority of the material was accounted for by the Arp2/3 complex3.

3.5.3 The analysis of the mixture of GST-VCA and Arp2/3—Three different

mixtures of Arp2/3 and GST-VCA were evaluated to allow the determination of the desired

parameters. Table 3 shows the sample parameters used in this analysis, and Table 4 shows

the total concentrations of these components in the three samples. As noted above, the

correct buffer parameters were used for these samples, but the v̄’s were set to 0.73 cm3/g for

all of the data sets, for the reasons discussed above. Because the concentrations of the

components (Table 4) in each sample were to be treated as refined parameters, it was

necessary to provide accurate extinction coefficients for the components

. Respectively, these values, which were determined elsewhere [32],

were 244,420 M−1·cm−1 and 92,416 M−1·cm−1.

Because the stoichiometry of these two proteins was known to be 1:1 [32], the model “A +

B ↔ AB Hetero-Association” was chosen to analyze these data. Component A was defined

as GST-VCA and Component B as Arp2/3. The values obtained in sections 3.5.1 & 3.5.2 for

molar mass and sedimentation coefficient of these two components were input and fixed.

The concentrations of both of the components in the samples were allowed to refine freely.

Previous experimentation [32] had established that the sedimentation coefficient of the AB

complex (“sAB” in the program's notation) was about 10.3 S, so this value was input. sAB

was allowed to refine because it was not known if this value was obtained under conditions

that allowed for the 100% saturation of the AB complex. The initial value for log(Ka) was

set to 7; a preceding study had indicated that the association between the two proteins was

on this order [25]. Finally, the koff was unknown. It was believed that it could be on the slow

side of discernible by SV, so the value of log(k−) was set to −4.

The problem of how to account for the minor contaminants present in the samples was then

considered. PBM could be utilized if there were enough resolution between the

contaminants and the components in r-space, but such resolution was impossible for

contaminants between 3.5 and 10 S, and not good for the contaminants with low

sedimentation coefficients or those with s-values between 10 and 15 S. SEDPHAT is

capable of modeling one “non-participating species” in LEq analyses. It was therefore

decided to use this method to analyze the most egregious contaminant outside the s-space of

the interaction: the 14–15 S species present in the Arp2/3 sample. SEDPHAT requires that

the mass, s-value, v̄, and local (i.e. sample-specific) signal concentrations be given for such a

species. Respectively, these values were set to 385,000 g/mol, 14 S, 0.73 cm3/g, 0.01 AU,

3It is justifiable to ask whether the continuous segments were necessary, given the small quantity of material (signal) detected in them.

A statistical analysis along the lines of that presented in 3.4.3 was carried out to test whether the segments improved the fit. The

amount of material present in them proved to be highly statistically significant. That is, the quality of the fit without the segments was

more than 2 σ worse than that with the segments.

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0.01 AU, and 0.01 AU. The mass value was based on an estimated frictional ratio of 1.36 for

that species. Except for the local concentrations, none of these values were allowed to refine

in the initial analysis.

After checking that the initial guesses were appropriate (see section 3.4.1), the Marquardt-

Levenberg minimization algorithm was used to arrive at a Global Fit of the model

parameters to the data. At this point, the quality of the fit was scrutinized as in section 3.4.2.

No pathologies were observed. The s-value of the non-participating species was allowed to

refine thenceforth. The mass of that species was not allowed to refine. Anecdotal evidence

suggested that the mass parameter for such a minority species is ill-defined by the data, and

refinement might allow that parameter to assume unphysical values.

Subsequent to further fitting utilizing both the Marquardt-Levenberg and Simplex

algorithms, a final fit was achieved. The final values for the fitted parameters are shown in

Table 4, and a graphical view of the fit to the data is shown in Fig. 7. After again examining

the results as in section 3.4.2, no problems with the fitted parameters were evident. The

refined value for Kd (i.e. log(Ka) transformed to Kd) was 74 nM, not far from the expected

value. Thus, despite the high concentrations of the components relative to the Kd, LEq

analysis allowed for a good estimate of the value. The 68.3% confidence interval for Kd

ranges from 60 to 91 nM, demonstrating that this property of the system is well determined

by the data. It is noteworthy that this parameter was, within error, the same as that

previously obtained from a single data set (84 nM; [25]). The refined s-value of the AB

complex, 10.42 S, also conformed to expectations and had a small error interval (Table 4).

This fit has a more tightly constrained value of koff than those described in section 3.4 for the

I95E system, but the 68.3% error interval still ranged over about an order of magnitude. All

indications are that the refined parameters adequately model the SV data for this system.

3.5.4 Multiple signals—Like SE [11,42–45], LEq hetero-analyses can benefit from the

presence of data acquired from different signals. The signals could be multiple light

wavelengths, or a light wavelength(s) and the interferometric concentration profiles

available from the Beckman centrifuge’s Rayleigh interferometer. For two interacting

molecules (A and B) and two signals (1 and 2), this principle might simplistically be viewed

with molecular signal increment ratios (e.g. the ratio of the molar signal increment of

molecule A at signal 1 to that molecule’s molar signal increment at signal 2). If the signal-

increment ratio of molecule A is significantly different from the signal-increment ratio of

molecule B, then the concentration of each component in the cosedimenting material can

more easily be calculated, facilitating a more accurate analysis.

4 Discussion

4.1 Recent successful implementations

Besides the two examples provided above, several recent articles have appeared using the

LEq approach as integrated into SEDPHAT. Schmeisser et al. [46] used this strategy to

analyze the self-association of interferon alpha 2c (IFN-α2c). They found that concentration-

dependent sedimentation characteristics of this protein could be described by a monomer-

dimer-tetramer model. Values for the equilibrium association constants for the monomer-

dimer and dimer-tetramer interactions were derived using LEq. This finding was significant,

because locally high concentrations of IFN-α2c at the cell membrane could induce the

formation of interferon-receptor oligomers [46].

Further, the laboratory of Cosgrove and coworkers have found LEq to be useful in the

dissection of interactions that occur in a complex of proteins that includes the mixed-lineage

leukemia protein-1 (MLL-1) [47,48]. This protein forms a complex with WD repeat

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protein-5 (WDR5), retinoblastoma-binding protein-5 (RbBP5), and absent small

homeotic-2-like protein (Ash2L). This core complex is required for methylation reactions

with histone H3 lysine 4 [49], and thus is important for transcriptional activation. Patel et al.

[48] first studied the interaction of a portion of MLL-1 with WDR5. Using LEq to determine

KA’s and kinetic off rates, they found that an arginine residue, R3765, was critical for the

interaction of the two proteins. In a follow-up study [47], they studied the pairwise

interactions of MLL-1 and WDR5, WDR5 and RbBP5, and RbBP5 and Ash2L. Equilibrium

association constants and koff’s were derived for all of these complexes using LEq. The

results led to a new model for the MLL-1 core complex, with significant implications for the

catalytic activity of this methyltransferase.

4.2 The utility of LEq

The results presented and reviewed herein demonstrate that LEq can successfully be used to

model SV data of interacting systems. Yet, compared to SV isotherm analysis [4,50], LEq

has some significant weaknesses: (1) The CPU time required for fitting the data ranges from

seconds to days, depending on clock speed of the CPU, number and size of the data sets, and

the type of interaction4. (2) One of the parameters that must be fitted in LEq analyses, koff, is

not particularly well determined by the data. (3) The presence of contaminants in the

macromolecular samples represents a significant challenge to the method. On the other

hand, deriving Kd and scomplex from isotherms based on the analysis of the data using the

c(s) distribution is several orders of magnitude faster, ignores koff, and can easily tolerate

contaminants outside the s-range of interest.

What, then, is the utility of LEq? A naïve examination of the GST-VCA/Arp2/3 data is

illustrative. The c(s) distributions of the three data sets used in section 3.5 show no

definition between the free B and AB species (not shown). That is, although there must be

three species present (free GST-VCA, free Arp2/3, and the complex of the two), there are

apparently only one5 or two boundaries. Thus, approaching the problem with no other

knowledge, the system would be judged to be appropriate for a Gilbert-Jenkins isotherm

analysis [4], i.e. the reaction kinetics must be fast on the time-scale of sedimentation. Three

types of isotherm can be fitted in a Gilbert-Jenkins analysis: the weighted-average s

isotherm of the entire distribution, the weighted-average s isotherm of the apparent reaction

boundary, and Gilbert-Jenkins population isotherms [4]. Treating the data thus for the GST-

VCA/Arp2/3 data sets allows the accurate determination of log(Ka), and sAB; they are 7.2

and 10.42, respectively (not shown). These values agree well with those obtained from LEq

(Table 4). However, examination of the error intervals of the Gilbert-Jenkins analysis shows

that the 68.3% error interval is larger for log(Ka) ([7.1, 7.4]) and sAB ([10.36, 10.49])6 (cf.

Table 4). Thus, the isotherm-based analysis led to essentially correct but less precise

parameter determinations, and all information regarding koff was necessarily discarded. In

this case, then, the utility of LEq over SV isotherm analysis is that the former can yield more

precise parameter estimates and information about koff that was not readily available from

the latter.

Another good example of the utility of LEq is when only one or a few data sets are at hand.

Obviously, it is good practice to explore a wide range of component concentrations when

4Importantly, there is a means in SEDPHAT to ameliorate the time-of-computation problem to some extent. The sampling of data

points in the SV concentration profiles can be modified such that fewer data points are considered in the analysis. This methodology

can be very useful, especially when multiple sets of interferometric data are globally analyzed.

5See Fig. 7C; there is apparently only one boundary. The c(s) analysis of these data (not shown) indicate a very small signal

concentration of free GST-VCA. Systems of three components exhibiting only a single boundary are described by Gilbert-Jenkins

theory and also by Effective Particle Theory (see reference [35]).

6Actually, given the erroneous assumption of instantaneous kinetics and the limited amount of data, it is remarkable that the Gilbert-

Jenkins analysis performs very well and has error intervals that are only slightly larger than those observed in the LEq analysis.

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studying macromolecular interactions. However, this strategy sometimes is impossible or

impractical. Both of the examples shown in sections 3.4 and 3.5 had only two or three data

sets available for analysis. In the latter, SV isotherm analysis could be reasonably carried out

because the three types of SV isotherms available to analyze resulted in a total of twelve

data points for the global fitting session. However, only one type of SV isotherm can be

derived from the I95E data (section 3.4): the sw isotherm. Fitting log(Ka), S(1), and S(2) to

such an isotherm would likely return unreliable results and unreasonably large error

intervals given the low number of degrees of freedom. By contrast, a single data set can

yield excellent estimates of log(Ka), S(1), S(2), and koff with meaningful error intervals

using LEq.

4.3 Conclusions

Herein, new approaches for the direct modeling of SV data using LEq have been described.

In the past, use of the methods put significant demands on sample purity, but the ability to

introduce non-participating species and to model only part of the boundary (PBM) hold the

promise of relaxing that strict requirement. The LEq algorithm as implemented in

SEDPHAT quickly and efficiently arrives at robust estimates of important parameters such

as KA, koff, and scomplex. LEq should be used in cases in which it is important to know an

approximate value for koff or when the number of data sets is limited. The results presented

in Section 3 as well as those garnered by others demonstrate that LEq has the potential to

tackle problems of biological import.

Supplementary Material

Refer to Web version on PubMed Central for supplementary material.

Abbreviations

Arp2/3

Actin related protein 2 – actin related protein 3 complex

AUC

Analytical ultracentrifugation

LE

Lamm equation

LEq

Lamm equation coupled to kinetic reaction fluxes q

MSSV

Multisignal sedimentation velocity

OD

Optical density

SE

Sedimentation equilibrium

SV

Sedimentation velocity analytical ultracentrifugation

ML

Marquardt-Levenberg

VCA

verprolin homology – central region – acidic region

PAS

PER-Arnt-Sim

PAS-A

The N-terminal PAS domain of KinA

EGTA

Ethylene glycol-bis(2-aminoethylether)-N,N,N',N'-tetraacetic acid

MLL-1

Mixed lineage leukemia protein-1

WDR5

WD repeat protein 5

RbBP5

Retinoblastoma-binding protein-5

Ash2L

Absent small homeotic-2-like protein

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r.m.s.d.

root-mean-square deviation

S(1)

the sedimentation coefficient of a monomer

S(2)

the sedimentation coefficient of a dimer

sAB

the sedimentation coefficient of the AB complex

TODA

time of data acquisition

Acknowledgments

The author thanks Drs. Sanjay Panchal, Michael Rosen, James Lee, and Kevin Gardner for providing the data sets

for the analyses in Section 3. Those data sets were collected with the support of a grant from the National Institutes

of Health (NIH) (R01-GM56322) to Dr. Michael Rosen. Also, support was provided to Dr. Kevin Gardner from the

NIH (R01-GM81875). The author extends his gratitude to Dr. Patrick H. Brown for a critical reading of the

manuscript and helpful comments thereon.

References

1. Harding, SE.; Chowdhry, BZ., editors. Protein-Ligand Interactions: Hydrodynamics and

Calorimetry. Oxford: Oxford University Press; 2001.

2. Harding, SE.; Chowdhry, BZ., editors. Protein-Ligand Interactions: Structure and Spectroscopy.

Oxford: Oxford University Press; 2001.

3. Brown, PH.; Balbo, A.; Schuck, P. Characterizing protein-protein interactions by sedimentation

velocity analytical ultracentrifugation, Current Protocols in Immunology. John Wiley & Sons; 2008.

p. 18.15.1-18.15.39.

4. Dam J, Schuck P. Sedimentation velocity analysis of heterogeneous protein-protein interactions:

sedimentation coefficient distributions c(s) and asymptotic boundary profiles from Gilbert-Jenkins

theory. Biophysical J. 2005; 89:651–666.

5. Dam J, Velikovsky CA, Mariuzza RA, Urbanke C, Schuck P. Sedimentation velocity analysis of

heterogeneous protein-protein interactions: Lamm equation modeling and sedimentation coefficient

distributions c(s). Biophysical J. 2005; 89:619–634.

6. Schuck P. On the analysis of protein self-association by sedimentation velocity analytical

ultracentrifugation. Anal. Biochem. 2003; 320:104–124. [PubMed: 12895474]

7. Schuck, P.; Braswell, EH. Measuring protein-protein interactions by equilibrium sedimentation. In:

Coligan, JE.; Kruisbeek, AM.; Margulies, DH.; Shevach, EM.; Strober, W., editors. Current

Protocols in Immunology. New York: Wiley; 2000. p. 18.8.1-18.8.22.

8. Scott, DJ.; Harding, SE.; Rowe, AJ., editors. Analytical Ultracentrifugation Techniques and

Methods. Norfolk, UK: RSC Publishing; 2005.

9. Vistica J, Dam J, Balbo A, Yikilmaz E, Mariuzza RA, Rouault TA, Schuck P. Sedimentation

equilibrium analysis of protein interactions with global implicit mass conservation constraints and

systematic noise decomposition. Anal. Biochem. 2004; 326:234–256. [PubMed: 15003564]

10. Ghirlando R. this volume.

11. Minton AP. Alternative strategies for the characterization of associations in multicomponent

solutions via measurement of sedimentation equilibrium. Prog. Colloid Polym. Sci. 1997; 107:11–

19.

12. Lamm O. Die Differentialgleichung der Ultrazentrifugierung. Ark. Mat. Astr. Fys. 1929; 21B:1–4.

13. Fujita, H. Foundations of Ultracentrifugal Analysis. New York: John Wiley & Sons; 1975.

14. Behlke J, Ristau Ol. Molecular mass determination by sedimentation velocity experiments and

direct fitting of the concentration profiles. Biophysical J. 1997; 72:428–434.

15. Demeler, B. UltraScan: a comprehensive data analysis software package for analytical

ultracentrifugation experiments. In: Scott, DJ.; Harding, SE.; Rowe, AJ., editors. Modern

Analytical Ultracentrifugation: Techniques and Methods. (UK): Royal Society of Chemistry;

2005. p. 210-229.

BrautigamPage 18

Methods. Author manuscript; available in PMC 2012 May 1.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 19

16. Schuck P, Perugini MA, Gonzales NR, Howlett GJ, Schubert D. Size-distribution analysis of

proteins by analytical ultracentrifugation: strategies and application to model systems. Biophysical

J. 2002; 82:1096–1111.

17. Stafford WF. Analysis of heterologous interacting systems by sedimentation velocity: curve fitting

algorithms for estimation of sedimentation coefficients, equilibrium and kinetic constants.

Biophys. Chem. 2004; 108:231–243. [PubMed: 15043932]

18. Schuck P. Size distribution analysis of macromolecules by sedimentation velocity

ultracentrifugation and Lamm equation modeling. Biophysical J. 2000; 78:1606–1619.

19. Schuck P. Sedimentation analysis of noninteracting and self-associating solutes using numerical

solutions to the Lamm Equation. Biophysical J. 1998; 75:1503–1512.

20. Brown PH, Schuck P. A new adaptive grid-size algorithm for the simulation of sedimentation

velocity profiles in analytical ultracentrifugation. Comput. Phys. Commun. 2008; 178:105–120.

[PubMed: 18196178]

21. Claverie J-M, Dreux H, Cohen R. Sedimentation of generalized systems of interacting particles. I.

Solution of systems of complete Lamm equations. Biopolymers. 1975; 14:1685–1700. [PubMed:

1156660]

22. Brown PH, Balbo A, Schuck P. On the analysis of sedimentation velocity in the study of protein

complexes. Eur. Biophys. J. 2009; 38:1079–1099. [PubMed: 19644686]

23. Schuck P, Demeler B. Direct sedimentation analysis of interference optical data in analytical

ultracentrifugation. Biophysical J. 1999; 76:2288–2296.

24. Lee J, Tomchick DR, Brautigam CA, Machius M, Kort R, Hellingwerf HJ, Gardner KH. Changes

at the KinA PAS-A dimerization interface influence histidine kinase function. Biochemistry. 2008;

47:4051–4064. [PubMed: 18324779]

25. Padrick SB, Cheng H-C, Ismail AM, Panchal SC, Doolittle LK, Kim S, Skehan BM, Umetani J,

Brautigam CA, Leong JM, Rosen MK. Hierarchical regulation of WASP/WAVE proteins. Mol.

Cell. 2008; 32:426–438. [PubMed: 18995840]

26. Laue, TM.; Shah, BD.; Ridgeway, RM.; Pelletier, SL. Computer-aided interpretation of analytical

sedimentation data for proteins. In: Harding, SE.; Rowe, AJ.; Horton, JC., editors. Analytical

Ultracentrifugation in Biochemistry and Polymer Science. Cambridge, UK: The Royal Society of

Chemistry; 1992. p. 90-125.

27. Pace CN, Vajdos F, Fee L, Grimsley G, Gray T. How to measure and predict the molar absorption

coefficient of a protein. Protein Sci. 1995; 4:2411–2423. [PubMed: 8563639]

28. Cole, JL.; Lary, JW.; Moody, TP.; Laue, TM. Analytical ultracentrifugation: sedimentation

velocity and sedimentation equilibrium. In: Correia, JJ.; Detrich, HWI., editors. Biophysical Tools

for Biologists. Volume One: In Vitro Techniques. Academic Press; 2008. p. 143-179.

29. Padrick SB, Deka RK, Chuang JL, Wynn RM, Chuang DT, Norgard MV, Rosen MK, Brautigam

CA. Determination of protein complex stoichiometry through multisignal sedimentation velocity

experiments. Anal. Biochem. 2010; 407:89–103. [PubMed: 20667444]

30. Solovyova A, Schuck P, Costenaro L, Ebel C. Non-ideality by sedimentation velocity of halophilic

malate dehydrogenase in complex solvents. Biophysical J. 2001; 81:1868–1880.

31. Balbo A, Minor KH, Velikovsky CA, Mariuzza RA, Peterson CB, Schuck P. Studying multiprotein

complexes by multisignal sedimentation velocity analytical ultracentrifugation. Proc. Natl. Acad.

Sci. (USA). 2005; 102:81–86. [PubMed: 15613487]

32. Padrick SB, Brautigam CA. this volume.

33. Brown PH, Balbo A, Schuck P. A bayesian approach for quantifying trace amount of antibody

aggregates by sedimentation velocity analytical ultracentrifugation. AAPS Journal. 2008; 10:418–

493.

34. Gilbert GA, Jenkins RCL. Boundary problems in the sedimentation and electrophoresis of complex

systems in rapid reversible equilibrium. Nature. 1956; 177:853–854. [PubMed: 13321982]

35. Schuck P. Sedimentation patterns of rapidly reversible protein interactions. Biophysical J. 2010;

98:2005–2013.

36. Brautigam CA, Wynn RM, Chuang JL, Chuang DT. Subunit and catalytic component

stoichiometries of an in vitro reconstituted human pyruvate dehydrogenase complex. J. Biol.

Chem. 2009; 284:13086–13098. [PubMed: 19240034]

Brautigam Page 19

Methods. Author manuscript; available in PMC 2012 May 1.

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Page 20

37. de la Torre JG, Huertas ML, Carrasco B. Calculation of hydrodynamic properties of globular

proteins from their atomic-level structures. Biophysical J. 2000; 78:719–730.

38. Rai N, Nöllmann M, Spotorno B, Tassara G, Byron O, Rocco M. SOMO (SOlution MOdeler):

Differences between X-Ray- and NMR-Derived Bead Models Suggest a Role for Side Chain

Flexibility in Protein Hydrodynamics. Structure. 2005; 13:723–734. [PubMed: 15893663]

39. Aragon SR. A precise boundary element method for macromolecular transport properties. J.

Comput. Chem. 2004; 25:1191–1205. [PubMed: 15116362]

40. Bevington, PR.; Robinson, DK. Data reduction and error analysis for the physical sciences.

Boston, MA: WCB/McGraw-Hill; 1992.

41. Houtman JCD, Brown PH, Bowden B, Yamaguchi H, Appella E, Samelson LE, Schuck P.

Studying multisite binary and ternary protein interactions by global analysis of isothermal titration

calorimetry data in SEDPHAT: application to adaptor protein complexes in cell signaling. Protein

Science. 2007; 16:30–42. [PubMed: 17192587]

42. Bailey MF, Davidson BE, Minton AP, Sawyer WH, Howlett GJ. The effect of self-association on

the interaction of the Escherichia coli regulatory protein TyrR with DNA. J. Mol. Biol. 1996;

263:671–684. [PubMed: 8947567]

43. Burgess BR, Schuck P, Garboczi DN. Dissection of merozoite surface protein 3, a representative of

a family of plasmodium falciparum surface proteins, reveals an oligomeric and highly elongated

molecule. J. Biol. Chem. 2005; 280:37236–37245. [PubMed: 16135515]

44. Ucci JW, Cole JL. Global analysis of non-specific protein-nucleic interactions by sedimentation

equilibrium. Biophys. Chem. 2004; 108:127–140. [PubMed: 15043926]

45. Yikilmaz E, Rouault TA, Schuck P. Self-association and ligand induced conformational changes of

iron regulatory proteins 1 and 2. Biochemistry. 2005; 44:8470–8478. [PubMed: 15938636]

46. Schmeisser H, Gorshkova I, Brown PH, Kontsek P, Schuck P, Zoon KC. Two interferons alpha

influence each other during their interaction with the extracellular domain of human type

interferon receptor subunit 2. Biochemistry. 2007; 46:14638–14649. [PubMed: 18027911]

47. Patel A, Dharmarajan V, Vought VE, Cosgrove MS. On the mechanism of multiple lysine

methylation by the human mixed lineage leukemia protein-1 (MLL-1) core complex. J. Biol.

Chem. 2009; 284:24242–24256. [PubMed: 19556245]

48. Patel A, Vought VE, Dharmarajan V, Cosgrove MS. A conserved arginine-containing motif crucial

for the assembly and enzymatic activity of the mixed lineage leukemia protein-1 core complex. J.

Biol. Chem. 2008; 283:32162–32175. [PubMed: 18829457]

49. Dou Y, Milne TA, Ruthenberg AJ, Lee S, Lee JW, Verdine G, Allis CD, Roeder RG. Regulation

of MLL1 H3K4 methyltransferase activity by its core components. Nat. Struct Molec. Biol. 2006;

13:713–719. [PubMed: 16878130]

50. Zhao H. This volume.

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Figure 1. Boundaries in sedimentation velocity experiments

(A) A typical sedimentation velocity concentration profile. The “top” of the centrifugation

cell is at the left of the figure, and the “bottom” is at the right. The boundary, plateau, and

solvent region are marked with a “b,” “p,” and “s,” respectively. Note the area of solute

buildup near to the bottom of the cell. (B) A concentration profile with two boundaries.

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Figure 2. The effect of concentration and off rate on c(s) distributions

An interacting system with a Kd of 1 µM was simulated. Parameters are given in Section

3.3. In all parts, the solid line represents the experiment with [A]tot = [B]tot = 2 µM. [B]tot

was held constant in all experiments. The dashed line shows the c(s) distribution from an

experiment with [A]tot = 5 µM, and the dotted line is the distribution from the experiment

with [A]tot = 20 µM. The kinetic off rates were (A) 10−6 s−1, (B) 10−4 s−1, and (C) 10−1 s−1.

The distributions were normalized by dividing all points by the total concentration of solute;

the extinction coefficients of A and B were identical.

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Figure 3. The c(s) distributions for 9 and 81 µM I95E

The distributions were normalized such that the highest c(s) value was 1.0 in both

distributions. The solid line is for the 9 µM sample; the dotted line is for the 81 µM sample.

The dashed vertical gray lines show the s-values of the monomer (labeled “sm, wt”) and the

dimer (labeled “sd, wt”) of wild-type KinA.

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Figure 4. The initial and final fits to the I95E samples

Conventions established in this figure hold for all figures in this paper depicting SV data.

(A) The effect of initial concentration on the fits. The circles represent the individual data

points, and the lines are the fit to the data points. Only every 3rd scan (concentration profile)

and every 3rd data point are shown. Time-invariant noise is not subtracted from the data in

this part. Here, the light lines show the initial fit with the input [I95E] = 81 µM. The heavy

lines show the corrected value of 85 µM. (B) The low-concentration data, fit, and residuals.

In the upper part, only every 6th scan is shown, because of the lower signal-to-noise ratio.

The lines show the final fit to the data. A plot of the residuals between the data and the fitted

line are shown in the lower part. In this part, and in all other figures depicting SV data in this

review, the time-invariant noise has been subtracted out. (C) The high-concentration data,

fit, and residuals.

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Figure 5. Hybrid discrete/continuous analyses for GST-VCA alone

(A) Data, fit, and residuals for the GST-VCA alone experiment. (B) The distribution and

discrete species used to fit the data in part (A). The distributions are shown as solid lines.

The discrete species is a bar; its height represents its refined signal concentration, and its x-

axis position represents its refined s-value.

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Figure 6. Hybrid discrete/continuous analyses for Arp2/3 alone

(A) Data, fit, and residuals for the Arp2/3 alone experiment. (B) Distribution and discrete

species used to fit the data in part (A). Conventions established in Figure 5B are followed.

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Figure 7. The final global fit to the GST-VCA + Arp2/3 mixture SV data

Data, fits, and residuals are shown for the final global fit. The refined parameter values are

given in Table 4.

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Table 1

Experimental Parameter Values for Both I95E Experiments

ParameterValue

vbar (cm3/g)

0.7443

density (g/cm3)

1.00177

viscosity (Poise) 0.009037

extinction coefficient A (M−1·cm−1)

11,460

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Table 2

Initial and Final Values of the Varied Global and Local Concentration Parameters for the I95E Analysis

Parameter Initial Value

Refined Value*

Monomer

s(1) (S) 1.41.67 [1.61, 1.72]

Dimer

log(Ka)5.1 3.96 [3.88, 4.05]

log(k−)

−4.0−3.7 [−4.1, −2.7]

s(2) (S) 2.2 2.20 [2.19, 2.25]

Low-Concentration Sample

[I95E] (µM) 9.0 9.266

High-Concentration Sample

[I95E] (µM) 85.087.364

*Where applicable, the limits of the 68.3% confidence interval are shown in brackets.

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Table 3

Experimental Parameter Values for All GST-VCA + Arp2/3 Experiments

Parameter Value

vbar (cm3/g)

0.73

density (g/cm3)

1.00079

viscosity (Poise) 0.010024

extinction coefficient A* (M−1·cm−1)

92416

extinction coefficient B* (M−1·cm−1)

244420

*Not applicable to the hybrid analyses.

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Table 4

Initial and Final Global Parameters and Local Concentration Parameters for the GST-VCA + Arp2/3 Data

Parameter Initial Value

Refined Value§

Component A*

Ma (g/mol)71,164N/A

sA (S)3.79 N/A

Component B*

Ma (g/mol)206,010 N/A

sB (S) 9.0N/A

Complex AB†

sAB (S) 10.3010.42 [10.37, 10.46]

log(Ka)7 7.13 [7.04, 7.22]

log(k−)

−4−4.1 [−4.8, −3.8]

non-participating species

M (g/mol) 385,000N/A

s‡ (S)

1414.9 [14.0, 16.2]

vbar (cm3/g)

0.73 N/A

Concentrations, Sample 1

[GST-VCA] (µM)3.1 3.004

[Arp2/3] (µM) 0.40.388

Concentrations, Sample 2

[GST-VCA] (µM)4.04.068

[Arp2/3] (µM) 0.50.474

Concentrations, Sample 3

[GST-VCA] (µM)1.00.805

[Arp2/3] (µM)1.2 1.101

*These values, obtained from earlier experiments, were fixed in this analysis.

†All values in this category were allowed to refine.

‡Only this parameter among those for the non-participating species was allowed to refine

§The 68.3% error intervals for these values are shown in brackets, where applicable.

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