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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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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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