Global Curve Fitting for Ka and Kd from Sedimentation Equilibrium Ultracentrifugation Data

 

Analytical ultracentrifugation involves the measurement of the radial concentration gradients of macromolecules created by the application of centrifugal force (1). In sedimentation equilibrium experiments, the concentration gradients are analyzed to determine molecular masses and equilibrium constants for reversibly associating complexes.

Global analysis of profiles measured at multiple absorbance wavelengths can be useful for samples containing two or more components with different absorption spectra. Dr. James L. Cole kindly provided sedimentation equilibrium data for the reaction between an enzyme and a nucleic acid activator at three activator concentrations and two absorption wavelengths (2). The following model was globally fit to these six data sets to determine the equilibrium constants for activator binding (Ka) and enzyme dimerization (Kd):

The dependent variable absorbance A(l,r) is measured as a function of cell radius r. The independent variable x = (r2 – r02)/2 is obtained from the cell radius r where r0 is an arbitrary reference radius. The six data sets consisted of (A, x) data pairs for the three activator concentrations and two absorbance wavelengths. The known or measured constants in this equation are the baseline offsets dl, the enzyme and activator extinction coefficients elE and elA and the enzyme and activator reduced molecular weights sE and sA.

The parameters to be determined from the curve fit are the two global parameters Ka and Kd and the six local parameters C0Ei and C0Ai where i corresponds to the three activator concentrations. See the enzyme kinetics article for more discussion of global and local parameters. The forms lnKa and lnKd were used to constrain Ka and Kd to be positive. Alternately the equation could be formulated without the logarithms and SigmaPlot constraints used to obtain the same result.

The single window dialog form of the SigmaPlot fit file for this global analysis is:

[Variables]

X = {col(1), col(3), col(5), col(7), col(9), col(11)} 'zero adjusted squared radius data

Y = {col(2), col(4), col(6), col(8), col(10), col(12)} ' absorbance data

[Parameters]

CE1 = 1e-8 ' initial estimate for local C0E for data 1B

CE2 = 2e-9 ' initial estimate for local C0E for data 1C

CE3 = 1e-9 ' initial estimate for local C0E for data 2B

CA1 = 8e-7 ' initial estimate for local C0A for data 1B

CA2 = 3e-6 ' initial estimate for local C0A for data 1C

CA3 = 1e-5 ' initial estimate for local C0A for data 2B

lnKa = -13 ' initial estimate for global lnKa

lnKd = -17 ' initial estimate for global lnKd

[Equations]

del1=-0.0387 ' baseline offset for 1B at 230 nm

del2=-0.19 ' baseline offset for 1C at 230 nm

del3=-0.0595 ' baseline offset for 2B at 230 nm

epsE230=729000 ' enzyme extinction coefficient at 230 nm

epsE260=80000 ' enzyme extinction coefficient at 260 nm

epsA230=16460 ' activator extinction coefficient at 230 nm

epsA260=55080 ' activator extinction coefficient at 260 nm

sigE=2.05 ' enzyme reduced molecular weight

sigA=0.48 ' activator reduced molecular weight

i1 = data(1, 1, size(col(1))) ' index values for 1B230

i2 = data(2, 2, size(col(3))) ' index values for 1C230

i3 = data(3, 3, size(col(5))) ' index values for 2B230

i4 = data(4, 4, size(col(7))) ' index values for 1B260

i5 = data(5, 5, size(col(9))) ' index values for 1C260

i6 = data(6, 6, size(col(11))) ' index values for 2B260

I={i1, i2, i3, i4, i5, i6} ' concatenated index values

CE={CE1, CE2, CE3, CE1, CE2, CE3}[I] ' vector of CEi parameters

CA={CA1, CA2, CA3, CA1, CA2, CA3}[I] ' vector of CAi parameters

del={del1, del2, del3, 0, 0, 0}[I] ' vector of baseline offsets

epsE={epsE230, epsE230, epsE230, epsE260, epsE260, epsE260}[I] ' E ext. coeff vector

epsA={epsA230, epsA230, epsA230, epsA260, epsA260, epsA260}[I] ' A ext. coeff vector

F1=del + epsE*CE*exp(sigE*X) + epsA*CA*exp(sigA*X)

F2=(epsE+epsA)*CE*CA*exp(-lnKa + (sigE+sigA)*X)

F=F1 + F2 + 2*(epsE+epsA)*(CE*CA)^2*exp(-lnKd - 2*lnKa +2*(sigE+sigA)*X)

fit F to Y

The six data sets are then concatenated in the [Variables] section to create X and Y. Parameter initial values are specified in the [Parameters] section. The constants are defined in the [Equations] section followed by the generation of the index variable I (see the enzyme kinetics article and the rabbit aorta concentration-response article for more on the index variable). Vector forms of local parameters and constants are then defined followed by the multiple-line form of the absorbance equation and the fit statement.

In the Regression Wizard select From Code for the data format to use the concatenated X and Y variables.

Performing the curve fit results in the following graphs for the two absorbance wavelengths where the fit lines are solid.

The fit of the model is excellent. The equilibrium parameters found are:

Ka = 1.55 µM

Kd = 16.5 nM

Download the files sedimentation_equilibrium_fit to obtain the SigmaPlot notebook with this example, the global curve fit equations and the fit line transform.

Cole, J.L., Carroll, S.S., Blue, E.S., Viscount, T. and Kuo, L.C., Activation of Rnase L by 2’,5’-Oligoadenylates, J. Biol. Chem. 272, 19187-19192 (1997)

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