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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 â€" 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 mM
Kd = 16.5 nM
Download the self-extracting file sedimentation_equilibrium fit.exe and double click on it to obtain the SigmaPlot notebook with this example, the global curve fit equations and the fit line transform.
- http://www.abrf.org/JBT/1999/December99/dec99cole.html
- 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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