The intrinsic binding energies and cooperativity energy are covariant parameters. As such, during the nonlinear regression process changes in the weighted least squared sum of residuals that result from a change in one parameter can be offset by changes in the covariant parameter. This means that it is not possible to determine a single set of values for these parameters from a single experiment titrating the wild type DNA template. A plot of the variance of fit vs free energy values for each parameter as analyzed for a single experiment readily illustrates this problem.
As indicated in the plot, values for dG1, dG2 or dG12 were individually constrained to arbitrary numbers (x-axis), and the remaining parameters were allowed to vary so that the minimal variance of fit (y-axis) given the defined constraint could be determined. (This provides a cross-section of the error surface of the experiment. One is looking for unique minima on this error surface. See Beechem reference for further discussion.) As can be seen in the plot, the rate of change in the variance of fit defines upper bounds on dG1 and dG2, but no lower bounds for either of these parameters emerges from the analysis. Likewise, there is a lower bound for dG12, but there is no upper boundary that can be inferred from the data. The data is thus inconclusive: the system may contain site heterogeneity, and or cooperativity. This limitation of interpreting data from a single experiment is not simply a quirk of the system being studied, but is typical of any system containing covariant parameters such as energies of instrinisc binding and cooperative interactions.
Only by studying additional DNA templates with "reduced valency" mutations can the confusion be clarified. In such studies, an independent set of data with different constraints on acceptable values for dG1 or dG2 can be obtained. When such data is analyzed together with the data for the wild type DNA template, the respective constraints "talk" to one another, resulting in bounds being present on both extremes of the free energy values. This is illustrated below, where data for both the wild type DNA template and for a mutant DNA template bearing a 5 bp insertion between the two sites were simultaneously subject to the NONLIN regression analysis. Now there is enough information content in the data to resolve unique minima in the experimental error surface, as viewed in 2-D cross sections for each parameter.
