The least-squares analysis of data from binding and enzyme kinetics studies weights, bias, and confidence intervals in usual and unusual situations.

The rectangular hyperbola, y=abx/(1+bx), is widely used as a fit model in the analysis of data from studies of binding, sorption, enzyme kinetics, and fluorescence quenching. The choice of this or its linearized versions--the double-reciprocal, y-reciprocal, or x-reciprocal--in unweighted least squares imply different assumptions about the error structure of the data. The rules of error propagation are reviewed and used to derive weighting expressions for application in weighted least squares, in the usual case where y is correctly considered the dependent variable, and in the less common situations where x is the true dependent variable, in violation of one of the fundamental premises of most least-squares methods. The latter case is handled through an effective variance treatment and through a least-squares method that treats any or all of the variables as uncertain. The weighting expressions for the linearized versions of the fit model are verified by computing the parameter standard errors for exactly fitting data. Consistent weightings yield identical standard errors in this exercise, as is demonstrated with a common data analysis program. The statistical properties of linear and nonlinear estimators of the parameters are examined with reference to the properties of reciprocals of normal variates. Monte Carlo simulations confirm that the least-squares methods yield negligible bias and trustworthy confidence limits for the parameters as long as their percent standard errors are less than approximately 10%. Correct weights being the key to optimal analysis in all cases, methods for estimating variance functions by least-squares analysis of replicate data are reviewed briefly.