Analytical methods for fitting integrated rate equations. A discontinuous assay

The integrated rate equation for reactions with stoichiometry A----P + Q is: e0t = -Cf . ln(1-delta P/A0) + C1 delta P + 1/2C2(delta P)2 where the coefficients C are linear or quadratic functions of the kinetic constants and the initial substrate and product concentrations. I have used the 21 progress curves described in the accompanying paper [Cox & Boeker (1987) Biochem. J. 245, 59-65] to develop computer-based analytical and statistical techniques for extracting kinetic constants by fitting this equation. The coefficients C were calculated by an unweighted non-linear regression: first approximations were obtained from a multiple regression of t on delta P and were refined by the Gauss-Newton method. The procedure converged in six iterations or less. The bias in the coefficients C was estimated by four methods and did not appear to be significant. The residuals in the progress curves appear to be normally distributed and do not correlate with the amount of product produced. Variances for Cf, C1 and C2 were estimated by four resampling procedures, which gave essentially identical results, and by matrix inversion, which came close to the others. The reliability of C2 can also be estimated by using an analysis-of-variance method that does not require resampling. The final kinetic constants were calculated by standard multiple regression, weighting each coefficient according to its variance. The weighted residuals from this procedure were normally distributed.