The Michaelis-Menten model of enzyme kinetics (Ratkowsky 1990, p. 59) relates a substrate’s concentration to its catalyzed reaction rate. The Michaelis-Menten model can be analyzed using a least squares estimation because it does not specify how the reaction rate is distributed around its predicted value. The relationship between reaction rate and substrate concentration is

where represents the concentration for n trials and is the reaction rate. The vector contains the rate parameters.

For this model, which has experimental measurements of reaction rate and concentration stored in the `enzyme`

data set, the following SAS statements estimate the parameters and :

proc hpnlmod data=enzyme; parms theta1=0 theta2=0; model rate ~ residual(theta1*conc / (theta2 + conc)); run;

The least squares estimation performed by PROC HPNLMOD for this enzyme kinetics problem produces the analysis of variance
table that is displayed in Figure 11.1. The table displays the degrees of freedom, sums of squares, and mean squares along with the model *F* test.

Figure 11.1: Nonlinear Least Squares Analysis of Variance

Finally, Figure 11.2 displays the parameter estimates, standard errors, *t* statistics, and 95% confidence intervals for and .

Figure 11.2: Parameter Estimates and Approximate 95% Confidence Intervals

In the enzyme kinetics model, no information was supplied about the distribution of the reaction rate around the model’s mean value. Therefore, the residual model distribution was specified to perform a least squares parameter fit.