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 10.1. The table displays the degrees of freedom, sums of squares, and mean squares along with the model *F* test.

Finally, Figure 10.2 displays the parameter estimates, standard errors, *t* statistics, and 95% confidence intervals for and .

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.