The MichaelisMenten model of enzyme kinetics (Ratkowsky, 1990, p. 59) relates a substrate’s concentration to its catalyzed reaction rate. The MichaelisMenten 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 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.
Figure 10.1: Nonlinear Least Squares Analysis of Variance
Analysis of Variance  

Source  DF  Sum of Squares  Mean Square  F Value  Approx Pr > F 
Model  2  290116  145058  88537.2  <.0001 
Error  12  19.6606  1.6384  
Uncorrected Total  14  290135 
An intercept was not specified for this model. 
Finally, Figure 10.2 displays the parameter estimates, standard errors, t statistics, and 95% confidence intervals for and .
Figure 10.2: Parameter Estimates and Approximate 95% Confidence Intervals
Parameter Estimates  

Parameter  Estimate  Standard Error  DF  t Value  Approx Pr > t 
Approximate 95% Confidence Limits 

theta1  158.1  0.6737  1  234.67  <.0001  156.6  159.6 
theta2  0.0741  0.00313  1  23.69  <.0001  0.0673  0.0809 
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.