Least Squares Model :: SAS/STAT(R) 13.1 User's Guide: High-Performance Procedures

Least Squares Model

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

$f(\mb {x}, \btheta ) = \frac{\theta _1 x_ i}{\theta _2 + x_ i}, \quad \mbox{for} \, \, i = 1, 2, \ldots , n$

where $x_ i$ represents the concentration for $n$ trials and $f(\mb {x}, \btheta )$ is the reaction rate. The vector $\btheta$ 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 $\theta _1$ and $\theta _2$ :

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

The HPNLMOD Procedure

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 $\theta _1$ and $\theta _2$ .

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.