Linearization of non-linear models

A classical non-linear model of chemical kinetics is defined by the Michaelis-Menten equation for rate-limited reactions, which has already been mentioned in Section 39.1.1  

The parameters of this model are the maximal rate of the process and the saturation constant The latter can also be defined as the amount of substrate which produces a rate V which is half the maximal rate as can be verified by substituting by X in the above relation. 

Usually, one plots the initial rate V against the initial amount X, which produces a hyperbolic curve, such as shown in Fig. 39.17a. The rate and amount at time 0 are larger than those at any later time. Hence, the effect of experimental error and of possible deviation from the proposed model are minimal when the initial values are used. The Michaelis-Menten equation can be linearized by taking reciprocals on both sides of eq. (39.114) (Section 8.2.13), which leads to the so-called Lineweaver-Burk form  

The parameters and can be obtained from the intercept and slope of the linear relationship between 1/V and 1/X as shown in Fig. 39.17b. 

Another linearized form of the Michaelis-Menten equation is defined by  

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Determination of confidence limits for non-linear models is much more complex. Linearization of non-linear models by Taylor expansion and application of linear theory to the truncated series is usually utilized. The approximate measure of uncertainty in parameter estimates are the confidence limits as defined above for linear models. They are not rigorously valid but they provide some idea about reliability of estimates. The joint confidence region for non-linear models is exactly given by Eqn. (B-34). Contrary to ellipsoidal contours for linear models it is generally banana-shaped. 

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