Be Cautious When Using Linearized Forms of Nonlinear Equations

Some nonlinear relationships can be converted into a linear form, thus allowing you to use LINEST for curve fitting rather than applying the Solver. You should avoid this approach, because the curve fitting coefficients you obtain can be incorrect. An example will illustrate the problem.

In biochemistry, the reaction rate of an enzyme-catalyzed reaction of a substrate as a function of the concentration of the substrate is described by the Michaelis-Menten equation,

where V is the reaction velocity (typical units mmol/s), Km is the MichaelisMenten constant (typical units mM), Vmax is the maximum reaction velocity and [S] is the substrate concentration. Some typical results are shown in Figure 1410.

Figure 14-10. Michaelis-Menten enzyme kinetics.

The curve is calculated using equation 14-9 with Fmax =50, K,„ = 0.5.

Figure 14-10. Michaelis-Menten enzyme kinetics.

The curve is calculated using equation 14-9 with Fmax =50, K,„ = 0.5.

Before desktop computers were available, researchers transformed curved relationships into straight-line relationships, so they could analyze their data with linear regression, or by means of pencil, ruler and graph paper. The MichaelisMenten equation can be converted to a straight-line equation by taking the reciprocals of each side, as shown in equation 14-8.

This treatment is called a double-reciprocal or Lineweaver-Burk plot. A Lineweaver-Burk plot of the data in Figure 14-10 is shown in Figure 14-11.

The parameters Fmax and Km can be obtained from the slope and intercept of the straight line (Vmm = 1/intercept, Km = intercept/slope). However, the transformation process improperly weights data points during the analysis (very small values of V result in very large values of 1IV, for example) and leads to incorrect values for the parameters. In addition, relationships dealing with the propagation of error must be used to calculate the standard deviations of Fmax and Km from the standard deviations of slope and intercept.

Figure 14-11. Double-reciprocal plot of enzyme kinetics. The curve is calculated using equation 14-10 with Fmax = 50, K,„ = 0.5.

Figure 14-11. Double-reciprocal plot of enzyme kinetics. The curve is calculated using equation 14-10 with Fmax = 50, K,„ = 0.5.

By contrast, when the Solver is used the data do not need to be transformed, _ycalc is calculated directly from equation 14-7, the Solver returns the coefficients Vmax and Km, and SolvStat returns the standard deviations of Fmax and Km.

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