## Determining Rate Parameters

Michaelis-Menten Parameters. In designing assays it is useful to know the maximum rate obtainable with a given amount of enzyme, Vmax, and the concentration of substrate that gives half that rate, KM. These are the fundamental parameters in the Michaelis-Menten (M-M) rate Figure 1. Nonlinearity of product formation with time, due to various factors. Initial rate for the theoretical reaction is 0.5. Fitting the nonlinear rates with a second degree polynomial, the calculated initial rates are: Substrate depletion, 0.51; Product inhibition, 0.51; Enzyme denaturation, 0.44. If the curves are fitted to a third degree polynomial, the calculated rates are 0.50,0.50, and 0.49, respectively. Source: Ref. 1, p. 93.

Figure 1. Nonlinearity of product formation with time, due to various factors. Initial rate for the theoretical reaction is 0.5. Fitting the nonlinear rates with a second degree polynomial, the calculated initial rates are: Substrate depletion, 0.51; Product inhibition, 0.51; Enzyme denaturation, 0.44. If the curves are fitted to a third degree polynomial, the calculated rates are 0.50,0.50, and 0.49, respectively. Source: Ref. 1, p. 93.

equation: v = Vmax[S]/(iCM + [S]>, where [S] is substrate concentration and v is the actual rate d[P]/dt. The usual procedure is to measure v at several concentrations [S] and then calculate Vmax and KM, either by applying a computer program (9) such as HYPER (Fig. 2a) or by fitting a straight line to one of the linear transforms of the M-M equation. The usual transform is the double reciprocal or Lineweaver-Burk plot: Uv = Wmax + (#M/VmaJ(l/[S]), where Uv is plotted versus 1/[S] (Fig. 2b). From statistical considerations, this is the least desirable transform to use (11).

A better plot is the Hanes plot of [S]/i> versus [S] (Fig. 2c): = KM/Vmax + (l/Vmax)[S]. The points are spaced along the x axis at the same intervals as in the M-M plot rather than being crowded together near the y axis. The larger experimental errors inherent in the smaller values of v (at low [S]) have less influence on the linear least squares regression line. The Hanes plot should be used in treating v, [S] data graphically; the use of the unsatisfactory Lineweaver-Burk plot should be discontinued.

The M-M equation is a differential equation, that is, v equals d[P]/dt at the instantaneous value of [¿>], usually taken as the initial substrate concentration. If v is only available as [P]/£ from a fixed-time assay, then the value taken for [S] for the above calculations should be the average of the substrate concentration at the beginning and end of the incubation period, ([S]0 + [Sj^/2. This approximation gives estimates of Vmax and KM that are much closer to the true values than if the initial value of [S] is used (12). 