$$W_f = \partial S / \partial y = -K_m \times S / (K_m + S) \quad (7)$$
It is also concerned which parameter is suitable for estimation by kinetic analysis of reaction curve. In theory, all parameters of an enzyme reaction system can be simultaneously estimated by kinetic analysis of reaction curve. However, there is unknown covariance among some parameters to devalue their reliability; there is the limited accuracy of original data for analyses and the estimation of some parameters with narrow working ranges will have negligible practical roles. $V_m$ is independent of all other parameters and so is $S_0$, and the assay of $V_m$ and $S_0$ are already routinely practiced in biomedical analyses. Therefore, $V_m$ and $S_0$ may be the parameters suitable for estimation by kinetic analysis of reaction curve. Additionally, $K_m$ is used for screening enzyme mutants and enzyme inhibitors; but $K_m$ estimated by kinetic analysis of reaction curve usually exhibits lower reliability and is preferred to be fixed for estimating $V_m$ and $S_0$. If $K_m$ is estimated as well, $S_1$ should be at least 1.5-fold $K_m$ and there should be more than 85% consumption of the substrate in the data selected for analysis (Atkins & Nimmo, 1973; Liao, et al., 2005a; Newman, et al., 1974; Orsi & Tipton, 1979). To estimate $K_m$, the initial datum ($S_1$) and its corresponding ending datum from a reaction curve for analysis should be tried sequentially till the requirements for data range are met concurrently. In this case, the estimation of $S_1$ has no practical roles. In general, the resistance of $V_m$ and $S_0$ to reasonable changes in ranges of data for analyses can be a criterion to select the optimized set of parameters that are fixed as constants.
In comparison to the low reliability to estimate $K_m$ independently for screening enzyme inhibitors and enzyme mutants, the ratio of $V_m$ to $K_m$ as an index of enzyme activity can be estimated robustly by kinetic analysis of reaction curve. Reversible inhibitors of Michaelis-Menten enzyme include competitive, noncompetitive, uncompetitive and mixed ones (Bergmeyer, 1983; Dixon & Webb, 1979; Marangoni, 2003). The ratios of $V_m$ to $K_m$ will respond to concentrations of common inhibitors except uncompetitive ones that are very rare in nature. Thus, the ratio of $V_m$ to $K_m$ can be used for screening common inhibitors. More importantly, the ratio of $V_m$ to $K_m$ is an index of the intrinsic activity of an enzyme and the estimation of the ratios of $V_m$ to $K_m$ can also be a promising strategy to screen enzyme mutants of powerful catalytic capacity (Fresht, 1985; Liao, et al., 2001; Northrop, 1983).
For robust estimation of the ratio of $V_m$ to $K_m$ of an enzyme, $S_0$ can be preset at a value below 10% of $K_m$ to simplify Equ.(2) into Equ.(9). Steady-state data from a reaction curve can be analyzed after data transformation according to the left part in Equ.(9). For validating Equ.(9), it is proposed that $S_0$ should be below 1% of $K_m$ (Meyler-Almes & Auer, 2000). The use of extremely low $S_0$ requires special methods to monitor enzyme reaction curves and steady-state reaction can not always be achieved with enzymes of low intrinsic catalytic activities. On the other hand, the use of $S_0$ below 10% of $K_m$ is reasonable to estimate the ratio of $V_m$ to $K_m$ (Liao, et al., 2001). To estimate the ratio of $V_m$ to $K_m$, the use of Equ.(9) to analyze data is robust and resistant to variations of $S_0$ if Equ.(9) is valid; this property makes the estimation of the ratio of $V_m$ to $K_m$ for screening reversible inhibitors superior to the estimation of the half-inhibition concentrations (Cheng & Prusoff, 1973).