one enzyme reaction system, but are just a few reports on enzyme-coupled reaction system (Atkins & Nimmo, 1973; Liao, et al., 2005; Duggleby, 1983, 1985, 1994; Walsh, 2010).
In theory, enzyme reactions may tolerate reversibility, the activation/inhibition by substrates/products, and even thermo-inactivation of enzyme. From a mathematic view, it is still feasible to estimate parameters of an enzyme reaction system by kinetic analysis of reaction curve if the roles of all those factors mentioned above are included in a kinetic model (Baywenton, 1986; Duggleby, 1983, 1994; Moruno-Davila, et al., 2001; Varon, et al., 1998). However, enzyme kinetics is usually so complex due to the effects of those mentioned factors that there are always some technical challenges for kinetic analysis of reaction curve. Hence, most methods for kinetic analysis of reaction curve are reported for enzymes whose actions suffer alterations by those mentioned factors as few as possible.
In practice, kinetic analysis of reaction curve usually employs nonlinear-least-square-fitting (NLSF) of the differential or integrated rate equation(s) to either the reaction curve per se or data set(s) transformed from the reaction curve (Cornish-Bowden, 1995; Duggleby, 1983, 1994; Orsi & Tipton, 1979). The use of NLSF rather than matrix inversion is due to the existence of multiple minima of the sum of residual squares with respect to some nonlinear parameters (Liao, et al., 2003a, 2007a). When a differential rate equation is used, numerical differentiation of data from the reaction curve has to be employed to derive instantaneous reaction rates. In this case, there must be intervals as short as possible to monitor reaction curves (Burden & Faires, 2001; Dagys, 1990; Hasinoff, 1985; Koerber & Fink, 1987). However, the instantaneous reaction rates from reaction curves inherently exhibit narrow distribution ranges and large errors; the strategy by numerical differentiation of data in a reaction curve is unfavourable for estimating $V_m$ and $S_0$ because of their low reliability and unsatisfactory working ranges. On the other hand, when an integrated rate equation of an enzyme reaction is used for kinetic analysis of reaction curve, there is no prerequisites of short intervals to record reaction curves so that automated analyses in parallel can be realized for enhanced performance with a large number of samples. As a result, integrated rate equations of enzymes are widely studied for kinetic analysis of reaction curve to estimate parameters of enzyme reaction systems (Duggleby, 1994; Liao, et al., 2003a, 2005a; Orsi & Tipton, 1979).
Due possibly to the limitation on computation resources, integrated rate equations of enzymes in such methods are usually rearranged into special forms to facilitate NLSF after data transformation (Atkins & Nimmo, 1973; Orsi & Tipton, 1979). In appearance, the uses of different forms of the same integrated rate equation for NLSF to data sets transformed from the same reaction curve can give the same parameters. However, kinetic analysis of reaction curve with rearranged forms of an integrated rate equation always gives parameters with uncertainty too large to have practical roles (Newman, et al., 1974). Therefore, proper forms of an integrated rate equation should be selected carefully for estimating parameters by kinetic analysis of reaction curve.
In the past ten years, our group studied chemometrics for kinetic analysis of reaction curve to estimate parameters of enzyme reaction systems; the following results were found. (a) In terms of reliability and performance for estimating parameters, the use of the integrated rate equations with the predictor variable of reaction time is superior to the use of the integrated rate equations with predictor variables other than reaction time (Liao, et al., 2005a); (b) the integration of kinetic analysis of reaction curve with other methods to quantify initial rates