Traditionally, the kinetic behavior of an enzyme is characterized by monitoring rates of product formation at various substrate concentrations. However, it is also possible to determine the kinetic parameters of an enzymatic reaction by monitoring depletion of substrate as a function of time, for use in situations where experimental considerations make it impossible or impractical to monitor product formation. Substrate depletion is increasingly widely used to characterize cytochrome P450 kinetics (Obach and Reed-Hagen, 2002; Isoherranen et al., 2004; Jones and Houston, 2004; Komura and Iwaki, 2005; Mohutsky et al., 2006). Obach and Reed-Hagen (2002) proposed the following empirical equation to obtain the Michaelis constant, K_M: where k_dep is the apparent first-order rate constant of substrate depletion, [S] is the substrate concentration and k_dep([S]→0) is the theoretical k_dep at infinitesimally low substrate concentrations. Since this is an empirical relationship and not derived from first principles, the K_M values obtained from the substrate depletion approach are not necessarily equivalent to those obtained from the traditional product formation approach. Here, we present a proof of this formula from the Michaelis-Menten equation and, thereby, demonstrate that kinetic parameters obtained from the substrate depletion approach can be meaningfully compared with those obtained by monitoring product formation.

Scheme 1 shows the conversion of a substrate S to a product P by enzyme E, via an intermediate enzyme-substrate complex ES. The following rate equations apply to this system: Note that [S]_t, [E]_t, [ES]_t, and [P]_t refer to the transient concentrations of each species at time t, whereas unsubscripted symbols refer to their initial concentrations. When substrate depletion displays apparent first-order kinetics, substrate concentration as a function of time can be described by the following single exponential: Taking the first derivative: We now assume that the concentration of the ES complex is constant. Under this steady-state approximation, Close to the start of a reaction, when [S]_t ≈ [S], eq. 4 gives us: Note that this assumption, that transient substrate concentration is almost equal to the initial substrate concentration, corresponds to the assumption in the traditional approach that the rate of product formation is linear at the beginning of a reaction (Segel, 1975). This assumption means that only data from the beginning of each reaction (where [S]_t is close to its initial value) should be used in the calculation of k_dep. Substituting eq. 3 into the rate equations gives us: where K_M = (k_–1 + k_cat)/k₁ (Segel, 1975). At infinitesimally low substrate concentrations, the transient free enzyme concentration will be equal to the initial concentration of total enzyme: The Michaelis-Menten equation typically takes the form: This can be rearranged to give a form analogous to eq. 1: Substituting eqs. 5 and 7 into eq. 8 gives us Obach and Reed-Hagen's (2002) original equation (eq. 1): This demonstrates that the substrate depletion approach to determining reaction parameters is equivalent to the traditional approach of monitoring product formation: therefore, values for K_M and V_max obtained from the two different approaches can be meaningfully compared. To verify this equivalence, we analyzed a simulated data set using the substrate depletion approach and the traditional product formation approach, and compared the recovered kinetic parameters. Using a simulated data set instead of experimental data allows the rigorous comparison of these two approaches in the absence of the complexities inherent in a real enzymatic reaction.

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Scheme 1.

The system under consideration: the conversion of a substrate S to a product P by enzyme E, via an intermediate enzyme-substrate complex ES.

Materials and Methods

Simulations of the system described in Scheme 1 were performed using a script in the Python programming language [G. van Rossum (2006) Python Reference Manual available at http://www.python.org]. Transient concentrations of the various species were governed by the differential rate eqs. 2, with k₁ = 10⁵ M^–1 s^–1, k_–1 = 1 s^–1, k_cat = 0.1 s^–1, and an initial enzyme concentration of 100 nM. The resulting (calculated) K_M and V_max for this system are 11 μM and 10 nM s^–1, respectively. Six substrate concentrations, from 10 nM to 1 mM, were simulated, and each simulation covered 4000 s.

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Fig. 1.

a, transient substrate concentrations (gray lines) for six simulations at varying initial substrate concentrations, along with fits (black dashes) of single exponentials to the first 100 s of each simulation. b, k_dep values (recovered from the exponential fits) as a function of substrate concentration, fit to eq. 1. c, velocity (black circles, obtained from the substrate depletion approach; gray squares, obtained from the traditional product formation approach) as a function of substrate concentration, fit to the Michaelis-Menten equation.

Substrate Depletion Approach. The equation [S]_t/[S] = e^k_dept was fit to the first 100 s of the transient substrate concentration curves (see Fig. 1a) to obtain k_dep values for the various initial substrate concentrations. (The transient substrate concentration was within 10% of the initial concentration for all six simulations over this time period.) Equation 1 was fit to the k_dep values to obtain k_dep([S]→0) and K_M (see Fig. 1b). V_max was calculated using eq. 7.

As an alternate method of analysis, eq. 5 was used to calculate velocity values for the various substrate concentrations. These data were then fit to the Michaelis-Menten equation to obtain K_M and V_max directly (see Fig. 1c, black circles).

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Fig. 2.

Error in the recovered values for K_M (squares) and V_max (circles) using the substrate depletion approach, as a function of the maximum amount of substrate depleted in the curves used to obtain k_dep. As data from later time points (when more substrate has been consumed) are included in the curve-fitting, the recovered kinetic parameters deviate from their true values.

Product Formation Approach. The slope of the transient product concentration curves over the first 100 s of each simulation was used to obtain the reaction velocities at the various substrate concentrations. These values were fit to the Michaelis-Menten equation to obtain K_M and V_max directly (see Fig. 1c, gray squares).

Results and Discussion

Substrate depletion curves can be used to obtain first-order substrate depletion rate constants (k_dep) for various initial substrate concentrations (see Fig. 1a). The relation between these rate constants and the substrate concentration (eq. 1) depends on the Michaelis constant, K_M, and the theoretical rate constant at infinitesimally low substrate concentrations, k_dep([S]→0). Substrate depletion rate constants at a range of substrate concentrations (Table 1) were used to determine K_M and V_max for a simulated data set. When k_dep data were fit to eq. 1 (see Fig. 1b), the recovered values for K_M and V_max were 11.4 μM and 10.2 nM s^–1, respectively (compared with the true values of 11 μM and 10 nM s^–1). When reaction velocities were calculated from substrate depletion rate constants (eq. 5) and then fit to the Michaelis-Menten equation (see Fig. 1 c, black circles), the recovered values for K_M and V_max were 11.0 μM and 10.0 nM s^–1, respectively. In comparison, when reaction velocities were obtained from product formation curves and fit to the Michaelis-Menten equation (see Fig. 1c, gray squares), the recovered values for K_M and V_max were 11.4 μM and 10.0 nM s^–1, respectively. For this simulated system, both the substrate depletion approach and the traditional product formation approach recover predicted kinetic parameters with about the same degree of fidelity.

There is one important constraint that must be kept in mind when using the substrate depletion approach: only time points for which transient substrate concentration is relatively close to the initial substrate concentration should be used to determine k_dep. To reiterate, this constraint corresponds to the assumption in the traditional approach that the rate of product formation is linear at the beginning of a reaction. As substrate is depleted, the assumption that [S]_t ≈ [S] (in eq. 5) becomes less and less accurate, and the resulting error will affect the values of the recovered kinetic parameters. The deviation of a substrate depletion curve from a single exponential is especially evident when the initial substrate concentration is close to the K_M. If data from later time points (where much of the substrate has been consumed) are used for fitting, the obtained k_dep values will be erroneously high, which in turn results in erroneously high K_M and V_max values (Fig. 2). As a rule of thumb, k_dep values should only be calculated from time points where no more than 10% of the substrate has been consumed to minimize this error; however, recovered kinetic parameters are only about 15% above their true values, even when the data used to obtain k_dep values include points where 50% of the substrate has been consumed.

Conclusion

We have shown that Obach and Reed-Hagen's (2002) method of kinetic analysis by monitoring substrate depletion is equivalent to the traditional product formation approach. Kinetic parameters obtained by one approach can indeed be compared with those obtained by the other, as long as the assumptions inherent in each method (that transient substrate concentration is close to its initial value, for the substrate depletion approach; that the rate of product formation is linear, for the traditional approach) are not violated.