Key words

1 Biphasic Kinetics

Reactions that follow Michaelis–Menten kinetics have saturable velocities. Thus, at substrate concentrations much greater than the K m value, the velocity approximates zero-order kinetics, and further increases in substrate concentration do not yield significant increases in velocity. In contrast, for some reactions, saturation cannot be achieved, even at very high substrate concentrations. In these cases, at high substrate concentrations, the reaction rate will continue to increase linearly in proportion to substrate concentration. These kinetic profiles are considered to be biphasic and may be described by Eq. 1 [1]:

$$ v=\frac{\left({V}_{\max 1}\left[\mathrm{ S}\right]\left)+\right({\mathrm{ CL}}_{\operatorname{int}}{\left[\mathrm{ S}\right]}^2\right)}{\left({K}_{\mathrm{ m}1}+\left[\mathrm{ S}\right]\right)} $$
(1)

In Eq. 1, K m1 and V max1 represent the kinetic parameters for the saturable component of the reaction. It is assumed that the K m for the second component is sufficiently high that saturation cannot be achieved due to experimental limitations, such as solubility. Therefore, individual K m and V max parameters for this component cannot be determined. Instead, the slope of this portion of the curve is represented by CLint [2]. When there is no second component, CLint is equal to zero, and Eq. 1 reduces to the Michaelis–Menten equation. Thus, larger CLint values will result in larger and more obvious deviations from Michaelis–Menten kinetics. An example of biphasic kinetics is shown in Fig. 1. The two components contributing to the velocity of the reaction are more clearly visualized in the Eadie–Hofstee plot (Fig. 1, Panel b).

Fig. 1
figure 1

Biphasic kinetics. Panels a and b show the linear plot and the Eadie–Hofstee transformation for biphasic data, respectively

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Biphasic kinetics may be observed when a reaction is carried out by more than one enzyme with different K m and V max values, e.g., in studies conducted in heterogenous systems, such as microsomes, cytosol, and hepatocytes. Biphasic kinetics have also been observed for reactions performed using purified enzymes. In this case, it is likely that the enzyme has multiple binding sites for the substrate with different K m and V max values [3]. For example, the O-demethylation of naproxen, which is mediated via CYP2C9, displays biphasic kinetics, even when evaluated using expressed enzymes [4]. Thus, it is assumed that the biphasic kinetics arise from two substrate molecules binding within the active site in different conformations rather than from catalysis by two different enzymes [1].

2 Multienzyme Kinetics

When heterogeneous enzyme systems, such as microsomes, S9, or hepatocytes are used to determine enzyme kinetic parameters, it is possible that more than one enzyme may contribute to a specific biotransformation. The overall velocity of the product formation will be the sum of the contributions of the individual enzymes. Assuming that each reaction follows Michaelis–Menten kinetics, the overall velocity can be defined as in Eq. 2:

$$ v={\displaystyle \sum_{i=1}^n\frac{V_{\max, i}\cdot\left[\mathrm{ S}\right]}{K_{\mathrm{ m},i}+\left[\mathrm{ S}\right]}} $$
(2)

In Eq. 2, n enzymes contribute to the catalysis of the substrate, with specific K m and V max values for each enzyme. A graphical representation for a 2-enzyme reaction is shown in Fig. 2.

Fig. 2
figure 2

A two-enzyme system. Panels a and b show the linear plot and the Eadie–Hofstee transformation for a two-enzyme system, respectively

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The K m and/or V max values for the various enzymes need to be sufficiently different in order to be able to detect the multiple contributions to the reaction [5]. If the kinetic parameters are not sufficiently different, the data may appear to conform to Michaelis–Menten kinetics even if contributions from multiple enzymes are present. If a hint of multienzyme kinetics is observed, experiments should be performed using individual expressed enzymes to determine whether multiple enzymes contribute to the metabolism of the substrate and, if so, to determine the individual kinetic parameters.

For example, the conversion of phenacetin to acetaminophen is catalyzed by multiple cytochrome P450 (CYP) isoforms in human liver microsomes [6]. It has been shown that CYP1A2 is the major contributor to phenacetin O-deethylation and that CYP2A6, CYP2C9, CYP2C19, CYP2D6, and CYP2E1 contribute to lesser extents. The K m values for these lesser contributors are more than tenfold greater than those for CYP1A2 [7]. Hence, at an appropriate substrate concentration, phenacetin is considered to be a selective probe for CYP1A2 activity.

3 Homotropic Cooperativity

Enzymes that possess single binding sites, or multiple independent binding sites, generally display Michaelis–Menten kinetics. However, if the binding of one substrate molecule induces conformational changes in the enzyme that result in altered affinities at other sites, the enzyme is considered to demonstrate allosteric, or cooperative, interactions [8]. Homotropic cooperativity arises when multiple molecules of the same substrate can bind to the enzyme simultaneously. Homotropic cooperativity is also referred to as autoactivation. The simplest equation for describing homotropic cooperative enzyme kinetics is the Hill equation (Eq. 3):

$$ v=\frac{V_{\max }\cdot{\left[\mathrm{ S}\right]}^n}{S_{50}^n+{\left[\mathrm{ S}\right]}^n} $$
(3)

In Eq. 3, S 50 represents the substrate concentration at which 50 % of V max is achieved and is analogous to the K m parameter in the Michaelis–Menten equation [5]. The parameter, n, describes the degree of cooperativity. When n is equal to one, Eq. 3 reduces to the Michaelis–Menten equation. Values of n greater than one indicate positive cooperativity, while values of n less than one indicate negative cooperativity. Negative cooperativity means that the binding of the first molecule makes it more difficult for the binding and/or catalysis of the second molecule to occur. In Fig. 3, positive and negative cooperativity curves are compared with Michaelis–Menten data. Positive cooperativity manifests itself as a sigmoidal shape in the plot of velocity versus substrate concentration. The deviations from the Michaelis–Menten curve due to cooperativity are much more apparent in the Eadie–Hofstee transformation than in the direct plot.

Fig. 3
figure 3

Curves generated from the Hill equation (Eq. 3) with different values of n. Panels a and b show the linear plot and the Eadie–Hofstee transformation for the Hill equation, respectively

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The basic Hill equation (Eq. 3) can only be used to describe homotropic cooperativity and does not provide any mechanistic insights into the interactions between the substrate molecules [5]. More sophisticated models have been proposed in order to account for multi-site kinetics. If sufficient data points are available, Eq. 4 can be used to estimate the K m and V max values at the individual binding sites [3, 9]. In Eq. 4, K m1 and V max1 represent the kinetic parameters for the first binding site, and K m2 and V max2 are the kinetic parameters for the second binding site:

$$ v=\frac{\displaystyle\frac{\left({V}_{\max 1}\cdot\left[\mathrm{ S}\right]\right)}{K_{\mathrm{ m}1}}+\frac{V_{\max 2}\cdot{\left[\mathrm{ S}\right]}^2}{K_{\mathrm{ m}1}\cdot{K}_{\mathrm{ m}2}}}{1+\displaystyle\frac{\left[\mathrm{ S}\right]}{K_{\mathrm{ m}1}}+\frac{{\left[\mathrm{ S}\right]}^2}{K_{\mathrm{ m}1}\cdot{K}_{\mathrm{ m}2}}} $$
(4)

In Eq. 4, the interactions between the sites are not explicitly defined. As shown in Fig. 4, alterations in the binding affinities as well as changes to the catalytic rate due to binding of a second substrate molecule can be described by incorporating interaction terms (α and β) into the kinetic scheme [5].

Fig. 4
figure 4

Schematic representation of a kinetic model for an enzyme with two binding sites and homotropic cooperativity

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The steady-state equation that results from Fig. 4 is presented as Eq. 5. In this model, it is assumed that the two substrate-binding sites are equivalent [5]. The terms α and β reflect interaction factors for binding and catalysis, respectively. Values of α that are less than one indicate an increased binding affinity for the second substrate. An enhancement in the catalytic rate constant is described by a positive value of β. Changes in α or β in the opposite direction describe negative cooperativity [5]. The plots generated with Eq. 5 are generally similar in shape to the curves that result from Eq. 3:

$$ v=\frac{V_{\max}\left(\displaystyle\frac{\left[\mathrm{ S}\right]}{K_{\mathrm{ S}}}+\frac{\beta {\left[\mathrm{ S}\right]}^2}{\alpha {K}_{\mathrm{ s}}^2}\right)}{1+\displaystyle\frac{2\left[\mathrm{ S}\right]}{K_{\mathrm{ s}}}+\frac{{\left[\mathrm{ S}\right]}^2}{\alpha {K}_{\mathrm{ s}}^2}} $$
(5)

A number of reactions catalyzed by CYP450 isoforms have been reported to display homotropic cooperativity. Several CYP3A4-mediated reactions, such as amitriptyline N-demethylation, diazepam 3-hydroxylation, nifedipine oxidation, and testosterone 6β-hydroxylation, display autoactivation in vitro [10]. Sigmoidal kinetics have also been observed for testosterone 16β-hydroxylation by CYP2B6 [11]. Autoactivation has also been observed for reactions catalyzed by UGT enzymes. For example, the kinetic profiles of estradiol 3-glucuronidation and acetaminophen O-glucuronidation were best characterized by the Hill equation [10, 12]. Additional examples of atypical kinetics for UGT enzymes are presented in Chapter 11.

4 Heterotropic Cooperativity

The result of substrate and effector molecules binding to the enzyme and altering each other’s binding and/or kinetics is referred to as heterotropic cooperativity. The effector molecule may or may not be biotransformed by the enzyme. There are a variety of kinetic models for describing heterotropic cooperativity. Equation 6 represents one model for heterotropic cooperativity [1]. In this equation [B], α, and β represent the effector concentration, the change in the K m value due to the effector binding, and the change in the V max value due to effector binding, respectively:

$$ v=\frac{V_{\max}\cdot\left[\mathrm{ S}\right]}{K_{\mathrm{ m}}\displaystyle\frac{\left(1+\displaystyle\frac{\left[\mathrm{ B}\right]}{K_{\mathrm{ s}}}\right)}{\left(1+\displaystyle\frac{\beta \left[\mathrm{ B}\right]}{\alpha {K}_{\mathrm{ s}}}\right)}+\left[\mathrm{ S}\right]\frac{\left(\displaystyle\frac{1+\left[\mathrm{ B}\right]}{\alpha {K}_{\mathrm{ s}}}\right)}{\left(1+\displaystyle\frac{\beta \left[\mathrm{ B}\right]}{\alpha {K}_{\mathrm{ s}}}\right)}} $$
(6)

In an experiment to determine the kinetic parameters for a heterotropic cooperative system, concentrations of both the substrate and the effector are varied and the reaction velocity will depend on both of these parameters. Thus, a 3-dimensional plot is useful to visualize the velocity as a function of substrate and effector concentrations (Fig. 5). The velocities can also be plotted as a function of substrate concentration for each effector concentration on a 2-dimensional coordinate (Fig. 6). Each individual curve is hyperbolic in nature, and when the data are transformed via the Eadie–Hofstee approach, no curvature is noted, as no terms are squared in Eq. 6.

Fig. 5
figure 5

Three-dimensional plot for heterotropic cooperativity. In this example, V max = 100, K m = 2, K s = 5, α = 2, β = 10

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Fig. 6
figure 6

Heterotropic cooperativity at five effector concentrations. Panels a and b show the linear plot and the Eadie–Hofstee transformation for the heterotropic cooperativity equation, respectively. Parameter values are the same as in Fig. 5

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X-ray crystal structure data have indicated that CYP3A4 has a particularly large active site, which contains multiple substrate-binding sites [13]. As a result, many CYP3A4 substrates have been reported to display atypical kinetics. Sigmoidal kinetics have been observed for diazepam and testosterone. Co-incubation of these two CYP3A4 substrates resulted in activation of diazepam 3-hydroxylation. These results led to the hypothesis that testosterone may bind at a distinct effector site and alter the enzyme conformation, thereby increasing the velocity of diazepam metabolism [14]. α-naphthoflavone (also known as 7,8-benzoflavone) is considered to be a classical activator of CYP3A4. This substance is typically utilized as an inhibitor of CYP1A2 in CYP reaction phenotyping experiments. However, numerous studies have also demonstrated its ability to activate the metabolism of CYP3A4 substrates, such as progesterone, diazepam, and 17β-estradiol [10]. Due to the large number of CYP3A4 substrates for which cooperative kinetics have been observed, specific multi-site models have been derived for the analysis of enzyme kinetic data from this isoform [15, 16].

Heterotropic cooperativity has also been reported for CYP2C9. A structural analog of dapsone has been shown to activate the 4′-hydroxylation of flurbiprofen [17]. Additionally, a change in the kinetic profile for naproxen demethylation in the presence of p-tolylsulfone has been observed. In the absence of p-tolylsulfone, naproxen demethylation displayed a biphasic kinetic profile. However, at increasing p-tolylsulfone concentrations, activation of naproxen metabolism was observed and the kinetic curve appeared more hyperbolic or Michaelis–Menten-like [17].

5 Substrate Inhibition

Substrate inhibition is observed when the reaction velocity is reduced at high substrate concentrations. Substrate inhibition is thought to arise from the binding of more than one substrate molecule in the active site, with the binding to subsequent sites reducing the turnover from the first site [3]. The simplest model for describing substrate inhibition is shown in Eq. 7:

$$ v=\frac{V_{\max }}{1+\displaystyle\frac{K_{\mathrm{ m}}}{\left[\mathrm{ S}\right]}+\frac{\left[\mathrm{ S}\right]}{K_{\mathrm{ i}}}} $$
(7)

In Eq. 7, K i is the dissociation constant for the binding of the substrate to the inhibitory site. A curve generated from Eq. 7 is shown in Fig. 7. Equation 7 does not provide any mechanistic explanation for the observed alterations in enzyme velocity. To provide additional information from the kinetic data, a more complex model of substrate inhibition has been derived and is shown in Eq. 8 [18]:

Fig. 7
figure 7

Substrate inhibition kinetics. Panels a and b show the linear plot and the Eadie–Hofstee transformation for the substrate inhibition equation, respectively

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$$ v=\frac{V_{\max}\left(\displaystyle\frac{1}{K_{\mathrm{ s}}}+\frac{\beta \left[\mathrm{ S}\right]}{\alpha {K}_{\mathrm{ i}}{K}_{\mathrm{ s}}}\right)}{\displaystyle\frac{1}{\left[\mathrm{ S}\right]}+\frac{1}{K_{\mathrm{ s}}}+\frac{1}{K_{\mathrm{ i}}}+\frac{\left[\mathrm{ S}\right]}{\alpha {K}_{\mathrm{ i}}{K}_{\mathrm{ s}}}} $$
(8)

In Eq. 8, K s is approximately equal to K m. The terms α and β are the factors by which the dissociation (K S and K i) of substrate at both sites, and the maximal velocity (V max), respectively, change when a second substrate is bound.

The K m value is typically much smaller than the K i value, as the substrate binds preferentially to the productive site [19]. If sufficiently high concentrations of substrate are not utilized in an in vitro experiment, the substrate inhibition component of the kinetics can be overlooked. In order to accurately characterize the enzyme kinetic profile and determine the kinetic parameters, the upper concentration of substrate in the experiment should be as high as possible within the limitations of solubility. The kinetic profiles for substrate inhibition and product inhibition will look similar. If a significant amount of product is formed during the course of the in vitro experiment, the product concentration and its inhibition constant (K i) should be considered in the analysis of the kinetic parameters [3, 9].

Substrate inhibition kinetics have been observed for a number of reactions catalyzed by CYP450 isoforms. Biotransformations of ethoxyresorufin by CYP1A2, celecoxib by CYP2C9, dextromethorphan by CYP2D6, and progesterone, testosterone, and benzyloxyresorufin by CYP3A4 have all shown substrate inhibition kinetics [18]. Substrate inhibition kinetics have also been observed for reactions catalyzed by UGT enzymes [20] (see Chapter 11).

6 Atypical Kinetics: Practical Considerations

The kinetic models for atypical kinetics are inherently more complex than the Michaelis–Menten equation. These more sophisticated equations contain more parameters to fit via nonlinear regression analysis. Thus, it is essential to evaluate more data points than would be required for typical Michaelis–Menten kinetics. For the more complex, multi-site models, 20–30 substrate concentrations should be evaluated in order to adequately describe the kinetic profile [3]. The data points should be spaced in such a way as to gain maximal information about the kinetic profile. For example, in order to capture the sigmoidicity observed with the homotropic cooperativity models, a sufficient number of data points at low substrate concentrations should be assessed.

The typical first approach in data analysis is to graph the reaction velocities as a function of substrate concentration. In the direct plot, a deviation from the hyperbolic Michaelis–Menten equation may not be obvious. However, the various atypical kinetic behaviors can be discerned quite readily when the Eadie–Hofstee transformation is applied [10]. The kinetic curves and Eadie–Hofstee transformations for a variety of atypical kinetics profiles are shown in Figs. 1, 2, 3, 6, and 7.

While the Eadie–Hofstee transformation is useful from a diagnostic perspective, kinetic parameters for atypical profiles cannot be easily derived from linear transformations, and nonlinear regression analysis should be used. In general, a more complicated model, with more fitted parameters, will produce a lower sum of squares because the data variance can be distributed over the additional parameters [21]. However, a lower sum of squares for a more complex model does not necessarily indicate that this model fits the data better than a simpler model. As described in Chapter 2, statistical methods can be used to compare goodness of fit between models.

Some of the atypical models can be considered to be nested with the Michaelis–Menten equation. For example, the multienzyme kinetics equation (Eq. 2) reduces to the Michaelis–Menten equation when n = 1. On the other hand, the Michaelis–Menten equation is not a simpler form of the substrate inhibition equation (Eq. 7). Whereas the F-test is only applicable to nested models, the AIC approach (see Chapter 2) can be used to compare multiple models which may or may not be nested. When atypical kinetics are suspected, it is recommended that the complex models are compared to the Michaelis–Menten equation in order to avoid using complicated models unnecessarily. It should also be noted that in some instances, more than one model may provide a suitable fit to a data set, particularly in the case of more complex kinetics, such as heterotropic activation. However, a good fit to the data does not ensure an accurate description of the enzyme–substrate–effector interactions [10].

A number of factors have been reported to artifactually contribute to the appearance of atypical kinetics. Any process that alters the available concentration of substrate could result in deviations from a Michaelis–Menten profile. Some examples of artifactual sources of atypical kinetics include substantial substrate depletion, saturable nonproductive binding of the substrate (or metabolites) within the incubation matrix, poorly defined analytical quantitation limits, and aqueous solubility limitations [10, 22] (see Chapters 7 and 16, and Case Studies 1 and 2). The contribution of multiple enzymes to formation of a single metabolite can also complicate the kinetic profile and may result in the observation of atypical kinetic behaviors (see Chapters 6, 8, and 11, and Case Study 7). Thus, it is recommended to confirm atypical kinetics using single-enzyme sources [10]. Concentrations of salts, solvents, and the accessory protein, cytochrome b5, may also contribute to the observation of atypical kinetic profiles [1].

Some examples of atypical kinetics in preclinical species have been reported. For instance, in monkey liver microsomes, diclofenac metabolism was stimulated by quinidine. Correspondingly, in rhesus monkeys, the clearance of diclofenac was enhanced by the co-administration of quinidine [23]. There is little definitive evidence of atypical kinetics of diclofenac in human clinical trials. Felbamate has been shown to heteroactivate the metabolism of carbamazepine, resulting in increased carbamazepine clearance by approximately 25 % [1, 5, 24].

Within drug discovery and early development, the kinetic parameters, K m and V max, are often generated in order to predict in vivo clearance of a new chemical entity (see Chapter 13). Although there are not many clinical examples of atypical kinetics, it is important to fit the data to the correct kinetic model in order to obtain reliable estimates of K m and V max. These parameters can then be used to calculate the intrinsic clearance, CLint, which subsequently can be scaled up to in vivo clearance (see Chapter 13).

7 Conclusion

Most enzyme-catalyzed reactions display Michaelis–Menten kinetics. However, atypical kinetic profiles have been reported for a number of drug biotransformations. When the observed kinetic profile does not conform to a hyperbolic profile, it is important to carefully assess the data so that the most appropriate model is applied. It is also critical to reevaluate the experimental design to ensure that the observed profile is not an artifact.

8 Questions

  1. 1.

    Upon performing an experiment to determine K m and V max values for a new chemical entity, you discover that at very high substrate concentrations, the velocity decreases. From this data set, what conclusions can you draw? What additional data would you need to confirm your hypothesis?

  2. 2.

    What are the advantages and limitations in using the simple Hill equation to describe activation kinetics? In what situations may a more complex activation model be applied? How can you evaluate whether the more complicated model is justified?