**Alternative Perspectives of Enzyme Kinetic Modeling**

Ryan Walsh *Chemistry Department, Carleton University, Ottawa Canada* 

#### **1. Introduction**

356 Medicinal Chemistry and Drug Design

Zufferey et al. 2003. Ether phospholipids and glycosylinositolphospholipids are not required

major, J Biol Chem 278: 44708–44718.

for amastigote virulence or for inhibition of macrophage activation by Leishmania

The basis of enzyme kinetic modelling was established during the early 1900's when the work of Leonor Michaelis and Maud Menten produced a pseudo-steady state equation linking enzymatic catalytic rate to substrate concentration (Michaelis & Menten, 1913). Building from the Michaelis-Menten equation, other equations used to describe the effects of modifiers of enzymatic activity were developed based on their effect on the catalytic parameters of the Michaelis-Menten equation. Initially, inhibitors affecting the substrate affinity were deemed competitive and inhibitors affecting the reaction rate were labelled non-competitive (McElroy 1947). These equations have persisted as the basis for inhibition studies and can be found in most basic textbooks dealing with the subject of enzyme inhibition. Here the functionality of the competitive and non-competitive equations are examined to support the development of a unified equation for enzymatic activity modulation. From this, a modular approach to pseudo-steady state enzyme kinetic equation building is examined. Finally the assumption that these equations, which stem from the Michaelis-Menten equation, are truly pseudo-steady state is also examined.

#### **2. Pseudo steady state enzyme kinetic**

#### **2.1 Michaelis-Menten kinetics**

Conventional views on how to handle enzyme kinetic data have remained essentially the same for nearly a century following the proposal of the Michaelis-Menten equation (1913; Equation 1).

$$\upsilon = V\_{\max} \frac{[\mathbb{S}]}{[\mathbb{S}] + K\_M} \tag{1}$$

The Michaelis-Menten equation was a large step forward in our ability to understand how biological systems control chemical processes. This equation linked the rate of enzymatic substrate catalysis to a mass action process relying on the fractional association between the substrate and the enzyme population. That is, the maximum conversion rate of substrate to product (Vmax) could be directly related to the concentration of the enzyme ([E]) present and the catalytic rate at which individual enzymes converted substrate molecules to product (kcat; Equation 2).

$$V\_{max} = k\_{cat} \text{[E]} \tag{2}$$

The second part of the equation describes the fractional association between the substrate and the enzyme population. Dependent on the Michaelis-Menten constant (KM; Equation 3), this part of the Michaelis-Menten equation partitions the binding of substrate to the enzyme population relative to the Michaelis-Menten constant.

$$\frac{[\text{S}]}{[\text{S}] + K\_M} \tag{3}$$

At substrate concentrations lower than the Michaelis-Menten constant, also known as the substrate affinity constant, less than half of the enzyme population would be expected to have substrate associated with it (Figure 1).

Fig. 1. Rectangular hyperbola plot of the Michaelis-Menten equation relating catalytic rate and substrate concentration.

At a concentration equal to the Michaelis-Menten constant, half of the enzyme population will have substrate associated with it. Therefore, the Michaelis-Menten constant itself is an inflection point. As substrate concentrations exceed the Michaelis-Menten constant, the fraction of the enzyme population interacting with substrate is pushed towards 100%. This term produces the characteristic rectangular hyperbolic profile associated with the Michaelis-Menten equation shown above.

#### **2.2 Linearization of the Michaelis-Menten equation**

The introduction of the reciprocal form of the Michaelis-Menten equation (Equation 4) in 1934 (Lineweaver & Burk) made the determination of the kinetic constants (KM, Vmax) of the Michaelis-Menten equation much simpler.

$$\frac{1}{\upsilon} = \frac{K\_M}{V\_{\max}} \frac{1}{[S]} + \frac{1}{V\_{\max}} \tag{4}$$

The second part of the equation describes the fractional association between the substrate and the enzyme population. Dependent on the Michaelis-Menten constant (KM; Equation 3), this part of the Michaelis-Menten equation partitions the binding of substrate to the enzyme

> [�] [�] + ��

At substrate concentrations lower than the Michaelis-Menten constant, also known as the substrate affinity constant, less than half of the enzyme population would be expected to

Fig. 1. Rectangular hyperbola plot of the Michaelis-Menten equation relating catalytic rate

**[Substrate]**

At a concentration equal to the Michaelis-Menten constant, half of the enzyme population will have substrate associated with it. Therefore, the Michaelis-Menten constant itself is an inflection point. As substrate concentrations exceed the Michaelis-Menten constant, the fraction of the enzyme population interacting with substrate is pushed towards 100%. This term produces the characteristic rectangular hyperbolic profile associated with the

The introduction of the reciprocal form of the Michaelis-Menten equation (Equation 4) in 1934 (Lineweaver & Burk) made the determination of the kinetic constants (KM, Vmax) of the

> 1 [�] + 1 ����

population relative to the Michaelis-Menten constant.

have substrate associated with it (Figure 1).

**Vmax**

**50% Vmax**

**Reaction rate (v)**

and substrate concentration.

Michaelis-Menten equation shown above.

Michaelis-Menten equation much simpler.

**2.2 Linearization of the Michaelis-Menten equation** 

**KM**

1 � <sup>=</sup> �� ����

���� = ����[�] (2)

(3)

(4)

The reciprocal form of the equation produced a straight line with intercept values on the Y axis of 1/Vmax and on the X axis of -1/KM (Figure 2). This advancement in analysis of the Michaelis-Menten equation allowed for a simplified way of analyzing the effect of compounds that altered the catalytic activity of enzyme systems. Changes in enzymatic activity were observed to result from changes in the substrate affinity or maximum velocity (Lineweaver & Burk 1934) resulting in the definition of inhibitory equations based on their effects on the kinetic constants of the Michaelis-Menten equation.

Fig. 2. Double reciprocal plot of the Michaelis-Menten equation indicating how the intercepts provide a simplified way of determining the kinetic constants of the equation.

#### **2.3 Modes of inhibition**

By far the most extensively documented form of interactions between modifiers of enzymatic activity and enzymes have been inhibitory, therefore, it is not surprising that the first mathematical models to be defined and accepted in the literature were the competitive (Equation 5) and non-competitive (Equation 6) modes of inhibition (Lineweaver & Burk 1934; McElroy 1947).

$$v = V\_{\max} \frac{[\text{S}]}{[\text{S}] + K\_M \left(1 + \frac{[I]}{K\_l}\right)}\tag{5}$$

$$\upsilon = V\_{\max} \frac{[\mathbb{S}]}{([\mathbb{S}] + K\_M) \left(1 + \frac{[I]}{K\_l}\right)} \tag{6}$$

The competitive inhibition (Equation 5) and non-competitive inhibition (Equation 6) equations model different inhibitory processes and are easily identified using Lineweaver-Burk double reciprocal plots (Figure 3). Competitive inhibition has been defined as a direct competition between the substrate and the inhibitor molecule for the active site of the enzyme. As inhibitor concentration is increased, the enzyme's substrate affinity is decreased. However, due to the competitive nature of this interaction, decreases in catalytic activity can always be overcome with sufficient increases in substrate concentration. In contrast, noncompetitive inhibition exclusively affects the catalytic velocity of the enzyme population. Shifts in the maximum velocity can be attributed to the inhibitor binding to the enzyme and shutting down its catalytic activity, such that the observed decrease in activity represents the percent of the enzyme population bound by inhibitor.

Fig. 3. Double reciprocal plots of a) competitive inhibition, where introduction of the inhibitor produces changes exclusively in the substrate affinity constant (KM) and b) noncompetitive inhibition, where inhibition is observed as a decrease in the maximum velocity of the enzyme catalyzed (Vmax) reaction.

Both of the competitive and non-competitive inhibition equations can be derived from rate and conservation of mass equations like the Michaelis-Menten equation. The derivation of the competitive (Equation 5) and non-competitive (Equation 6) inhibition equation also results in a similar inhibitory term (Equation 7).

$$\left(1 + \frac{[I]}{K\_l}\right) \tag{7}$$

This type of equation derivation, which segregates modes of inhibition based on inhibitory effect on kinetic constants of the Michaelis-Menten equation, has formed the basis for equation derivation in modern enzyme kinetics. However the use of the inhibitory term (Equation 7) found in the competitive (Equation 5) and non-competitive (Equation 6) inhibition equations may be regarded as an incomplete derivation that obscures the relationship between inhibitor binding and kinetic effect (Walsh et al., 2007).

competition between the substrate and the inhibitor molecule for the active site of the enzyme. As inhibitor concentration is increased, the enzyme's substrate affinity is decreased. However, due to the competitive nature of this interaction, decreases in catalytic activity can always be overcome with sufficient increases in substrate concentration. In contrast, noncompetitive inhibition exclusively affects the catalytic velocity of the enzyme population. Shifts in the maximum velocity can be attributed to the inhibitor binding to the enzyme and shutting down its catalytic activity, such that the observed decrease in activity represents

Fig. 3. Double reciprocal plots of a) competitive inhibition, where introduction of the inhibitor produces changes exclusively in the substrate affinity constant (KM) and b) noncompetitive inhibition, where inhibition is observed as a decrease in the maximum velocity

Increasing [I]

ΔVmax

1/v

Both of the competitive and non-competitive inhibition equations can be derived from rate and conservation of mass equations like the Michaelis-Menten equation. The derivation of the competitive (Equation 5) and non-competitive (Equation 6) inhibition equation also

1/[S] 1/[S]

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relationship between inhibitor binding and kinetic effect (Walsh et al., 2007).

This type of equation derivation, which segregates modes of inhibition based on inhibitory effect on kinetic constants of the Michaelis-Menten equation, has formed the basis for equation derivation in modern enzyme kinetics. However the use of the inhibitory term (Equation 7) found in the competitive (Equation 5) and non-competitive (Equation 6) inhibition equations may be regarded as an incomplete derivation that obscures the

� (7)

Increasing [I]

the percent of the enzyme population bound by inhibitor.

a) b)

of the enzyme catalyzed (Vmax) reaction.

ΔKM

1/v

results in a similar inhibitory term (Equation 7).

In the competitive inhibition equation (Equation 5), the inhibitory term, as written, directly affects the Michaelis-Menten constant. This might be expected as competitive inhibition exclusively alters substrate affinity. In the non-competitive equation (Equation 6), the inhibitory term is inversely related to the maximum velocity (Equation 8).

$$\frac{V\_{\text{max}}}{1 + \frac{[I]}{K\_l}}\tag{8}$$

A rearrangement of this term (Equations 9-12) demonstrates the similarities of the inhibitory term and the term relating the fractional association between substrate and enzyme population (Equation 3).

$$\frac{V\_{\text{max}}}{[I] + K\_l} \tag{9}$$

$$\frac{V\_{\text{max}}}{\frac{[I]}{[I]} + K\_{\ell}}\tag{10}$$

$$\frac{\frac{V\_{\text{max}}}{1}}{1 - \frac{[I]}{[I] + K\_l}}\tag{11}$$

$$V\_{\max} - V\_{\max} \frac{[I]}{[I] + K\_l} \tag{12}$$

Therefore the inhibitory term of the non-competitive inhibition equation directly equates shutting down of enzymatic activity with the fraction of the enzyme population bound by the inhibitor. This on off analog behaviour provides a simplistic way looking at enzymatic activity and limits the usefulness of this equation for describing anything other than complete inhibition of the enzyme upon inhibitor binding. However, the addition of a governor term (*Vmax-Vmax2*) changes the non-competitive term such that it can be used to account for changes in catalytic activity, other than complete inhibition (Equation 13; Walsh et al., 2007; Walsh et al., 2011a).

$$(V\_{max} - (V\_{max} - V\_{max2})\frac{[I]}{[I] + K\_l} \tag{13}$$

This rearrangement, or insertion of a governor term, allows for the description of inhibitory effects ranging from just greater than 0% to 100% and has the potential to describe activation as well if the secondary maximum velocity is greater than the initial velocity. It is convenient to classify compounds with the potential to activate as well as inhibit as modifiers, denote here as X (Equation 14).

$$(V\_{max} - (V\_{max} - V\_{max2})\frac{[X]}{[X] + K\_X} \tag{14}$$

Even without the addition of the governor term to the non-competitive equation, this rearrangement (Equation 9-12) accounts for the rectangular hyperbolic change in maximum velocity produced by the non-competitive inhibition equation (Figure 4). This change is identical to the mass action binding observed between the substrate and the enzyme population in the Michaelis-Menten equation (Figure 1).

Fig. 4. Rectangular hyperbolic changes in the maximum velocity produced by modifiers. Here the mass action binding between the enzyme population and modifier results in the characteristic shape of the curve but the change in activity depends on the change induced by single binding events between the enzyme and the modifier. For example the four lines represent stimulation (Vmax2 > Vmax), binding without catalytic effect (Vmax2 = Vmax), partial inhibition (Vmax2 = 0.5x Vmax) and complete inhibition (Vmax2 = 0) as would be observed with the classical non-competitive equation.

With the clear way non-competitive inhibition mimics the kinetics observed with the Michaelis-Menten equation, the manner in which competitive inhibitors affect enzyme activity becomes obscure. This can be demonstrated through the same rearrangement of the inhibitory term which directly affects the substrate affinity (Equations 15-19).

$$K\_M \left( 1 + \frac{[I]}{K\_l} \right) \tag{15}$$

$$K\_{\mathcal{M}}\left(\frac{[I]+K\_l}{K\_l}\right) \tag{16}$$

$$K\_{\mathcal{M}}\left(\frac{[I] + K\_l}{[I] + K\_l - [I]}\right) \tag{17}$$

Even without the addition of the governor term to the non-competitive equation, this rearrangement (Equation 9-12) accounts for the rectangular hyperbolic change in maximum velocity produced by the non-competitive inhibition equation (Figure 4). This change is identical to the mass action binding observed between the substrate and the enzyme

[�] � ��

Vmax2 > Vmax

Vmax2 = Vmax

Vmax2 = 0.5x Vmax

Vmax2 = 0

(14)

���� <sup>−</sup> ����� − ������ [�]

Fig. 4. Rectangular hyperbolic changes in the maximum velocity produced by modifiers. Here the mass action binding between the enzyme population and modifier results in the characteristic shape of the curve but the change in activity depends on the change induced by single binding events between the enzyme and the modifier. For example the four lines represent stimulation (Vmax2 > Vmax), binding without catalytic effect (Vmax2 = Vmax), partial inhibition (Vmax2 = 0.5x Vmax) and complete inhibition (Vmax2 = 0) as would be observed with

**[Modifier]**

With the clear way non-competitive inhibition mimics the kinetics observed with the Michaelis-Menten equation, the manner in which competitive inhibitors affect enzyme activity becomes obscure. This can be demonstrated through the same rearrangement of the

�� �� � [�]

�� �

�� � [�] � �� [�] � �� − [�]

��

[�] � �� ��

� (15)

� (16)

� (17)

inhibitory term which directly affects the substrate affinity (Equations 15-19).

population in the Michaelis-Menten equation (Figure 1).

**Observed Vmax**

the classical non-competitive equation.

$$K\_M \left( \frac{1}{1 - \frac{[I]}{[I] + K\_l}} \right) \tag{18}$$

$$\frac{K\_M}{1 - \frac{[I]}{[I] + K\_l}}\tag{19}$$

As can be seen in equation 19, the inhibitory term in the competitive inhibition equation (Equation 5) actually describes a situation where the substrate affinity term is divided by the percent of the enzyme population free of the competitive inhibitor. This implies that modifiers that affect the substrate affinity exclusively produce a linear increase in the substrate affinity with increasing inhibitor concentration (Figure 5). However, as substrate binding specificity and affinity result from three point binding (Ogston, 1948), changes in the ability of an enzyme to bind substrate are more likely to result from inhibitor interactions that shift the enzyme's ability to do this away from its native state. Such perturbations would follow the same mass action mode of interaction as observed with non-competitive inhibition (Equation 6). These changes in substrate affinity would be finite and the overall observable effect would result from individual interactions between inhibitor and enzyme which would shift the binding affinity form that of the native enzyme (KM) to an affinity produced by the inhibitor (KM2) (Equation 20; Walsh et al., 2007; Walsh et al., 2011a).

$$w = V\_{max}\frac{[S]}{[S] + K\_M - (K\_M - K\_{M2})\frac{[I]}{[I] + K\_l}}\tag{20}$$

While true competitive inhibition may exist, the criteria for identifying an inhibitor as truly competitive needs to include a linear shift in substrate affinity resulting from increase in inhibitor concentration (Figure 5d; Walsh et al., 2007; Walsh et al., 2011a). This should be examined with global data fitting to confirm the inhibitory effect on substrate affinity.

#### **3. Modular enzyme kinetic equation building**

#### **3.1 Unified modifier equation**

By recognizing that changes in the substrate affinity and maximum velocity result from stoichiometric interactions between the modifier and the enzyme and that the effects of the modifier can be regulated with a governor term, a single equation for describing these effects can be generated (Equation 21; Walsh et al., 2007).

$$\upsilon = \frac{[S]}{[S] + K\_{S1} - (K\_{S1} - K\_{S2})\left(\frac{[X]}{[X] + K\_{X1}}\right)} V\_{S1} - (V\_{S1} - V\_{S2})\left(\frac{[X]}{[X] + K\_{X1}}\right) \tag{21}$$

Here the maximum velocity term has been abbreviated as VS1, and the substrate affinity term KS1, for simplicity. Of note, the modifier binding constant (KX1) is the same in the term modifying the substrate affinity and the term modifying the maximum velocity. This is in contrast to the mixed non-competitive equation (Equation 22) which has been used to describe similar dual effects on substrate affinity and maximum velocity but requires two separate inhibitor binding constants to accommodate the effects of a single inhibitor.

$$v = V\_{max} \frac{[S]}{[S] \left(1 + \frac{[I]}{\alpha K\_l}\right) + K\_M \left(1 + \frac{[I]}{K\_l}\right)} \tag{22}$$

Fig. 5. Double reciprocal plots of the a) competitive inhibition equation (Equation 5), representing a continuous change in substrate affinity with increasing inhibitor concentration, b) Equation 20 representing a hyperbolic change from one substrate affinity to another as the inhibitor binds in a stoichiometric way with the enzyme, c) an overlay of the two plots and d) a plot of the shift in substrate affinity at different concentrations of the inhibitor.

$$w = \frac{V\_{\ $1}[\$ ]}{[\ $] + K\_{\$ 1}} \times \frac{1 + b[\ $]/K\_{\$ \ $1}}{1 + [\$ ]/K\_{\ $\$ 1}} \tag{23}$$

$$\upsilon = \frac{[\mathbb{S}]}{[\mathbb{S}] + K\_{\mathbb{S}1}} V\_{\mathbb{S}1} - \frac{[\mathbb{S}]}{[\mathbb{S}] + K\_{\mathbb{S}\mathbb{S}1}} V\_{\mathbb{S}1} + \frac{[\mathbb{S}]}{[\mathbb{S}] + K\_{\mathbb{S}\mathbb{S}1}} V\_{\mathbb{S}\mathbb{S}1} \tag{24}$$

$$\boldsymbol{\nu} = \frac{[\mathbb{S}]}{[\mathbb{S}] + \mathbb{K}\_{\mathbb{S}1}} \boldsymbol{V}\_{\mathbb{S}1} - \frac{[\mathbb{S}]}{[\mathbb{S}] + \mathbb{K}\_{\mathbb{S}\mathbb{S}1}} \boldsymbol{V}\_{\mathbb{S}1} + \frac{[\mathbb{S}]}{[\mathbb{S}] + \mathbb{K}\_{\mathbb{S}\mathbb{S}1}} \boldsymbol{V}\_{\mathbb{S}\mathbb{S}1} - \dots - \frac{[\mathbb{S}]}{[\mathbb{S}] + \mathbb{K}\_{\mathbb{S}n-1}} \boldsymbol{V}\_{\mathbb{S}n-1} - \frac{[\mathbb{S}]}{[\mathbb{S}] + \mathbb{K}\_{\mathbb{S}n}} \boldsymbol{V}\_{\mathbb{S}n-1} + \frac{[\mathbb{S}]}{[\mathbb{S}] + \mathbb{K}\_{\mathbb{S}n}} \boldsymbol{V}\_{\mathbb{S}n} \tag{25}$$

$$(V\_{S1} - (V\_{S1} - V\_{S2})\left(\frac{[X]}{[X] + K\_{X1}}\right) + (V\_{S1} - V\_{S2})\left(\frac{[X]}{[X] + K\_{X2}}\right) - (V\_{S1} - V\_{S3})\left(\frac{[X]}{[X] + K\_{X2}}\right) \tag{26}$$

$$V\_{S1} - (V\_{S1} - V\_{SX})\left(\left(\frac{[X]}{[X] + \mathbb{K}\_{X1}}\right) - \left(\frac{[X]}{[X] + \mathbb{K}\_{X1}}\right)\left(\frac{[Y]}{[Y] + \mathbb{K}\_{Y1}}\right)\right) - 1$$

$$(V\_{S1} - V\_{SY})\left(\left(\frac{[Y]}{[Y] + \mathbb{K}\_{Y1}}\right) - \left(\frac{[X]}{[X] + \mathbb{K}\_{X1}}\right)\left(\frac{[Y]}{[Y] + \mathbb{K}\_{Y1}}\right)\right) - (V\_{S1} - V\_{SXY})\left(\frac{[X]}{[X] + \mathbb{K}\_{X1}}\right)\left(\frac{[Y]}{[Y] + \mathbb{K}\_{Y1}}\right) \tag{27}$$

$$A = A\_o e^{-kt} \tag{28}$$

$$A = A\_{\alpha}e^{-k\mathbf{1}}\tag{29}$$

$$\frac{A}{A\_o} = e^{-k} \tag{30}$$

$$\frac{A}{A\_o} = k\_1 \tag{31}$$

$$A = A\_o k\_1^{\ t} \tag{32}$$

$$A = A\_o(1 - k\_2)^t\tag{33}$$

$$A = A\_o \left(1 - \frac{v}{A\_o}\right)^t \tag{34}$$

$$A = A\_o \left( 1 - \frac{[S]}{[S] + K\_{S1}} V\_{S1} \right)^t \tag{35}$$

$$A = A\_o \left( 1 - \frac{[S]}{1 - \frac{[S] + K\_{S1} - (K\_{S1} - K\_{S2}) \left(\frac{[X]}{[X] + K\_{X1}}\right)}{A\_o} \frac{V\_{S1} - (V\_{S1} - V\_{S2}) \left(\frac{[X]}{[X] + K\_{X1}}\right)}{A\_o} \right)^{\varepsilon} \tag{36}$$

$$k\_a \frac{[X]}{[X] + K\_{X1}} \tag{37}$$

$$A\_o \left(\frac{\frac{\upsilon}{A\_o}}{\frac{\upsilon}{A\_o} + k\_a \frac{[X]}{[X] + K\_{X1}}}\right) \tag{38}$$

$$A\_o \left(\frac{k\_a \frac{[X]}{[X] + K\_{X1}}}{\frac{\upsilon}{A\_o} + k\_a \frac{[X]}{[X] + K\_{X1}}}\right) \tag{39}$$

$$A = A\_o \left(\frac{\frac{\upsilon}{A\_o}}{\frac{\upsilon}{A\_o} + k\_a \frac{[X]}{[X] + K\_{X1}}}\right) \left(1 - \frac{\upsilon}{A\_o \left(\frac{\upsilon}{\overline{A\_o}}\right)}\right)^t + A\_o \left(\frac{k\_a \frac{[X]}{[X] + K\_{X1}}}{\frac{\upsilon}{A\_o} + k\_a \frac{[X]}{[X] + K\_{X1}}}\right) \quad (40)$$

#### **5. Conclusions**

#### **5.1 Modular enzyme kinetic equations**

The modular method of equation generation discussed here does not necessarily require derivation from initial conservation of mass and rate equations that were used in the generation of classical pseudo steady state enzyme kinetic equations. Rather, by clearly distinguishing between the mass action binding terms and the governor terms, which describe the kinetic effect of modifiers, a general method to characterize the effect of inhibitors and activators on enzymatic activity is suggested (Equation 21). The structure of this modified version of the Michaelis Menten equation allows for its modular expansion to describe multiple substrate binding interactions (Equation 24), multiple modifier binding interactions (Equation 26) and the effects of more than one modifier binding to the same enzyme (Equation 27). The modular way in which these equations can be expanded to describe the bulk kinetic properties associated with enzyme kinetic modeling suggests that they may neglect processes such as modifier and substrate binding order. However there are several possibilities which may result from such processes. For example, an inhibitor may bind to an enzyme only in the absence of the substrate or only in its presence, in both of these instances the inhibition would most likely manifest as a rectangular hyperbolic change in the catalytic constants influencing enzymatic activity. Alternatively if the inhibitor binds both forms of the enzyme, the affinity for each form may be quite different resulting in a term similar to that proposed with equation 26. While there are undoubtedly many more possibilities, as have been outlined in texts such as Enzyme Kinetics by Segel (1993), the derivation of these equations have neglected the division between mass binding and modifier effect proposed here.

This distinction between mass binding and modifier effect combined with the modular equation construction described herein represents a new way of addressing enzyme kinetic modelling which permits the simple adaptation of kinetic models for data analysis. This allows for a simplified comparative global data fitting to discriminate between competing kinetic models using nonlinear regression. A helpful guide to nonlinear data fitting in excel has recently been published in Nature Protocols (Kemmer & Keller, 2010).

#### **5.2 Pseudo-steady state equations in time course modeling**

Integral forms of the Michaelis Menten equation have been proposed for use in time course analysis for many years, with more complex mathematical models appearing with time (Russell & Drane, 1992; Goudar et al., 1999). Integral forms of the Michaelis Menten equation however have been found to be limited in their usefulness for time course models

�

The modular method of equation generation discussed here does not necessarily require derivation from initial conservation of mass and rate equations that were used in the generation of classical pseudo steady state enzyme kinetic equations. Rather, by clearly distinguishing between the mass action binding terms and the governor terms, which describe the kinetic effect of modifiers, a general method to characterize the effect of inhibitors and activators on enzymatic activity is suggested (Equation 21). The structure of this modified version of the Michaelis Menten equation allows for its modular expansion to describe multiple substrate binding interactions (Equation 24), multiple modifier binding interactions (Equation 26) and the effects of more than one modifier binding to the same enzyme (Equation 27). The modular way in which these equations can be expanded to describe the bulk kinetic properties associated with enzyme kinetic modeling suggests that they may neglect processes such as modifier and substrate binding order. However there are several possibilities which may result from such processes. For example, an inhibitor may bind to an enzyme only in the absence of the substrate or only in its presence, in both of these instances the inhibition would most likely manifest as a rectangular hyperbolic change in the catalytic constants influencing enzymatic activity. Alternatively if the inhibitor binds both forms of the enzyme, the affinity for each form may be quite different resulting in a term similar to that proposed with equation 26. While there are undoubtedly many more possibilities, as have been outlined in texts such as Enzyme Kinetics by Segel (1993), the derivation of these equations have neglected the division between mass binding and

This distinction between mass binding and modifier effect combined with the modular equation construction described herein represents a new way of addressing enzyme kinetic modelling which permits the simple adaptation of kinetic models for data analysis. This allows for a simplified comparative global data fitting to discriminate between competing kinetic models using nonlinear regression. A helpful guide to nonlinear data fitting in excel

Integral forms of the Michaelis Menten equation have been proposed for use in time course analysis for many years, with more complex mathematical models appearing with time (Russell & Drane, 1992; Goudar et al., 1999). Integral forms of the Michaelis Menten equation however have been found to be limited in their usefulness for time course models

has recently been published in Nature Protocols (Kemmer & Keller, 2010).

**5.2 Pseudo-steady state equations in time course modeling** 

� ��

> [�] [�] � ���

� �

� � � � � �

� �� �

��

� �� � ��

[�] [�] � ���

> [�] [�] � ���

� (40)

���� �

**5. Conclusions** 

� �� � ��

modifier effect proposed here.

� ��

**5.1 Modular enzyme kinetic equations** 

[�] [�] � ��� �

�

� � � � �

1 −

�� �

� �� � �� which has spurred further research (Liao et al., 2005). Integral forms of the Michaelis Menten equation also predominately model the Michaelis Menten equation and do not deal with modifier interactions. This may be in part due to the problems associated with pseudosteady state modifier equations, such as lack of governor terms on the effects of modifiers in enzyme systems, as outlined in the first section of this chapter. To attempt to address these issues a new way of using pseudo-steady state equations in time course modeling has been proposed (Walsh et al., 2010). The proposed methodology inserts the pseudo-steady state equations directly into the exponential decay equation (Equation 35) allowing for the same degree of equation flexibility outlined with the methods for modular expansion of pseudosteady state equations described in section 3.2.

The direct use of so called pseudo-steady state equations in exponential equations relies on several assumptions. Primarily, the development of pseudo-steady state equations has been based on experimental data generated in closed systems. That is, even if preformed in conditions where the rate of substrate hydrolysis is taken as linear or is linearized through the use of tangential slope lines, the observed rates are actually exponentially decreasing. Additionally, single substrate enzymes, which are not subject to conditions that would alter their catalytic activity, such as substrate or product modulation, as catalysts follow first order kinetics in closed systems (Equation 35). Due to this, time course modeling has the advantage of being able to identify a variety of kinetic situations, such as strong substrate activation or inhibition, for which initial rate analysis is not optimal (Shushanyan et.al., 2011). This sort of modeling can also be used to detect the influence of irreversible inhibition as deviation of the exponential curve away from the predicted initial exponential rate in substrate hydrolysis are more apparent with time course models than models using initial rates. For example, in our initial examination of the inhibition of β-galactosidase with imidazole with initial rates the irreversible inhibition of β-galactosidase was not apparent (Walsh et al., 2007), however it became quite apparent using time course models (Walsh et al., 2010).

The ease with which this method allows the integration of pseudo-steady state and time course kinetic equations holds the promise of making time course kinetic modeling a more prominent part of modifier kinetic analysis. Additionally, the modular compilation of kinetic components outlined in this chapter and their application to time course modeling suggest this form of modeling may be particularly useful for in-depth characterization of enzymatically regulated pathways which is directly applicable to systems biology.

#### **6. References**


Menten rate equations of different predictor variables. *J. Biochem. Biophys. Methods*, 62, 13–24.

