**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 � = �� ����

���� = ����[�] (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]

�� � [�] ��

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)

���� − ����� − ������ [�]

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.


*2.3. Transesterification*

Transesterification is a cheaper way of converting the branched, large molecules of oil to smaller molecules [26]. The extracted *Azadirachta indica* oil and its fractions were converted into biodiesel and glycerol (byproduct) using methanol through the catalytic transesterification process. The transesterification process was carried out by base (potassium hydroxide), enzyme (lipase Novozymes) and acid (hydrochloric acid) catalysts. Each catalyst was used at five different concentration levels for the optimization of the catalyst dose required for the transesterification process. The concentration of catalyst was measured in percentage (weight of catalyst/weight of oil). The base catalyst (KOH) was used at five different concentrations levels (0.2%, 0.4 %, 0.6%, 0.8% and 1% of the *Azadirachta indica* oil weight). The mixture of KOH (of respective concentration), *Azadirachta indica* oil or its fractions (10 g) and methanol (3 g) was stirred at 60 ◦C for 150 min. For the acid-catalyzed transesterification, HCl concentrations were 20%, 40%, 60%, 80% and 100% of oil weight. The used

concentration of methanol was 5%. A mixture of oil, methanol and acid was stirred at 75 ◦C for 270 min. Five concentrations (1%, 2%, 3%, 4% and 5%) of lipase were added to the mixture of oil (10 g) and methanol (50 g) under constant stirring at 100 rpm for 24 h. The water insoluble biodiesel layer and glycerol (water soluble) layer were separated using a 250 dm3 separatory funnel. The biodiesel layer was washed with an excess quantity of warm deionized distilled water (DDW) to remove soap and other water-soluble contaminants. Biodiesel was dried over anhydrous sodium sulfate before further use.

## *2.4. Biodiesel Quality Testing*

The chemical composition of *Azadirachta indica* oil and its fractions was analyzed by gas chromatographic and mass spectrometric (GC-MS) analysis. A Perkin Elmer Clarus 600 GC System was used for the GC-MS analyses fitted with a capillary column (30.0 m–0.25 mm) at the 350-◦C optimum temperature. As carrier gas, ultra-high (99.99%) pure helium was used with 0.2 dm3/min constant flow. The transfer line, ion source and injection temperatures were 200, 200 and 220 ◦C, respectively. There was 70 eV ionization energy in the system for the breakdown of fragments. The data collection was done by mass spectra ranging from 10 to 600 m/z. Then, 0.1 dm3 of the injected sample was used with 50:1 split ratio. The identification of obtained compounds was done by comparison with mass spectrum libraries (Wiley, 9th edition) and retention times.

The physico-chemical properties such as densities (kg/lit), saponification values (mg KOH/g), iodine values, cetane number, cloud and pour points were tested by following standard procedures [2,27]. The cold filter plugging point (CFPP) was determined in accordance with the EN 116 standard procedure for CFPP testing, as specified in the EN 590 standard [28]. Biodiesel pH was measured using a Hanna pH meter (model, HI 8010), and a densitometer (Wilnos LCD 51) was used for density (gram per dm3) determination. The oxidation stability was measured by the Induction Period (IP) according to EN14112 [29].

For iodine value (I.V) determination, 0.05 g of *Azadirachta indica* biodiesel was taken in an iodine flask with a capacity of 250 dm3. Then, 12.5 dm3 of Wijs solution and 10 dm3 of carbon tetrachloride were dissolved in biodiesel. Subsequently, contents of mixture were shaken vigorously and kept in a dark place for almost 30 min followed by the addition of 10 dm3 of potassium iodide solution (15%) and 50 dm3 of DDW. The obtained mixture was titrated against Na2S2O3.5H2O (0.1 N) until the disappearance of the color of the iodine using starch as an indicator. For the blank sample, a similar procedure was adopted. The following formula (Equation (1)) was used for the determination of the iodine value of biodiesel samples [2]:

$$\begin{array}{l}\text{Iodium value} \\ = \frac{(\text{Blank titration} - \text{Sample titration}) \times \text{Normalality of Na}\_2\text{S}\_2\text{O}\_3,5\text{H}\_2\text{O} \times 12}{\text{Sample weight (g)} \times 100} \end{array} \tag{1}$$

To determine the saponification value, 10 dm3 KOH (alcoholic) solution and 0.25 g *Azadirachta indica* oil were taken in a round bottom flask of 250 dm3 capacity. A condenser was attached with this flask and heated smoothly till transparent solution was obtained. The appearance of a transparent solution indicates the completion of the saponification reaction. After cooling the reaction mixture to room temperature, a phenolphthalein indicator (2–3 drops) was added to it and the whole mixture was titrated against HCl (0.5 N) until the disappearance of the pink color. An appropriate reagent blank sample was also prepared, and a reading was determined for it. Using the following formula (Equation (2)), the saponification value in mg KOH/g of oil sample was calculated [2,30]:

$$\text{Saponification value (SV)} = \frac{(\text{B} - \text{S}) \times \text{N} \times 56.1}{\text{w}} \tag{2}$$

where S is the volume used of titrant for biodiesel sample, B is the volume used of titrant for the blank sample, 56.1 is MW of KOH in mg per mmol, N is the normality of HCl in mmol per dm3 and w shows sample mass (gm).

For the determination of acid value, 1g of *Azadirachta indica* oil, 20 dm3 of ethanol and 2–3 drops of phenolphthalein were added to a 250-dm3 titration flask. This solution was titrated against NaOH (0.1 N) solution until there was a pink appearance. Using the following equation (Equation (3)), free fatty acid contents (oleic acid) were determined [2]:

$$\text{Free fatty acid (FFA)}, \,\%= \frac{(\text{V} \times \text{N} \times 282)100}{\text{w}} \tag{3}$$

where V stands for volume of titrant sodium hydroxide (dm3), N for Normality of sodium hydroxide (mol/1000 dm3), 282 is the molecular weight of oleic acid in gram per mole. The obtained value was converted into acid value using the following equation (Equation (4)):

$$\text{Acid value (AV)} = 1.989 \times \% \text{ FFA} \tag{4}$$

The following formula (Equation (5)) was used to determine the Cetane number (CN) of all biodiesel samples:

$$\text{Cetane number (CN)} = 46.3 + \frac{5458}{\text{SV}} - 0.225 \times \text{IV} \tag{5}$$

## **3. Results and Discussion**

#### *3.1. E*ff*ect of Various Catalysts on the Biodiesel Yield Percentage (%)*

The effect of different catalysts (acid, base and enzyme) on the percentage yield of biodiesel is shown in Figure 2a–c. Transesterification using base catalysts is found to be cost effective because of its reusability, wide availability, easy separation from product and longer lifetime [31,32]. In base-catalyzed transesterification of *Azadirachta indica* oil and its fractions, the maximum biodiesel yield of 99.9% was shown by fraction F4. In another study, the biodiesel yield obtained by whole date seed oil using base catalysts was approximately 80% [2]. The optimized catalyst level for pure oil, F1 and F4, was 1% KOH (wt. of catalyst/wt. of oil) and for F2 and F3 was 0.2% KOH (wt. of catalyst/wt. of oil). Thus, the result obtained clearly indicated that, for better biodiesel yields, higher alkaline catalyst doses are required in the case of *Azadirachta indica* oil, F1 and F2. However, fractions F2 and F3 produced the highest quantity of biodiesel at the lowest KOH dose of 0.2% KOH (wt. of catalyst/wt. of oil). This indicates that after fractionation biodiesel could be produced at much lower levels of base catalyst using some specific fractions. This will be helpful in cutting costs on base catalysts and will also be helpful in reducing environmental pollution caused by base catalysts. The maximum biodiesel yields of acid catalyzed transesterification reactions were 99%, 85%, 97.25%, 76.25% and 94.25%, respectively, for *Azadirachta indica* oil, fraction F1, fraction F2, fraction F3 and fraction F4. The most suitable level of HCl for obtaining maximum biodiesel from *Azadirachta indica* oil, F3 and F4 was 80% (wt. of HCl/wt. of oil). However, F1 and F2 have shown maximum biodiesel yields at 100% and 60%, respectively. The acid catalysis reaction rate was much slower than base catalysis. The acid-catalyzed transesterification reaction was completed in 4.5 h as compared to the 1.5 h taken by the base-catalyzed reaction. Transesterification is the replacement of original ester groups into desired esters. The best way for transesterification is base-catalyzed as it is a very facile reaction. The enzyme is inhibited in the presence of methanol. Acidic catalysts have low acid site concentration, low micro porosity, and high cost compared with basic types. Acids catalyzes both esterification and transesterification simultaneously and is insensitive to FFA and water. In a previous study, biodiesel productivities of used vegetable oils by acidic catalysis at 25 and 100 ◦C, respectively, were as follows: Safflower (84.7% and 94.3%), Soybean (85.9% and 94.2%), Sunflower (83.4% and 95.2%), Canola (80.8% and 93.7%), Corn (83.2% and 83.3%), Olive (84.3% and 85.3%), Hazelnut (82.5% and 83.4%) and Waste sunflower (84.3% and 90.4%) [33].

**Figure 2.** (**a**) Biodiesel yield using HCl as the catalyst; (**b**) biodiesel yield using KOH as the catalyst; (**c**) biodiesel yield using lipase enzyme as the catalyst.

(**c**)

Enzyme-catalyzed biodiesel production was completed in 24 h. For the biodiesel synthesis, five different lipase concentrations (1–5%) were used. The results obtained show that *Azadirachta indica* oil and its four fractions have the highest biodiesel yield at different enzyme concentrations. F2, F4, F3, pure oil and F1 produced the maximum quantity of biodiesel at 1%, 2%, 3%, 4% and 5% of lipase, respectively, and the respective maximum yields were 97.75%, 93.75%, 97.25%, 99.1% and 89.75%. The *Azadirachta indica* oil fractionation process yielded distillates with a variable composition as confirmed by gas chromatographic and mass spectrometric analysis. The variance in composition of fractions and *Azadirachta indica* oil was responsible for the variable optimized yield. Enzymes have a three-dimensional structure and are large protein biomolecules that can take part in many chemical reactions. Enzymes are highly specific; they function with only one reactant to produce specific products. Comparing acid, base and enzyme catalyst-optimized yields, it is apparent that the highest biodiesel yield was shown by base-catalyzed transesterification. The biodiesel yields of acid and enzyme-catalyzed reactions were comparable.

#### *3.2. Gas Chromatographic and Mass Spectroscopic (GC-MS) Analysis*

The identification and quantification of the fatty acids present in biodiesel was conducted by GC-MS. The major fatty acids present in *Azadirachta indica* oil were found to be oleic acid (67%), palmitic acid (14.7%) and lignoceric acid (8.23%). The fuel and physical properties of biodiesel are dependent on chain length, degree of unsaturation, and branching of the chain. In a previous study, the quality of biodiesel produced from shorter fatty acids was superior than longer chain fatty acids [34]. The longer carbon chain length leads to decreased ignition delay. Longer chain fatty acids are present in all oils. However, their percentages in various oils widely di ffer. Lignoceric acid, or tetracosanoic acid, is the saturated fatty acid with the formula C23 H47COOH. It is present in almost 8.23% of *Azadirachta indica* oil. Its removal from *Azadirachta indica* oil was necessary to avoid its negative e ffects on the biodiesel centane number, cloud point, pour point and freeze point. The vacuum fractional distillation process was used to carry out lignoceric acid removal from biodiesel. Lignoceric acid was removed from oil and collected in the first fraction F1. The other major components of F1 were erucic acid (22.4%), oleic acid (20.9%), phthalic acid (13.4%) and linoleic acid (11.7%). Fraction F2 contained oleic acid (C18, 61.3%) as a major fatty acid and all other fatty acids were present at less than 10%. Capric acid (C10) was 43.6% present in the fraction F3. Fraction F4 was found to have highest quantities of oleic acid (71.8%). Linoleic acid (C18) was present 16.91% in F4 (Table 2).

**Table 2.** Fatty acid composition of *Azadirachta indica* oil and its fractions.

#### *3.3. Assessment of Fuel Quality Parameters*

The acid value (AV) is defined as "the weight of KOH in mg needed to neutralize the organic acids present in 1 g of fat and it is a measure of the free fatty acids (FFA) present in the fat or oil" [31]. An increment in the amount of FFA in a sample of oil or fat indicates hydrolysis of triglycerides. The acid value (AV) of *Azadirachta indica* oil, F1, F2, F3 and F4 were 1.23, 2.91, 6.61, 8.89 and 3.36, respectively. The results obtained show that there was breakdown of fatty acids from triglyceride chains during the molecular distillation process of fractionation. The observed density of all biodiesel samples is given in Table 3. According to American standards (ASTM), there is no specified range of density of produced biodiesel. However, in European standards (EN), the recommended range of densities of biodiesel should range from 0.86 to 0.90 kg/liter. The density of *Azadirachta indica* oil and fraction biodiesel samples ranged from 0.70–0.91 kg/liter. The density value has great influence on some properties of the biodiesel concerning the engine e fficiency including injection timing, injection system and spray properties. Lower density biodiesel has lower engine efficiency and higher density value exhibits greater mass of biodiesel introduced into the injection system [35].

**Table 3.** Densities (kg/dm3) of biodiesel produced from *Azadirachta indica* seed oil and its fractions.

The saponification value or saponification number of *Azadirachta indica* biodiesel synthesized by using several catalyst concentrations is given in the Table 4. The saponification value or number relies on fatty acid concentration present in the biodiesel and the molecular weight. The existence of fatty acids with short chains of alkyl groups in biodiesel is also indicated by the saponification number. Using a KOH catalyst, higher saponification values of *Azadirachta indica* biodiesel were observed. This indicated high volatilities of the synthesized biodiesel. This might be supportive in burning *Azadirachta indica* biodiesel easily and efficiently to avoid the backfire. This happens in diesel engines when unburned fuel meets hot parts that are close enough to the tip, where oxygen is present, to explode spontaneously. However, it is quite rare for this to happen. This property is reflected in the existence of the small chained fatty acids or methyl esters.

**Table 4.** Saponification values (mg KOH/g) of biodiesel from *Azadirachta indica* seed oil and its various fractions.

Furthermore, a high saponification number also articulates the presence of carboxylic groups in high amounts in the biodiesel. In the present study, the maximum saponification value was 376 mg KOH/g of biodiesel sample synthesized by the pure *Azadirachta indica* oil using 1% KOH catalyst (w/w of oil), while the minimum saponification value (136.6 mg KOH/g) of biodiesel sample produced by the F4 fraction. Using the HCl catalyst, a maximum saponification value (281.4) of pure *Azadirachta indica* oil biodiesel was shown using 100% HCl catalyst (w/w of oil) and the minimum value (101.9) for the F3 fraction was observed using 60% and 80% HCl catalyst w/w of oil. Using the lipase enzyme catalyst, a maximum saponification value (226.3) was observed for the F3 fraction using 1% lipase enzyme catalyst (w/w of oil), while the minimum SP value (101.9) was observed for the F4 fraction using 3% and 4% lipase enzyme catalyst (w/w of oil).

The iodine values (IV) of *Azadirachta indica* biodiesel are shown in Table 5. The minimum iodine value (37.2) for the *Azadirachta indica* biodiesel was found for the biodiesel synthesized using 20% HCl catalyst w/w of oil, while the maximum iodine value (218.4) was shown by pure *Azadirachta indica* oil transesterified using KOH 0.8% w/w of oil. Iodine value relies on the unsaturated fatty acid and indicate double bound present in biodiesel sample [36]. According to European standards, iodine values equal to or less than 120 is preferable for biodiesel produced for consumption. However, in American standards, no iodine value is specified. Unsaturation represented by the iodine value reflects the chance of biofuel solidification [37]. The greater the unsaturation, the greater the iodine value will be. High unsaturation in the biofuel may cause deposition as breakage of some weak bonds results in polymerization or leads to epoxide formation in the engine at high temperatures. A careful examination of Table 5 shows that biodiesel produced from *Azadirachta indica* oil had higher iodine values. However, fractionation of *Azadirachta indica* oil into various components was useful for producing better quality biodiesel with lower iodine values. The production of biodiesel from oil fractions (F1–F4) as compared to whole oil was advantageous, as evidenced by the great oxidative stabilities of oil fractions (F1–F4). Cloud point and pour point are important physical properties of any liquid fuel. Some fractions produced under vacuum produced high-quality biodiesel with comparatively much better cloud, pour points and Cold Filter Plugging Point than *Azadirachta indica* oil (Table 6). In fractionation, oil was separated into various fractions. It was expected that some fractions will have better properties than oil and others. As oil comprises all components so it displayed intermediate properties.

Cetane number (CN) is a biofuel quality parameter. CN is one of the factors that determines the ignition capability and depends on the fatty acid or methyl ester composition in the biodiesel. It influences the engine e fficiency and ignition power after injection of biofuel [38]. In the present study, the CN value ranged from 9.1–87.3 (Table 7). None of biodiesel sample produced from pure *Azadirachta indica* oil met standard biodiesel cetane number requirements. However, the fractionation process improved the centane number of produced biodiesels. Cetane number of all biodiesel samples produced from *Azadirachta indica* oil fractions was in the recommended range. Biodiesel o ffers the advantage of a direct replacement for ordinary diesel. In the present study, we propose a much cheaper and simpler method named high vacuum fractional distillation as an alternative to refining to produce high quality biodiesel with desired fuel properties. The cetane number of commonly occurring fatty acids methyl esters are as follows: palmitic (85.9), palmitoleic (51.0), stearic (101.0), oleic (59.3), and linoleic (38.2).

*Energies* **2020**, *13*, 2858

A previous study [39] was undertaken with similar objectives such as development of cost-effective processing technology, and elimination of degumming, esterification and dehydration steps from the oil purification processes by conducting a supercritical extraction using CO2 [39].

However, when capital investment and processing cost of high vacuum fractional distillation (HVFD) used in the present study is compared with a supercritical extraction using CO2, HVFD is a much cheaper and simpler alternative. HVFD does not add to the cost of biodiesel as it is an alternative route for oil purification. In another previous study [40], supercritical fluid carbon dioxide was used to separate out biodiesel fraction from impure fatty acid methyl ester (FAME) solution mixes. The disadvantage of doing fractionation after production of methyl esters is the production of a number of useless byproducts, which does not happen in HVFD. Secondly, the prior methodology does not bring any significant improvements in the cetane number, cloud point or pour point of biodiesel.