Kinetics Mathematical Modeling And Thermodynamic Study

Published Date: 02 Nov 2017

The present chapter will be focused on the understanding of steady state kinetics catalyzed by biocatalyst especially lipases to explain the theory and experimental approaches employed.

3.1 Introduction

Biocatalyst catalyzes reaction with specific reactants that are called substrates ad products. In any chemical reaction, there can be two kinds of enzymes forms called stable and transitory. An enzyme is said to be stable if, when isolated from the rest of the reaction components, it has a long half-life with respect to the assay time scale. An enzyme form is termed as transitory if, when isolated from the reaction system, its half life is short with respect to assay time scale. The latter is also known as enzyme-reactants complex. There are two types of transitory enzyme complexes, central and non-central. Those in which active site is fully occupied by reactants are known as central whereas those in which it is partially filled, termed non-central. For example, E is a stable enzyme whereas EA, EP and EI are transitory enzymes in equations (3.1) and (3.2) respectively.

(3.1)

(3.2)

Kinetic mechanisms, that is, order of addition of reactants to and release of products from the enzyme active site, fall into two classes.

Sequential, where all reactants bind to enzyme before any reaction occurs as given in equation (3.3). It can be termed as ordered, when there is a definite order of addition of reactants or release of products to the enzyme. Otherwise, it is random.

(3.3)

Ping-pong, in which a product is released between the additions of two substrates. In the ping-pong mechanism, a reactant binds to the enzyme active site, undergoes chemical transformations, and leaves a part of substrate on the enzyme prior to dissociating as a product. The second reactant then binds, undergoes a second chemical transformations involving the fragment of the first substrate, and dissociate as the second product as in equation (3.4).

(3.4)

3.1.1 Steady state enzyme kinetics

The enzyme catalyzed reaction (see equation 3.5) is assumed to be in steady state when the concentration of intermediate along the reaction pathway i.e. E, EA and EP change very slowly than the reactant concentration so that it can be assumed dE/dt = d(EA)/dt = d(EP)/dt = 0. In addition, equilibrium is assumed to be established rapidly. The steady state is achieved whenever A >Et; under these conditions, dA/dt >> dE/dt.

(3.5)

The steady state rate is measured as close to time zero as possible such that little change in the concentration of added substrate occurs and the final and initial concentrations of substrate are the same. The initial rate is obtained as a function of the concentration of substrate as it is varied around its Michaelis constant (Km). The Km values for the majority of substrate of enzyme-catalyzed reactions are in the range 1 µM to 1 mM, while the assay concentration of most enzymes in the range of 10-8-10-9 M (Cook and Cleland, 2007).

3.1.1.1 The Michaelis-Menten/Briggs-Haldane Equation

The initial rate for a reaction is described mathematically by a rate equation that describes the initial rate of the reaction under any given set of conditions. The simplest treatment of the steady state rate equation was provided by Henri Michaelis and Maude Menten (Bailey and Ollis, 1986) using assumptions that an enzyme-catalyzed reactions could be described by the mechanism given in equation (3.6).

(3.6)

The rate equation shows that the initial rate would be a saturable function of substrate concentration and retains their name. The rate equation is given as:

In the equation (3.7), (k3Et) and (k2/k1) were called the maximum rate (Vmax) and the Km respectively. Where, Km is also equal to the dissociation constant (Kd) for EA.

Briggs and Haldane modified the treatment of Michaelis-Menten and assumed that a steady state was reached in which the concentration of enzymes forms was determined by the rates of formation and breakdown and did not change during the initial velocity phase. Therefore, the expression of Km modified to (k2 + k3)/k1 (Michaelis and Menten, 1913; Briggs and Haldane, 1925).

3.1.1.2 Initial reaction velocity or rate of reaction

A determination of the initial rate as a function of the concentration of one reactant, maintaining the level of all others fixed, and then repeating this at several additional fixed levels of the second reactant. Data are displayed graphically by use of the linear transform of the Michaelis-Menten equation devised by Linewaver and Burk, equation (3.8) and known as Lineweaver-Burk or double reciprocal plot. The reciprocal of the initial velocity is plotted against the reciprocal of the reactant concentration (Linweaver and Burk, 1934). The slope of it represents (Km/Vmax) and vertical intercept represents (1/Vmax) as given in Fig. 3.1. The Km value can also be obtained by the definition, from the value [A] at v = (2/Vmax).

Fig. 3.1:Lineweaver-Burk or double reciprocal plot of initial velocity and substrate concentration.

3.1.2 Study of initial reaction velocity without inhibition

The study of initial reaction velocity in the absence of added inhibitors, using differences in the overall initial velocity rate equation, depending on the kinetic mechanism of the enzyme-catalyzed reactions.

3.1.2.1 Uni Bi enzyme reactions

There are two possibilities exist for Uni Bi mechanism: steady state ordered and rapid equilibrium ordered. In both cases, substrate A binds to the enzyme active site and is converted to the products P and Q with possibilities. The first is an ordered release of P prior to Q (see Equation 3.9), while other allows release of products either P or Q first (see Equation 3.10).

(3.9)

(3.10)

Therefore, the initial rate of reaction for the two mechanism in equation 3.9 and 3.10 are given assuming P and Q to be zero for ordered mechanism with equation 3.11 and A tends to zero for random mechanism with equation 3.12. Thus, the equations are:

3.1.2.2 Bireactant enzyme reactions

3.1.2.2.1 Ordered Sequential mechanism

The ordered Bi reactant mechanism has three common variants that represents a continuous spectrum from rate limiting inter conversion of the central complexes, a rapid equilibrium ordered, steady state ordered and Theorell-Chance. A common features of all the ordered mechanism is the implication that no site for B exits initially. Thus, the binding of A to enzyme must induce a conformational change in E that allows B to bind or A, once bound to enzyme, actually forms a portion of the binding site for B.

Equilibrium ordered mechanism

The rate equation is asymmetric (see equation 3.13) i.e. gives a different algebraic form dependent on which reactant concentration is varied. Because of equilibrium addition of A, the kinetic mechanism has a different rate equation which is given as (3.14):

Theorell-Chance mechanism

This mechanism have been named after the investigator who developed the theory to explain the kinetics of alcohol dehydrogenase from equine liver. The reaction equation is shown in (3.15). There are no ternary complex formation, and reaction of EA and B gives EQ and P directly. The initial velocity rate equation for the Theorell-Chance mechanism is identical to steady state mechanism with slight modifications as given in equation (3.16):

Random sequential mechanism

In a rapid equilibrium random mechanism the formation of EA and EB complexes is at equilibrium. But the off rate constants of A and B from the EAB complex do not have to be greater than kcat in order to observe the simple rate equation that corresponds to rapid equilibrium mechanism. The rates of the two pathways from E to EAB often differ significantly in rate. The general equation for steady state random rate equation is given in assuming constant B (3.17), where V, p, q and r combinations of coefficients a, b, c etc.

3.1.2.2.2 Ping-pong mechanism

In a ping pong reaction, the first substrate binds to enzyme, transfers a piece of itself to the enzyme, and dissociates as the first product before the second reactant binds, picks up the transferred piece and dissociates as the second product. There are two kinds of ping pong mechanisms. The first is called a one-site or classical, while second is a two sites or nonclassical ping pong mechanism.

An enzyme with a classical ping pong mechanism has a single site in which each reactant binds and thus can only bind one at a time. The rate equation is as shown in equation (3.18)

The non-classical ping pong mechanism has an initial velocity equation that is identical to that of the classical one, but there are two sites, with A and P being adsorbed at one and B and Q at the other. The physical connection between the two sites is usually made up by a carrier. The difference between the two kinds of mechanism will become evident when product and dead-end inhibition are considered.

Table 3.1: Summary of kinetic mechanism without inhibition assuming reactant [B] constant

Mechanism

Conditions

Reciprocal plot

Slope

Intercept

Steady state random (SSR)

No restriction on any rate constant

Rapid equilibrium random (RER)a

Dissociation of binary complexes fast; central complex inter-conversion may be slow or fast

Steady state ordered (SSO) a

Compulsory order of addition of A and B

Same as RER

Equilibrium ordered (EO)b

Off rate constant for A much greater than kcat

Theorell-Chance (TC) a

Second product releases very slow; appears that no central complex exists.

Same as RER and SOS

Ping Pong (PP)c

Product released between additions of substrates.

aIf Ka > Kia, the double reciprocal plot appear parallel.

bIf strong synergism of binding occurs between A and B, then RER degenerates to EO.

cRER, SSO, and TC could give this pattern if Ka > Kia.

3.1.3 Study of Initial reaction velocity with inhibition

3.1.3.1 Types of inhibition

There are three kinds of inhibitors, called competitive, non-competitive and uncompetitive. The kinds of inhibitors are distinguished qualitatively on the basis of initial velocity patterns observed in the presence of the inhibitors.

3.1.3.1.1 Competitive inhibition

A competitive inhibitor is one that competes with the varied substrate for enzyme, generally suggesting that the inhibitor and the varied substrate combine with the same form of the enzyme.

A scheme describing competitive inhibition by I versus A in a unireactant mechanism is provided in equation (3.19):

The rate equation for equation (3.19) is given in equation (3.20) as:

The apparent Km obtained as s function of I increases linearly, reflecting the increased concentration of substrate required to attain the maximum rate in the presence of I.

3.1.3.1.2 Non-competitive inhibition

A non-competitive inhibition is one which binds to an enzyme form different than that to which the varied substrate binds, and there is a reversible connection between the addition of varied substrate and that of the inhibitor. Thus, increasing the reactant to an infinite concentration will not eliminate all of the inhibition by the inhibitor. A scheme representing non-competitive inhibition by I versus B in an ordered Bi Bi reaction with A fixed and B varied is provided in equation (3.21) an the rate is described with equation (3.22).

Where appV = V(1+Ka/[A]), appK = Kb[(1+Kia/[A])/(1+Ka/[A])], Kis = Ki(1+[A]/Kia), and Kii = Ki(1+[A]/Ka). As can be seen, both the B terms, representing EAB and part of E, and the appK term, representing EA and part of E, are affected by the presence of inhibitor.

3.1.3.1.3 Uncompetitive inhibition

An uncompetitive inhibitor combines with enzyme forms that result from the combination of varied substrate with substrate. Thus, inhibition is observed when the varied substrate is saturating, conditions that favor production of the enzyme form to which inhibitor binds. A scheme describing uncompetitive inhibition by I versus A in an order Bi Bi kinetic mechanism with B maintained constant is provided in equation (3.23) and the rate of reaction is given with equation (3.24).

Where appV = V/(1+Ka/A), appK = Ka[(1+(KiaKb/Ka[B]))/(1+(Kb/[B]))], and Kii = Ki(1+([B]/Kb)); Ki is the dissociation constant for I from EAI. As can be seen, only the A term representing the EA form of the enzyme is affected by the presence of inhibitor.

Table 3.2: Summary of Reciprocal plots for kinetic mechanism with inhibition

Mechanism

Rate of reaction with [A]& [Q] ([P] = 0)

Rate of reaction with [B]& [P] ([Q] = 0)

Slope

Intercept

Slope

Intercept

Uni Bi Steady state ordered

Uni Bi Rapid Equilibrium Random (RER)a

Bireactant Steady state ordered (SSO) a

Equilibrium ordered (EO)b

Theorell-Chance (TC) a

N/A

N/A

Ping Pong (PP)c

3.2 Mathematical models for kinetics of lipase catalyzed hydrolysis reactions

The lipase catalyzed hydrolysis of triglycerides is a complex reaction which involves the breakdown of tri-, di- and mono- glycerides through consecutive and reversible steps. Several mathematical models have been developed for the hydrolysis of triglyceride using lipases in both the forms free and immobilized.

For the hydrolysis of oil's triglycerides in the presence of excess water with lipases, a three step mechanism have been suggested by various researchers in literature (Alenezi et al., 2009) which can be compared with the overall mechanism of enzymatic hydrolysis (Al-Zuhair et al., 2004 and Tsai & Chang 1993) as given below. The fish oil triglycerides (TG) act as a substrate (S) for enzyme (E) action to liberate free fatty acids as product (P) with the formation of glycerol (G) after complete hydrolysis. Moreover, the immobilized lipase enzymes can be recovered from enzyme-substrate complex (ES) for its recycling after every run (Cheong et al., 2012).

Step-I

Step-II

Step-III

Overall Step

The simplest reaction mechanism for hydrolysis of oils is Michaelis-Menten kinetics which was first proposed by Herni in 1903. The details of this model have already been discussed in the starting of this chapter in general. In order to increase the reaction velocity, reactions have been performed in different reaction systems such as emulsion, biphasic solvent and reverse micelle. The different types of kinetic models and modifications of this basic mechanism have been proposed. This has been studied by various researchers such as Palsson B.O., 1987; Ekiz and Caglar have studied the hydrolysis of tributyrin by Candida cylindracea using this model at the n-heptane -water interface (Ekiz and Caglar, 1989). Brown and Holtzapple, have used this mechanism and developed three rate expressions using assumptions for three different model such as exact solution, the pseudo-steady state and the amount of free substrate is approximately equal to the total amount of substrate (Brown and Holtzapple, 1990).

Mukataka et al., 1985 proposed a model to simulate the interfacial reaction of lipases by considering the effects of agitation and volume ratio between organic and aqueous phases. Since the substrate concentration used in their experiments was low, the validity of their model according to the scheme of Michaelis-Menten equation was questionable.

Han and Rhee, 1987 suggested that lipase reaction in reverse micelle system should be regarded as a two substrate reaction due to the variation in degree of hydrolysis and the rate of hydrolytic reaction that depend on water content (Han and Rhee, 1987). To derive the rate equation, the following scheme for two substrate reactions forming two products was considered:

Where [A] is the ester bonds which can be attacked by lipase; [W] is the water molecule and [P] and [Q] are the two-substrate, second order reversible kinetics can be described as:

Tsai et al., 1993 modified the analysis of Mukataka to a high substrate concentration in a biphasic system to propose a kinetic model (Tsai and Chang, 1993). According to this model the reaction rate was given as

Where [S] is the total substrate concentration in the organic phase, A is total interfacial area, Et is total mass enzyme, Va is the volume of aqueous phase, Vmax is apparent maximum velocity and is the apparent Michaelis constant. Where Vmax = K2Et/[=(1+Va/b1A); = Km/b2; b1 = [(E)i + (ES)i]/(E)1 and b2 = (S)i/(S)0. Where (E)i, (S)i and (ES)i represents the interfacial concentration of enzymes, lipid structure and enzyme-substrate complex respectively. (E)t and (S)0 are the enzyme concentration in the aqueous phase and the total substrate concentration in the organic phase and b1 and b2 are the proportionality constants.

This kinetic model was verified for short term hydrolysis. The model does not deal about substrate and product inhibition and hence the validity of this model in long term hydrolysis is doubtful.

Al-Zuhair et al., 2002 has developed a mathematical model considering the hydrolysis reaction and effect of interfacial area between the oil phase and the aqueous phase (Al-Zuair et al., 2002) similar to that proposed by Tsai an Chang (1993). The concentration of enzyme-substrate complex and the adsorbed enzyme are both assumed constant, and the interfacial product concentration is assumed to be proportional to the free product concentration. It was also assumed that the interfacial product concentration is low and hence it occupies negligible fraction of the total interfacial area. With these assumptions, a simplified rate equation was given as:

Where Ke = (kcat + k-1)/k1 and k*cat = kcat/C*. This rate equation agrees in the basic form with the previous model. Later same Al-Zuhair et al. in 2004 have modified their previous model given in 2002 and predicted the hydrolysis mechanism of oils by lipases assuming interfacial saturation at high enzyme concentration (Al-Zuair et al., 2004)

Knezevic et al., 1998 reported the hydrolysis of palm oil in a lecithin-isooctane reverse miceller system followed one substrate first order reversible Michaelis-Menten kinetics, when the initial substrate concentration is less. A disagreement with this model was found when the initial substrate concentration was higher. In order to study the progress of lipase hydrolysis over time, they adopted the kinetic model developed by Kosugi and Suzuki (1983). On the basis of this model, the mechanism of lipid hydrolysis may represent as follows:

Where [S] is the concentration of the ester bonds, [W] is the water concentration and [P] and [Q] are product concentration. This product concentration is given by

Where [S0] is initial concentration of the ester bond. The rate equation for one-substrate, first order reversible is expressed as follows:

Where k1 is the rate constant of ester bond decomposition and k-1 is the rate constant of ester bond formation.

A kinetic model based on mechanism of lipase catalyzed in oil-aqueous system using Michaelis-Menten equation was used to determine the rate constant of Vmax and Km, and it was found to be 370.37 mol/min mg-enzyme and 1.23 g/ml, respectively by Serri et al., 2008.

The hydrolysis rate in the reverse micellar system was interpreted by the interfacial reaction model, in which desorption of fatty acids from the interface was the rate determining step. The desorption rate constants of fatty acids produced were independent of the AOT concentration at a fixed W, and were of nearly the same order of magnitude as that obtained in the emulsion system (Shriomori et al., 1996).

The kinetics of the hydrolysis of corn oil in the presence of a lipase from Pseudomonas sp. immobilized within the alls of a hollow fiber reactor can be modeled in terms of a three-parameter rate expression. This rate expression consist f the product of a two-parameter rate expression for the hydrolysis reaction itself (which is of the general Michaelis–Menten form) and a first-order rate expression for deactivation of the enzyme. Analyses of the three kinetic models associated with the ping-pong bi–bi mechanism show that Model B provides the best fit of the experimental data (Sehanputri and Hills, 1999).

A lipase from Candida cylindracea immobilized by adsorption on micro porous polypropylene fibers was used to selectively hydrolyze the saturated and monounsaturated fatty acid residues of menhaden oil at 40°C and pH 7.0. Rate expressions associated with a generic ping-pong bi-bi mechanism were used to fit the experimental data for the lipase catalyzed reaction. Both uni- and multi-response nonlinear regression methods were employed to determine the kinetic parameters associated with these rate expressions. The best statistical fit of the uniresponse data was obtained for a rate expression, which is formally equivalent to a general Michaelis–Menten mechanism. After the parameterization, this rate expression reduced to a pseudo-first-order model. For the multiresponse analysis, a model that employed a normal distribution of the ratio of Vmax/Km with respect to the chain length of the fatty acid residues provided the best statistical fit of the experimental data (Rice et al., 1999).

The kinetics for the tributyrin hydrolysis using lipase (Pseudomonas fluorscenes CCRC-17015) was investigated in the liquid–liquid and liquid–solid–liquid reaction systems in a batch reactor. The kinetic parameters in the reaction system were also obtained for two reaction systems. The turnover numbers calculated for free lipase and immobilized lipase were 29 and 5.7 s−1, respectively. The parameters of k and km obtained using Lineweaver-Burk plot method were 26.2 mol/ (mg min) and 1.35 mol/dm3 for free lipase, 5.2 mol/(mg min) and 0.2 mol/dm3 for immobilized lipase, respectively (Wu and Tsai, 2004).

Chew et al., 2008 reported lipase catalyzed hydrolysis of palm oil in aqueous-organic phase followed ping pong bi bi model with substrate inhibition by water. Two assumptions were made while performing the material balance: mass transfer limitation in the reaction system was negligible and all the reactions are elementary (Ref.).

According to a kinetic model proposed by Prazeres et al., 1993, the rate of reaction (rP) was determined by equation (5) assuming a multi order nonlinear product inhibition with a maximum three molecules of product binding to the enzyme.

Where, rP = rate of reaction (µmoles of FFAs formed/ml.min); Et = amount of immobilized CAL-B used in reaction (mg); S = substrate concentration (total FFAs present in oils, µmoles of FFAs formed/ml); P = product formed (free fatty acids/ml); KM = 1/K1 (K1 is rate constant for first stage hydrolysis; µmoles of FFAs formed/ml); Ki1 = rate constant for product inhibition after first stage product formation (µmoles of FFAs formed/ml); Ki2 = rate constant for product inhibition after second stage product formation (µmoles of FFAs formed/ml) and Ki3 = rate constant for product inhibition after third stage product formation (µmoles of FFAs formed/ml).

A mathematical model describing the triglyceride hydrolysis with Candida rugosa lipase was constructed by Hermansyah et al., 2006. As stepwise hydrolysis at the oil-water interface based on the ping pong bi bi mechanism and the inhibition by a fatty acid were considered in addition to the differences in the interfacial and bulk concentrations of enzyme, substrate and product (Hermansyah et al., 2006). This model was found effective for predicting the appropriate conditions for the efficient production of desired fatty acids.

The kinetics of lipase-catalyzed hydrolysis of olive oil in AOT/isooctane reversed micellar media was studied. It was shown that the deactivation of lipase had a great influence on the reaction kinetics. Based on whether the enzyme deactivation and influences of both product and substrate on enzyme stability were included or not, four different kinetic models were established. The simulating results demonstrated that the kinetic model, which including product inhibition, enzyme deactivation and the improvements of lipase stability by both product and substrate, fit the experimental data best with an overall relative error of 4.68% (Yao et al., 2005).

The model proposed by Prazeres et al., 1993 for reverse micellar system with third order product inhibition appear to be most describing the long term reaction kinetics.

3.3 Mathematical models for kinetics of lipase catalyzed esterification reactions

The lipase catalyzed esterification of fatty acids with alcohol is generally considered as a bireactant reaction using lipases which involves the formation of esters either with random or ordered bi bi kinetic mechanism. Various kinetic mechanisms have been developed for lipase catalyzed esterification for both soluble and immobilized lipases under different reaction conditions.

Lipase-catalyzed esterification has previously been described by a Ping-Pong kinetic model. Competitive inhibition by the alcohol has usually been found to be significant, leading to the use of the following equation:

Where v is the initial reaction rate, Vm is the maximum reaction rate, KA and KB are the Ping-Pong constants for the fatty acid A and the alcohol B, and KIB is the inhibition constant for the alcohol. Equation (3.33), describes the initial reaction rate in the absence of any product. In esterification reactions, water is one of the products and is usually present in the reaction medium since some water is essential for enzyme activity. This indicates that equation (3.33), is not the correct equation to describe the kinetics of an esterification reaction. The full equation to be used when the first product [P] is present at significant levels is

Where = (Vm.KQ)/(Vm,r.Keq).KQ. Keq is the equilibrium constant, and Vm,r is the maximum reaction rate for the reverse reaction. KiP is the inhibition constant for the product [P] in equation (3.34) and shows that two extra parameters are necessary to describe the reaction rate if the product water is present. This means that the differences between the kinetic constants are to a greater extent explained by substrate solvation when using the more correct model (Janssen et al., 1999).

The magnitude of various kinetic constants based on a factorial design and ternary complex mechanism for a kinetic model suggested by Oliveira et al., 2001 were determined assuming esterification reaction without substrate inhibition. For both substrates (DHA rich FFAs and Lauryl alcohol) , formation of ternary complex was studied assuming the reaction mechanism as of the Bi-Bi order using straight lines with an intersection point at the second quadrant. According to this model, the two substrates can bind to immobilized lipases either in a specific or a random order to form a ternary complex to form products.

The rate of reaction for this ternary complex mechanism assuming no product inhibition conditions can be given with equation-(3.35):

Where, ro is initial rate of reaction from a model given by Oliveira et al., 2001; [A] and [B] are the concentrations of DHA rich FFAs and lauryl alcohol respectively; Vmax (= kcat.[E]) is the maximum velocity; Km(A) and Km(B) are Michaelis-Menten constants for DHA rich FFAs and lauryl alcohol respectively; and K(B) is the dissociation constant of the alcohol-lipase complex (Oliveira et al., 2001).

A kinetic model for the lipase catalyzed esterification in a biphasic organic aqueous system has been proposed by Shintre et al., 2002. Based on the interfacial substrate concentration, an analytical rate equation for inital rate of reaction was derived and confirmed with the experimental data. The esterification reaction was simulated with the ping pong bi bi model without substrate and product inhibition, taking into consideration the interfacial substrate concentration (Shintre et al., 2002).

The kinetics of the lipase-catalyzed (Pseudomonas cepacia) ethanolysis of fish oil has been studied in a batch reactor using menhaden oil, tuna oil, and acylglycerol mixtures derived from menhaden oil. Multiresponse models derived from a generalized Michaelis–Menten mechanism were developed to describe the rates of formation of ethyl esters of the primary fatty acids present in the precursor oil. A first-order model for deactivation of the lipase was fit simultaneously to one of the data sets (Torres et al., 2003). The rate expressions re,j, of the multi-response model are represented by

where M represents the number of fatty acid residues under consideration in the model (M = 7; j = 1, myristic; j = 2, palmitic; j = 3, palmitoleic; j = 4, stearic; j = 5, oleic; j = 6, eicosapentaenoic; and j = 7, docosahexaenoic acid); [Gj], [Qj], and [Pj] are the molar concentrations of the parent acylglycerol of interest, the fatty acid ethyl ester formed by reaction, and the lower acylglycerol of the jth type, respectively. [B] denotes the concentration of ethanol, the vmax f,j are the maximum rates of reaction at a particular glyceride bond [Gj] at a saturating concentration of the substrate (and in the absence of the other glycerides); the vmax r,j are the maximum rates of reaction between the fatty acid ethyl esters [Qj] and the lower glyceride [Pj] at a saturating concentration of the substrate (and in the absence of other glycerides); the Km,j are the Michaelis–Menten constants for the glycerides, and the Ki,j are the inhibition constants for the corresponding ethyl esters. The lumped parameters, vmax,j, vmax r,j, Ki,j, and Km,j are defined as follows:

A comprehensive kinetic study on esterification of lauric acid with lauryl alcohol catalyzed by commercial porcine pancreatic lipase (PPL) in the form of cross-linked enzyme crystals (CLEC) using glutaraldehyde as the cross linker. The kinetics of the esterification reaction conformed with the so-called Ping-Pong_Bi-Bi mechanism with alcohol inhibition. For this mechanism the initial velocity equation is represented as:

where [Acid] and [Alcohol] represents the initial molar concentrations of lauric acid and lauryl alcohol respectively. Km(acid) and Km(alcohol) are the respective affinity constants, Ki is the inhibition constant for lauryl alcohol and Vmax is the maximum reaction rate (Gogoi et al., 2006).

Al-Zuhair et al., 2006 determined the kinetics of the esterification of butyric acid with methanol catalyzed by a free R. miehei lipase in a micro aqueous system containing enzyme in a suspension in hexane and a biphasic system. Ping pong model with competitive inhibition system has been described in n-hexane micro aqueous system but not for the biphasic system (Ref.).

Romero et al., 2007 proposed a reaction mechanism and developed a rate equation for the synthesis of isoamyl acetate by acylation of the corresponding alcohol with acetic anhydride using the lipase Novozym 435 in n-hexane. The analysis of the initial rate data showed that reaction followed a Ping-pong mechanism with inhibition by acetic anhydride. (Ref.)

The kinetics of esterification of oleic acid with 1-butanol catalyzed by free Rhizomucor miehei lipase in a biphasic system was studied in a batch reactor. The reaction appeared to proceed via a ping pong mechanism with 1-butanol inhibition. The model was used to simulate the batch concentration profiles of the product as well as the initial reaction rates. The following rate equation was proposed for the given mechanism:

Where is the rate of reaction for oleic acid, and are concentrations of oleic acid and 1-butanol respectively. is concentration of enzyme, the enzymatic constant, and are Michaelis-Menten constant for oleic acid and 1-butanol, respectively, is inhibition constant for 1-butanol (Kraai et al., 2008).

3.3 Thermodynamics study for lipase catalyzed reactions

3.3.1 Temperature dependence of rate constant

The dependence of temperature on the rate constant for a lipase catalyzed hydrolysis and esterification reaction can be explained in terms of Activation energy (E). The energy required to proceed from the reactant state to the transition state, known as the activation energy or energy barrier of the reaction, is the difference in the free energy between these two states (Reactant to product). The activation energy is represented by a symbol E. This energy barrier is important because height of activation energy barrier can be directly related to the rate of reaction. Lipase accelerates the rate of hydrolysis or esterification reaction by stabilizing the transition state, hence lowering the activation energy barrier to product formation. In general, a linear decrease in activation energy results in exponential increases in reaction rate.

The magnitude of activation energy (E) for enzymatic reactions is always lower than the reactions which are taking place in the absence of enzymes. The activation energy (E) for lipase catalyzed reactions can be determined by studying Arrhenius plot as given below in equation (3.41).

Where, k = rate constant; Ao = pre-exponential factor; E = activation energy (J/mol); R = gas constant and T = temperature (Kelvin). The linear form of Arrhenius equation is given in equation (3.42).

According to equation (3.42), a linear plot between ln k versus (1/T) can be drawn to get the activation energy (E) from the slope and pre-exponential factor (A0) from the intercept of the curve. For evaluating the activation energy, the rate constant (k) has be calculated first which depends on the type of reaction being catalyzed at varied temperature conditions. Various researchers have suggested that percentage conversion of substrate into product when plotted against reaction time, the rate constant can be obtained from the slope of the plot.

Gogoi et al., 2006 have studied the esterification of lauric acid with lauryl alcohol using cross linked enzyme crystals over temperature range 30-50 ºC and estimated the value of activation energy was 183.81 J/mol using Arrhenius rate equation. They have plotted initial rate of esterification reaction against temperature to record the rate constant for Arrhenius linear plot (Gogoi et al., 2006). Similar observation was made by Harikrishna et al. (2000), for Rhizomucor miehei lipase catalyzed synthesis of isoamyl butyrate (Harikrishna et al., 2000).

3.3.2 Determination of thermodynamic constants for hydrolysis reaction

The thermodynamics is concerned with the change in energy and similar factors as s biochemical process takes place.

3.3.2.1 Gibbs free energy (G), Enthalpy (H) and Entropy (S)

Thermodynamic functions such as Gibbs free energy, enthalpy and entropy, are functions of state. This means that they depend only on the state of the system being considered and not on how that system came into being. Changes in the functions of state between two states depend only on the initial and final states and not on the route between them (Fersht A., 1999). For a chemical reaction, the change in the Gibbs free energy function (ΔG) is the energy which is available to do work as the reaction proceeds from the given concentrations of reactant and products to chemical equilibrium. The enthalpy change (ΔH) is defined as the quantity of heat adsorbed by the system under the given conditions. Whereas the increase in the entropy of surroundings is represented by (-ΔH/T) and the increase in the entropy of system is (ΔH/T). For any spontaneous process at constant temperature and pressure, J. Willard Gibbs in 1878 defined the increase in the free energy of the system (ΔG) as

A plot of ln k against 1/T, will give a straight line whose slope will be equal to ΔH/RT and intercept will be ΔS/R (Pogaku et al., 2012).

Table 3.3: Physical significance of thermodynamic constants according to their magnitude (Meena and Chauhan, 2009)

Thermodynamic constants

Positive magnitude

Negative magnitude

Enthalpy (ΔH, KJ/mol)

Endothermic reaction (ΔH > 0)

Exothermic reaction (ΔH < 0)

Entropy (ΔS, KJ/kelvin)

Natural and disordered reactions (ΔS > 0)

ordered reactions (ΔS < 0)

Gibbs free energy (ΔG, KJ/mol)

Nonspontaneous and unfavorable reaction (ΔG > 0)

Spontaneous and favorable reaction (ΔG < 0)

3.3.2.2 Turnover number (kcat)

The thermodynamic catalytic constant of an enzyme which is a measure of its catalytic efficiency and is defined as given below:

This quantity is also known as the turnover number of an enzyme because it is the number of reactions processes that each site catalyses per unit time.

The term kcat/Km is the catalytic efficiency. A high value indicates that the limiting factor for the overall reaction is the frequency of collisions between enzyme and substrate molecules. A low value indicates the equilibrium assumptions. A comparison of kcat/Km for alternative substrates or lipases can be used as a measure of the specificity of an enzyme (Liese et al., 2006). When using expensive catalyst, the turnover number should be as high as possible so as to reduce the cost of the product (Price and Stevens, 1999).

The kcat is a first order rate constant that refers to the properties and reaction of the enzyme-substrate, enzyme-intermediate and enzyme-product complexes. In the simple Michaelis-Menten mechanism in which there is only one enzyme-substrate complex and all binding steps are fast, kcat is simply a first order rate constant for the chemical conversion of [ES] complex to [EP] complex. For more complicated reactions, kcat is a function of all first order rate constants. For example, when in equation (3.46) the k2 is comparable k-1 it is known as Briggs-Haldane mechanism. According to this, when the dissociation of the [EP] complex is fast, kcat is equal to k2 (see equation). But if dissociation of [EP] is slow, the rate constant to this process contributes to kcat and for very slow dissociation of [EP] kcat will become equal to dissociation rate constant.

In the other words, some similar definitions are also available in the literature to define a turnover number (kcat) as given below (Fersht A., 1999):

3.3.2.3 Dissociation constant (Kd)

Immobilized enzymes lose their catalytic activity upon reuse over time. The dissociation constant value shows the deactivation nature given by the equation (3.48)

A plot of ln Ea/Ea(t) against 1/t will give a straight line whose slope will be Kd (Pogaku et al., 2012).

Alternatively the magnitude of Kd can also be determined according to a thermodynamic study conducted by Gitin et al. in 2006 for the thermal deactivation of Novozym 435 assuming enzyme may be reversible, irreversible or a combination of the two. The constant of deactivation (Kd) was calculated with equation (3.49) where Tmax is the maximum thermodynamic temperature (in kelvin). In this study, it was concluded that the smaller values of Kd is a indicative for the more active enzyme (Gitin et al., 2006 and Primozic et al., 2003).