Enzyme Kinetics and Modeling of Enzymatic Systems

C H A P T E R

3 Enzyme Kinetics and Modeling of Enzymatic Systems Emmanuel M. Papamichael*, Haralambos Stamatis†, Panagiota-Yiolanda Stergiou*, Athanasios Foukis*, Olga A. Gkini* *

University of Ioannina, Department of Chemistry, Ioannina, Greece †University of Ioannina, Department of Biological Applications & Technology, Ioannina, Greece

3.1 INTRODUCTION Enzymes are specific biocatalysts. The importance of in vitro enzyme kinetics has been long considered important to understanding the chemistry of their complex systems. Enzyme kineticists did a great deal of work to reach the knowledge available now on the catalytic tools of enzymes. The enzymatic reaction rates are functions of various entities, such as the reactants’ concentrations, the pH, and temperature values of reaction media, mass transport effects, and so forth [1]. Currently, many attempts are focused on the modeling of enzymatic systems based on existing and evolving information due to enzyme kinetics, for optimal implementations and yields [2]. Mechanistic models and equations assimilate all the information regarding each individual enzymatic reaction; however, due to inherent difficulties, various approaches, including rational approximations, have been considered to facilitate satisfactory solutions. The modeling of steady enzymatic systems, in vitro, is easier due to fewer parameters, which are described by easily analyzed models [3]. Mathematical modeling investigates effectively optimal functioning and productivity conditions of enzymatic catalysis, providing information on the reactants’ and products’ concentrations, on enzymatic mechanisms, and on the parameter estimates. The detailed study of enzymatic reactions through in silico modeling is useful, not only in enzyme kinetics [4].

Advances in Enzyme Technology https://doi.org/10.1016/B978-0-444-64114-4.00003-0

71

# 2019 Elsevier B.V. All rights reserved.

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3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

3.2 PRELIMINARIES 3.2.1 The Concept of Kinetics and Its Fundamentals Chemical and enzymatic reactions are subject to the laws of thermodynamics. Reaction rates and pathways, rate equations, equilibria, mechanisms, composition, and structure of reactants, chemical species detected by in silico calculations, estimated parameter values, quantum mechanics calculations, and so forth, are analyzed through chemical kinetics [5]. In the simplest kinetics, the rate v of a homogeneous reaction is a function of the concentrations of all reactants and products, due to the law of mass action. The probabilities of collisions among the reactants are related to their concentrations, and account for the reaction rates [6]. Scheme (3.1) describes a simple example, where the rate is given by Eq. (3.1), rv, rvfr, rvbr, k1 and k
1 are the: reaction rate, forward and backward rates, and kinetic rate constants, respectively. In general, when more reactants and products are involved, Scheme (3.2) and Eq. (3.2) describe the case, where a number of n reactants R are transformed into m products P; the subscript eq ½C2eq stands for equilibrium. In Eq. (3.1), the equilibrium constant Keq is defined as Keq ¼ , ½Aeq ½Beq Qmj Pj, eq whereas in Eq. (3.2), the equilibrium constant may be generally defined as Keq ¼ Qnj¼1 . i k1

Scheme ð3:1Þ : A + B !  2C, k
1

i¼1

Ri, eq

k1

Scheme ð3:2Þ : R1 + R2 + ⋯ + Rn !  P1 + P2 + ⋯ + Pm k
1

rv ¼ rvfr
rvbr ¼ k1 ½A½B
k
1 ½C2 rv ¼ rvfr
rvbr ¼ k1

ni Y i¼1

Ri
k
1

mj Y

Pj

(3.1) (3.2)

j¼1

3.2.2 Chemical and Biochemical Catalysis: The Enzymes Catalysts accelerate the reactions by decreasing the Gibbs energy of activation without affecting equilibrium and spontaneity, modify the reaction mechanisms, and remain unaffected and ready for the next reaction cycle. Reactions, from reactants to products, pass through energy barriers where activated intermediates, that is, transition states (TS), are formed, possessing higher Gibbs free energy than reactants; the lower the energy barrier, the higher the reaction rate. Catalysts do this, as illustrated in Fig. 3.1. The energy conservation and spontaneity of reactions (ΔS > 0) are key issues. The total entropy is the measure of a systems’ disorder, and if it is not measured, it tends toward a maximum. Entropy changes (ΔS) are estimated through relation ΔG ¼ ΔH
TΔS for finite variations at constant T. Spontaneous exergonic reactions reveal ΔG < 0, while ΔG > 0 characterize nonspontaneous endergonic reactions; at equilibrium, ΔG ¼ 0 is valid. The relation ΔG ¼
RT ln(Keq), and its integrated form Keq ¼ exp(
ΔG/RT) ¼ exp(ΔS/R
ΔH/RT), are also valid (R is the gas constant 8.314 J mol
1 K
1). The activation thermodynamic parameters are described by the previous relations [7].

3.2 PRELIMINARIES

73

FIG. 3.1 Gibbs free energy changes of uncatalyzed (ΔG‡u) and catalyzed (ΔG‡c ) reactions.

Enzymes are functional proteins catalyzing various reactions in vivo and in vitro. Enzyme kinetics investigate the rates and mechanisms of enzymatic reactions. Enzymes are chemo-, regio- and enantio-selective biocatalysts, and are active in aqueous, nonaqueous, and nonconventional media, including surface interfaces, depending also on the features of their substrates [8–11]; the efficient catalysts enzymes are applied in industry and biotechnology [12]. Enzymatic catalysis is based on the structural complementarities among enzymes and substrates due to hydrogen, hydrophobic, π–π stacking, and other bonding, which increase their binding; the TS theory is obviously valid in enzyme kinetics [13].

3.2.3 Modeling Approach as a Key to Enzyme Kinetics Enzyme kinetic modeling offers essential information for the enzymatic reactions through reactivities of the participating species, which are crucial in designing improved reaction devices, and estimating their performance. Mass conservation requirements, inhibition, mass transfer effects, and so forth should be provided in the establishment of reacting systems and the development of suitable kinetic equations. In addition, the values of pH and temperature, the reaction media composition, the use of immobilized enzymes, the interfacial tension, and others are key factors affecting enzyme kinetic modeling [13].

3.2.4 Selected Enzymatic Catalytic Motifs and Related Mechanisms Specific hydrolases are used extensively in biotechnology and industry due to their catalytic modes, which support the enzyme kineticists with valuable operative sources. An example of subsites in hydrolases is depicted in Fig. 3.2A [14]; subsites contribute essentially in the enzyme–substrate binding by possessing particular structural conformations, undergoing dynamic reorganization during the substrate binding, and promoting the catalysis [13]. Fig. 3.2B represents the active site of α-amylase (EC 3.2.1.1) [15]. Glycosidases catalyze the hydrolysis of glycosidic bonds by means of two acidic residues (E35 and D52), which constitute its catalytic site, as depicted in Fig. 3.2C [16]. Serine (EC 3.4.21), cysteine (EC 3.4.22), and aspartic proteases (EC 3.4.23) are gaining increasing interest for applications in pharmaceuticals, foods, peptide synthesis, recombinant

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3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

FIG. 3.2

(A) subsites of papain (as modified from I. Schechter, and A. Berger, On the size of the active site in protease: I. Papain, Biochem. Biophys. Res. Commun. 27 (1967) 157–162): (B) Polysaccharide hydrolysis by α-amylase with active site including 11 subsites: (I) initial, (II) hydrolysis of one glycoside bond (O and Ø are glucose and reduced glucose, respectively). As modified from A.K. Mazur, and H. Nakatani, Multiple attack mechanism in the porcine pancreatic alpha-amylase hydrolysis of amylose and amylopectin, Arch. Biochem. Biophys. 306 (1993) 29–38. (C) The hydrolysis mechanism of glycosidases; the hydrogen bonds are indicated by bold dotted lines and either the x bond or the y bond can occur due to the reversion of the α-anomeric or the preservation of the initial configuration, respectively, and where the carboxyl O of residue D52 (lysozyme numbering) plays the role of oxyanion hole due to hydrogen bonding on the nucleophilic water molecule in order to maintain the proper position. As modified from A.J. Kirby, The lysozyme mechanism sorted after 50 years. Nat. Struct. Mol. Biol. 8 (2001) 737–739.

protein techniques, biorefineries, and so forth [17]. Examples of mechanisms of serine, cysteine, and aspartic proteases are depicted in Figs. 3.3A–C, respectively [18, 19]. Lipases (triacylglycerol acylhydrolases, EC 3.1.1) are active in water–lipid interfaces and in heterogeneous biphasic systems (micelles or finely dispersed macromolecules). In nonaqueous or in low water-containing systems, lipases catalyze synthetic reactions, provided that the formed water is continually removed [10, 20]. Lipases share similar catalytic motifs with serine proteases, that is, a catalytic triad S162, H263 and D176 (porcine pancreatic lipase numbering) assembles their catalytic site; occasionally an E residue replaces D residue [8]. Mobile domains in lipases’ surface, the lid or the flap, cover and/or protect their catalytic site, and are crucial for their catalytic efficiency; several lipases lack a lid [21]. Fig. 3.4A illustrates a full mechanism of action of lipases, both hydrolytic and synthetic [8, 22]. On the other hand, lipases act markedly dissimilar in relation to the hydrolysis of fatty acid esters in lipid– water interfaces, small aggregates, and/or emulsions [23]. Within these systems, a simple equilibrium is shown in Fig. 3.4B. Catalytic motifs and active sites should be considered, rather, as dynamic entities of the enzyme molecule. Hence, glycosidases, proteases, and lipases do not perform the catalysis through standard catalytic units combined in dyads, triads,

3.2 PRELIMINARIES

FIG. 3.3 Examples of catalysis by proteases: (A) Catalysis by serine proteases via a charge relay system; the values of fractionation factors (ϕs) and of pKa,s, with the corresponding enzymatic species and steps are referred. (B) Illustration of the catalytic mechanism of cysteine proteases (papain, chymopapain, and bromelain) whose the reactive ion-pair S
/ImH+ is in equilibrium with its tautomer neutral S/Im form; a general acid/base catalysis is performed by the catalytic residues C25, H159, D158, and/or N175, along with a hydrogen bond between D158 and/or N175 and the positively charged H159 which promote the catalysis. Continued

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76 3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

FIG. 3.3, Cont’d (C) The catalytic mechanism of aspartic protease Rhizomucor pusillus Rennin (MPR); by black and light gray colors are referred components related to MPR and to substrate, respectively, whereas by light gray dotted lines are indicated the hydrogen bonds and by black color H the protons in transfer. As modified by P.-Y. Stergiou, A. Foukis, O.A. Gkini, E. Barouni, P.S. Georgoulia, M. Kanellaki, A.A. Koutinas, M. Papagianni, E.M. Papamichael, Novel FRET-substrates of Rhizomucor pusillus rennin: activity and mechanistic studies, Food Chem. 245 (2018) 926–933.

3.2 PRELIMINARIES

FIG. 3.4 Catalysis by Candida antarctica lipase-B (CALB): (A) The more likely full mechanism of action of lipases, where black, gray, and light gray colored species are referred to lipase (CALB), to a fatty acid and its synthesized ester, to a different fatty acid ester and its hydrolysates, respectively. (B) Schematic representation of a lipase catalyzed hydrolysis in a water–lipid interface; physical adsorption and desorption of lipase between an aqueous/lipid interface and a water phase leads to its activation and the formation of products, which are released into water phase, where the subscripts Wat, Int, and Wat/str denote water, interface, and water/stressed forms of the lipase molecule, respectively.

77

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3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

and so forth; the enzymatic catalysis proceeds by means of astonishing grouping and a function of more catalytic units.

3.2.5 Important Kinetic Features 3.2.5.1 Transition-States and Transition-State Analogs The transition-state (TS), or the activated complexes theory, associates kinetics and thermodynamics based on Planck’s relation E ¼ ℏv, and it is applicable in enzymatic catalysis [24]. Transition-states are activated intermediates, with short lifetimes, that is, <10
13 s, where bonds break and new bonds are formed, submolecular species rotate, and protons are transferred; in Fig. 3.1 reactants and stable intermediates pass through TSs, where the species are unstable. Enzymes are more reactive with ligands (substrates, inhibitors) whose structural components match the TS. Inhibitors are TS-analogs, and block entirely the catalytic sites of enzymes, as several substrates do, and both are useful tools in the investigation of enzymatic mechanisms [13]. The kinetic isotope effects are important complementary tools for prediction of TS and design of TS-analogs [25]. The term “virtual transition state” is used to describe processes in enzymatic catalysis where physical and chemical steps compete and contribute to the development of the next enzymatic species toward the completion of the reaction [9]. 3.2.5.2 Low Harrier Hydrogen Bonds Hydrogen bonding is common in enzymatic catalysis, and occurs under particular conditions as a result of: (a) enzyme–substrate conformational interactions, (b) pH-value, (c) interactions among the enzyme catalytic residues and other catalytic elements, and (d) variations in the pKa-values of the side chains of the enzyme residues. Shorter hydrogen bonds are stronger depending on the pKa matching of the electronegative atoms, which share the hydrogen atoms; enzymes convert ordinary hydrogen bonds into strong ones by modifying the pKa-values [14, 18]. Ordinary hydrogen bonds are switched to low harrier hydrogen bonds (LBHB) by being shorter and stronger, when the electronegative atoms that share the hydrogen atom show almost equal pKa values; LBHB exhibit low deuterium oxide fractionation factors. As the distances of the electronegative atoms decrease, the hydrogen bond takes on the form of a single well, as depicted in Fig. 3.5 [26, 27]. In three relative articles, a LBHB was proposed as effective for catalysis by cysteine proteases, as well as a key feature for the elucidation of their kinetic mechanism [28–30]. 3.2.5.3 Oxyanion Holes Despite the catalytic motifs of enzymes and their classification similarities, there are more structural tools that promote catalysis, and are occasionally prerequisites, as the oxyanion holes; their function should be taken into account. It is widely accepted that serine proteases carry out catalysis only via the support of an oxyanion hole, as shown in Fig. 3.3A; this is not a prerequisite for catalysis by cysteine proteases. Lipases require the function of oxyanion holes to perform catalysis in aqueous media; similarly in glycosidases, an oxyanion hole is found in their mechanism of action, as depicted in Fig. 3.2C [8, 9, 31].

3.3 UNDERSTANDING THE ENZYME KINETICS

79

FIG. 3.5 Typical hydrogen bonds: (A) normal, (B) LBHB, and (C) single well.

3.2.5.4 The Catalytic Water The presence or absence of water directly affects the enzymatic reactions, as an ordinary reactant, and contributes significantly to enzymatic catalysis. Examples of the catalytic value of water are found in the mechanisms of hydrolases; their catalytic sites (dyads, triads) are enlarged by water molecules, as depicted in Figs. 3.2C, 3.3, and 3.4A. In glycosidases, the catalytic waters acquire additional functions during either the retaining or the inverting mechanism, depending on the ionizations of residues E35 and D52, according to Fig. 3.2C. Water molecules have been detected bound to inhibitors, under transient states, promoting the development of hydrogen bridges [31].

3.3 UNDERSTANDING THE ENZYME KINETICS 3.3.1 A Dynamic Reaction System: Single-Substrate Enzymatic Reactions Enzymatic reaction pathways pass across a series of intermediates where enzymes and substrates undergo transformations affecting the molecularity of reaction [13]. Singlesubstrate enzymatic reactions start as bimolecular, and become unimolecular; then, in vitro, the simplest model is the Henri–Michaelis–Menten (H.M.M.), according to Scheme (3.3) [32]. The following assumptions are required: (a) an equilibrium or quasi-steady-state (QSS) is established, (b) the development and breaking down of a stable intermediate (ES complex) precedes the product formation, and (c) the relation [E]t << [S]t is valid throughout the reaction. In Scheme (3.3), E, S, and t are enzyme, substrate, and total, respectively. The QSS approximates the equilibrium for k2 << k
1; if k2 >> k
1, then equilibrium is not achieved, and the ES breaks to E + P, while for comparable values of k
1 and k2, an equilibrium is established [7]. k1 k2 Scheme ð3:3Þ : E + S !  ES ! E + P k
1

d½S d½P k2 ½Et ½S k2 ½Et ½S Vmax ½S ¼ ¼ k2 ½ES ¼ v¼
¼ ¼ k
1 + k2 dt dt Km + ½S Km + ½S + ½S k1

(3.3)

The aforementioned hypotheses and Scheme (3.3) lead to the Henri, Michaelis, and Menten rate equation (3.3) [32]. An additional assumption is the linearity versus time, of the entire process, that is, v ¼
Δ[S]/dt ¼ Δ[Ρ]/Δt; then it follows that v ¼ k2[ES], k2[E]t ¼ Vmax (the k
1 + k2 limiting maximal velocity where all enzymes are bound as ES), and ¼ Km . The plot k1

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3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

of Eq. (3.3) is a rectangular hyperbola passing through the origin of Cartesian axes with asymptotes; with the parameters Vmax (parallel to abscissas) and
Km (parallel to ordinates). The Km takes on different meanings according to the values of k
1 and k2; for k2 << k
1, or k2  k
1 and/or k2 << k
1 the Km acquires the property of an instability constant, or of a composite equilibrium constant, and/or of a dynamic or a pseudo-equilibrium constant, respectively. Nevertheless, and in all cases, Km equals [S], which provides half of Vmax of the enzymatic reaction. Two more important parameters should be considered according to kcat k2 k1 k2 Scheme (3.3), that is, kcat ¼ k2, and ¼ ¼ . At very low [S] << Km, where Km k
1 + k2 k
1 + k2 k1 [E]t and [S] can be characterized as comparable, the enzymatic reaction is of the 2nd order, k2 ½Et ½S k2 ½Et ½S k2 ) ka[E]t[S] ¼ ) ka ¼  whose rate is expressed as v ¼ ka[E]t[S] ¼ Km + ½S Km + ½S Km + ½S k2 kcat ¼ . On the contrary, at high [S] >>Km where is valid that [E]t << [S] the Km Km k2 ½Et ½S ) kb[E]t enzymatic reaction is of the 1st order, whose rate is expressed as v ¼ kb[E]t ¼ Km + ½S k2 ½Et ½S k2 ½S  ¼ k2 ¼ kcat. In cases of pre-steady state kinetics, a quasi-steady-state ) kb  ¼ Km + ½S ½S approximation (QSSA) can be considered; by applying a perturbation expansion with k2 << k
1 estimates of [E]t, [S], [ES]t and [P] can be achieved as time functions [33]. 3.3.1.1 Alternative Formulations of Enzyme Kinetic Equations In homogeneous media, the rates of enzymatic reactions rely on the reacting molecules’ contacts, which are proportional to the product of concentrations of reactants; then, multiparametric equations describe the enzyme kinetics. Hence, a lack of fit experimental data by conventional rate equations, as Eq. (3.4), leads to the introduction of alternative formulations, where n m, and a1, a2, … an, b1, b2, … bm, are conditional parameters. The Hill equations 3.5(a) and/or 3.5(b) is applied in cases of positive cooperativity, that is, for enzymes with quaternary structure, where n is the number of subunits, KnH has a dissimilar meaning, as compared with Km and excluding the case of n ¼ 1; the rate constants k2 and k/2 in Schemes (3.3) and (3.4), respectively, are similar [34]. Other attempts were based on Mandelbrot’s idea that by introducing a fractal dimension D leading to Eq. (3.6), which for D ¼ 1 is converted to eff Eq. (3.3); Veff max and Km are effective H.M.M. parameters, whose meaning depends on the fractal dimension D [35]. = k1 k2 Scheme (3.4): E + nS !  ESn ! E + nP k
1

v¼

a1 ½S + a2 ½S2 + … + an ½Sn 1 + b1 ½S + b2 ½S2 + … + bm ½Sm Vmax ½Sn n + ½Sn KH

(3.5a)

Vmax en½ ln ð½S
ln ðKH Þ 1 + en½ ln ð½S
ln ðKH Þ

(3.5b)

v¼ v¼

(3.4)

v¼

eff Vmax ½S2
D eff + ½S Km

(3.6)

3.3 UNDERSTANDING THE ENZYME KINETICS

  Vmax ½S 1 + A2 ½S + A3 ½S2 + …   v¼ Km + ½S 1 + A2 ½S + A3 ½S2 + …

81 (3.7)

Starting from different concepts, general equations can be formulated for many cases of enzyme kinetics. A virial expansion of the H.M.M. equation is Eq. (3.7), as different numbers and types of contacts among enzymes and substrate molecules establish diverse microenvironments within the enzymatic reaction medium [36]. Hence, the dependence of the reaction rate on [S] is proportional to virial expansion of the product [E][S] ([E] is the unbound enzyme). A2, A3, and so forth, are the coefficients related by statistical thermodynamics on the enzymatic system, and express positive or negative contributions to the product [E][S] in dyads, triads, and so forth. Eq. (3.7) can fit different kinetic data and mechanisms. The H.M.M. equation was based on the relation [E]t << [S]t, during in vitro reactions. In contrast, [E]t is comparable to [S]t in reactions in vivo; then, the mass balance equation [S]t ¼ [S] + [ES] should be considered, leading to the quadratic equation [ES]2
{[E]t + [S]t + Km}[ES] + [E]t[S]t ¼ 0, whose negative root solution produces the rate equation (3.8) [37]. ffi3 2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ½Et + ½St + Km
½Et + ½St + Km
4½Et ½St 5 (3.8) v ¼ Vmax 4 2½Et

3.3.1.2 Detailed Analysis of Experimental Data in Enzyme Kinetics The availability of computer programs allows the fitting of kinetic data by a chosen equation, and the subsequent estimation of its parameters, which does not guarantee statistical robustness of results, as most computer programs operate by parametric statistics. Only a limited number of data and replicates are available from enzyme kinetic experiments, which are described by nonlinear and multiparametric equations. Nonparametric statistics in enzyme kinetics have been introduced through the concept of a direct linear plot, where the requirement for normal distribution of errors has been replaced by the condition for equal positive Vmax Km and/or negative errors [38]. Hence, as v  [S] 6¼ 0 is always valid, equation
¼ 1 has v ½S replaced the H.M.M.; the experimenter has to draw straight lines in a (Vmax, Km) plot, with intercepts at
[S] (abscissa) and v (ordinate), respectively. For n experiments, the lines are intercepted by two, in n(n
1)/n intersections (in errorless experiments lines are intercepted in a single point), where the projections of medians on both axes are evaluated as Km (in abscissas) and Vmax (in ordinates), respectively; for even n are collected the mean values of the two medians. Therefore, the analysis of enzyme kinetic data is complex; equations with reasonable parameters should be chosen, whereas the erroneous experimental data do not allow the use of least squares as convergence criterion. This drawback is ill-resolved by fitting _ yi
y i _ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi the normalized residuals nr¼v 2 per point; n, yi, and y i , are the number of data uP  u n y
_ t i¼1 i y i n
1 points, their experimental and calculated values, respectively [39].

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3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

Despite the preceding suggestion, the problem still exists, as it is unlikely that a limited number of enzyme kinetic experiments will show normally distributed errors. Insensitive sensors devices employed for measurements, complicated and multistage enzymatic processes, sampling, and other kinds of errors related to the nature of enzymes are sources of observed outliers; the latter constitute the problem when not many measurements, each with limited replicates, are available. The fitting of experimental data containing outliers by multiparametric equations, and the unavailability of initial parameters values, leads to the use of the statistically robust nonparametric fitting method; the latter, however, has been mainly introduced for the H.M.M. equation; however, it has already been extended for use in almost all cases of equations that could potentially be employed in enzyme kinetics. The only requirement is an appropriate linearization of the working equation in order to reP1 P2 Pn ceive the general form: + + … + ¼ 1, where Pi (i ¼ 1, 2, …, n) and xi (i ¼ 1, 2, …, n) are x1 x2 xn the unknown terms, and the known terms, respectively; then equation can be introduced into the program list. Proper conversions of equations useful in biotechnology, including the program listing, are available [40, 41]. 3.3.1.3 The Effect of pH and Absolute Temperature on the Michaelis–Menten Parameters The effects of pH and absolute temperature are meaningful only when referring to the parameters of the used equations; the meaningless “% relative activity,” and so forth, should be rejected. Ionizations of significant groups of the involved enzyme are influenced by the changes in the pH-value of reaction media. The equilibrium constants of these ionizations are known as Ka ¼ [A
][H+]/[HA] and/or Ka ¼ [B:][H+]/[B:H], and are formulated as Henderson–Hasselbalchequations (3.9) and (3.10). The pH-value of reaction media reversibly affect the reaction rates due to interactions on the: (a) ionized groups of enzyme’s catalytic site, affecting the kcat/Km, (b) ionization of the ES complex, affecting the kcat, and (c) affinity of enzyme and substrate affecting the Km. The shape of the resulting curves is important. 
½A  (3.9) pH ¼ pKa + log ½HA  ½B : pH ¼ pKa + log (3.10) ½B : H +  By taking into account the concept aforementioned in Section 3.3.1.1, then it is valid that [13, 19, 24, 31]: (a) The dependencies of kcat/Km and 1/Km versus the pH-value of the reaction medium kcat =Km

1=Km

follow the ionizations of E and S, that is, E + S ! E + P and E + S ! ES, respectively. (b) The dependencies of kcat and Km versus the pH-value of the reaction medium follow the kcat

ionizations of ES (i.e., ES ! E + P and E + S slow

Km

 ES, respectively).

(c) If the Km is unaffected by the changes in the pH-value of the reaction medium, whereas k
1 the kcat is affected, then Km equals to , that is, it is the instability constant of the k1 ES complex.

3.3 UNDERSTANDING THE ENZYME KINETICS

83

Frequently, the profiles of kcat/Km and kcat versus the pH-values are bell shaped, and have been explained in terms of two opposite ionized groups of enzymes; as the steepest is the acid limb of profiles as much as acid ionized groups are active, whose pKas can be estimated under steady-state conditions. The general equation (3.11) can describe all specific cases where kobs, klim i , n, and Bij are, the value of the parameter under investigation (e.g., kcat/Km, etc.), its corresponding maximum value, the number of active hydrogenic sites, and a description of the quantity KXHp[H+]m, respectively. The maximum value of ki (i.e., the klim i ) corresponds to the hydrogenic state EHi
1, whereas m, and p are elements of two relative matrices, which correspond to the indices i and j, respectively [29]. kobs ¼

n X i¼1

0

klim i

@1 +

n X

1

(3.11)

Bij A

j¼1

As an example could be the Scheme (3.5), in which the enzyme possesses four active hydrogenic states, and one nonactive (EH4); whereas for more active hydrogenic states, more factors, for example, EHn (n ¼ 1, 2, …, n) should be added to Scheme (3.5). In Scheme (3.6), the enzyme possesses only one active hydrogenic state (EH), and it is a specific case of the previous one. More examples of specific cases of Eq. (3.11), showing symmetric bell-shaped profiles, are describe by Eqs. (3.12) and (3.13) whose the acid limb comprises either two pKas or one pKa, respectively, and their basic limbs comprise one pKa. Scheme (3.5):

EH4

KEH

4

KEH KEH KEH 2 3 EH3 EH EH2 E kEH kEH kEH kEH 4 2 3 Products

Scheme (3.6):

EH4

KEH4

KEH3

EH3 kEH4

KEH2 KEH EH2 EH E kEH2 kEH3 kEH Products

A nonsymmetric, nonbell-shaped profile with two peaks is described by Eq. (3.14), comprising one pKa per acid and/or basic limb, where the pKa value of the basic limb of the first curve equals the pKa value of the acid limb of the second curve; there are also cases in which all four pKa values differ. kobs ¼

ðkÞlim 1 + 10pKa1 + pKa2
2pH + 10pKa2
pH + 10pH
Ka3 kobs ¼

kobs ¼

1 + 10

ðkÞlim + 10pH
pKa2

pKa1
pH

ðk1 Þlim ðk2 Þlim + 1 + 10pKa1
pH + 10pH
pKa2 1 + 10pKa2
pH + 10pH
pKa3

(3.12) (3.13) (3.14)

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3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

The basic relations associating the changes of Gibbs free energy, enthalpy, and entropy to the values of equilibrium constants through absolute temperature have already been introduced. As the reaction rates are influenced by temperature, then the Arrhenius’ equation k ¼ Ae–Ea/RT ¼ Ae–Ea/kBT is applicable, as based on the Planck’s hypothesis, where k, A, Ea, R, T, kB, are the: rate constant, preexponential factor, or frequency of correct orientation collisions, activation energy, universal gas constant, absolute temperature, and Boltzmann’s constant, respectively. An increase in temperature positively affects the rate of

homogeneous  
Ej 1 1
  reactions. Alternatively, the Arrhenius’ equation can be written as k ¼ k e R T T0 , j

j 0

where kj, (kj)0, T0, T, Ej, are: the rate constant of occurring reaction, the same rate constant at reference absolute temperature, the reference absolute temperature, the absolute temperature, and the activation energy, respectively. Furthermore, the Eyring equation (3.15) is more useful, and it can be transformed to Eq. (3.16); the quantity ln(kB/ℏ) equals 23.76 [7]. k ¼ ðkB =ℏÞ  T  e
ΔG‡=RT ¼ ðkB =ℏÞ  T  e
ΔH‡=RT  eΔS‡=R

(3.15)



T  ½ ln ðk=T Þ ¼ T  ln ðkB =ℏÞ + ΔS‡ =R
ΔH‡ =R

(3.16)

Chemical and enzyme kinetics follow the same laws; however, enzyme kineticists should consider that in either relatively high or relatively low temperatures, enzymes may denaturize, where a sudden decrease of the occurring reaction rate is observed. Thus, the thermal stability of enzymes should be confirmed within a temperature range under the reaction conditions. Eqs. (3.17) and (3.18) are useful, and refer to Scheme (3.7), where (k1)0, E1, (k
1)0, E
1, (k2)0 E2, (k3)0 and E3 are the rate constants; their activation energies in the reference absolute temperature T0, whereas Eα ¼ E
1
E2 and α0 ¼ k2/k
1, in the reference absolute temperature, too. Eq. (3.16) is useful in the estimation of the activation thermodynamic parameters ΔG‡, ΔH‡ and ΔS‡. 8

  9

  Ea 1 1 > > > > >
>
E1 1 1 > > =
kcat < a0 e R T T0

  ðk1 Þ0 e R T T0 ¼ > Ea 1 1 Km > > > > >
> > ; : T T R 0 1 + a0 e

 
ð E1 + E 1 Þ 1 1
R T T0 ðk Þ ðk Þ e

1 0 2 0 

  ¼ E
1 1 1
E2 1 1


R T T0 + ð k Þ e R T T0 ðk Þ e
1 0

(3.17)

2 0

 
ð E2 + E 3 Þ 1 1
R T T0 ðk2 Þ0 ðk3 Þ0 e

 

  kcat ¼ E2 1 1
E3 1 1


+ ð k 3 Þ 0 e R T T0 ðk2 Þ0 e R T T0

(3.18)

3.3 UNDERSTANDING THE ENZYME KINETICS

85

3.3.1.4 Breakdown of the Simple Theory The dependencies of enzyme kinetic parameters, on both the pH and the absolute temperature values, was based on the “hidden” assumption of the validity of Schemes (3.3) and (3.7). In practice, and from the enzyme–substrate complex ES up to the development of products, the enzymatic reaction passes through more than one intermediate. The simplest case is the three-step mechanism of Scheme (3.7), where (a) is a general case and (b) the case of most of the hydrolases. Hence, the Michaelis–Menten parameters are more complicated versus those corresponding to Scheme (3.3), that is: KS ¼ k
1k+1 k2 6¼ Km ,

Scheme (3.7a)

Scheme (3.7b)

kcat Km

¼

E+S

k2

k
1 + k2 k1 k1 k–1

ES

k1 E+S

k-1

ES

k 2 k3 k1 k2 k2 ¼ k
1 + k2 ¼ KS , kcat ¼ k2 + k3 , and …

k2

EA

k3

H2O

k2

Eacyl P1

E+P

k3

E + P2

Simply, intermediates EA and Eacyl have no effect on the dependencies of kcat/Km and 1/Km due to pH-value of the reaction medium, as concerning changes of E and S; the estimated pKa values also match the E and S. In contrast, the pH dependencies of kcat and Km of the reaction medium are highly affected by the transformations of species EA and/or Eacyl; then, these pH dependencies provide estimates of pKa that correspond to the dominant intermediate enzyme-bound species.

3.3.2 Reversible Enzymatic Inhibition In enzymatic reactions, substrates other than species can be involved, as the inhibitors. Herein, we refer only to reversible enzymatic inhibitors that are bound onto enzymes, to reduce their reaction rates according to Scheme (3.8), and are valuable in detecting enzyme– ligand binding functional relations. Hence, the parameters Km, KI andK/I , denote the establishment of the corresponding equilibria. /

Scheme (3.8) E

+ +

S

Km

I + ES

KI k2

ESI E+P

KI

I

EI

Suitable equations have been established that describe different kinds of inhibitions ½E ½S ½E ½I ½ES ½I = , KI ¼ and KI ¼ ½ESI . Furthermore, Eq. (3.19) was desthrough the relationsKm ¼ ½ES ½EI ignated as the mixed inhibition one, comprising all the three reactions of Scheme (3.8). Eq. (3.20) describes one partial case of Scheme (3.8) where K/I ! ∞, that is, [ESI] ¼ 0, and it is designated as that of the completive inhibition; then, as the Km is multiplied by a quantity >1, the apparent enzyme–substrate affinity is decreased. Eq. (3.21) describes the

86

3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

case where KI ! ∞, that is, [EI] ¼ 0, and it is designated as that of the uncompetitive inhibition; then, both Vmax and Km are divided by a quantity >1, that is, the apparent maximum velocity of the enzymatic reaction and the enzyme–substrate affinity are decreased. In the case where KI  K/I , then Eq. (3.22) is derived and designated as that of the noncompetitive or nonlinear inhibition, where only the apparent maximum velocity of the enzymatic reaction is decreased. v¼

Vmax ½S !  ½I ½I ½S 1 + = + Km 1 + KI KI v¼

(3.19)

Vmax ½S  ½I ½S + K m 1 + KI

(3.20)

Vmax ½S ½I 1+ = KI Vmax ½S ! v¼ ¼ Km ½I ½S 1 + = + Km ½S + ½I KI 1+ = KI Vmax ½S ½I 1+ Vmax ½S KI  ¼ v¼ ½I ½S + Km ð½S + Km Þ 1 + KI

(3.21)

(3.22)

All four Eqs. (3.19)–(3.22) are complex, comprising four parameters, and appear as rectangular hyperbolas. Therefore, it is impossible to distinguish which one could best fit inhibition experimental data, as all of them can do it. Dixon and Cornish-Bowden contributions resolved this problem by means of two complementary diagnostic tests [42, 43], expressed by Eqs. (3.23), (3.24) that are linear transformations of Eq. (3.19). The diagnostic plots of Eqs. (3.23), (3.24) are illustrated in Fig. 3.6. Scheme (3.9) illustrates a particular case of inhibition, designated as substrate inhibition, which is formulated by Eq. (3.25), and it is occasionally observed as [S] >> Km. !  1 1 Km 1 Km 1 ¼ +1 + + ½I (3.23) v Vmax ½S Vmax ½SKI K= I ! ½S 1 1 Km ½S ¼ ðKm + ½SÞ + + ½I (3.24) v Vmax Vmax KI K= I

Scheme (3.9) E+S

Km

S + ES

KSS k2

SES E+P

3.3 UNDERSTANDING THE ENZYME KINETICS

87

FIG. 3.6 Dixon (A1, B1, C1, D1), and Cornish-Bowden (A2, B2, C2, D2) plots, as diagnostic tool of the kind of inhibition; the conclusions on the kind of inhibition is unambiguous. The curved arrows show the increasing of successive [S]. The experimenter should measure initial velocities by varying the [I] at constant [E]t and [S]; the measurements should repeated at no less than two different higher [S], and the results are plotted according to both the Dixon equation (3.23) and the Cornish-Bowden equation (3.24).

v¼

Vmax Km ½S + 1+ ½S KSS

(3.25)

3.3.3 Bi-substrate Enzyme Kinetics (Examples) The bi-bi systems comprise one enzyme, two different substrates, and two products; whereas functional groups are transferred reversibly among reactants through specific intermediates. Complex mechanisms and multiparametric equations describe these systems; thus, specific nomenclature is necessary [44]. Key issues are the order and the kind of substrate binding to the enzyme, and the release of products. In sequential bi-bi mode, both substrates bind onto the system before the reaction starts, either randomly or ordered; both cases proceed by means of a ternary complex. In the ping-pong bi-bi mode, the first substrate binds onto the enzyme and forms an acyl-enzyme; then, the first product is released. The second substrate reacts with acyl-enzyme and forms the second product. The Theorell-Chance is a specific sequential bi-bi ordered mode whose central complex decomposes very fast. The mechanisms in Figs. 3.7 and 3.8 were developed via Cleland’s nomenclature, and comprise horizontal lines for enzymes and enzyme–substrate complexes, and vertical arrows for the consecutive substrate inputs and product releases. Suitable rate equations (3.26)–(3.28) were developed according to significant requirements by the King-Altman method. Fig. 3.7 illustrates the sequential bi-bi mode (types random and ordered), while Fig. 3.8 presents the types of Theorell–Chance, which is described by an equation similar to (3.27), and of ping-pong bi-bi, respectively; Eqs. (3.26–3.28) are associated to Figs. 3.7 and 3.8.

88

3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

FIG. 3.7 Graphic illustration of the sequential bi-bi mode with the corresponding reaction schemes, where (I) is the random type and (II) is the ordered type.

FIG. 3.8 Graphic illustration of (I) the Theorell–Chance type, and (II) the ping-pong bi-bi type; their corresponding reaction schemes ate included.

 ½P½Q V1 V2 ½A½B
Kc v¼ V1 V2 ðKIA KmB + KmB ½A + KmA ½B + ½A½BÞ + ðKmQ ½P + KmP ½Q + ½P½QÞ Kc

(3.26)

V1 ½A½B (3.27) KmA KAB + KAB ½A + ½A½B  ½P½Q V1 V2 ½A½B
K  c v¼  KmA ½B½Q V1 KmQ ½A½P V2 KmB ½A + KmA ½B + ½A½B + KmQ ½P + KmP ½Q + ½P½Q + + KIQ KIA Kc (3.28) v¼

89

3.3 UNDERSTANDING THE ENZYME KINETICS

3.3.4 Kinetics of Immobilized Enzymes Enzymes immobilized on natural or artificial matrices of various types and dimensions have found widespread use in heterogeneous catalytic biochemical and biotechnological processes. In these nonaqueous and/or organic systems, the established dynamic equilibrium differs substantially versus the aqueous ones. Further important factors are the: (a) polarity of solvents, (b) structural and electrochemical characteristics of matrices, and (c) surface charge and configuration of the catalytic site of enzymes. Helmholtz-Stern double layers are formed around solid immobilization matrices, and immersed in aqueous media, where substrates and/or products move toward the enzyme surface and/or to solution only via diffusion effects. Similar microenvironments are observed when immobilization matrices approach the colloid scale, and are more complicated when porous matrices are used [45]. Therefore, the development of rate equations describing the aforementioned complicated systems is unrealistic without rational approximations. It should be assumed that immobilized enzymes exhibit H.M.M. kinetic behavior before and after immobilization, whereas the Fick’s laws of diffusion are valid; thus, the actual concentration of reactants is vague. Additionally, steady-state conditions and single-substrate reactions are also assumed. Hence, any observed change in the substrate concentration [S] is due to two processes, that is, the diffusion of substrate through the double layer, and its consumption by the immobilized enzyme. Eq. (3.29) andits transform equation (3.30) describe the change in [S] in terms of par ∂½S tial derivatives under ¼ 0, where the amount of [S]l depends on the thickness x of dou∂t t ble layer. Subscripts t, dif, E, s, and l denote: total, diffusion, enzymatic, surface (of immobilized enzyme), and layer, respectively, while [S] is the substrate concentration in the solution. The partial derivatives of 2nd degree and 2nd order differential equation (3.30) are transformed to ordinary, as diffusion coefficient D is assumed equal in three dimensions. Systematically, there is not a general solution for Eq. (3.30), and any approach to solve it could be achievable under two distinct conditions, that is, [S]s << Km or [S]s >> Km.     ∂½S ∂½S ∂½S ∂ ∂½S kcat ½Et ½Ss D ¼ + ¼ ¼0
∂t t ∂t dif ∂t E ∂x ∂x ½Ss + Km  d2 ½S‘ ∂ ∂½S kcat ½Et ½Ss kcat ½Et ½Ss  D or ¼  ¼ 2 ∂x ∂x ½Ss + Km D ½Ss + Km dx Subsequently, for [S]s < < Km Eq. (3.30) degenerates to

d2 ½S‘ 2

dx

¼

(3.29) (3.30)

kcat ½Et ½Ss ¼ C1 ½Ss , where D ðK m Þ

kcat ½E0 d2 ½S‘ kcat ½Et ¼ C2 , where ¼ constant, whereas for [S]s >> Km degenerates to ¼ DKm D dx2 kcat ½Et ¼ constant. Partial solutions of the simplified forms are achieved by assuming that C2 ¼ D for x ¼ ‘ (on the surface of immobilized enzyme) is valid that [S] ¼ [S]s, whereas for x ¼ 0 (in the solution) is valid that [S] ¼ [S]t. Therefore, substrate concentrations are related by means of the p pffiffiffiffi pffiffiffiffi ffiffiffiffi pffiffiffiffi ½Ss eð‘ + xÞ C1
eð‘
xÞ C1 + ½St eð2‘
xÞ C1
ex C1 fC2 ‘ðx
‘Þ + 2½Ss gx
2ðx
‘Þ½St p ffiffiffi ffi ¼ : the relations ½S‘ ¼ 2‘ C1 2‘ C1 ¼

e


1

90

3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

Nevertheless, to estimate the values of [S]s, [S]‘, and [S]t, one of them should be known, and this is not easy. Alternatively, an assumption has been introduced that the jth constituent of the catalyzed reaction by the immobilized enzyme is distributed between the enzyme’s sur½ js ½ j and equilibrium [j]s > [j]sln; sln deface and solution according to the coefficient Ps,sln ¼ ½ jsln app k ½Et ½S notes solution [46]. Next, by introducing more assumptions, the rate equation v ¼ cat app is ½S + K m delivered, which fits easily into the experimental data, as [E]t is known, and [S] is well monapp itored. Additionally, the system is represented by the entities kapp cat and Km , which are comprehended by considering the flux rate per unit surface of substrate from solution to the matrix surface, the electrostatic valence of immobilization matrix, and the electrochemical valence of charged reactants [45].

3.3.5 Enzymes in Non-aqueous Organic and/or Multiphasic Media Methods of treating enzymes in nonaqueous and/or organic media have been reported. A first approach was the direct dispersion of lyophilized enzymes and substrates in the chosen solvent, whereas newer alternatives were targeted to increase the effectiveness of enzymes as catalysts in organic media. Enzymes in reversed micelles, chemically modified, coated, and immobilized enzymes are further approaches that have been extensively applied; the enzymes in reversed micelles found a lot of biotechnological applications. In contrast, various reports argue for the increased, rather than decreased, enzymatic activity in organic solvents; this latter is not a rule, and it should be considered according to the examined reaction, and by taking into account the mass transfer effects [47,48]. 3.3.5.1 Catalysis by Immobilized Enzymes The synthetic enzymatic reactions are favored in nonaqueous organic media, providing advantages to industrial biotechnology. Focus should be given to proper substrate(s), for a successful catalysis in nonconventional media, only due to solvation effects. Immobilized enzymes offer much in the improvement of a substrate’s and product’s reactivity, specifically in continuous reactors, where fast handling of reactants—immersion and/or withdrawing—is crucial. The catalytic efficacy of enzymes is based on their structure, depending on many hydrogen and hydrophobic bonds, and other interactions in which the solvent participates. Therefore, technologically, enzymes are less useful in aqueous media than in organic solvents, where they show increased stability and perform catalytic reactions impossible in aqueous environments. These are irrelevant to whom is considering that proteins denaturize in media containing high percentages of organic solvents; however, in neat and/or dehydrated organic solvents, enzymes are stable due to their rigidness in contrast to their flexibility in hydrated media [49]. In hydrophobic organic solvents, enzymes are thermally stable. In single-phase organic reaction media, traces of water molecules may be present on the enzymes’ surface, which have been found to be exceptionally effective as enzymes maintain their catalytic activity. It is advantageous to use immobilized enzymes as: (a) immobilization matrices holding enzymes, which are separated easily from the reaction mixtures, while there is control on the reaction time and reduction of enzyme loss; (b) these matrices can

3.3 UNDERSTANDING THE ENZYME KINETICS

91

be reused in multiple reactions, and thus minimize the cost of catalysis; (c) purer products are easily separated from the reaction mixtures that are manufactured; (d) proper design of the reaction system may enhance the enzymatic activity; and (e) diffusional limitations can be better managed [22].

3.3.5.2 The Crucial Role of Water Activity A sense of the water’s impact on enzymes acting in nonaqueous organic media was given in 3.5.1. A few water molecules can resist careful dehydration, they remain attached onto enzymes’ surfaces, and cannot be totally managed; a property like that is useful in cases of lipase catalyzed reactions in organic media. In several enzyme-catalyzed synthetic reactions in organic solvents, in either free or immobilized form, water’s accumulation affects the composition of the medium with consequences on the reaction rate and the percentage product yield. Another important point is the development of water/organic solvent interfaces, which are potentially formed in the presence of an adequate quantity of water in the reaction medium, and affect the enzymatic reactions. However, what an “adequate quantity of water” means depends on numerous factors, such as the enzyme in free or immobilized form, the presence of substrate(s) or of surfactants, the concentrations of all reactants and products, and so forth. In the nonaqueous media, the water is quantitatively measured by its activity aw ¼ p/po, a thermodynamic entity, where p and po are the water vapor pressures over a substance and over pure water, respectively. The estimation of aw is not an easy task, especially for reactions in organic solvents. Simple methods comprising the use of molecular sieves or pervaporation membranes are used to continually remove water from nonaqueous organic reaction media, when it is necessary [10].

3.3.5.3 Nature and Influence of the pH-Value (Catalysis in Nonaqueous/ Multiphasic Media) The concept of pH-value is obscure in nonaqueous organic systems, as the Brønsted pH scale is meaningless in the absence of water. On the other hand, the pH-values of reaction media influence enzyme functional groups and rates by means of appropriate ionizations, and in organic solvents, according to the well-known phenomenon of pH memory. Only in aqueous solutions is it valid that pH ¼
log10[α(H+,aq)], and it is dissimilar to any determined pH-value in nonaqueous media; α(H+,aq) denotes the activity of protons in aqueous homogenous systems. The IUPAC-Commission of Electroanalytical Chemistry has defined Eq. (3.31) in water/organic solvents mixtures where m and γ m are the molality and the proton activity coefficient in the liquid phase under consideration, respectively. Unified quantitative scales that compare acidities among different media have been established, and are based on the absolute chemical potential μabs(H+,slv) of protons in the liquid phase independently of the medium; slv denotes the solvent liquid phase [50]. In these relations, α(H+) is the activity of protons, and μ‡abs(H+) is the standard chemical potential that is considered as a reference of gaseous protons at zero point, and it was set axiomatically to 0 kJ/mol, according to Eqs. (3.32) and (3.33). The μabs(H+,g) in liquid phases is lowered by its ΔslvG‡(H+,g), due to proton transfer from an ideal gas phase (1 Atm) to an ideal 1 M solution (pH ¼ 0). Under standard conditions, the absolute acidities of solutions are comparable,

92

3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

as μabs(H+,slv) ¼ ΔslvG‡(H+)
[pH  (5.71 kJ/mol)] whose division by
5.71 produces Eq. (3.34), that is, the assignment of pHabs-values [51]. pH ¼
log 10 ½αðH + Þ ¼
log 10 ½mðH + Þ  γ m ðH + Þ

(3.31)

μabs ðH + Þ ¼ μðH + Þ
μ‡ ðH + , g298:15 K Þ ¼ 0 kJ=mol

(3.32)

  Δslv G‡ H + , g ¼ μ‡ abs ðH + , slvÞ

(3.33)

pHabs ¼

μabs ðH + , slvÞ
5:71 kJ mol
1

(3.34)

3.3.6 Computational Part The fitting of experimental data into suitable equations is more demanding as complex biochemical and biotechnological processes are developed; then, rather complicated nonlinear multiparametric equations can fit the data and obtain parameter estimates. The parameter values of nonlinear differentiable equations are customarily estimated using gradient algorithms. However, for equations with discontinuous derivatives, it is necessary to use search and/or nonparametric methods. Gradient and search methods are available by means of many publications and suitable integrated packets; however, nonparametric methods are less available. In its simplest idea, as simplex is designated a geometric scheme with n + 1 vertices, where n is the number of parameters of the fitting equation; for two parameters the simplex is an equilateral triangle. The search originates through an initial simplex, by introducing guessing of parameter values, which correspond to the n simplex vertices; then a mirror image of the initial simplex is formed, and new parameter values are evaluated. If new parameters are unsatisfying, then a new trial forms a third simplex, which is a mirror image of the previous, but lengthwise to a different direction. The iterations are continued by testing various directions and/or simplexes with varying edges to reach a set of parameters satisfying the experimental data. The pattern search method starts with a local search, using the introduced guessing parameter values, and it depicts equally spaced points that are connected to a minimum, testing one parameter at a time; the estimated minimum, that is, the base, is stored, and a new pattern search starts. Then the algorithm searches for new bases to reach a set of parameters satisfying the experimental data [52, 53]. Nonparametric fitting methods are extremely useful, and may be the only alternative when a high occurrence of outliers is detected in a set of experimental measurements. This problem is increased when a small number of replicates are available, and the fitting equation is complex, that is, the case in biochemistry and biotechnology, where the experimental errors are not normally distributed. The concept, which was developed by Cornish-Bowden, has been extended and systematically applied to numerous useful equations [38]. The program, in the case of nonparametric fitting, finds, develops, and evaluates all possible intersections of the formed hyperplanes [40, 41]. The lists of all referred programs, including the appropriate statistical treatment, are available on request.

3.4 ADDITIONAL OPERATIVE METHODOLOGIES

93

3.4 ADDITIONAL OPERATIVE METHODOLOGIES 3.4.1 H/D Solvent Isotope Effects, Kinetic Isotope Effects, and Proton Inventories Deuterium isotope (D) can potentially replace the protium isotope (H) in water species (HOD, D2O, H+3O, H+2OD, etc.); similar replacements are observed on susceptible sites of enzymes and substrates under particular conditions. These phenomena are termed as H/D solvent isotope effects (S.I.E.), and in most cases, influence kinetic and equilibrium constants of enzymatic reactions, which are associated with isotopic solvents, and affect ground and transition states. Therefore, sensitive hydrogenic sites that contribute significantly in enzymatic mechanisms can be identified by means of S.I.E., and quantified using the isotopic fractionation factors ϕ. Generally, the hydrogenic sites are sorted as internal (framework of solutes) or external (isotopic water molecules), and interact with the solutes [18, 28]. H/D kinetic isotope effects (K.I.E.) are observed when reactions are receptive to hydrogen isotopic exchange, and are robust tools in investigating enzymatic mechanisms where steps of hydrogen transfers are involved. The K.I.E. provide information on important features of the enzymatic reaction, such as the rate-limiting steps, the organization of transition states, and the changes in the catalytic site [54]. The K.I.E. can be estimated also using NMR spectroscopy as functions of the deuterium atom fraction n in the exchangeable hydrogenic sites, and by means of the study of kinetic isotopic exchange reactions. Proton inventories (P.I.) originate from the preceding phenomena, they have been proven as reliable probes for the elucidation of enzymatic mechanisms, and comprise S.I.E. studies in a series of isotopic water mixtures. Subsequently, reaction rate and/or equilibrium constants that are expressed as kn(n) and/or Kn(n) functions of the deuterium atom fraction n 5 [D2O]/ ([H2O] + [D2O]) in the solvent are related to the corresponding fractionation factors according to the Gross-Butler-Kresge equation (3.35), where k0, kn, φTi , φG j and n are the reaction rates in H2O and D2O, the isotopic fractionation factors of ith transition state and of jth ground state protons, and the deuterium atom fraction, respectively. Simplified forms of Eq. (3.35) can be formulated based on the previously described approaches in order to fit various enzyme kinetic experimental data appropriately. Eq. (3.36) is a specific form of (3.35) applicable in situations when the substrate is binding onto the enzyme, or the formation and breakdown of acyl-enzyme are regulated by a virtual transition state that contributes two steps, a physical and a chemical one. Z1 is a composite fractionation factor related to the solvent reorganization, Cph, CCh, ϕT,Ch are the fractionation factors of physical and chemical steps, and of the transition state of chemical step, respectively, while μ are the transition state protons that are transferred during the chemical step. μ  Q

kn ¼ k0

i¼1

1
n + nφTi



 ν  Q 1
n + nφG j

(3.35)

j¼1

Zn1

kn ¼ k0 CPh +

CCh μ ð1
n + nφT,Ch Þ

(3.36)

94

3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

3.4.2 Liquid/Solid-State and Heteronuclear NMR Spectroscopy in Hydrogen Transfer Reactions NMR spectroscopy supplies the enzymologists by structures of proteins and their complexes at atomic resolution, and is informative about conformational dynamics and exchange processes of enzymes at timescales up to picoseconds, which are significant in enzyme kinetics. The energy that is absorbed by the variation of the spin orientation of a nucleus within a magnetic field results in an NMR spectrum; protons are commonly used due to their high natural abundance and gyromagnetic ratio. Common NMR interactions are the through-bond spin–spin and the through-space nuclear Overhauser effects, which both contribute in the determination of three-dimensional structures and dynamics of enzymes in solution. The NMR spectroscopy of deuterium in solutions derives analogous spectra as those of protium, with broader peaks due to quadrupole interactions. Currently, the NMR spectroscopy deals mainly with 1H, 13C, 15N and 31P and several other nuclei, as concerns the biochemistry and biology; the nucleus 13C NMR spectroscopy, which was considered complicated due to the high incidence of 13Cd1H coupling, where many protons are involved, has been improved through wide-band proton decoupling. Additionally, the 17O NMR spectroscopy in solutions has been developed into a useful technique that solves structural drawbacks of small organic molecules; on the contrary, the 19F NMR spectroscopy has more favorable characteristics, while the 15N NMR spectroscopy, and despite its natural abundance, is very useful in cases of interacting sites of biomolecules (basic residues as histidine, amide nitrogens of peptide bonds, etc.) due to recent instrumentation. Enzymatic catalysis occurs within microseconds to milliseconds; thus, the detection of dynamic processes is performed through various techniques, as the fluorescence resonance energy transfer (FRET), the stopped flow fluorescence, and so forth, which deal with the dynamics of individual sites. The protein NMR spectroscopy can detect simultaneous motions of atomic sites carried out within picoseconds [13, 19, 24, 31].

3.4.3 Circular-Dichroism Circular dichroism (CD) is a useful tool in the research fields of proteomics and structural genomics, and depends on the differentiation between the absorptions of left and right circularly polarized radiation of chromophores due to their intrinsic chirality, which generates appropriate CD signals. The method is informative in evaluating conformations and stability of enzymes owed to temperature, ionic strength, and other changes, contributing to the comprehension of protein folding procedures. This becomes important because the folding-unfolding pathways of enzymes have an immediate practical impact on the elucidation of their mechanisms. Therefore, CD spectroscopy is well applied in estimating the secondary structure of the asymmetric molecules of enzymes. The CD spectra of enzyme–ligand complexes could develop extra CD bands and changes in ellipticity θ, in the UV range, which provides important insights about the enzyme molecule folding. Fig. 3.9 is informative in this respect [13, 19, 24, 31].

3.5 MODELING OF ENZYME REACTIONS

95

FIG. 3.9 CD spectra of characteristic structures. Modified from E.M. Papamichael, P.-Y. Stergiou, A. Foukis, M. Kokkinou, and L.G. Theodorou, Effective kinetic methods and tools in investigating the mechanism of action of specific hydrolases, in: InTech-Medicinal Chemistry and Drug Designs, Chapter 12, Rijeka, Croatia, 2012, pp. 235–274.

3.4.4 Mass Spectroscopy Α rapidly expanding research tool, regarding its sensitivity and accuracy, is mass spectroscopy (MS), which offers exceptional advantages in the study of biochemical and biological systems, allowing in-depth investigation of proteins, protein–ligands interactions, and enzyme catalysis; the techniques of electrospray ionization (ESI) and of matrix-assisted laser desorption (MALDI) are nowadays necessary complements of the conventional spectrophotometric ones. MS techniques are promising for enzyme kineticists as neither chromophoric nor radioactive substrates are required, and are also helpful in cases of pre-steady-state kinetic measurements, as well as in H/D exchange experiments. Currently, the research in genomics and proteomics finds MS as the basis in identifying and characterizing cellular proteins, expression levels, posttranslational modifications, and products of the metabolism [55].

3.4.5 Rate-Limiting Steps It is not uncommon to regard as synonymous the terms rate-determining step, rate-limiting step, and rate-controlling step; however, kineticists should avoid confusion. The term “ratedetermining step” fits rather to a specific case of the rate-controlling step, in which an initial slow step is followed by successive rapid steps. The rate-limiting step is the status at which maximum activation energy is required, that is, a TS, which possesses the highest free energy and is essential to realizing what the reaction rate is. Therefore, as concerns multi step reactions, their slowest step is the rate-limiting or the rate-determining step; this is based on the detection of the highest energy transition state, and it may be incorrect in the case of multistep irreversible reactions with more stable intermediates than their reactants [28].

3.5 MODELING OF ENZYME REACTIONS The construction of a reaction model is based on the collection of a series of data, their organization, and illustration, as well as on reasonable hypotheses; this is more complicated in the case of enzyme catalyzed reactions. In its common concept, a reaction model is thought

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3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

of as the formalized description of what is known, and it is defined through mathematical expressions. A schematic representation of the previous sentences is depicted in Fig. 3.10. Within the different steps illustrated in Fig. 3.10, various choices and decisions should be considered, depending on both the objectives of modeling and the previous steps; besides, the details of individual modeling build up differently according to the case; in addition, a number of these steps require several iterations before completion of the model [56]. Additionally, the formulae of models, during their building processes, more likely undergo evolutions due to the fact that new elements are added, while other are removed or changed; thus, the estimated values of the model parameters should be reconsidered as additional analyses that may be required. The potential procedures leading to enzyme kinetic modeling are summarized in brief as follows, along with their aspects, and according to Fig. 3.10. The aim of the modeling procedure should be first defined including hypothesis testing, and understanding the interactions of different components of the system under consideration, which are hard access experimentally; hence, the complexity of the modeling problem, and its influence on all the subsequent steps of modeling should be known. The system configuration comprises the accessible biochemical data that are required and can be translated into mathematical expressions; a schematic depiction of the latter may be the significant viewpoint that leads to an approved system configuration. To accomplish analogous tasks uniformly, verbal communication is necessitated, such as the Systems Biology Graphical Notation (SBGN) [57]. The reader should recall that Section 3.3 discusses the rate equations, which are expressions describing the reactions through interactions involving their reactants, within the system configuration, as well as the law of mass action and its usefulness (Section 3.2.1). There, the derivation of reaction mechanisms and of multiparametric rate equations were performed by means of exploitation of specified experimentation and modeling, where the estimation of parameters was not easy work. It should also pointed out that enzymatic reaction rates are strongly dependent on issues that were not taken into account during modeling, such as the pH and temperature values, the ionic strength of the reaction mixture, and so forth. The smaller the number of model parameters, the easier is their realistic physical meaning and estimations. Nevertheless, rate equations described by parameters characterized as apparent is not uncommon; the physical meaning of these latter entities (apparent parameters) depends on the system configuration. The vast majority of the model equations in enzyme Rate equations

Aim

Equations (linear, nonlinear)

Parameters' estimation

Validation

Use of the model

System configuration

FIG. 3.10 Schematic representation of the different steps leading to a kinetic modeling operation. Modified from J. Almquist, M. Cvijovic, V. Hatzimanikatis, J. Nielsen, M. Jirstrand, Kinetic models in industrial biotechnology—improving cell factory performance, Metab. Eng. 24 (2014) 38–60.

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kinetics are nonlinear, nonpolynomial, and multiparametric, as it has been described in detail in Section 3.3, where various useful methods were discussed for fitting the corresponding experimental data and estimating its parameters similarly in the cases of awkward equations with noncontinuous derivatives. After estimating the parameter values, the quality of the model could be assessed by comprising both qualitative reasoning and formal statistical investigation. The establishment of the models’ quality should precede their use, and a priori available estimates of their parameter values, as well as additional features, should be taken into account. Although these aforementioned controls stand as model validation, in general, models cannot describe and/or fit all similar experimental data; the more sets of data that are fitted by a model, the more rational it is, and it can be reused. All employed models should continually be validated before and after the estimation of their parameters; a fast qualitative judgment can be achieved by simple visualization of the figure comprising the experimental points and the fitting curve [58]. Other formal statistical methods for model validation have been given in detail in Section 3.3.1.2. There are cases in which the experimenter should distinguish among rival models that best describe the data points and provide values for parameters with physical meaning. Except for the criteria given in Section 3.3.1.2, the following F-test likelihood ratio test [56], Akaike’s information criterion-AIC, and Akaike Weights [59], are in common use. A validated and well established model can be generally useful in similar multiple enzyme kinetic systems, whereas the prediction, estimation, and investigation of various hypotheses show its importance and acceptance.

3.6 IN SILICO ENZYME MODELING The remarkable progress over the past few years on the comprehension of enzymatic reactions and mechanisms has challenged the development of novel biotechnological products. In many cases, the recognition of enzymatic kinetic mechanisms has been proven extremely difficult, based only on experimental work. Therefore, computational molecular modeling and simulation methods were found as excellent complementary tools of the experimental techniques, as we have mentioned herein. Thorough and reliable computational methods are outlined in this section, which contributes significantly to the study and identification of catalytically important issues of enzyme kinetics, of their structures, interactions and transition state intermediates. Various aspects and insights about the enzymatic action can be elucidated, as well.

3.6.1 Molecular Mechanics Methods Molecular mechanics (MM) methods are applied for molecular dynamics simulations of enzymes, or in combination with quantum mechanical methods in QM/MM calculations (see the following) [60]. These methods are limited to model processes in which details at the atomic level are required; that is, in conformational changes or in enzyme–ligand binding,

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and are not suitable in reactions. Similarly, dynamic systems of several thousand of atoms can be simulated for long time-scales (nano- to micro-seconds) by MM methods using simple energy functions (force fields) and van der Waals interactions through simple Lennard-Jones functions; electrostatic interactions are also accounted for by using fixed atom-centered point charges, whereas electronic polarization is not included. This is a crucial point, and currently is a subject for further research. Nevertheless, intermediates and transition states cannot be examined by standard MM force fields due to their ineffectiveness in cases of electronic reorganization, as well as in modeling of bond making and breaking [61]. However, MM functions may be potentially developed for particular enzymatic reactions, where reparameterization is mandatory.

3.6.2 Quantum Mechanical Methods Quantum mechanical methods (QM), or electronic structure or quantum chemical methods, are suitable in modeling enzymatic systems and studying the chemical steps of catalysis; the electron density of these systems may also be computed. Most QM methods are based on wave functions, and are used to find solutions of the Schr€ odinger equation through reasonable approximations; the Schr€ odinger equation cannot be solved exactly for molecular systems comprising more than one electron [61]. The QM methods are grouped on the basis of approximation used to treat the Schr€ odinger equation, as follows [61, 62]: (a) ab initio QM methods based on the assumption that electrons’ spatial distribution does not depend on instantaneous motions of other electrons; the electron’s correlation by applying the Hartree–Fock (HF) theory is neglected. Hence, HF calculations are prone to significant errors in the estimated values of the reaction’s total energy. An interesting development is the use of highly correlated ab initio methods to obtain more accurate energies over HF calculations whose application is limited to large systems due to higher computational cost. (b) Density functional theory (DFT) can model large systems with sufficient accuracy at substantially lower computational expense. From the modeling point of view, groundstate energies of molecules are estimated directly through their known electron density distributions. The density function in DFT is much simpler than the ab initio wave function, with the hybrid functional B3LYP2 being particularly prevalent. (c) Semi-empirical methods which can be applied to larger systems than that the DFT or the correlated ab initio methods could be applied; molecular dynamics simulations could also be applied to larger systems providing a good balance between computational cost and accuracy. It is important to underline that semi-empirical methods are typically the less accurate QM, excluding the cases where they are specifically parameterized for a particular property. The QM methods are dealing with a small part of the enzyme molecule as its entire modeling is computationally costly. In contrast, the accurate modeling of an enzymatic reaction is achievable through quantum mechanics/molecular mechanics (QM/MM) approaches and/ or cluster approaches.

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3.6.3 Quantum Mechanics/Molecular Mechanics Methods The QM/MM approach is a powerful method for modeling enzymatic reactions, as they combine the benefits and the strength of both the QM and MM methods. The basic idea of this methodology is the partition of the system under investigation in two regions: a small part, that is, the QM reacting region of the system, and the rest of the enzyme molecule which is treated with a MM force field (MM region) [60–62]; this concept is represented in Fig. 3.11A. An important issue is how the atoms of the QM and MM regions interact. Generally, three classes of interactions are accounted between atoms: (i) in QM region, (ii) in the MM region, and (iii) in QM and MM regions. The interactions within the QM and MM regions are treated easily at the QM and MM level, respectively; whereas the more complicated interactions between the two subsystems are accounted for by using several approaches [60]. There are two main approaches to QM/MM calculations, that is, the subtractive and the additive coupling ones. In a subtractive approach (ONION method in Gaussian), the total energy (QM/MM energy) of the system is calculated stepwise. First, the total energy of the system (QM and MM regions) is evaluated using the MM methodology, followed by the incorporation of the isolated QM energy. Consequently, the MM energy of the QM subsystem is calculated and subtracted; a last step corrects the calculations by including the interaction within the QM subsystem twice. Eq. (3.37) and Fig. 3.11 are explanatory. Despite this coupling approach, the quantum chemistry and molecular mechanics routines are independent, and a force field that is sufficiently flexible is required for the QM subsystem. In the additive approach, the QM system is incorporated in the larger MM subsystem, and the potential energy of the total system is a sum of MM and QM energy terms, as well as QM/MM coupling terms, according to Eq. (3.38). VQM=MM ¼ VMMðMM + QMÞ + VQMðQMÞ
VMMðQMÞ Abstractive

(3.37)

VQM=MM ¼ VQMðQMÞ + VMMðMMÞ + VQM
MMðQM + MMÞ Additive

(3.38)

Hence, it is important to note that only the interactions within the MM region are described at the force field level VMM(MM); the interactions between the two subsystems are treated explicitly. VQM
MM(QM+MM), in contract to the subtractive coupling approach, can be described using sophisticated approaches, and therefore for any analogous figure, it would be more or less confusing [62, 63].

3.6.4 The Quantum Chemical Cluster Approach The quantum chemical cluster approach (or all-QM approach) is another extremely valuable method for elucidating enzymatic reaction mechanisms, as well as other specific properties of enzymes [63–65]. The principle of this method is based on the treatment of a small part of the enzyme molecule around the catalytic site (minor parts of the active site could be included) using quantum chemical methods; the rest of the enzymatic system is accounted for by two simple approximations. In order to model the steric influence of the enzyme molecule in a region around its catalytic site, a coordinated locking diagram is used, comprising a number of atoms whose locations are crystallographically rigid. Generally, one approach is to

100 (A) An example of application of QM idea to an advanced level in a part of a region, where a reaction is taken place, as it not likely to be expressed through the force field theory. (B)–(D) Example of subtractive QM/MM coupling, by assuming that the QM/MM energy of an enzymatic system equals to the energies of the isolated QM region, plus that of the complete system, and minus the energy of the isolated QM region; another assumption is that a force field is available for the QM region, while the latter energy term has been used in order to correct the inclusion of a double contribution of the QM region to the total energy.

3. ENZYME KINETICS AND MODELING OF ENZYMATIC SYSTEMS

FIG. 3.11

3.7 CONCLUSIONS AND PERSPECTIVES

101

construct large models, as for too small models, the process could lead to the wrong energy profiles due to artificial strains. On the contrary, in larger models, multiple minima errors are more likely, and thus a good practice is to start with a relatively small model, increasing it gradually in size. The electrostatic influence of the enzyme environment is accounted for by using implicit solvation models. Due to this approximation, it is assumed that the enzyme environments are homogeneous and polarized, comprising proper dielectric constants, which, along with the model size, are crucial in yielding reliable results. Unraveling the mechanism of an enzymatic reaction by the cluster approach relies on the use of electronic structure methods; the (DFT) method, and in particular, the B3LYP2 functional method are extensively used in the cluster approach, providing a good correlation between speed and accuracy. An interesting recent development in this context is the DFTD technique, which, in combination with the use of highly correlated ab initio methods, results in significant improvement in accurate energy calculations [63]. The estimated values of energies are usually compared with the experimental rates by means of the classical transition state theory. In cases where kinetic isotope effects are prerequisites to establishing an enzymatic reaction mechanism, they can be included in the cluster approach by other approximations. It is important to point out that the cluster approach starts from the enzyme–substrate complex (chemical step of enzymatic reaction), where the entropy effects are rather small. So, a good practice is to approximate the free energy with the entropy, only in cases where a gas molecule doesn’t appear during the reaction. Another technical point is to include in the cluster approach the experimentally estimated values of redox potentials, or pKa values (if available), in cases where protons or electrons are included in the model that describes the region around the catalytic site; the environment strongly affects the energies under calculation. Over the years, the cluster approach has become a robust, reliable, and highly versatile technique that has been applied to a broad variety of enzyme families and contributed enormously to their reaction mechanism [63–65].

3.7 CONCLUSIONS AND PERSPECTIVES Although the leading concept of theories introduced by Henri, Michaelis, Menten, and others sounds simple nowadays, as based on elementary assumptions and simple mathematic relations; they have proven to be pioneering, due to their widespread use in enzyme kinetics more than 100 years ago. On the other hand, the development of new, efficient laboratory instrumentation, the novelties in statistical treatment of experimental data, the sophisticated algorithms, the progress in applied mathematics, and the tremendous evolution of computers could not leave the enzyme kineticists unconcerned. These latter tools became powerful, and thus the modeling of enzymatic reactions provided advantages in the elucidation of a plethora of enzymatic mechanisms. Herein, we have presented and considered systematically significant methods, including the modern in-silico approaches, which can be applied for efficient enzyme kinetics and modeling of enzymatic systems in which the dynamics will develop and evolve continuously toward the future.

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APPENDIX: SUPPLEMENTARY MATERIAL Supplementary material related to this chapter can be found on the accompanying CD or online at https://doi.org/10.1016/B978-0-444-64114-4.00003-0.

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