Reaction Kinetics and Stoichiometry

2.05Reaction Kinetics and StoichiometryK Miyanaga and H Unno,Tokyo Institute of Technology,Yokohama,Japan©2011Elsevier B.V.All rights reserved.2.05.1Introduction33 2.05.2Enzyme Kinetics34 2.05.2.1Michaelis–Menten Equation34 2.05.2.2Initial Rate35 2.05.2.3Experimental Determination of Kinetic Parameters in the Michaelis–Menten Equation35 2.05.2.4Complex Enzyme Kinetics36 2.05.2.4.1Allosteric enzyme36 2.05.2.4.2Two substrate reactions37 2.05.3Factors Affecting Reaction Kinetics38 2.05.3.1Inhibited Reaction Kinetics38 2.05.3.1.1Inhibition of reversible inhibitor38 2.05.3.1.2Substrate inhibition39 2.05.3.1.3Product inhibition40 2.05.3.2Deactivation of enzyme40 2.05.3.3pH Dependency of Reaction Rate41 2.05.4Biochemical Reaction Rate Related to Cellular Systems42 2.05.4.1Growth Rate43 2.05.4.2Substrate Uptake Rate43 2.05.4.3Product Formation Rate44 2.05.5Stoichiometry44 2.05.5.1Yield Coefficient based on Mass44 2.05.5.2Biochemical Stoichiometry45 2.05.5.3Heat of Reaction46 2.05.5.4Yield Coefficient based on ATP Synthesis46 References46Glossarykinetics Generally means a quantitative study of phenomenological speed(rate)in relation with relevant factors.In this article,kinetics refers to chemical reaction speed relevant to enzyme and/or cell.material/energy balance In any chemical reaction, elemental amount is preserved during the reaction process.To confirm this principle,material balance is evaluated in any chemical reaction process.Energy is also preserved in any chemical reaction process,where input energy and/or reaction energy must be taken into consideration for taking the balance.rate Changing speed of items is expressed as rate. Growth rate is increasing speed of biomass or number of cell,such as microbial,animal,and plant cells, under appropriate culture environment.Substrate uptake rate and product formation rate implies speed of relevant affairs.stoichiometry Generally means quantitative relationships among the affairs,like amount of materials and/or energy, of reactants and products in chemical reactions.In this article,quantitative consideration of the material and energy balance of biochemical reactions are discussed. yield Yield is the ratio of amount of target affair changed during any event period of time to any reference affair. Yield does not include time-course change.Growth yield is the increased biomass referred to concerned item,such as uptaken substrate amount and uptaken oxygen amount.Unit of yield depends on what unit is used for the relevant items.In the similar manner,product yield is defined,which is used for quantitative analysis for the process design.2.05.1IntroductionOne of the aims of biotechnology is to apply biofunction(bioreaction)occurring in living organisms as biocatalyst to produce materials and/or systems useful for human beings.Bioreaction is categorized into two,single enzyme reaction and cell reaction,the latter occurring in cells such as microbial,animal,and plant cells.The cell reaction is also catalyzed by enzyme,but the reaction involves many enzymes like a consortium of many enzyme reactions,which is different from the single enzyme reaction.3334 Underlying PrinciplesGenerally in the engineering application of any reaction, the reaction rate or kinetics, the material and energy flow have to be quantified. In addition, for the materialization of the reaction process, bioreactor and further bioprocess, an integra­tion of the bioreactor and the up-and downflow processes has to be designed. For the design, material and energy balances are made. In this procedure, the material and energy flow must be appropriately evaluated for the relevant bio-reaction. These flows for single enzyme reaction are relatively simple, because chemical component involved in the reaction is clear and the activation energy can be evaluated definitely. In addition, the reaction rate can be measured based on any definite base.On the other hand, the material and energy flow in cell reaction is complicated due to the involvement of many and frequently unidentified reactions. Even though the raw material is unified from the target product, the elemental balance, say carbon balance, cannot necessarily be completed only by the raw material and the product because of the involvement of many reaction schemes that may take carbon from another source. For the reaction energy, the situation will be more complicated. A reaction will release or absorb energy according to the reconstruction of substances in the reaction. Reaction energy from the process is a resultant of all reactions involved. When considering cell reaction, all elemental reactions are impossible to be quantified completely. Therefore, it is necessary to establish methods for evaluating such situation which is practicable to the design of the bioreactor and/or process and to unify the operational parameters. Kinetics and stoichiometry are the fundamental and inevitable knowledge for the above consideration.2.05.2 Enzyme Kinetics2.05.2.1 Michaelis –Menten EquationIn the usual single-substrate-enzyme-catalyzed reaction, the relationship between initial rate of reaction and the substrate concen­tration assumes the form of saturation curve. A mathematical model to describe the kinetics was developed by Henri in 1902 [8] and Michaelis and Menten in 1913 [9]. Furthermore, Briggs and Haldane [5] generalized this model using the quasi-steady-state assumption in 1925. A simple schematic diagram is described in Figure 1.Henri and Michaelis and Menten assumed a production reaction of enzyme –substrate (ES) complex derived from free enzyme and substrate rapidly reaches equilibrium state. In equilibrium state, the following equation is derived:k 1E ½ ¼k − 1 ES ½½ S ½1The dissociation constant (K s ) of ES complex isk − 1E ½ ½ S K S ¼¼½2k 1ES ½ On the other hand, in the quasi-steady-state, it is assumed that the initial substrate excessively exists in the batch system comparedwith the initial enzyme, and the concentration of ES complex rapidly reaches steady state under the condition where substrate concentration [S] is fixed. It means that time course of ES complex concentration is assumed as zero. In addition, we must be conscious of the limitation that this perspective can be applicable only under the condition where ES complex concentration is sufficiently low.d ES ½¼k 1E ½ ½ ¼0 ½½ S −ðk − 1 þk 2ÞES 3 d tK m ¼E ½ S ½¼k − 1 þk 21 ½4Since total enzyme concentration, [E 0], is constant in the batch reaction systemE 0½ ¼E ½ þES ½The concentration of ES complex isES ½ ¼E 0½ S ½ m þS ½½5k 1 k2++k –1Enzyme SubstrateEnzyme–subtrate (ES)Enzyme Product complex E+SESE+PFigure 1 Schematic diagram of enzyme reaction.Reaction Kinetics and Stoichiometry 35V(=k 2m m ′Figure 2 Relationship between substrate concentration and initial rate in the Michaelis –Menten-type enzyme reactions.The product formation rate, v , is defined as follows:d P ½v ¼¼k 2 ES ½ d tk 2 E 0½ ½ ½ S V m S ¼¼½6K m þ½ S K m þS ½where V m (maximum reaction rate) is k 2[E 0]. So far, in many enzyme reactions it is revealed that k −1 is much larger than k 2(k −1 >> k 2). Therefore, K m is corresponding to K s (K m ≈ K s ). Generally, eqn 6 and K m are often called the Michaelis –Menten equation and the Michaelis constant, respectively. As shown in Figure 2, the Michaelis constant can be determined from the substrate concentration which gives a half of the maximum reaction rate. Smaller K m indicates that the enzyme can easily combine with its substrate, implying high affinity to substrate.2.05.2.2 Initial RateInitial rate of reaction is defined as the reaction rates under the assumption that substrate consumption and reaction product do not affect the reaction. The initial rate is calculated by a correlation between substrate concentration or product concentration change and reaction period insofar as their time course is estimated as a linear relationship. The time-course data, in the later period, are affected by a variety of factors. Generally, in the case of irreversible reaction, the initial rate of reaction can be calculated by change in the concentration gradient within 5–10% of reaction ratio. In the case of reversible reaction or irreversible reaction with significant product inhibition, the concentration change is subject to deviate from linearity.2.05.2.3 Experimental Determination of Kinetic Parameters in the Michaelis –Menten EquationTo experimentally determine kinetic parameters V m and K m , Michaelis –Menten equation is modified as described in Figure 3. Each equation derived from Michaelis –Menten equation has been suggested to determine these kinetic parameters with high precision [4]. That is, suitable method should be used to minimize the effect of the error of data. Experimental data obtained from enzymatic reaction experiment in a batch reactor are subject to include the error. Usually, Lineweaver –Burk plot (L –B plot) plot is applied to determine the kinetic parameters due to its simplicity [11]. However, we must pay attention to accuracy of parameter in applying least-square analysis to data points when the variability of initial rate is not negligible at low substrate concentrations. In such a case, Hanes –Woolf plot or Eadie –Hofstee plot is applied to determine the parameters more accurately rather than L –B plot. It is highly possible that the error in initial rate is observed in measuring the change of concentrations. Therefore, it is important not to expand the error by applying the initial rate to arithmetic operation. The Eadie –Hofstee plot does not emphasize the plots at low substrate concentrations, therefore recommendable as a suitable method.The Eisenthal –Cornish –Bowden (ECB) plot [6, 7] is the method that there are not any calculations to determine parameters. This method can also exclude outliers and provide unbiased estimates of the kinetic parameters because each pair of experimental data (v , [S]) does not interfere with other pairs. When it is an ideal case without any data error, all lines intersect at one fixed point, giving V m and K m . When the data include errors, individual intersections of plot are dispersed around the fixed point. Therefore, in the ECB plot, the error of data can be simultaneously evaluated besides the estimation of the kinetic parameters.Instead of the above four types of plots, kinetic parameters can be determined by plotting time-course data of ([S 0]–[S])/t versus (ln([S 0]/[S])/t . However, the quality of data in this integration method is different from those of other plots. The data in this method contain the reaction time and substrate concentration, while the initial reaction rate and substrate concentration are used in other four plots.Lineweaver–Burk (L–B) plot1 1 K m 1 =+v V m V m [S]1− 1 Km[S]Hanes–Woolf plot= + [S] v V m V m[S] K m 1[S]v v =V m −K mEadie–Hofstee plotm V [S]Eisenthal–Cornish–Bowden (direct linear) plotK m m [S]vV m v=[S ]+ K m [S]m−[S]Integration methodln [S 0] [S] 1 [S 0]−[S] V m=− +t K mt K m m0]−[S]t36 Underlying PrinciplesFigure 3 Various types of plot for determination of kinetic parameters in the Michaelis –Menten type enzyme reactions.2.05.2.4 Complex Enzyme Kinetics 2.05.2.4.1 Allosteric enzymeAmong enzymes, some of them are multimeric enzymes. They are mostly multisite-type enzymes that have more than one binding site to their substrate. In the multisite-type enzyme, after binding substrate to the enzyme, a steric structure of protein changes and its conformation change often results in alteration in overall reaction rate. Such kind of enzyme is known as allosteric enzyme. When the first substrate binding to the enzyme facilitates the binding of second substrate to same enzyme, this regulation is defined as a positive cooperativity. The reaction rate with the positive cooperativity is characterized by a sigmoidal curve of reaction rate versus time. If it is assumed that n -molecules of substrate bind one enzyme simultaneously, the reactions are described as follows:E þ n S ⇄ES n½7Reaction Kinetics and Stoichiometry37The dissociation equilibrium constant of ES n complex and the reaction rate in this case are described:nE ½ ½ S K m ¼½8 ½ES n V m S ½ nv ¼½9½ nK m þS where n is the Hill ’s coefficient which indicates the extent of allosteric effectiveness. Concerning the value of n, n > 1 and n <1 indicate positive and negative cooperativity, respectively (Figure 4).From eqn (9), the Hill ’s coefficient can be determined by plotting the relationship between log[S] and log(v /(V m −v )) (the Hill ’s plot):vlog ¼n log S −log K m½10½ V m −v2.05.2.4.2 Two substrate reactionsIn the enzyme reaction, multisubstrates are usually responsible for its reaction in the system. The simplest case is a two-substrate reaction. The reaction to be considered isA þB ↔P þQwhere A and B are substrates and P and Q are products, respectively. There are four main reaction mechanisms based on binding order between enzyme and substrate, and production order of products: (1) ordered Bi Bi mechanism, (2) Theorell-chance mechanism, (3) rapid equilibrium random Bi Bi mechanism, and (4) ping pong Bi Bi mechanism. Focusing on the initial rate of forward reaction, these four reaction mechanisms are classified as follows:1. Ordered Bi Bi mechanism . The binding orders of substrate A, B to enzyme and the detachment orders of products P, Q are fixed. That is, binding of substrate B to enzyme occurs after substrate A binding. Subsequently, detachment of product P from enzyme occurs and then product P desorbs from enzyme.2. Theorell-chance mechanism . First, substrate A attaches to enzyme. Then, substrate B reacts with complex EA and it results in production of product P and subsequently product Q.3. Rapid equilibrium random Bi Bi mechanism. The substrate binding and product detachment occur at random and they are in the dissociation equilibrium. The complex EAB is formed not only from the complex EA and substrate B but also from the complex EB and substrate A.4. Ping pong Bi Bi mechanism. First, after substrate A binds to enzyme, product P is produced. Subsequently, the binding of substrate B to enzyme results in the production of product Q.Substrate (S) (mM)R e a c t i o n r a t e (V ) (m M m i n –1)Figure 4 Dependence of initial reaction rate on substrate concentration in Michaelis –Menten (n = 1) and allosteric enzyme kinetics. The value of n indicates the Hill ’s coefficient. V m =1mM min −1, K m =10 mM.38 Underlying Principles2.05.3 Factors Affecting Reaction KineticsChemical substances other than the substrate can combine with enzymes and reduce their catalytic activity. Such substances are called enzyme inhibitors. The combination between enzyme and the inhibitor may be reversible, partially reversible, or irreversible. For example, heavy metals are well known as irreversible inhibitors to form stable complex with enzyme.There are three major types of reversible inhibition, such as competitive, uncompetitive, and noncompetitive inhibitions, respectively.2.05.3.1 Inhibited Reaction Kinetics 2.05.3.1.1 Inhibition of reversible inhibitorThe inhibition types by reversible inhibitor is typically classified into four categories (competitive, uncompetitive, non competitive, and mixed). Competitive inhibition usually occurs due to existence of substrate analogs that compete with substrate for the active site of the enzyme. The competitively inhibited enzyme reaction can be described asK mK 2E þ S ⇄ES E þ P þ→ I→ E ½↓↑K EI→ EI½11where K EI is the dissociation constant of enzyme –inhibitor complex (EI):½ I K EI ¼In uncompetitive inhibition, there are two binding sites both to substrate and to inhibitor in the enzyme. Uncompetitive inhibitor binds to the ES complex only but not to the free enzyme. Besides this property, ES-inhibitor (ESI) complex does not form product because of the conformation change of enzyme. The uncompetitively inhibited enzyme reaction can be described asK mk 2E þ S ⇄ES E þ PþI↓↑K ESIESI½12where K ESI is the dissociation constant of ESI complex:½ES ½I K ESI½ESIIn noncompetitive inhibition, there are also two binding sites both to substrate and to inhibitor in the enzyme. The inhibitor can bind not only to the free enzyme but also to the ES complex equivalently and they form EI and ESI complexes, respectively. Substrate also can bind both to free enzyme and to EI complex. Moreover, ESI complex cannot form the product:K mk 2E þ S ⇄ES E þ P þþI I ↓↑K EI ↓↑K ESIEI þS ⇄ESIK m ½E ½S ½EI ½S ½E ½I ½ES ½IK m ¼¼ K EI ¼ K ESI ¼¼ ½13½ES ½ESI ½EI ½ESIReaction Kinetics and Stoichiometry 39In mixed-type inhibition, there are also binding sites both to substrate and to inhibitor in the enzyme. The inhibitor can bind not only to the free enzyme but also to the ES complex. However, the binding abilities are not equivalent (K EI is different from K ESI ):K mk 2E þ S ⇄ES → E þ P þþI I ↓↑K EI ↓↑K ESI EI þS ⇄ESIK EIS ½E ½S ½E ½I ½ES ½I ½EI ½SK m ¼; K EI ¼ ; K ESI ¼ ; K ESI ¼ ;½ES ½EI ½ESI ½ESI½14½E ½S ½IK EIS K EI ¼ K ESI K m ¼½ESIIn the above four types of inhibition scheme, each reaction rate equation is developed by dissociation equilibrium equations of complexes, the assumption that total enzyme concentration are conserved before and after enzyme reaction, and the definition that reaction rate is described as k 2[ES]. The reaction rate for each inhibition scheme can be expressed by eqn 15:V m ; app ½S ν¼½15m ; app þ½Swhere V m, app and K m, app indicate apparent maximum velocity and apparent Michaelis constant, respectively, which are summarized in Table 1.2.05.3.1.2 Substrate inhibitionSubstrate may inhibit the reaction and deteriorate the reaction rate due to the change of enzyme structure or solution properties, which is known as substrate inhibition. The reaction scheme for uncompetitive substrate inhibition isK mK 2E þ S ⇄ES → E þ PþS↓↑K ESIESS½16with the definition of ESS complex dissociation constantES ½ ½ S K ES1 ¼½ESSIn addition, under the assumption that product can be formed from only ES complex not from ESS complex. The reaction rate can be developed as follows:½ V m S v ¼½17S ½½ 1 þK m þ S K ES1Table 1 Apparent kinetic parameters in the various inhibited reactions with reversible inhibitors Inhibition type V m,app K m,appCompetitive V mV mUncompetitive½I1 þ K ESIV mNoncompetitive½I 1 þ KEIV mMixed½I K m1 þ K EIK EIS½IK m 1 þ KEIK m½I1 þ K ESIK m½IK m 1 þ KEI½I K m1 þ K EIK EIS40 Underlying PrinciplesR e a c t i o n r a t e (V ) (m M m i n –1 )Substrate (S) (mM)Figure 5 Dependence of substrate inhibition on initial reaction rate in Michaelis –Menten enzyme kinetics. K ES1 indicates ESS (enzyme –substrate –subtrate) complex dissociation constant. V m =1mM min −1, K m = 10 mM.The substrate concentration that gives the maximum reaction rate, [S]m , can be determined by setting d v /d[S] = 0. The substrate concentration [S]m is given bypffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS m ¼ K ES1 ½18 ½ K m Dependence of substrate inhibition on initial reaction rate in Michaelis –Menten kinetics are shown in Figure 5.To be more general form, reaction rate can be described as follows:V m ½Sv ¼!½19i ½S K m þ½S 1 þ∑ni ¼1K ESiwhere n -molecule substrates can bind to the one enzyme.2.05.3.1.3 Product inhibitionSome products are structurally analogous to its substrate. Therefore, they are apt to bind the active site of enzyme. In product-inhibition scheme, the maximum velocity does not change but apparent Michaelis constant is increased like a competitive inhibition:K mk 2E þS ⇄ES → E þP↓↑K EPEP½20 ν¼ V m ½SK m 1 þ ½PK EPþ½S½212.05.3.2 Deactivation of enzymeGenerally, the reaction rate increases with temperature via reaction rate constant according to the Arrhenius equation (empirically suggested by S. Arrhenius in 1889) [1] as follows:k ¼A exp ð−E a =RT Þ½22where E a , R , and T indicate the activation energy (J mol −1), gas constant (8.3 J K −1 mol −1), and temperature, respectively. It gives the quantitative basis of the relationship between the activation energy and the rate at which a reaction proceeds. This formula has a background of physicochemical absolute kinetics that the molecules with higher thermal energy than activation energy are shifted to transition state and then transformed to products; further by statistical –mechanical approach, the number of these molecules among total molecules with thermal motion can be calculated.Reaction Kinetics and Stoichiometry 41In many enzyme-catalyzed reactions, the reaction rate increases also with temperature according to the Arrhenius equation described ask 2 ¼ A exp ð−E a =RT Þ½23The reaction rate increases with temperature up to a certain limit. Over this critical temperature, enzyme activity decreases with increase of temperature because of enzyme denaturation. Enzyme activity can be affected by various factors, such as temperature, pH, and so on. Especially, heat can significantly deteriorate the enzyme activity, known as thermal deactivation. This inactivation rate constant also changes with temperature according to the Arrhenius equation:k d ¼ A d exp ð−E d =RT Þ½24where E d indicates inactivation energy (E d > 0), which is about 10-fold larger than the activation energy E a . That is, enzyme inactivation is much faster than enzyme activation by temperature. For example, assuming that E a and E d are 40 and 300 kJ mol −1,respectively, when the reaction temperature rises from 30 to 40 °C, the enzyme inactivation rate becomes 45-fold faster, while the enzyme activation rate becomes 1.7-fold faster.In general, enzyme activity is considered to be all-or-none. Therefore, the inactivation rate of enzyme can be applicable to first-order reaction rate scheme where it is proportional to the concentration of intact enzyme [E]. The inactivation rate can be described as follows:d ½E¼ −k d ½E d t ½E In ¼ −k d t ½25½E 0where k d is the inactivation rate constant.From eqn 25, the half-life (t 1/2), amount of time required to be half of initial concentration, can be determined and it is commonly used as a criterion of enzyme stability:In 2 t 1=2 ¼½26k d2.05.3.3 pH Dependency of Reaction RateEnzyme activity and stability depend strongly on the pH value of the reaction solution. The active sites of enzyme consist of amino acids with some ionic groups, such as carboxyl group, amino group, thiol group, imidazole group, phenolic hydroxyl group, and so on. The dissociation state of amino acid residue significantly affects the structure of active site, the affinity to substrate, or the catalytic activity. Based on the differences of dissociation state, we can assume that the enzyme has three different forms, EH 2+, EH, and E −. Three different forms indicate EH 2+ (–COOH, –NH 3+), EH (–COO − , –NH 3+), and E − (–COO − , –NH 2), respectively:K E1½EH ½H þ EH þ Hþ⇄EH þ2 ;K EI ¼ ½EH þ2 ½E − ½H þK E2E − þ H þ ⇄EH ; K E2 ¼½EHK S ′EH þþ S ⇄EH þ2SE H þ22K E1↑↓ K ES1↑↓↑↓H þ H þ H þ þ þþK Sk 2EH þ S ⇄EHS →EH þ PK E1↑↓ K ES1↑↓ ↑↓H þ H þ H þ þþþK E − þ S ⇄S ″E −SE −k 2½E 0 ½Sv ¼ k 2½EHS ¼ ½27K s pH E1 − pH pH þp K E2p K ES1 − pH pH þp K ES2where p K E n =−log K E n (n = 1, 2), p K ES n = −log K ES n (n = 1, 2), and pH = −log [H +].The optimum pH is usually determined experimentally because of the difficulty in characterizing the active sites of enzyme. The profiles of enzyme reaction rate for each pH often become bell shaped. The typical pH dependence of enzyme reaction rate is shown in Figure 6.v k +2 e0 0.60.50.4 0.3 0.20.10 1234567891011121314pH42 Underlying PrinciplesFigure 6 pH dependence of enzyme reaction rate. p K E1 =p K ES1=3 and p K E2 =p K ES2 =9, K s = 10, s = 10.Assumed to be p K E1 =p K ES1 and p K E2 =p K ES2, the optimum pH that provides maximum reaction rate is (p K E1 +p K E1)/2 and pH that provide a half of maximum reaction rate are p K E1 and p K E2.2.05.4 Biochemical Reaction Rate Related to Cellular SystemsIn microbial, animal, and plant cells, there are various intracellular enzyme reactions. To produce any material by utilizing these intracellular reactions, all biochemical reactions involved in the production should be considered in principle. However, each individual cell is a complicated multicomponent system, and quantitative clarification of the enzyme reactions occurring in the microbial cell is much more complicated and difficult than that of the single enzyme reaction. The multi-and single-component representations are the concepts in structured model and in unstructured model, respectively. In unstructured model, balanced growth in which all intracellular components simultaneously proliferate is approximated. Furthermore, there are diversities of different cells in the population in regard to cell phase. The cells in different cell phase often show different features of metabolic functions and activities. From this point of view, the entire cell is hitherto dealt with as a black box and the individual cells in the microbial population are considered to be homogeneous without respect to their variability in shape, size, physiological function, and so on. The heterogeneous individual cells and homogeneous (average) cells are segregated and unsegregated viewpoints, respectively. In the actual case of cell, cellular representations should be described by a structured and segregated model. Generally, however, an unstructured and unsegregated model, the most idealized case, isapplied to analyze cell population with a determi­nistic approach. These perspectives are summarized in Figure 7 [2].Figure 7 Various types of cell population models.。