TABLE 7.1. Enzyme Databases
Web Site URL
ENZYME DB: General information http://www.expasy.ch/enzyme/
Enzyme structure database: Structures http://www.biochem.ucl.ac.uk/bsm/enzyme/index.html LIGAND: Enzyme reactions http://www.gebine.ad.jp/dbget/ligand.html
Brenda: General enzyme data http://www.brenda.uni-koeln.de/
EMP: General, literature summary http://wit.mcs.anl.gov/EMP/
Esther: Esterases http://www.ensam.inra.fr/cholinesterase/
Merops: Peptidases http://www.bi.bbsrc.ac.uk/Merops/Merops.htm
Protease http://delphi.phys.univ-tours.fr/Prolysis
CAZy: Carbohydrate active enzymes http://afmb.cnrs-mrs.fr/:pedro/CAZY/db.html REBASE: Restriction enzymes http://rebase.neh.com/rebase/rebase.html Ribonuclease P database http://www.mbio.ncsu.edu/RnaseP/home.html
PKR: Protein kinase http://pkr.sdsc.edu
PlantsP: Plant protein kinases and http://PlantP.sdsc.edu phosphatase
Aminoacyl-tRNA synthetases http://rose.man.poznan.pl/aars/index.html MDB: Metalloenzymes http://metallo.scripps.edu/
Promise: Prosthetic group/Metal enzymes http://bmbsgi11.leads.ac.uk/promise/
Aldehyde dehydrogenase http://www.ucshc.edu/alcdbase/aldhcov.html G6P dehydrogenase http://www.nal.usda.gov/fnic/foodcomp/
2-Oxoacid dehydrogenase complex http://qcg.tran.wau.nl/local/pdhc.htm
Figure 7.1. Diagram for random bi bi kinetic mechanism. The random addition of substrates, A and B to form binary (EA and EB) and ternary (EAB) complexes. The two ternary complexes EAB and EPQ interconvert with the rate constant of k and k. The release of products P and Q also proceeds in a random manner. Ks are dissociation constants where KaKab:KbKbaand KpKpq:KqKqp.
the rate of forming E; P (rate constant of k). Because enzyme kinetic studies are carried out with excess concentrations of substrate — that is, [S] [E] — the conservation equations [E]:[E]; [ES] and [S] :[S] ;[ES]5[S] apply.
The overall rate of the reaction is limited by the breakdown of the enzyme—substrate complex, and the velocity is measured during the very early stage of the reaction.
Because the quasi-equilibrium treatment expresses the enzyme—substrate complex in terms of [E], [S], and KQ— that is, [ES]:[E][S]/(KQ ;[S])— the kinetic express-ion is obtained if we insert the expressexpress-ion for the enzyme—substrate complex into the rate expression:
v: k[ES]:k[E][S]/(KQ;[S]):V [S]/(KQ;[S])
This is known as the Michaelis—Menten equation, where there are two kinetic parameters, the maximum velocity V: k[E] and the Michaelis constant KQ(KK):
k/k.The general rule for writing the rate equation according to the quasi-equilibrium treatment of enzyme kinetics can be exemplified for the random bisubstrate reaction with substrates A and B forming products P and Q(Figure 7.1), where K?K?@ : K@K@? and KNKNO:KOKON.
1. Write the initial velocity expression: v: k[EAB] 9 k[EPQ], where the interconversion between the ternary complexes is associated with the rate constants k and k in the forward and reverse directions, respectively.
2. Divide the velocity expression by the conservation equation for enzymes, [E]:[E]; [EA];[EB];[EAB];[EPQ];[EP]; [EQ], that is, v/[E]:(k[EAB]9k[EPQ])/([E];[EA];[EB];[EAB];[EPQ];[EP];[EQ])
3. Set the [E] term equal to unity, that is, [E]: 1.
4. The term for any enzyme-containing complex is composed of a numerator, which is the product of the concentrations of all ligands in the complex, and a denominator, which is the product of all dissociation constants between
KINETICS OF ENZYMATIC REACTIONS 127
the complex and free enzyme, that is, [EA]: [A]/K?, [EB]:[B]/K@, [EP]:[P]/KN, [EQ]:[Q]/KO, [EAB]:([A][B])/(K?K?@), and [EPQ]:
([P][Q])/(KOKON).
5. The substitution yields the rate expression:
v: k([A][B])/(K?K?@)9k([P][Q])/(KOKON)[E]
1;[A]/K?;[B]/K@;([A][B])/(K?K?@);[P]/KN;[Q]/KO;([P][Q])/(KOKON) The rate expression for the forward direction is simplified to
v: k([A][B])/(K?K?@)[E]
1; [A]/K?;[B]/K@;([A][B])/(K?K?@)
: V [A][B]
K?K?@;K?@[A];K@?[B];[A][B]
7.2.2. Steady-State Assumption
The steady-state treatment of enzyme kinetics assumes that concentrations of the enzyme-containing intermediates remain constant during the period over which an initial velocity of the reaction is measured. Thus, the rates of changes in the concentrations of the enzyme-containing species equal zero. Under the same experi-mental conditions (i.e., [S] [E] and the velocity is measured during the very early stage of the reaction), the rate equation for one substrate reaction (uni uni reaction), if expressed in kinetic parameters (V and KQ), has the form identical to the Michaelis—Menten equation. However, it is important to note the differences in the Michaelis constant that is, KQ:k/k for the quasi-equilibrium treatment whereas KQ:(k;k)/k for the steady-state treatment.The general rule for writing the rate equation according to the steady-state treatment of enzyme kinetics by King and Altman method(King and Altman, 1956) can be illustrated for the sequential bisubstrate reaction. Figure 7.2 shows the ordered bi bi reaction with substrates A and B forming products P and Q. Each enzyme-containing species is associated with two rate constants, k in the forward direction and k in the reverse direction.
The steady-state rate equation is obtained according to following rules (King and Altman method):
1. Write down all possible basic patterns with n9 1 lines (n : number of enzyme forms that is, enzyme-containing species), in which all lines are connected without closed loops — for example,
2. The total line patterns equals m!(n 9 1)!(m 9 n ; 1)!, where m is the number of lines in the patterns.
128 DYNAMIC BIOCHEMISTRY: ENZYME KINETICS
Figure 7.2. Diagram for ordered bi bi kinetic mechanism. The free enzyme, E, binds to A (first substrate) to form a binary complex, EA, which then interacts with B (second substrate) to form a ternary complex, EAB. The two ternary complexes EAB and EPQ interconvert. The release of P (first product) forms EQ, which then dissociates to E and Q (second product) in an ordered sequence. k1to k10are rate constants.
3. Write a distribution equation for each enzyme form — for example, E/E : NC/D, where NC and D are the numerator (for E) and denominator terms, respectively.
4. Numerator terms(e.g., NC) are written:
(a) Follow along the lines in the basic pattern in the direction from other enzyme forms (i.e., EA, EQ, EAB, and EPQ) leading toward the free enzyme form(i.e., E) for which the numerator is sought.
(b) Multiply all the rate constants and concentration factors for this di-rection.
(c) Repeat the process for all the basic patterns.
(d) The numerator terms are the sum of all the products of rate constants and concentration factors.
5. Write the numerator terms for all the other enzyme forms by repeating the process — for example, NC?, NCO, NC?@, and NCNO for EA/E, EQ/E, EAB/E, and EPQ/E, respectively.
6. Denominator terms are the sum of all numerator terms, that is D: NC; NC?; NCO ;NC?@;NCNO.
7. Substitute appropriate distribution equationsthe initial velocity expression: (e.g., EPQ/E and EQ/E) into
v: dP/dt : k[EPQ]9k[EQ][P]
: kEPQ/E[E]9kEQ/E[E][P]
: (kkkkk[A][B]9kkkkk[P][Q])[E])/k(kk;kk;kk)k
; k(kk;kk;kk)k[A];kkkk[B];kkkk[P]
; kk(kk;kk;kk;kk)[A][B]
; (kk;kk;kk;kk)kk[P][Q]
; kkkk[A][P];kkkk[B][Q]
; kkk(k;k)[A][B][P] ;kkk(k;k)[B][P][Q]
KINETICS OF ENZYMATIC REACTIONS 129
TABLE 7.2. Cleland Nomenclature for Bisubstrate Reactions Exemplifieda Mechanism Cleland Representation Forward Rate Equation
Order bi bi v: VAB
KG?K@; K@A; K?B;AB
Random bi bi v: VAB
KG?K@; K@A; K?B;AB
Ping pong bi bi or Uni uni uni uni ping pong
v: VAB
K@A;K?B;AB
?Three common kinetic mechanisms for bisubstrate enzymatic reactions are exemplified. The forward rate equations for the order bi bi and ping pong bi bi are derived according to the steady-state assumption, whereas that of the random bi bi is based on the quasi-equilibrium assumption. These rate equations are first order in both A and B, and their double reciprocal plots(1/v versus 1/A or 1/B) are linear. They are convergent for the order bi bi and random bi bi but parallel for the ping pong bi bi due to the absence of the constant term (KG?K@) in the denominator. These three kinetic mechanisms can be further differentiated by their product inhibition patterns(Cleland, 1963b).
Note: V, K?, K@, and KG? are maximum velocity, Michaelis constants for A and B, and inhibition constant for A, respectively.
This steady-state equation expressed with rate constants can be converted into the rate equation expressed with kinetic parameters according to Clelend(Cleland, 1963a, 1963b):
v: VV(AB9PQ/K)
(KG?K@;K@A;K?B;AB)V;(KNQ;KOP;PQ)V/K;(K?BQ/KGO;ABP/KGN)V;(KOAP/KG?;BPQ/KG@)V/K
where the equilibrium constant, K:(VKNKGO)/(VK@KG?). V and V are maxi-mum velocities for the forward and reverse reactions. K?, K@, KN, and KO are Michaelis constants, while KG?, KG@, KGN, and KGO are inhibition constants associated with substrates(A and B) and products (P and Q), respectively. The rate equation for the forward reaction can be simplified(Table 7.2) to
v: VAB
KG?K@;K@A;K?B;AB 7.2.3. Cleland’s Approach
The Cleland nomenclature(Cleland, 1963a) for enzyme reactions follows:
1. The number of kinetically important substrates or products is designated by the syllables Uni, Bi, Ter, Quad, Pent, and so on, as they appear in the mechanism.
130 DYNAMIC BIOCHEMISTRY: ENZYME KINETICS
2. A sequential mechanism will be one in which all the substrates must be present on the enzyme before any product can leave. Sequential mechanisms will be designated ordered or random depending on whether the substrate adds and the product releases in an obligatory sequence or in a nonobliga-tory sequence.
3. A ping pong mechanism will be designated if one or more products are released during the substrate addition sequence, thereby breaking the sub-strate addition sequence into two or more segments. Each segment is given an appropriate syllable corresponding to the number of substrate additions and product releases.
4. The letters A, B, C, D designate substrate in the order of their addition to the enzyme. Products are P, Q, R, S in the order of their release. Stable enzyme forms are designated by E, F, G, H, with E being free enzyme.
5. Isomerization of a stable enzyme form as a part of the reaction sequence is designated by the prefix Iso, such as Iso ordered, Iso ping pong.
6. For expressing enzymatic reactions, the sequence is written from left to right with a horizontal line or group of lines representing the enzyme in its various forms. Substrate additions and product releases are indicated by the down-ward( ) and upward (!) vertical arrows, respectively.
7. Each arrow is associated with the corresponding reversible step — that is, one rate constant each with the forward and reverse directions. Generally, odd-numbered rate constants are used for the forward reactions whereas even-numbered ones are used for the reverse direction.
Examples of the bisubstrate reactions according to Cleland nomenclature are listed in Table 7.2.
For multisubstrate enzymatic reactions, the rate equation can be expressed with respect to each substrate as an n:m function, where n and m are the highest order of the substrate for the numerator and denominator terms respectively(Bardsley and Childs, 1975). Thus the forward rate equation for the random bi bi derived according to the quasi-equilibrium assumption is a 1:1 function in both A and B(i.e., first order in both A and B). However, the rate equation for the random bi bi based on the steady-state assumption yields a 2:2 function (i.e., second order in both A and B).
The 2:2 function rate equation results in nonlinear kinetics that should be differen-tiated from other nonlinear kinetics such as allosteric/cooperative kinetics(Chapter 6, Bardsley and Waight, 1978) and formation of the abortive substrate complex (Dalziel and Dickinson, 1966; Tsai, 1978).
7.2.4. Environmental Effects
The rate of an enzymatic reaction is affected by a number of environmental factors, such as solvent, ionic strength, temperature, pH, and presence of inhibitor/activator.
Some of these effects are described below.
Presence of Inhibitors: inhibition Kinetics. The kinetic study of an enzy-matic reaction in the presence of inhibitors is one of the most important diagnostic procedures for enzymologists. The inhibition (reduction in the rate) of an enzyme
KINETICS OF ENZYMATIC REACTIONS 131
TABLE 7.3. Types of Enzyme Inhibitions
Type of Inhibition Complex Formation Forward Rate Equation
Control E; S : ES ; E ; P v: VS
KQ ;S
Competitive E; I: EI)GQ v: VS
KQ(1;I/KGQ);S
Uncompetitive ES; I: ESI)GG v: VS
K?s;S(1;I/KGG) Noncompetitive or
mixed competitive E; I: EI)GQ v: VS
KQ(1;I/KGQ);S(1;I/KGG) ES; I: ESI)GG
Note: KGG and KGQ are inhibition (dissociation) constants for the formation of the inhibitor complexes in which the subscripts denote the intercept effect and slope effect, respectively.
reaction is one of the major regulatory devices of living cells and offers great potentials for the development of pharmaceuticals. An irreversible inhibitor forms the stable enzyme complex or modifies the enzyme to abolish its activity, whereas a reversible inhibitor (I) forms dynamic complex(es) with the enzyme (E) or the enzyme substrate complex(ES) by reducing the rate of the enzymatic reaction (see Table 7.3).
Temperature Effect: Determination of Activation Energy. From the tran-sition state theory of chemical reactions, an expression for the variation of the rate constant, k, with temperature known as the Arrhenius equation can be written
k: Ae\#? 02 or
ln k: ln A 9 E?/RT
where A, R, and T are preexponential factor(collision frequency), gas constant, and absolute temperature, respectively. E? is the activation energy that is related to the enthalpy of formation of the transition state complex, H‡, of the reaction E?:H‡; RT. The lowering of the activation energy of an enzymatic reaction is achieved by the introduction into the reaction pathway of a number of reaction intermediate(s).
pH Effect: Estimation of pKaValue(s). Some of the possible effects that are caused by a change in pH are:
1. Change in the ionization of groups involved in catalysis
2. Change in the ionization of groups involved in binding the substrate
132 DYNAMIC BIOCHEMISTRY: ENZYME KINETICS
TABLE 7.4. pH Effects on Enzyme Kinetics
Diagnostic pH-Rate
Profile Rate Expression Low pH High pH
Full bell K:KK/(H/K;1;K/H) logK:logKK9pK;pH logK:logKK;pK9pH Left half bell K:KK/(H/K;1) log K:log KK9pK;pH
Right half bell K:KK/(1;K/H) log K:log KK;pK9pH
Full bell V:VK/(H/K;1;K/H) log V:logVK9pK;pH log V:log VK;pK9pH Left half bell V:VK/(H/K;1) log V:log VK9pK;pH
Right half bell V:VK/(1;K/H) log V:log VK;pK9pH
Note: K and K or K and K are ionizing group(s) in the free enzyme or the enzyme—substrate complex, respectively.
3. Change in the ionization of substrate(s)
4. Change in the ionization of other groups in the enzyme 5. Denaturation of the enzyme
The pH effect on kinetic parameters (pH-rate/binding profile) may provide useful information on the ionizaing groups of the enzyme if the kinetic studies are carried out with nonionizable substrate in the pH region(pH 5—9) where enzyme denaturation is minimum. If the Michaelis constant (K) and/or the maximum velocity(V ) vary with pH, the number and pK values of the ionizing group(s) can be inferred from the shape of pH-rate profile(pH versus pK and pH versus log V plots), namely, full bell shape for two ionizing groups and half bell shape for one ionizing group(see Table 7.4).
The initial rate enzyme kinetics uses very low enzyme concentrations (e.g., 0.1M to 0.1 pM) to investigate the steady-state region of enzyme-catalyzed reac-tions. To investigate an enzymatic reaction before the steady state (i.e., transient state), special techniques known as transient kinetics (Eigen and Hammes, 1963) are employed. The student should consult chapters of kinetic texts (Hammes, 1982;
Robert, 1977) on the topics. KinTekSim (http://www.kintek-corp.com/kintek-sim.htm) is the Windows version of KINSIM/FITSIM (Frieden, 1993) which analyzes and simulate enzyme-catalyzed reactions.
7.3. SEARCH AND ANALYSIS OF ENZYME DATA