Journal of Biosciences and Medicines, 2014, 2, 24-29
Published Online June 2014 in SciRes.
How to cite this paper: Imperial, S. and Centelles, J.J. (2014) Enzyme Kinetic Equations of Irreversible and Reversible Reac-
tions in Metabolism. Journal of Biosciences and Medicines, 2, 24-29.
Enzyme Kinetic Equations of Irreversible
and Reversible Reactions in Metabolism
Santiago Imperial, Josep J. Centelles
Departament de Bioquímica i Biologia Molecular (Biologia), Facultat de Biologia, Universitat de Barcelona,
Barcelona, Spain
Received March 2014
This paper compares the irreversible and reversible rate equations from several uni-uni kinetic
mechanisms (Michaelis-Menten, Hill and Adair equations) and bi-bi mechanisms (single- and double-
displacement equations). In reversible reactions, Haldane relationship is considered to be iden-
tical for all mechanisms considered and reversible equations can be also obtained from this rela-
tionship. Some reversible reactions of the metabolism are also presented, with their equilibrium
constan t.
Adair E qua ti on, Enzyme Kinetics, Equilibrium Constant, Haldane Relationship, Hill Equati on ,
Metabolism, Michaelis Menten Equatio n, Reversible Reactions
1. Introduction
Thermodinamical considerations in a metabolic pathway include different aspects like kinetic analysis, and
identification of reversible steps in this pathway [1]. Although most of the reactions are reversible, it is usual in
general Biochemistry textbooks to present to students kinetic irreversible equations. For instance, the irreversi-
ble Michaelis-Menten equation is a well-known example and it is presented in this way in general Biochemistry
books (either to simplify the mechanism, or because this reaction is used for an in vitro study in absence of the
product of the reaction). Nevertheless, when performing an in vivo study, or when using a biochemical mathe-
matical model presenting several reactions of a metabolic pathway, reversible equations should be considered.
In this paper, we present several reversible equations and we compare them with the irreversible ones.
Haldane relationship, an equation which can only be used for reversible reactions, connects biochemical
thermodynamics and biochemical kinetics. Thus, for a reversible uni-uni reaction A = P, Haldane relationship
connects equilibrium constant Keq with kinetic parameters for both irreversible reactions, A P (Vf and KmA)
and P A (Vr and KmP). Haldane relationship is in this case: Keq = Vf KmP/Vr KmA. This is a general relationship
that is also valid for several other mechanisms, including Hill equation (although [P]0.5 and [S]0.5 should replace
the values of KmP and KmA, respectively). Several reversible equations are obtained from the Haldane relation-
ship considering that in equilibrium total velocity should be zero, and that v = vAP – vPA. Nevertheless, this
relationship is not considered universal, as when considering a bi-bi reaction with two reactions: A = Q, fol-
S. Imperial, J. J. Centelles
lowed by B = P, Haldane relationship will be the product of the two equilibrium constants:
Keq = Keq(1) Keq(2) = (Vf KmQ/V r KmA ) (Vf KmP/Vr KmB) = Vf2 KmP KmQ/Vr2 KmA KmB.
In general, Haldane relationship for a bi-bi mechanism is an equation more similar to the uni-uni equation:
Keq = Vf KmP KmQ/Vr KmA KmB.
2. Kinetic Equations of Reversible Reactions
2.1. Uni-Uni Mechanisms
The easiest mechanism for uni-uni enzyme kinetics is the Michaelis-Menten mechanism. The best known equa-
tion is the irreversible equation, which is used for a reaction with one substrate, independently on if the obtained
products are one, two, or several. For a uni-uni mechanism, it can be observed a competitive inhibition by the
product, as both the substrate and the product bind to the active site. This is the reason for the presence of a term
depending on [P] at the denominator of the reversible equation, which is not considered for an irreversible equa-
tion, as [P] = 0.
Figure 1 shows the different kinetic equations obtained for several uni-uni mechanisms. Reversible Michae-
lis-Menten equation is considered as a uni-uni reaction, where α and π are the relative concentrations for the
substrate and the product respectively: α = [A]/KmA and π = [P]/KmP.
Although irreversible Hill equation is presented as a general equation (where h is the Hill coefficient), the
Adair irreversible equation is presented only for an enzyme with two active centers. Figure 1 also shows the
equation obtained for two enzymes (with different kinetic parameters) acting in the same reaction. This could be
the case of two isoenzymes. This equation for two enzymes is the same as Adair equation for two sites, as two
enzymes have also two active sites. Reversible Hill equation is very similar to Adair equation [2], as it can be
calculated from Figure 1 for h = 2.
It should be noted that the reversible equations should be always converted into irreversible equations consi-
dering zero the products concentrations. Thus, Michaelis-Menten and Hill irreversible equations are obtained
from the reversible equations on Figure 1 considering π = 0, and Adair irreversible equation is obtained from
Figure 1. Comparison between the reversible and the irreversible kinetic equations of an uni-uni reaction.
S. Imperial, J. J. Centelles
the reversible equation by considering [P] = 0.
Similar equations should be also obtained for the inverse irreversible reaction (P A) by considering [A] = 0
(or α = 0) although in these cases a negative equation is obtained, as the velocity is considered in the inverse
Haldane relationship from these reversible equations can be solved by considering that equilibrium concentra-
tions would led to v = 0 (for Michaelis-Menten and Hill equations, Vf αeq = Vr πeq), and Keq = [P]eq/[ A] eq. Thus,
resulting Haldane relationship to be: Keq = Vf KmP/Vr KmA.
2.2. Bi-Bi Mechanisms
Although there are several possible mechanisms, the most common bi-bi mechanisms include the ternary com-
plex mechanism (random or ordered bi-bi) and the substituted-enzyme mechanism (ping-pong bi-bi). As it can
be seen in Figure 2, the main difference between both mechanisms is the independent term present in the deno-
minator of the ternary complex mechanism. This independent term is present either for the irreversible or the
reversible equation.
These equations are more complex than those for the uni-uni mechanism, but it should be observed that both
irreversible equations can be simplified to a Michaelis-Menten uni-uni equation by considering saturated the
concentration of one of the substrates. For example, for a high β, v = Vf αβ/(β + αβ), and simplifying, v = Vf α/(1
+ α). Similarly, for a high α, the equation obtained would be v = Vf β/(1 + β).
The reversible equation, when considering high β, is transformed to an irreversible Michaelis-Menten equa-
tion, with inhibitions of products (P and Q). These inhibitions depend on the bi-bi mechanism considered, and
they follow the Cleland laws.
Haldane relationship from these equations can be solved by considering that equilibrium concentrations
would led to v = 0 (in both cases, Vf αeq βeq = Vr πeq ρeq), and Keq = [P]eq[Q]eq/[ A] eq[B]eq. Thus, resulting Haldane
relationship would be: Keq = Vf KmP KmQ/Vr KmA KmB.
2.3. Multisubstrate Mechanisms
We have considered until now uni-uni and bi-bi mechanism. Nevertheless, some reactions can be also uni-bi, or
bi-uni. Irreversible equations from uni-uni and from uni-bi are identical, as both consider only one substrate. But
reversible reactions present other factors in the numerators, as it can be seen in Figure 1 and Figure 2. Thus, for
uni-bi reactions, numerators of the equations can be easily corrected by using the uni positive factor and the bi
negative factor in the numerator: Vf α – Vr π ρ. And in the same way, the bi-uni equation should have the fol-
lowing factor as numerator: Vf α β – Vr π. In general, numerators can be deduced from the previous Figures. In
fact, if the reaction mechanism is known, the King and Altman method [3] can be used to deduce the enzymatic
equation. This method can be easily performed from the web page m/king-a lt man/i nd ex. ht ml .
3. Some Examples of Reversible Reactions in Metabolism
Equilibrium constant can be used to calculate kinetic parameters by using Haldane relationship. For this reason,
we present in Table 1 a brief summary of some reversible reactions extracted from Barman [4]. Equilibrium
constants from the table are not taken all in the same conditions of pH and temperature, and the substrate or
products concentrations in the cell would indicate whether the reaction is far or near equilibrium. Reversible
Figure 2. Comparison between the reversible and irreversible kinetic equations of a bi-bi reaction.
S. Imperial, J. J. Centelles
Table 1. Some reversible reactions of the most common pathways in metabolism (extracted from [4]).
En z ym e
(E.C. number)
Reaction Keq
Alcohol dehydrogenase
(EC alcohol + NAD = aldehyde or ketone + reduced NAD 8.0 × 1012
Gl yc ero l-3 -phosphate
dehydrogenase (EC L-glycerol-3-P + NAD = dihydroxyacetone phosphate + reduced NAD 1.0 1012
Lactate dehydrogenase
(EC L-lactate + NAD = pyruvate + reduced NAD 2.76 × 106
Malate dehydrogenase
(EC L-malate + NAD = oxaloacetate + reduced NAD 6.4 × 1013
Glucose 6-phosph ate
dehydrogenase (EC D-glucose 6-phosphate + NADP = D-glucono-δ-lactone 6-phosphate + reduced NADP 6.0 × 107
dehydrogenase (EC D-gl yc e ra l d ehyde-3-phosphate + Pi + NAD = 1,3-diphospho-D-glycerate + reduced NAD 0.5
Butyryl-CoA dehydrogenase
(EC butyryl-CoA + FAD = crotonoyl-CoA + FADH2 0.22
Alanine dehydrogenase
(EC L-alanine + H2O + NAD = pyruvate + NH3 + reduced NAD 6.98 × 1014
Glutamate dehydrogenase
(EC L-glutamate + H2O + NAD = 2-oxoglutarate + NH3 + reduced NAD 4.5 × 1014
Tetrahydrofolate dehydrogenase
(EC 5,6,7,8-tetrahydrofolate + NADP = 7,8-dihydrofolate + reduced NADP 1.79 1012
NAD(P) transhydrogenase
(EC reduced NADP + NAD = NADP + reduced NAD 1.43
Glutathione reductase (EC reduced NAD(P) + oxidized glutathione = NAD(P) + 2 glutathione 9.8 × 106
(EC L-serine + tetrahydrofolate = glycine + 5,10-methylenetetrahydrofolate 10.2
Me t h yl ma l onyl-CoA
(EC methylmalonyl-CoA + pyruvate = propionyl-CoA + oxaloacetate 0.526
Ornithine carbamoyltransferase
(EC carbamoylphosphate + L-ornithine = Pi + L-citrulline 1 × 105
Transketolase (EC sedoheptulose 7-P + D-glyceraldehyde 3-P = D-ribose 5-P + D-xylulose 5-P 0.95
Transaldolase (EC sedoheptulose 7-P + D-glyceraldehyde 3-P = D-erythrose 4-P + D-fructose 6-P 1.05
Choline acetyltransferase
(EC acetyl-CoA + choline = CoA + acetylcholine 5.1 × 103
Carnitine acyltransferase
(EC acetyl-CoA + carnitine = CoA + acetylcarnitine 1.67
Phosphate acyltransferase
(EC acetyl-CoA + Pi = CoA + acetylphosphate 1.35 × 102
Ace t y l-CoA acetyltransferase
(EC 2 acetyl-CoA = CoA + acetoacetyl-CoA ~2 × 105
(EC AMP + pyrophosphate = adenine + 5-phospho-α-ribosyl-pyroph os phat e 0.1
Aspartate aminotransferase
(EC L-aspartate + 2-oxoglutarate = oxaloacetate + L-glutamate 0.16 - 0.17
Alanine aminotransferase
(EC L-alanine + 2-oxoglutarate = pyruvate + L-glutamate 2.2
Glucokinase (EC ATP + D-glucose = ADP + D-glucose-6-ph os phate 3.86 × 102
S. Imperial, J. J. Centelles
Conti n ue d
Galactokinase (EC ATP + D-galactose = ADP + D-galactose-1-phosphate 26
Pyruvate kinase (EC ATP + pyruvate = ADP + phosphoenolpyruvate 1.55 × 104
Acetate kinase (EC ATP + acetate = ADP + acetylphosphate ~8 × 103
Carbamate kinase (EC ATP + NH3 + CO2 = ADP + carbamoylphosphate 4 102
Phosphoglycerate kinase
(EC ATP + 3-phosph o-D-glycerate = ADP + 1,3-diphosph o-D-glycerate 2.9 × 104
Creatine kinase (EC ATP + creatine = ADP + phosphocreatine 7.2 × 109
Adenylate kinase (EC ATP + AMP = 2 ADP 2.26
Glutaminase (EC L-glutamine + H2O = L-glutamate + NH3 320
(EC 4,5-dihydrouracil + H2O = 3-ureidopropionate 0.67
Dihydro-orotase (EC L-4,5-dihydro-orotate + H2O = N-carbamoyl-L-aspartate 1.9
cyclohydrolase (EC 5,10-methyltetrahydrofolate + H2O = 10-formyltetrahydrofolat e 2.4 × 108
Phosphopyruvate carboxylase
(EC GTP + oxaloacetate = GDP + phosphoenolpyruvate + CO2 0.372
Fructosediphosphate aldolase
(EC fructose-1,6-diphosphate = dihydroxyacetone-P + D-glyceraldehyde-3-P 8.1 × 105
Citrate lyase (EC citrate = acetate + oxaloacetate 0.325
Citrate synthase (EC citrate + CoA = acetyl-CoA + H2O + oxaloacetate 1.2 × 104
ATP citrate lyase (EC ATP + citrate + CoA = ADP + Pi + acetyl-CoA + oxaloacetate 1.0 - 1.5
Carbonic anhydrase (EC
H2CO3 = CO2 + H2O 2.51 × 106
Fumarate hydratase (EC
L-malate = fumarate + H2O 0.23
En o yl-CoA hydratase
(EC L-3-hydroxyacil-Coa = crotonoyl-CoA + H2O 16.2
Argininosuccinate lyase
(EC L-argininosuccinate = fumarate + L-arginine 1.14 × 102
Adenylosuccinate lyase
(EC adenylosuccinate = fumarate + AMP 6.8 × 103
Glutamate racemase
(EC L-glutamate = D-gluta mate ~1
Hydroxyproline epimerase
(EC L-hydroxyproline = D-allohydroxyproline 0.99
3-epimerase (EC D-ribulose 5-phosphate = D-xylulose 5-phosphate 1.5 - 3.0
UDP-glucose epimerase
(EC UDP-glucose = UDP-galactose 0.284
4-epimerase (EC L-ribulose 5-phosphate = D-xylulose 5-phos phate 1.86
Me t h yl ma l onyl-CoA
racemase (EC D-meth y lmalonyl-CoA = L-m eth yl m a l onyl-CoA 1.0
Triose phosphate isomerase
(EC D-glyceraldehyde 3-phosphate = dihydroxyacetone phosphate 22
Arabinose isomerase
(EC D-arabinose = D-ribulos e 0.179
L-arabinose isomerase
(EC L-arabinose = L-ribulose ~0.11
S. Imperial, J. J. Centelles
Conti n ue d
Xylose isomerase
(EC D-xylose = D-xylulose 0.16
Ribosephosphate isomerase
(EC D-ribose 5-phosphate = D-ribulose 5-phosphate 0.30
Mannose isomerase
(EC D-mannose = D-fructose 2.45
Mannosephosphate isomerase
(EC D-mannose 6-phosphate = D-fructose 6-phosphat e 1.78
Glucosephosphate isomerase
(EC D-glucose 6-phosphate = D-fruc t ose-6-phosphat e 0.298
Glucuronate isomerase D-glucuronate = D-fructuronate 0.82
Arabinosephosphate isomerase
(EC D-arabinose-5-phosphate = D-ribulose 5-phosph ate 0.295
L-rhamnose isomerase
(EC L-rhamnose = L-rhamnulose 1.5
(EC 2-phospho-D-glycerate = 3-phosphoglycerate 5.0
L-meth ylmalonyl-CoA mutase
(EC L-methylmalonyl-CoA = succinyl-CoA ~20
Muconate cycloisomerase
(EC (+)-4-carboxymethyl-4-hydroxyisocrotonolactone = cis-cis-muconate 4.03 × 102
Va l yl-sRNA synthetase
(EC ATP + L-valine + sRNA = AMP + PPi + L-valyl-sRNA 0.32
Ace t y l-CoA synthetase
(EC ATP + acetate + CoA = AMP + PPi + acetyl-CoA 0.86
Ac yl -CoA synthetase
(EC ATP + an acid + CoA = AMP + PPi + an acyl-C oA ~1.5
Succinyl-CoA synthetase
(EC ATP + Succinate + CoA = ADP + Pi + Succinyl-CoA 0.27
Glutamine synthetase
(EC ATP + L-glutamate + NH3 = ADP + Pi + L-glutamine 1.2 × 103
(EC GTP + IMP + L-aspartate = GDP + Pi + adenylosuccinate 2.9 - 10
Propi onyl-CoA carboxylase
(EC ATP + propionyl-CoA + CO2 + H2O= ADP + Pi + methylmalonyl-CoA 5.7
reactions are usually considered non-controlling reactions in a pathway, but they can be interesting for antagonic
metabolic pathways (i. e. glycolysis and gluconeogenesis), as depending on the intermediate concentrations, they
can be redirected to the products or the substrates.
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zymes into Metabolic Models. CABIOS, 13, 377-385.
[3] King, E.L. and Altman, C. (1956) A Schematic Method of Deriving the Rate Laws for Enzy me -Catalyzed Reactions.
The Journal of Physical Chemistry, 60, 1373-1378.
[4] Barman, T.E. (1969) Enzyme Handbook. Springer-Verlag, Berlin. -9