a mathematical model for glucose oxidase kinetics, including inhibitory, deactivant and diffusional...

Enzyme and Microbial Technology 34 (2004) 513–522

A mathematical model for glucose oxidase kinetics, including inhibitory,deactivant and diffusional effects, and their interactions

Jesús Miróna, Ma Pilar Gonzáleza, José A. Vázqueza, Lorenzo Pastranab, M.A. Muradoa,∗a Instituto de Investigacións Mariñas (CSIC), r/Eduardo Cabello 6, Vigo 36208, Galicia, Spain

b Área de Bioqu´ımica e Biolox´ıa Molecular, Facultade de Ciencias de Ourense, Universidade de Vigo, As Lagoas s/n, Ourense 32004, Galicia, Spain

Received 9 August 2003; accepted 15 December 2003

Abstract

Glucose oxidase (GOD) kinetics comprises a group of inhibitory, deactivant and diffusional phenomena interacting with each other in acomplex way. These phenomena have given rise to differing interpretations, and their description is hindered by studies at moderate glucoseconcentrations (usual condition in many applications of the enzyme), the presence (or absence) of catalase in the enzymatic preparationand the biases associated to the frequent use of linear transformations for kinetic characterizations. The objective of this work is to achievea kinetic model that integrates all the mentioned effects and allows satisfactory descriptions in conditions (frequent in microbiologicalcontext) of wide intervals of substrate concentration, at long reaction periods and in presence of catalase.© 2004 Elsevier Inc. All rights reserved.

Keywords:Glucose oxidase kinetics; Inhibitory, deactivant, diffusional effects

1. Introduction

In glucose oxidase (GOD) kinetics, especially when oneworks on a wide range of substrate concentrations, it is pos-sible to detect a complex group of inhibitory effects dueto the substrate (glucose) and product (gluconic acid), thathave received little attention.

Substrate inhibition (SI) has been described by Nicoland Duke[1] for the free enzyme fromAspergillus niger,although only at oxygen concentrations less than 2% rel-ative to saturation (2× 10−5 M). The conclusion of theseobservations, cited in the review by Barker and Shirley[2],was confirmed by Tse and Gough[3], who specified thatthe effect is weak with the reduced form of the enzyme, andalso by Kozhukharova et al.[4], working with immobilizedGOD and glucose concentrations above 150 g/l. Much morenumerous, however, are the studies which, in diverse condi-tions of operation, have either found no evidence of SI[5–9]or avoided the allusion to the phenomenon when describingthe kinetic properties of the enzyme[10–14], or providedkinetic constants without including the corresponding tothis effect[15–19].

On the other hand, the structural similarity between glu-cose and gluconic acid allows to suspect the possibility

∗ Corresponding author. Tel.:+34-86-23-1930; fax:+34-86-29-2762.E-mail address:[email protected] (M.A. Murado).

of competitive product inhibition. Nevertheless, the allu-sions to such mechanism as well as the work specificallydirected to verify it are scarce. With a correct experimen-tal approach, although nowadays deemed unconventional,Nakamura and Ogura[20] concluded that inhibition bygluconolactone (precursor of gluconic acid) was due togluconic acid binding with the active site of the enzyme(competitive inhibition). Rogers and Brandt[21] describedthe same effect with glucal (analogue of glucose with a dou-ble bond between carbons 1 and 2). Additionally, althoughthe phenomenon is not directly transferable to enzymologicdomain, it is worth noting that Velizarov and Beschkov[22]found substrate and product inhibition in the conversion ofglucose to gluconic acid in cultures ofGluconobacter oxy-dans. Nevertheless, neither the review of Barker and Shirley[2] nor the most recent one of Crueger and Crueger[23]include gluconic acid among the GOD inhibitors, and Tseet al.[24], working with GOD and immobilized catalase onsynthetic membranes, do not corroborate this effect.

These discrepancies could be due to structural differencesin enzymes of different origin, such as those noted by sev-eral authors[6,13,25,26]. However, other causes may alsoplay a role, for example, moderate glucose concentrations,which are usual conditions in many applications of GOD.Another may be the frequent use of linear transformations(e.g. Lineweaver-Burk) for the kinetic description. This ap-proach is very sensitive to experimental error and the cause

0141-0229/$ – see front matter © 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.enzmictec.2003.12.003

514 J. Miron et al. / Enzyme and Microbial Technology 34 (2004) 513–522

Nomenclature

ATP (and ADP) adenosin triphosphate (andadenosin diphosphate)

FAD flavin adenine dinucleotideGOD glucose oxidaseNADP+ nicotinamide adenine

(and NADPH+ H+) dinucleotide phosphate(and reduced form)

PO peroxidaseSI substrate inhibition

Simulationg kinetic parameter (second-order)GOD concentration of GOD (with GOD0

as initial value)Km Michaelis constantOX hydrogen peroxide formed after a time

interval tS substrate concentrationuGOD rate of GOD deactivationv GnH rate of gluconate productionv OX rate of hydrogen peroxide productionVm maximum rate of product formation

(Michaelis–Menten model)

Michaelian approachesIa concentration of uncompetitive inhibitorIc concentration of competitive inhibitorIn concentration of noncompetitive inhibitorKa uncompetitive inhibition coefficientKc competitive inhibitor coefficientKn noncompetitive inhibition coefficientKs substrate inhibition coefficient

Dynamic modelCATt catalase concentrationg1 kinetic parameter (first-order)g2 kinetic parameter (second-order)g3 kinetic parameter (second-order)GnHt gluconate formed after a time intervaltGODt GOD concentrationHt difference between pH of maximum

stability of GOD and real pHk1 kinetic parameter (first-order)m Michaelian functionOXt hydrogen peroxide formed after a time

interval tS substrate concentrationuGODt rate (total) of GOD decompositionu1 GODt rate of spontaneous (first-order)

decomposition of GODu2 GODt rate of GOD deactivation (second-order)u3 GODt rate of GOD deactivation (second-order)uOXt rate (total) of hydrogen peroxide

formation in a time instantt

u1 OXt rate of spontaneous (first-order)decomposition of hydrogen peroxide

u2 OXt rate of hydrogen peroxide decompositionby catalase (first-order)

v GnHt rate of gluconate production in a timeinstantt

v OXt rate of hydrogen peroxide production in atime instantt

of substantial biases, particularly when applied to cases withsubstrate inhibition. Finally, the frequent presence of cata-lase in the preparations of GOD tends to obscure a kineticprofile which, although due to partial inactivation of the en-zyme by the hydrogen peroxide co-product of the reaction,contributes to focus the attention towards the possibility ofinhibitory effects.

The aim of this work is to study in more detail thesephenomena and propose corresponding descriptive models.

2. Materials and methods

2.1. Enzymes employed

Two preparations of GOD (EC 1.1.3.4) were studied, bothfrom A. niger. The first was the product SIGMA G-7141.Limits of activity: 15,000–25,000 units/g of solid, withoutadded oxygen. Unit definition: required amount of enzymeto oxidize 1�mol of �-d-glucose tod-gluconolactone andH2O2 per minute at 35◦C, pH 5.1. The second was obtainedfrom a submerged culture ofA. nigerCBS 554-65[27] bythe following procedure. When the maximum GOD activ-ity was reached, the filtered medium (50 l) was concentratedby membrane ultrafiltration (100 kDa) to a final volume of2.5 l. The retentate was washed by diafiltration, continu-ously adding water (up to∼10 l) at an equivalent rate to thatpermeate outflowing. Ultrafiltration was continued to makethe washed retentate up to∼0.4 l and this final volume wasfreeze-dried to obtain a dry powder.Table 1shows the maincharacteristics of both preparations.

The catalase (EC 1.11.1.6) used, as an additive andas a standard to determine its presence in the GOD

Table 1Analytical details of the GOD preparations used in the kinetic experiments

Raw preparation SIGMA G-7141

GOD (EU/mg solid) 16.7 172.0GOD (EU/mg protein) 34.79 236.52Catalase (EU/mg solid) 7.93 trCatalase (EU/mg protein) 16.52 trCatalase/GOD 0.475 trFAD (mg/mg solid) 0.002 0.007Total carbohydrates (mg/mg solid) 0.22 0.11Protein (mg/mg solid) 0.48 0.73

J. Miron et al. / Enzyme and Microbial Technology 34 (2004) 513–522 515

preparations, was the product SIGMA C-3515. Limits of ac-tivity: 4000–8000 units/mg protein. Unit definition: requiredamount of enzyme to decompose 1�mol of H2O2 per minat 25◦C, pH 7.0, while the H2O2 concentration (followedby means of the decrease in absorbance at 240 nm) fallsfrom 10.3 to 9.2 mM.

2.2. Analytical methods

All reagents used were of analytical quality (SIGMA).

2.2.1. GOD and hydrogen peroxideIn the method of Lloyd and Whelan[28] for the deter-

mination of GOD, the reaction catalyzed by this enzymeis linked to the decomposition of hydrogen peroxide withperoxidase (PO) and oxidation of the chromogenic reagento-dianisidine:

glucose+ O2GOD→ gluconolactone↔ gluconate+ H2O2

H2O2 + o-dianisidinePO→o-dianisidine(OX) + H2O

This sequence of reactions is applicable to several objec-tives, among them are (a) to quantify the enzymatic activityof a sample with an unknown amount of GOD and (b) toquantify, in a typical Michaelian approach, the amount ofhydrogen peroxide (and, consequently, gluconate) that pro-duces a known activity of GOD in the presence of variableconcentrations of glucose.

In both cases, analysis was performed according to themethod of Fiedurek et al.[29]. In the basic method for deter-mination of GOD, the reagents used were—A: 18 mg/ml glu-cose, 16 EU/ml peroxidase, and 0.53 g/ml glycerol, in 0.2 Mcitric–phosphate buffer (pH 5.1);B: 2 mg/ml o-dianisidinein distilled water;C: sample with an unknown amount ofGOD. The reaction was carried out by mixing 1.9 ml ofA and 0.1 ml ofB, maintaining the mixture for 10 min at30◦C, and then adding 0.1 ml ofC. After shaking for 15 minat 200 rpm and 30◦C, the reaction was stopped by adding3 ml of 5N HCl and absorbance was measured at 525 nm(oxidizedo-dianisidine). As a standard a series of dilutionsof GOD SIGMA G-7141 was used (a suitable interval is0.1–1.0 EU/ml) in place of solutionC.

This method permits a variation (recommended bySIGMA) which consists of eliminating the glycerol fromsolutionA, making the reaction mixture in a spectrophoto-metric cell and measuring absorbance at 435 nm every 30 sover a period of 5 min maximum, taking as the reactionrate the slope of the stretch in which the variation of theabsorbance is linear.

Although both procedures provide equivalent results, thefirst was preferred in the Michaelian experiments. The mainreason was the need to work under controlled agitation con-ditions, an important factor when it became necessary tostudy the possible diffusional restrictions due to gluconate.In this case, the basic method was applied with the follow-ing variations: (1) In solutionA the glycerol was eliminated,

the concentration of peroxidase was increased to 300 EU/ml,the concentration of glucose was varied within the desiredrange, and, when the effects of gluconate were studied, thedesired concentrations of the acid were added. (2) The re-action time was reduced to 4 min. (3) The absorbance wasmeasured at 525 nm. (4) As a standard a series of dilutions ofH2O2 was used (a suitable interval is 0.5–5.0 mM) in placeof solutionC.

One alternative which was also assayed in the Michaelianapproach was the determination of gluconate formed. Todo this, the reaction was stopped with 4 ml of 1N HCland, after 2 h, the solution was taken to pH 10–11 with 2NKOH, and gluconate determined by means of the methoddescribed below. The results did not show significant dif-ferences from those of theo-dianisidine reaction which,simpler and more economical, was the method finallyadopted. For long reaction periods (∼75 h) the necessarysterile conditions were achieved by filtering all solutionsthrough a 0.22�m membrane (the usual method of sodiumazide creates interferences with the reaction catalyzed bythe GOD).

2.2.2. CatalaseThe method proposed by SIGMA was followed: 100�l of

the enzyme solution was mixed in a 1 cm cell with 2.9 ml ofa substrate solution (100�l of 30% H2O2 in 50 ml phosphateof buffer 0.05 M; pH 7.0). The disappearance of H2O2 wasmeasured as the decrease in absorbance at 240 nm. Underthese conditions (initial value >0.450) the reproducibilityis good in the period of time required for the absorbanceto decrease between 0.450 and 0.400, which correspondsto the decomposition of 3.45�M H2O2. Accordingly, theconcentration of catalase in the sample was 3.45/(min×0.1) EU/ml, where 1 EU is the quantity of catalase whichdecomposes 1�M H2O2/min.

2.2.3. Glucose4-Aminoantipyrine method[30].

2.2.4. Gluconic acidSpecific method of Möllering and Bergmeyer[31] based

on the measurement of NADPH+ H+ formed in the pro-cesses:

d-gluconate+ ATP + gluconate kinase

→ d-gluconate-6-P+ ADP

d-gluconate-6-P+ NADP++6-P-gluconate dehydrogenase

→ ribulose-5-P+ NADPH+ + H+ + CO2

Gluconolactone potentially present is quantitatively con-verted to gluconate by taking the sample pH to 10–11 withaddition of 2N KOH, and maintaining this pH for 15 min.Thereafter, prior to analysis, the pH was adjusted to 7–8with 0.1N HCl.


Table 2Initial models applied to the analysis of the kinetic results

Without inhibition : V = VmS

Km + S(1)

Substrate inhibition : V = VmS

Km + SFs= VmS

Km + S + KsS2(2)

Competitive inhibition : V = VmS

KmFc + S(3)

Noncompetitive inhibition : V = VmS

(Km + S)Fn(4)

Uncompetitive inhibition : V = VmS

Km + SFa(5)

Vm: maximum rate;Km: Michaelis–Menten constant;S: substrate con-centration. In all cases the factorFi takes the formFi = Ki I i , whereI i isthe inhibitor concentration andKi is the inhibition constant (dimensions1/I i ). In substrate inhibition,Ii = S.

2.3. Other methods

The initial equations of kinetic descriptions are shown inTable 2. To avoid the biases associated with linearizationprocedures, the calculation of the coefficients was performedwith nonlinear methods. The quadratic differences betweenexperimentally determined and model-predicted values wereminimized (quasi-Newton). The experimental plan for de-scription of the interactions between GOD and catalase willbe described below.

3. Results and discussion

3.1. Preliminary results

In Fig. 1 the results of two conventional Michaelian ex-periments with the two GOD preparations (Table 1) arecompared at a concentration of 1 EU/ml in the incubation

0,0

0,4

0,8

1,2

1,6

0 25 50 75 100

S (g/L)

0 25 50 75 100

Fig. 1. Kinetics of the preparations of GOD described inTable 1 (left:impure preparation; right: SIGMA product).S: substrate concentration;V: rate of production of gluconic acid. The experimental results (points)show fits to the Michaelis–Menten model, without (dotted line) and with(continuous line) substrate inhibition.

mixture and substrate concentration between 1 and 100 g/l.In both cases, the fits (quasi-Newton) toEqs. (1) and (2)in Table 2support the hypothesis of SI existence, which ismost prominent in the purest preparation.

The isoelectric focusing of both preparations showed nodifferences in the four bands of GOD, and supplementationof the purest preparation with FAD until the FAD/GOD ra-tio of the least pure preparation did not change its activ-ity. Therefore, the ratio catalase/GOD was suspected to beinvolved in the apparent kinetic differences. It should benoted that hydrogen peroxide, with deactivating action onGOD, is a co-product of the enzymatic reaction, and there-fore the high substrate concentrations imply high rates ofperoxide production. Accordingly, it may be assumed thatcertain combinations of kinetic parameters cause a decreasein the effective GOD from a certain substrate level, produc-ing profiles showing a rate decrease from a maximum. Thesituation can be schematically simulated on the basis of thefollowing assumptions.

The molar rates of formation of the products (gluconicacid:v GnH; hydrogen peroxide:v OX) are the same and canbe modeled by means of the Michaelis–Menten equation ateach time instant. WhereSand GOD are the concentrationsof substrate and enzyme:

v GnH = v OX =(

VmS

Km + S

)GOD (6)

OX formed after a time intervalt is obtained by numericalintegration ofEq. (6)with respect to time:

OX =t∑

t=0

[(VmS

Km + S

)GOD

](7)

Accepting as the simplest kinetic hypothesis that peroxidedeactivates the enzyme in a second-order reaction (first-orderin each of the species; kinetic rate constant= g), the rate ofdeactivation of GOD in each time instant will be

u GOD = g OX GOD (8)

and the active enzyme after a time intervalt, where GOD0is the initial concentrations, is

GOD = GOD0 −t∑

t=0

(g OX GOD) (9)

Fig. 2 shows a simulation of this type of system, usingEqs. (6)–(9)with arbitrary parameters (specified in the fig-ure) for initial concentrations and coefficients. Although amore realistic model ought to include other phenomena aswill be seen later, the schematism of such a simulation doesnot affect the conclusions of interest here: (1) in a typicalMichaelian experiment, the effect of the hydrogen peroxidecan produce a decreasing rate from a maximum value, withsuperficially similar profiles to those due to SI; however,(2) the fall from a maximum only occurs when working atrates far from the initial rate and (3) the fit of those profiles


S (g/L) S (g/L)

t1

t2

t3

Fig. 2. Simulation of the interference by hydrogen peroxide in the ki-netic profiles obtained with increasing reaction periods. Simulation wasperformed usingEqs. (6)–(9)with the following parameters:Vm = 0.8;Km = 2; GOD(t=0) = 1; S0 = 100; g = 0.005; t1 = 75; t2 = 100;t3 = 110. Left: expected results (points) and their fit (lines) toEq. (2) inTable 2(note the similarity of casest2 and t3 with the true substrate in-hibition). Right: residuals of the corresponding fits (note the pronouncedautocorrelation).

to Eq. (2) in Table 2(true substrate inhibition) is poor andpresents clear autocorrelation in the residuals.

Accordingly, the discrimination between the effects of SIand hydrogen peroxide tend to become more problematicas the reaction period increases and, naturally, as the ex-perimental error increases. These factors therefore acquiremore importance than usual in this type of test. In fact, com-paring the proportions of product formed and activity ofGOD remaining in buffered media, at long reaction periods(Fig. 3) the apparent stability of the most impure prepara-tion is greater, and addition of a catalase supplement equatesthe results of both preparations. In a Michaelian approach,however, the impure supplemented preparation reproducesthe kinetic profile ofFig. 1, without contributing to the dis-appearance of the effect attributed to SI.

3.2. Substrate inhibition

Considering the fact that the presence of catalase elimi-nates the interference of hydrogen peroxide in the evaluationof SI, it was decided to substantiate the extent of this effectwith the most impure preparation in a new Michaelian ex-periment. The conditions were the same as previous, but in-creasing the domain of the substrate concentration to 250 g/l.The experiment was carried out by triplicate, and confidenceintervals (α < 0.05) are represented inFig. 4. Again, thefit of the values (quasi-Newton, parameters inTable 3) toEqs. (1) and (2)in Table 2suggested SI. The revised conclu-sion, now with no possible interference by peroxide, mustconsider the following factors.

As Tan et al.[32] have shown in terms applicable to thiswork, Eq. (2) in Table 2can be derived from the existencein the enzyme of specific points, different from the active

0

50

100

0 25 50

Time (h)

0

50

100

0 25 50

Act

ivity

GnH

(%

)A

ctiv

ity G

OD

(%

)Fig. 3. Production of gluconic acid and GOD remaining in the impurepreparation (left) and the SIGMA product (right), without (�) and with(�) a supplement of 100 EU/ml catalase. The solution was buffered with0.2 M citrate–phosphate; pH 5.1. Initial GOD: 2 EU/ml; initial substrate:25 mg/ml.

site, whose binding to the substrate determines inhibition.This may justify the discrepancies alluded to in the intro-duction in terms of the differences between enzymes of dif-ferent origin. However,Eq. (2)also adequately describes thephenomena derived from diffusion restrictions in the reac-tion mixture. The effect is clear with polymeric substrates,

0

0,4

0,8

1,2

1,6

0

2

4

6

8

0 50 100 150 200 250S (g/L)

Fig. 4. Kinetics of the impure preparation of GOD, adjusted to theMichaelis–Menten model without (fine line) and with (thick line) SI. Thebars indicate the confidence interval (n = 3; α < 0.05) of the productionslopes of gluconic acid, calculated from five intervals of maximum lengthof 4 min.


Table 3Parameters for models (1) and (2), applied to the experimental results ofFig. 1

Without substrateinhibition

With substrateinhibition

Km (g/l) 2.613 3.523Vm (mM/min) 8.052 9.003Ks (1/[S]) – 0.0010Correlation (r) observed versus

expected values0.974 0.996

whereby the effect increases with the degree of polymeriza-tion [33,34], and generates kinetics formally indistinguish-able from SI[35]. Accordingly, and in absence of evidencefor the existence of specific points of SI, the results inFig. 4could translate this second type of phenomena, which areprobably more independent of the enzyme origin. Althoughfor GOD it is unreasonable to attribute significant diffusionaleffects to the substrate, the high viscosity of gluconic acidcould make a contribution, in addition to its possible role asa competitive inhibitor.

3.3. Product inhibition

Product inhibition was studied by comparing the ratesof a control under the previous conditions (substrate up to150 g/l), with those obtained with 2.5, 5, 10, and 20 g/l glu-conic acid (12.75, 25.49, 50.98, and 101.96 mM, respec-tively). Fig. 5shows the resulting fits to the models discussedbelow.

With the most moderate levels of substrate tested here, thecharacteristic profile of SI in the control without initial glu-

0

2

4

6

8

0 50 100 150

S (g/L)

Fig. 5. Kinetics of GOD in absence (�) and presence of gluconic acid(() 12.75; (�) 25.49; (�) 50.98; (�) 101.96 mM), fitted (continuouslines) to model (14).

conic acid was less prominent, despite the fact that the bestfits were again obtained withEq. (2) in Table 2. Therefore,this was taken as the starting point in which other possibleinhibitory effects were introduced:

competitive inhibition :

V = VmS

Km(1 + KcIc) + S(1 + KsS)(10)

uncompetitive inhibition :

V = VmS

Km + S(1 + KsS)(1 + KaIa)(11)

noncompetitive inhibition :

V = VmS

[Km + S(1 + KsS)](1 + KnIn)(12)

Since under these conditions the classic methods of lin-earization were inadequate, the evaluation criterion wasbased on the ability of the proposed models to simultane-ously solve (quasi-Newton) the five series of results.

A superficial examination ofFig. 5suggests that inhibitionby gluconic is not, or not only, competitive, since it producesa decrease in the maximum rate. In fact, the fit toEq. (10)ispoor: at high levels of gluconic acid the rate is overestimated,and low levels create SI as an artefact. It may be concluded,therefore, that the effect of gluconic acid is more importantthan that proposed by the model, whereby the fitting method(in absence of the adequate term) put too much emphasis onthe coefficientKs. Additionally, models (11) and (12) wereunsatisfactory, although the latter produced profiles betweenobserved and expected results somewhat more approximatedand correlations close to linearity (r = 0.982;Fig. 6A).

Accordingly, the noncompetitive model can be acceptedas a base for improvement with additional assumptions,guided by the analysis of the residuals generated by thecorresponding equations. Such residuals demonstrate a cur-vature in the noncompetitive model which suggests theabsence of a second-order term (Fig. 6A). Therefore, inthis case the noncompetitive term can be transformed from(1 + KnI) to(1 + KnI

2):

V = VmS

[Km + S(1 + KsS)](1 + KnI2n)

(13)

The modification raises the randomness of the distributionof the residuals, reduces the curvature of the correspondingfunction (Fig. 6B), and improves the correlation betweenobserved and expected values (r = 0.993). However, thecriterion for introducing this modification must also validatecompetitive mechanism (r = 0.997; Fig. 6C) which, aloneincapable of explaining the results, is now the only assump-tion which improves the fit of theEq. (13). Table 4resumesthe characteristics of the equations discussed, leading to theproposition of the following descriptive model:

V = VmS

[Km(1 + KcIc) + S(1 + KsS)](1 + KnI2n)

(14)


0

5

10

-1

0

1

0

5

-1

0

1

0

5

0 5 10

-1

0

1

0 40 80

observed values gluconic acid (mM)

(A)

(B)

(C)

Fig. 6. Correlation between expected and observed results (left) andresidual distributions (right) as a function of the concentration of gluconicacid (right), usingEqs. (12)–(14)(A, B and C, respectively).

Although competitive inhibition of GOD by gluconic acidis not problematic, the justification of the qualified term of“second-order noncompetitive effect” seems more obscure.Accordingly, it should be noted that it is difficult to find analternative toEq. (14), with other reasonable solutions beingvery far from a satisfactory fit. In fact, the hypothesis of anytwo or three simultaneous inhibitory effects generates unac-ceptable residuals and biases. An additional uncompetitiveterm inEq. (14)produces a negativeKa coefficient, and thecalculation with the restrictionKa ≥ 0 reproducesEq. (14)with Ka = 0. Finally, if the concentration of gluconic acid isconsidered as the sum of the initial plus the average concen-tration produced during the incubation, again one obtainsEq. (14), with variations lower than 10−5 in Kn andKc.

To suggest an underlying physical phenomenon of thequadratic term inEq. (14), a direct interpretation of the

Table 4Parameters and fits of the specified models for the kinetic description ofGOD

Model (2) Model (12) Model (13) Model (14)

Km (g/l) 3.523 5.888 5.708 4.364Vm (mM/min) 9.003 11.091 9.526 9.350Ks (l/g) 0.0010 0.0016 0.0015 0.0017Kc (l/g) – – – 0.015Kn (l/g) – 0.014 0.00015 0.00011r (observed versus

expected)0.996 0.982 0.993 0.997

expression would suggest that there is a retarding effectof enzyme–substrate binding. Moreover, the effect ap-pears not to involve blocking of the active site and is ofgreater intensity (thus second-order) than a conventionalnoncompetitive inhibitor. Furthermore, this phenomenonmay also be related to the cited restrictions on diffusiondue to gluconic acid viscosity, and in fact, the effect de-creases when increasing the stirring rate of the reactionmixture, a factor, however, that is not possible to includedirectly in a Michaelian model. Thus, it can be speculatedthat the viscosity retards the enzyme–substrate binding andthe enzyme–product separation, by mechanisms not strictlyagreeable to those of noncompetitive mechanism or SI.Therefore, they cannot be described adequately with thesefunctional forms.

3.4. Effect of catalase: a dynamic model

To study the effect of catalase on GOD activity, two seriesof tests were performed with GOD (2 EU/ml) and glucose(150 mg/ml) in the absence and presence of 100 EU/ml cata-lase. The incubation period was prolonged for 70 h at 30◦C.The experimental results, fitting to the model discussed be-low, are shown inFig. 7.

The model is based on arguments which complete thosefor the simulation inFig. 2, and is of dynamic character fortwo reasons. Firstly, hydrogen peroxide, a product of thereaction, deactivates the GOD; moreover, catalase decom-poses hydrogen peroxide, at the same time as it contributesto deactivate catalase. Accordingly, the presence of catalasecauses the following sequence of phenomena: (a) favors theaction of GOD; (b) increases the rate of hydrogen perox-ide production; (c) increases the rate of catalase deactiva-tion; (d) contributes to the decrease in GOD activity. Thesituation, therefore, can be described from the instantaneousrates of formation (v) and decomposition (u) of the speciesinvolved, and thereafter their concentrations remaining aftera time interval�t.

3.4.1. Gluconic acid (GnH)WhereS is the concentration of substrate (glucose),m a

Michaelian function, with or without inhibition, and wheresubscriptt refers to time

v GnHt = m(St, GODt) (15a)

GnHt = GnHt−1 + v GnHt−1 �t (15b)

3.4.2. Hydrogen peroxide (OX)As a co-product of the reaction, its formation rate (in

molar units) is given by the same equation as for gluconicacid:

v OXt = v GnHt (16a)

OXt = GnHt (16b)


0

50

100

150

0 20 40 60

Time (h)

0

1

2

0 20 40 60 80

Fig. 7. Gluconic acid produced (left) and remaining GOD (right), in incubations of the impure preparation, without (�) and with (�) catalase supplement,under the conditions described inFig. 3. The continuous lines represent the corresponding fits to the model proposed inEqs. (15a)–(23b).

Hydrogen peroxide decomposition can be considered asthe result of two processes, one spontaneous, for whichfirst-order kinetics can be assumed as the simplest hypoth-esis:

u1 OXt = k1 OXt = k1 GnHt (17)

and another catalyzed by catalase (CAT) and, therefore,Michaelian:

u2 OXt = m(OXt , CATt) = m(GnHt , CATt) (18)

In view of the fact that both processes, along with the for-mation of OX, are simply additive

u OXt = u1 OXt + u2 OXt (19a)

OXt = OXt−1 + �t(v OXt−1 − u OXt−1) (19b)

3.4.3. Glucose oxidaseIs assumed to be affected by three types of processes:

1. Spontaneous decomposition, with first-order kineticsas the simplest hypothesis:

u1 GODt = g1 GODt (20)

2. Deactivation by hydrogen peroxide, with second-orderkinetics, first in each species, again as the simplesthypothesis:

u2 GODt = g2 OXt GODt (21)

3. Deactivation by the difference (H) between the pH ofmaximum stability and the system pH:

u3 GODt = g3 Ht GODt (22)

Summing the three rates:

u GODt = GOD(g1 + g2 OX + g3 Ht) (23a)

GODt = GODt−1 − u GODt−1 �t (23b)

With these principals, the rate of formation of gluconicacid in a time instantt, equivalent to the formation of hy-drogen peroxide during the same instant, is modified inthe following time instant,t + 1, in agreement with theMichaelis–Menten equation. This arises in virtue of the vari-ation in substrate concentration and also by the fall in theconcentration of GOD during the time period�t.

Accordingly, numerical integration of the net rate equa-tions (16a) and (23a) with respect to time allows to cal-culate the concentration of product and enzyme activity.Thereafter minimizing the quadratic differences betweenthese and the experimental values, it is possible to estimatethe parameters involved inEqs. (16a)–(23b). The solutionobtained (Fig. 7) presents a good fit to the experimentalresults (correlation coefficients of 0.978 and 0.995). ThepH of maximum stability (5.49) agrees with the literaturedata and the data confirmed by direct measurement. Thekinetic parameters, including the corresponding inhibitoryeffects, coincide with the results described in the previoussection (the noncompetitive inhibition does not appear lessimportant).

Finally, attention should be drawn to a problem whichis not easy to solve in the context of the system studiedhere and already discussed in a precedent work[29]. Re-cent estimates[33] of the kinetic parameters of the purifiedcatalase fromA. niger provide values of 322 mM H2O2 and3.62×105 mol O2/heme/s, respectively, forKm andVm. Butthe conditions studied here are those of a more complex sys-tem that does not necessarily guarantee a behavior of thecatalase that allows the rigorous calculation of its kinetic pa-rameters. Moreover, since catalase is an enzyme with a highconversion rate, even a very low concentration is enough toreduce the concentration of hydrogen peroxide in the sys-tem to values that are difficult to quantify with a reasonabledegree of error (i.e. the rapid disappearance of the peroxideis guaranteed starting from a value of the maximum rate thatcan be inferior to the real one).


4. Conclusions

Oxidation of glucose catalyzed by GOD can only be satis-factorily adjusted to a Michaelian model if diverse inhibitoryeffects of the substrate and products (gluconic acid and hy-drogen peroxide), as well as possible interferences due tothe catalase (often present in partially purified preparationsof GOD), are taken into account. Many kinetically focusedstudies concentrate on the behavior of the immobilized en-zyme, applied to the development of glucose probes. Dueto the very low substrate concentrations used in these stud-ies, detection of the above-mentioned effects becomes diffi-cult. In the microbiological field, on the contrary, the usualconditions involve higher concentrations of glucose, but thesystems studied contain many components that vary withtime, hindering the kinetic approach (it should be pointedout, however, that the indications of inhibitory effects orig-inate basically from this area).

In this work, instigated by previous observations in micro-bial cultures, the complexity of these systems is simplifiedwithout reducing their habitual glucose levels, and maintain-ing the presence of catalase. In this way, it was possible toshow that the kinetics of the GOD is affected by (1) substrateinhibition, (2) competitive inhibition by gluconic acid, (3) adecrease of the reaction rate due to diffusional restrictionsdetermined by the viscosity of the gluconic acid, and (4) adecrease of the reaction rate due to enzyme deactivation byhydrogen peroxide, a feature (with some phenomenologicalresemblance to true substrate inhibition) which disappearswhen catalase is present. Working within a wide interval ofglucose concentrations, or in the context of microbial cul-tures for production of GOD, the above effects must nec-essarily be considered. The models proposed in this paperallow us to evaluate them.

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a mathematical model for glucose oxidase kinetics, including inhibitory, deactivant and diffusional...

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