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Process Biochemistry 37 (2001) 267–274

A mechanistical mathematical model to predict lactose hydrolysisby -galactosidase in a permeabilized cell mass of Kluy eromyceslactis : validity and sensitivity analysis

Edimar A.F. Fontes a , Flavia M.L. Passos b, *, Frederico J.V. Passos a

a Departamento de Tecnologia de Alimentos , BIOAGRO , Uni ersidade Federal de Vicosa , 36571 -000 Vicosa , MG , Brazil b Departamento de Microbiologia , BIOAGRO , Uni ersidade Federal de Vicosa , 36571 -000 Vicosa , MG , Brazil

Received 3 November 2000; received in revised form 10 April 2001; accepted 13 May 2001

Abstract

The kinetics of lactose hydrolysis by intracellular -galactosidase in a preparation of Kluy eromyces lactis cells permeabilizedwith ethanol was assessed in the presence of lactose and its hydrolysis reaction products galactose and glucose. Enzyme inhibitionby the reaction products was tested using different concentrations of galactose (1, 5, 10, 20, 30 and 40 mM) and glucose (10, 30and 50 mM) and a combination of both. Galactose was a competitive inhibitor. The competitive or non-competitive nature of theinhibition was dened by the smallest value presented by the residual sum of squares of regression as determined using theMarquardt iterative method of the non-linear regression (NLIN) program of the statistical analysis system (SAS). The kineticconstants V max , K m and K i, with values corresponding to 0.291 0.01 mM min − 1 ; 2.536 0.412 mM and 10.88 8.86 mM,respectively, were also estimated by the same program. Enzyme activity was not affected by glucose at the concentrations tested.Glucose affected enzyme activity only when galactose was present. A mechanistic mathematical model was developed to describe

lactose hydrolysis, taking into consideration the inhibitory effect of galactose. Sensitivity analysis of the coefcients of the modelproposed was performed. Varying estimated K m and K i values by 20% had no effect on lactose hydrolysis kinetics. However,with a 20% increase in the estimated V max value, the model more closely approximated the experimental data for initialconcentrations of 30 mM glucose and 10 mM galactose. Model simulation closely approximated experimental data from lactosehydrolysis in a 5% phosphate-buffered lactose solution as well as in skim milk. © 2001 Elsevier Science Ltd. All rights reserved.

Keywords : Milk; Galactose; Enzymatic inhibition

www.elsevier.com / locate /procbio

1. Introduction

One of the commercial sources of -galactosidase for

processing milk and its products is the yeastKluy eromyces lactis whose natural habitat is the dairyenvironment. This enzyme is used by the industry tohydrolyze lactose to galactose and glucose to producemore digestible, sweeter and smoother dairy products[1,2].

Because it is an intracellular enzyme, yeast -galac-tosidase must be extracted from the cell to be isolated.Alternatively, yeast cells can be permeabilized to allow

substrate access to the cytoplasmic enzyme. Ethanolpermeabilization of yeast cells has been tested andproven to be an economical, easy and safe process

[2–5]. Permeabilization produces a crude but very con-venient enzyme preparation for the regional dairy in-dustry, which could be obtained it at low cost or evenproduced in its own process line.

Enzyme activity in the permeabilized cell mass is lowand thus must be compensated by adopting a biopro-cess with high volumetric productivity. To select andoptimize this process, it is important to characterize theenzyme kinetics of the permeabilized cell mass, espe-cially with respect to product inhibition. TheMichaelis–Menten kinetic model has been used to de-scribe the hydrolysis reaction, taking into account theproduct inhibition effect of galactose [6–10].

* Corresponding author. Tel.: + 55-31-899-2958; fax: + 55-31-899-

2864.E -mail address : [email protected] (F.M.L. Passos).

0032-9592 /01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved.PII: S0032-9 592(01)00 211-4



E .A.F . Fontes et al . / Process Biochemistry 37 (2001) 267 – 274 268

A kinetic model equation must not only be de ned,but also validated. In fact, the model equation is thestarting point for validity and sensitivity analysis stud-ies, through which changes in the equation parameterscan be made. In order to be validated the equationmust describe satisfactorily the actual system underdened conditions. The next step in validation is sensi-tivity analysis, whereby the sensitivity of the solution

for different values of model coef cients is determined[11]. There are many reasons to perform sensitivityanalysis; (1) to nd the most sensitive equation coef -cients; (2) to predict how any modi cation on thesystem would improve overall system operation; (3) toidentify the effect of coef cient imprecision on systemvariation and (4) to suggest the effect of uncontrolledvariation on extreme values of the coef cients. There-fore, after establishing a system solution, a detailedsensitivity analysis has greater value than the solutionitself [11].

This paper is concerned with the preparation of alcohol permeabilized K . lactis cells and the evaluationof -galactosidase kinetics in this crude preparation.The individual and combined inhibition effects of galactose and glucose on lactose hydrolysis in phos-phate buffer were estimated. The kinetic constants wereestimated by nonlinear regression and a mechanisticmodel was proposed, which described lactose hydrolysisusing permeabilized K . lactis as the biocatalyst. A sensi-tivity analysis of the proposed mathematical model wasperformed. We also veri ed the accuracy of the modelin describing lactose hydrolysis in a 5% buffered lactose

solution as well as in skim milk.

2. Materials and methods

2 .1. Cell mass production and permeabilization

K . lactis was originally obtained from the FoodScience Department of the University of California atDavis. The stock culture was maintained in 20% glyc-

erol at − 80 °C. Cells were activated and cultured inultra ltered cheese whey (UFW) prepared using a10 000 Da exclusion limit membrane (Desalination Sys-tem) and autoclaved at 121 °C for 10 min after addi-tion of 4 g l − 1 sodium citrate. At the maximum

-galactosidase activity [12], i.e. after cultivation for 18h at 30 °C and 150 rpm in UFW ( A600 3.5), cellsfrom 1000 ml of the culture medium were harvestedand permeabilized. Cells were resuspended in 100 ml50% ethanol, stirred for 15 min, washed twice withdistilled water and the precipitate dried at 30 °C for 1h under a ow of air. The permeabilized cell mass wasmascerated and stored at 4 °C.

2 .2 . Effect of enzyme concentration on lactosehydrolysis

A suspension of permeabilized cell mass (12 mgml − 1 ) was prepared in phosphate buffer [13], withoutmercaptoethanol, as an enzyme stock solution. Theeffect of enzyme concentration on reaction rate wasevaluated at lactose concentrations of 1, 40, 100 and

140 mM. At each substrate concentration, permeabi-lized cell concentrations of 0.1, 0.5, 1.0, 1.5, 2.0, 2.5 and3.0 mg ml − 1 were assayed. The activity of -galactosi-dase was determined using the GOD-PAP ® (Merck)enzyme system for glucose analysis. The reaction mix-tures (2.5 ml) consisted of 2 ml of GOD-PAP ® reagent,0.25 ml of substrate and 0.25 ml of enzyme. Thereaction rate was determined by linear regression of theplot of absorbance at 510 nm ( A510 ) versus time foreach enzyme and lactose concentration after convertingA510 to mM of glucose using a standard curve.

2 .3 . -Galactosidase unit

Enzyme activity was determined from the linear re-gression of a plot of reaction rate (velocity) versus dryweight of permeabilized cells. A linear relationship wasdened between A600 measurements of permeabilizedcell suspensions and dry weight. One enzyme unit wasdened as ‘milligram of permeabilized cell mass, whichproduces 1 mM of glucose from lactose per min, underconstant reaction conditions (pH 7.1; 25 °C; phosphatebuffer) ’.

2 .4 . Effect of galactose and glucose concentrations on- galactosidase acti ity

An enzyme stock solution of 6 mg ml − 1 of permeabi-lized cells was prepared as previously described. Tostudy the effect of product inhibition, the enzyme con-centration chosen was 1 mg ml − 1 , which produced alinear relationship at all substrate concentrations previ-ously tested. Solutions of substrate and the hydrolysisproduct galactose were prepared from an initial stock

solution of 480 mM of the sugar in phosphate buffer.The concentrations assayed were 0, 1, 5, 10, 20, 30 and40 mM of lactose or galactose and all possible combi-nations thereof. The reaction was conducted in a spec-trophotometer cuvette with a nal volume of 3 ml,consisting of 0.5 ml of enzyme solution, 0.25 ml each of lactose and galactose and 2 ml of GOD-PAP ® reagent.The reaction kinetics were followed in a Beckman DU ®

spectrophotometer at 510 nm. The reaction rate wasdetermined by linear regression of the plot of A510

versus time for each combination of substrate andinhibitor concentration after converting A510 to mM of glucose.




To evaluate glucose inhibition, the absence and pres-ence of glucose at 10, 30 and 50 mM concentrations,with a xed lactose concentration of 100 mM, weretested. To test the combined effect of glucose andgalactose, the same concentrations, i.e. 10, 30 and 50mM for each sugar, in a total of nine combinationswere tested. All solutions were prepared in phosphatebuffer and the assays were performed in a reaction

volume of 30 ml. The reaction was conducted at 25 °Cwith magnetic stirring. Samples were withdrawn fromeach reaction mixture at 15-min intervals. The reactionwas stopped by adding 500 l of 4.5% trichloroaceticacid. Samples were analyzed for glucose, lactose andgalactose using a Hewlett Packard HP 1050M highpowered liquid chromatography (HPLC), equippedwith an HPX 87 H (Bio Rad) column and a refractiveindex detector. The carbohydrates were eluted from thecolumn with 0.005 M H 2 SO 4 at a ow rate of 0.7 mlmin − 1 and 60 °C.

2 .5 . Kinetic constant estimations

The kinetic values for V max , K m and K i were esti-mated using the non-linear regression (NLIN) programof the Statistical Analysis System (SAS), by the iterativeMarquart method [14]. The following equations wereapplied in the program, Michaelis – Menten (Eq. (1))and its derivative equations for competitive (Eq. (2))and non-competitive inhibition (Eq. (3)).

V =V max S

K m + S (1)

V = V max S

S + K m (1 + (I /K i)) (2)

V =V max S /(1 + (I /K i))

K m + S (3)

The best estimate of the kinetic constants and themodel of galactose inhibition on lactose hydrolysis weredened by the smallest value presented by the residualsum of squares of the regression model.

2 .6 . Numerical solution and sensiti ity analysis of model coefcients

The dynamic system with ordinary differential equa-tions was solved by the 4.5 order Runge – KuttaFehlberg method with step-wise variations, in the Win-dows operating system [15]. The ordinary differentialequations were

d(Lactose)dt

= − V max L

L + K m (1 + (G /K i))

d(Glucose)dt = −

d(Lactose)dt

d(Galactose)dt

= −d(Lactose)

dt − K [G ]

where L stands for lactose concentration (mM); G denotes galactose concentration (mM); and K is rate of galactose consumption by the transferase reaction (mMmin − 1 ).

A program was developed for input of the K , reac-tion time and the initial concentrations of lactose,

glucose and galactose, generating a spread sheet con-taining variation in sugar concentration with time.Data on glucose and galactose liberated from hydrol-

ysis of 100 mM lactose by a 1 mg ml − 1 permeabilizedyeast biomass suspension were used in order to estimatethe constant K . The reactions were conducted at roomtemperature (25 °C) for 3 h. Samples were withdrawnat 15-min intervals.

2 .7 . Model alidation

Hydrolysis reactions in a 0.5 g l − 1 phosphate-buffered lactose solution and in skim milk containing0.48 g l − 1 lactose were performed to validate themodel. The reactions were carried out in a 125-ml askwith a 50-ml reaction volume using a 1 mg ml − 1

enzyme preparation. Samples were withdrawn from thereaction at 15-min intervals for 7.5 h. The reaction ineach sample was stopped by adding 500 l TCA (pH1.0). Glucose concentration was determined by twomethods, liquid chromatography and spectrophotome-try at 510 nm (Beckman DU ® ), using the GOD-PAPkit (Merck) [7].

Data generated by the model were compared withexperimental data obtained from the hydrolysis reac-tion using two substrates, a 5% lactose solution andmilk.

3. Results and discussion

3 .1. Effect of enzyme concentration on lactosehydrolysis

The effect of enzyme concentration on lactose hy-

drolysis rate (Fig. 1) indicated that 1 mg ml− 1

was themaximum enzyme concentration at which the Michael – Menten model could be applied when the substrateconcentration was in the range of 1 – 140 mM. Whenthe ratio enzyme /substrate was high ( 1 mg ml − 1

enzyme) the relationship between hydrolysis rate andsubstrate concentration was no longer linear.

3 .2 . -Galactosidase unit

The A600 values of the biomass suspension wereconverted to dry weight using the equation in Fig. 2. Itwas estimated that 1 mg of dried permeabilized cell




Fig. 1. Effect of permeabilized K . lactis cells on the kinetics of lactosehydrolysis.

mass hydrolyses 0.32 mM of lactose per min. It wasdeduced that 3.2 mg of permeabilized cell mass corre-sponded to one enzyme unit. As expected, this enzymeunit was about ve times lower than that of a commercialsource of -galactosidase [16]. This re ects the fact thatthe enzyme studied was a crude preparation, which wasnot submitted to extraction or puri cation. However, if this enzyme can be recycled it still may be economically

competitive.

3 .3 . Effect of galactose and glucose concentration on- galactosidase acti ity

The effect of galactose concentration on the -galac-tosidase activity of K . lactis permeabilized cell mass isillustrated in Fig. 3. Each point represents the hydrolysisrate for the combined concentration of lactose andgalactose. The lines shown represent the competitiveinhibition model generated using Eq. (2) described inSection 2. The estimates for the kinetic constants V max ,K m and K i are shown in Table 1. Under the conditionsdescribed in this study, these values were estimated to be0.29 mM min − 1 ; 2.54 and 10.88 mM for V max , K m andK i, respectively. The inhibition model of galactose onlactose hydrolysis was de ned by minimizing the residualsum of squares. Galactose is a competitive inhibitor (Fig.3). It effects the apparent K m but leaves the maximumvelocity unchanged, typical of competitive inhibition.

As can be seen in Table 2, large variations in the kineticconstant values are reported in the literature. Thisvariation could be due to differences in experimental

conditions, including temperature, pH, substrate andproduct concentrations but also to the use of lineariza-tion methods to determine kinetic constants.

The advantage of using statistical programs such asSAS to estimate kinetic constants by non-linear regres-sion is that they generate more precise values than thosecalculated from the linearization methods of theMichaelis – Menten model and its derivatives (Table 1).Furthermore, it is possible to de ne a model, which betterts the experimental data by the minimum residual sumof squares of the regression models tested.

In selecting an enzyme bioprocess it is important to

consider product inhibition. Models based on competi-tive inhibition require increased substrate concentrationin the bioreactor; however, in the case of milk or cheesewhey where lactose concentration is around 140 mMcomplete hydrolysis will never be achieved. Completehydrolysis of lactose requires a reaction time ve timeslonger than required to achieve 50% hydrolysis [17].

No effect of glucose on lactose hydrolysis rate wasobserved. It also was noted that glucose did not affectlactose hydrolysis by -galactosidase from Aspergillus foetidus [18]. On the other hand, it has been reported thata high glucose concentration inhibits -galactosidaseactivity in a non-competitive way [19 – 22].

Fig. 2. Linear relationship between A 600 of permeabilized K . lactis cellsuspensions and cell dry weight.

Fig. 3. Effect of substrate concentration on the kinetics of lactosehydrolyses by permeabilized K . lactis whole cell -galactosidase in thepresence of different galactose concentrations. Competitive InhibitionModel. Symbols represent experimental data and lines the SASnon-linear regression model.




Table 1Estimated values of the kinetic constants of lactose hydrolysis by permeabilized whole cell -galactosidase from K . lactis , by linearization of themodels or by non-linear regression using NLIN from SAS (iterative method Marquardt)

V max (mM min − 1 ) K m (mM) K i (mM) Residual square sum

Michaelis – Menten model : Lineawear – Burk 2.5250.278 – 0.0081

2.721 – 0.281 0.0074Langmuir0.275Eadie – Hosftee 2.551 – 0.0079

0.280Non-linear regression 2.609 – 0.00062.536 10.8800.291 0.0256Competitive inhibition*

Non-competitive inhibition 0.320 4.538 91.393 0.0479

* , Constants were adjusted by non-linear regression.

Table 2Kinetic constants of -galactosidase from K . lactis

Enzyme preparation Substrate and reaction condition Kinetic constant Reference

V max K m K i(mM min − 1 ) (mM) (mM)

0.29 2.54Lactose; phosphate buffer; pH 7.1 and 10.88 aEthanol permeabilized cells This work25 °C

43.6Semipuri ed-commercial 51.9Lactose; phosphate buffer; pH 6.86 [6]0.12 17.3Lactose; phosphate buffer; pH 7.3 and 42 aPuri ed [23]

37 °CONPG; phosphate buffer; pH 6.5 and 3.8 b 4.2 – Chloroform-ethanol permeabilized [24]

cells 30 °CLactose; phosphate buffer; pH 6.5 and 0.33 30Puri ed 47 a [20]30 °C

a K i for galactose.b U ONP mg− 1 (DW).

The combined effect of several concentrations of galactose and glucose on lactose hydrolysis is shown inTable 3. Considering V = f (galactose) f (glucose) and themodel proposed here to estimate the effect of galactose,it is possible to estimate the independent effect of glucose, f (glucose) = V ob / f (galactose). Again in thesetests, there was no inhibition effect on the reaction ratedue to glucose alone, while the presence of galactose inthe reaction mixture de nitively affected the hydrolysisrate. The mechanistic model proposed to describe thelactose hydrolysis is

d(L )dt

= − V max L

L + K m (1 + (G /K i))

where L is lactose concentration (mM) and G is galac-tose concentration (mM).

Product inhibition studies are an important part of the characterization of the mechanism of catalyzedreaction. Different kinetic patterns are associated withdifferent mechanistic patterns. In competitive inhibitionsubstrate and product bind to the same enzymeform. Either substrate or product can complex withfree enzyme. This con rms the evidence that -

galactosidase from K . lactis presents a sequential mech-anism in its catalysis of hydrolysis [16]. This also ex-plains the involvement of -galactosidase on transferasereactions .

Table 3In uence of glucose and galactose concentration on the lactosehydrolysis rate by permeabilized whole cell -galactosidase from K .lactis

f (Glucose) bGalactose f (Galactose) aGlucose V ob (mM(mM) (mM) min − 1 )

1.083010 10 0.3005 0.27751.10500.27750.306630 10

10 0.2775 1.079050 0.299410 0.95400.26560.253430

0.26560.2411 0.908030300.2702 0.2656 1.01703050

50 0.229810 0.2548 0.901950 0.254830 0.2044 0.8022

0.225350 0.254850 0.8843

a , f (Galactose) = (V max LK i)/(K iL+ K m K i+ GK m ). The kinetic con-stant values for competitive inhibition from Table 1 were used with

L = 100 mM.b , f (Glucose) = V ob / f (galactose).




Fig. 4. Determination of the rate ( K ) of galactose utilization in thetransferase reaction during the lactose hydrolysis study.

3 .5 . Numerical solution and sensiti ity analysis of themodel coef cients

Fig. 5 presents lactose hydrolysis in the presence of glucose and galactose. The concentration of lactosehydrolyzed was calculated from glucose produced. Thesymbols indicate the experimental data and the line theproposed model.

Variations in K m and K i values on the order of 20%do not affect lactose hydrolysis, as shown in Fig. 5Band C. On the other hand, at initial concentrations of

Fig. 5. Effect of variations in V max (A), K m (B) and K i (C) on lactosehydrolysis in the presence of glucose and galactose. ( — — ) V max ,+ 20% and ( … ) V max , − 20%. The line represents the values esti-

mated by the model and symbols represent experimental data ( )glucose and ( ) galactose.

3 .4 . Model alidation

An equimolar reaction was assumed for the HPLCanalysis, that is 1 mM lactose producing 1 mM glucoseand 1 mM galactose. However, it was observed thatlactose hydrolysis did not actually generate equimolarconcentrations of glucose and galactose. One possibleexplanation was that galactose was partially consumedin the transferase reaction, producing oligosaccharides.An unidenti ed peak in the chromatograms, whichincreased as galactose concentration decreased, sup-ports this idea. The galactose molecule could havereacted with another galactose molecule, resulting in a

disaccharide. Similar results were reported with -galac-tosidase from Bacillus circulans where, in addition toglucose and galactose, the presence of di- and trisaccha-rides were detected [25].

Besides the ability to hydrolyze lactose into itsmonosaccharides, -galactosidase also catalyses thetransferase reaction between lactose and its hydrolysisproducts, especially galactose, to yield a family of oligosaccharides rich in galactose moieties called galac-toligosaccharides [26].

The formation of oligosaccharides when lactose washydrolyzed by lactase from Saccharomyces lactis (Max-ilact ® ) was reported [27]. Transferase activity of theenzyme is especially noted at high lactose concentra-tion. Hydrolysis occurs at low lactose concentration,while the production of oligosaccharides by the trans-ferase reaction increases with increasing lactose concen-tration. Both hydrolysis and transferase reactions canoccur simultaneously [28].

Fig. 4 illustrates the data used to calculate the K value, used to correct the estimated galactose concen-tration values, by taking into account loss of galactosethrough the transferase reaction. A K value of 0.0554mM min − 1 was estimated by regression analysis of thedata.




Fig. 6. Lactose hydrolysis in a 5% lactose solution. ( ) representsglucose concentration obtained by HPLC and ( ) by spectrophoto-metry. The line represents the values estimated by the model with a30% increase in V max .

tration, after hydrolysis of 75% of the lactose, was 4%.When the same test was performed with milk, a 13%variation between measured and predicted glucose con-centrations was found after hydrolysis of 85% of thelactose. The measured V max value was 62.5% higherthan that predicted.

Lactase from yeast presents higher activity in skimmilk then in buffered solution. It has been suggested

that milk components, in particular whey proteins andcasein, increase the activity of -galactosidase from K . fragilis [29]. This may also be true for -galactosidasefrom K . lactis , where the V max value was twice as highin milk as in buffered solution, evidence that milkcomponents increase the enzyme ’s activity.

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3 .6 . Veri cation of the model alidity

Figs. 6 and 7 show lactose hydrolysis in a 5% lactosesolution and in skim milk, respectively. The symbolsindicate the experimental data and the lines the pro-posed model for lactose hydrolysis as a function of glucose concentration in the solutions.

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