Determination of kinetics parameters for free and immobilized β-galactosidase

Process Biochemistry Vol.31, No. 3, pp. 243-248, 1996 Copyright O 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0032-9592/96 S 15.00 + 0.00

0032-9592(95)00056-9

ELSEVIER

Determination of Kinetics Parameters for Free and Immobilized fl-Galactosidase C. R. Carrara* & A. C. Rubiolo Instituto de Desarrollo Tecnol6gico para la IndustriaQuimica (U.N.L.,CONICET) Santa Fe, Rep. Argentina (Received 21 November 1994; revised manuscript received 19 June 1995 and accepted 27 June 1995)

The kinetic parameters of soluble and immobilized fl-galactosidase of

Kluyveromyces fragilis were determined using integrated reaction rate equations. Three reaction rate models were used to correlate experimental values by means of a multi-response non-linear computer program. The deviations of the fitted models to the data were compared using a residual sum of squares and F-test for nested models. The Michaelis-Menten equation predicted the best values with product inhibition by galactose and provided the most reasonable approach to describe the enzymic hydrolysis of lactosefor low and high conversion in a range of initial concentrations from 2"5 to 15<9%.

NOMENCLATURE

ks

Mass transfer coefficient (cm/

Co dc Ds

P Re

Galactose concentration (M) Reynolds number Re = (6~//z~)

S Sc Sh

Substrate concentration (M) Schmidt number Sc = (/zs/6~ Ds) Sherwood number Sh-- 2 + 0.6 Sc~/3 ReU2 Initial substrate concentration

Eo E (ES) (EP) g

Drag coefficient Bead diameter (cm) Substrate diffusion coefficient in water (cm2/min) Initial enzyme concentration (mg protein/ml or support weight) Enzyme concentration (mg protein/ml or support weight) Enzyme-lactose complex Enzyme-galactose complex Gravity acceleration constant (9.81 m/seg 2--- 3.53 x 106 cm/

min) edc

So

(M) T Ut V

Temperature (K) Terminal velocity (cm/min) Liquid velocity relative to bead

Vm x

Maximum reaction rate Conversion of lactose

rain2)

(cm/min)

GI Glucose concentration (M) kl, k_ 1, k2, k3, k- 3Specific rate constant k~ Inhibition constant (M) gm Michaelis constant (M) kp ( = k2/k3, M)

Greek letters tZs

*To w h o m c o r r e s p o n d e n c e should b e a d d r e s s e d .

6c

243

Solution viscosity (g/cm min) Solution density (g/ml) Catalyst density (g/ml)

244

C. R. Carrara, A. C. Rubiolo

INTRODUCTION fl-Galactosidase catalyses the hydrolysis of flgalactosidic linkages such as those present in lactose. This enzyme has been isolated from a wide range of microorganisms and has found applications in the hydrolysis of milk lactose. This process is nutritionally necessary for consumers that suffer from lactase deficiency and technologically important for the production of high sugar concentrations, since the mixture of glucose and galactose is sweeter and more soluble than lactose. 1 In cheese manufacture, by-products of the dairy industry including whey and permeate cause environmental problems, which can be reduced when lactose is removed. The neutral fl-galactosidase derived from yeast such as Kluyveromyces fragilis has optimal activity in the pH range 6"5-7"5 and is normally used for the hydrolysis of lactose in milk and sweet whey systems. 2 The kinetic model widely used to describe the reaction of this enzyme is the Michaelis-Menten equation with product inhibition by galactose. 3,4 Another proposed model is one of multiple inhibition; competitive inhibition by D-galactose and non-competitive inhibition by D-glucose. The effect of D-glucose only appears above a threshold concentration?,6 A more complex kinetic system has been proposed by Fhaschel for the Aspergillus niger enzyme] This involves the mutarotation of ogalactose in which a stronger competitive inhibitor, a-galactose, results. Yang, 8 has used a reversible reaction kinetic and a model with a small variation from the Michaelis-Menten equation with product inhibition in the chemical intermediate, while Bernal 9 has used the simpler Michaelis-Menten kinetic model. Usually the Lineweaver-Burk linear transformation of the Michaelis-Menten equation is used to determine the velocity (Vm) and Michaelis-Menten constant (Kin) and the Dixon plot for calculating inhibition constant (ki). These involve using the initial rate data to determine kinetic parameters and neglect the effects of high conversion. Other methods use integrated rate equations to fit the model with a set of values that have been obtained from low to high conversions.10-~2 The K. fragilis enzyme has been immobilized on chitosan beads. ~3The influence of immobilization on the activity of the enzyme can be obtained

by determining the intrinsic kinetics parameters of the immobilized enzyme and comparing them with those of the soluble enzyme. In this paper, a multi-response non-linear computer regression method was used to fit experimental results in order to determine the kinetic constants for soluble and immobilized enzymes of K. fragilis. The values obtained were compared with published data for enzyme preparations of different purity and used to indicate the reaction behaviour.

Kinetics of lactose hydrolysis by fl-galactosidase Michaelis-Menten kinetics without product inhibition are derived from the following reaction sequences: S + E k-L (ES) k-~2E+P+GI.

k-1 Considering a steady-state for the enzyme-lactose complex (ES), the reaction rate is described by: dS

E°k2 S

dt

Km+S '

(1)

where S, P, GI, and E are lactose, galactose, glucose and enzyme, respectively. This expression can be integrated considering the conversion as x = S - So~So, where SO is the initial substrate concentration.

x-

tn(a-x).

(2)

Equation (2) is used to determine the kinetic constants with Model A and its initial values were determined using the Lineweaver-Burk's graphical method. The reaction mechanism of the Michaelis-Menten kinetics model with competitive product inhibition by galactose can be described by the following equations. S+E ~

(ES) k--~2E+P+GI

k-1 k~ (EP) ~ (E +P).

k- 3

Kinetic parameters for fl-galactosidase Following a similar procedure to the previous samples, if k i is the inhibition constant and P - GI, the equation is:

dS --= -

E°k2S

(;)

gm 1+

(3)

+S

Integrating this equation, Model B is obtained:

(4) A different model is obtained with the mechanism postulated by Yang 12 which assumes that the glucose molecule is the first to leave the active site of the enzyme, leaving a galactosyl group joined as an enzyme-galactosyl complex from which the galactose is released. This mechanism of lactose hydrolysis is described by the following equations S+E &

(ES)k--~e(EP)+GI

k-i k~ (EP) ~ (E + P). k_ 3 Operating in a similar mode as the earlier model, the expression of reaction rate is:

dS --=

E°k2S -

,

K m 1+

(5)

+(1 + kp)S

where the reaction rate constant is kp = k2/k3, and the integrated expression for Model C is:

x In(1 - x).

hi] (6)

The graphical method of Yang ~2can be used to determine the constant values for Models A and B, involving two successive linear regressions, which can introduce large uncertainty to the determination of constant values. Due to its simplicity it is used with the Levenberg-Marquardt

245

algorithm as an initial estimation in a multiresponse non-linear computer regression method.

MATERIALS AND M E T H O D S

Kluyveromyces fragilis fl-galactosidase (Lactozym 3000) was obtained from NOVO (Bagsvard, Denmark). The enzyme solution in 4.75% lactose had a specific activity of 3000 LAU/cm 3 (NOVO lactase unit). This commercial preparation was used without purification and had a protein concentration of 35"09 mg/ml. Crab shell chitosan and sodium triphosphate of practical grade were obtained from Sigma Chemical Co. (St Louis, MO). All other chemicals, lactose, dibasic sodium phosphate, monobasic potassium phosphate and (glacial) acetic acid, were analytical reagents from Mallinckrot (St Louis, USA)or Merck (Buenos Aires, Argentina). Kits for the enzymic determination of glucose and the Biuret method of protein estimation were obtained from Wiener Lab (Rosario, Argentina). Preparation of immobilized enzyme fl-Galactosidase was immobilized on chitosan beads by glutaraldehyde as described earlier. ~3 The beads obtained had an average diameter (de) of 2"2 mm and a density (6c) of 1.102 g/cm 3. This support has a nonporous structure. 14The value of protein concentration (E °) determined immediately after immobilization was 20.86 mg protein/g support weight. Determination of kinetic parameters for soluble and immobilized ~-galactosidase The kinetic parameters were determined with the use of varying concentrations of lactose as substrate in potassium phosphate buffer of pH 6.86 (0-025 M KHzPO4, 0"025 M Na2HPO4 and 1 mM Mg 2+ ), in a stirred reactor with temperature control. Following lactose hydrolysis, samples were taken at 60 min to evaluate the glucose concentration. Each lactose concentration was studied at different concentrations of soluble and immobilized enzyme. The glucose concentration was determined enzymicaUy using glucose oxidase)5 In determining the kinetic constants for the immobilized enzyme, it was important to know the influence of the external mass transfer resistance. This was estimated considering the fluid characteristics and differences between the rela-

246

C R. Carrara, A. C. Rubiolo

tive velocity of the surrounding liquid to the particle. If the Sherwood number is larger than 20 (Sh>20), l < R e < 1 0 4 and Sc>0"7, the mass transfer resistance is negligible.16 The Sherwood number was calculated by: Sh

=

2 + 0-6 Scl/3Re 1/2.

where Sh is the Sherwood number (Sh = ksddDs) , Sc is the Schmidt number (Sc = l.ts/t~sDs), and Re is the Reynolds number (Re = (6s/~) Vd,:). The liquid velocity relative to the particle (V) was calculated as V--5 Ut, 16 where Ut, the terminal velocity of a particle falling under gravitation forces in a fluid phase, was determined as:

Table 1. Residuals of the three models with soluble (SE) and immobilized enzyme (IE)

Model Number of Equation parameters number ASE BSE CSE Am Bm Cm

2 3 4 2 3 4

/ 4 g de(6c - 6s)

Co is the drag coefficient, with a value of 1 for a sphere under the experimental conditions used. 17 Variables were calculated with Weast data, TM correcting for temperature changes with the known equation Cte = Dslus/T. A temperature of 43°C, at which both soluble and immobilized enzyme were stable, was used for all assays. Comparison of models The incidence of the parameter number in the different models was evaluated using the following criterion: F(1,n-(i+

l);c)>

sai SSi+ --

1

SSi + i l[ n - ( i - 1 ) ] '

where iis the number of parameters in the simpler model, SS i is the residual sum of squares for the model with i parameters and F ( 1 , n - ( i + 1);c) is an F-test with 1 and n - ( i + 1 ) degrees of freedom at c, confidence level (95%), n is the number of experimental values, used by adjusting the models and equals 234 for the soluble enzyme and 110 for the immobilized enzyme. The parameters of all the models were estimated using a FORTRAN program with a Levenberg-Marquardt non-linear regression subroutine (IMSL) implemented in a VAX 780. RESULTS AND DISCUSSION The results shown in Table 1 were obtained after performing the experiments described carlier in order to evaluate the proposed rate expressions.

2 4 6 2 4 6

1"7248 1"0645 1"0645 0"0174 0"0105 0"0105

j0. j 1.0

.

-y-

1.0 0.8

,//•

0.4 + '!'" . . . . /;~ ~" 0.2

,, •

So 5.0% So 10 0% ModolA 6

I ...... "*d~e

0.e'-

0"985 0"988 0"988 0"996 0"998 0"998

•

,~,.-

/

=v YCc

Coefficient of determination

(R9

0.8 .......

Ut

Residual of squares

. .10. . . .20. . . 30 ..

40 Time (nVrO

50

I 0o

0.6

~50%1 /,~

0.4

•

0.2

I

~.,o.0% I

...... M m I ~ C

0.0

10

0

a)E°: O.OU n.~ p m ~ n ~

20 30 Time (n~n.)

40

[

50

b)E°: 0.070mg pm~n/ml

1.0

0.8

~. f ~ - - -

o,

v/~

i

•

s.

5.0%

~1

•

~* 5,0%1

g~,~J

li " ~','*'~'1°.0" o., e~J'4 /t • ~'°0% '.'*,~". tI

o.2

~

i__uk~.

0.0

. . . . . . . . . . . 10 20 30 40 50 "time (mirt) c)E*: 0.0~5 mo promln/m~

¢/ I 60

o.o

0

/-

"**~"1

............. I 10 20 30 40 50 60 70 Time (rain.)

d)E*: o.noJ mo pm~n/rnl

Fig. 1. Experimental and calculated conversion of lactose by the three models for different initial soluble enzyme

concentrations. The three mechanistic models of the MichaelisMenten type considered in the analysis of the kinetic data for the soluble enzyme showed that the simplest, which did not contain a term to account for competitive inhibition by galactose (eqn 2), presented the greater residual sum of squares. This indicates that Model A did not represent the experimental data as adequately as either Model B or Model C. Figure 1 shows the comparison of the three models with experimental values from initial lactose concentrations of 5"0 and 10.0%. Model A showed the biggest difference from the experimental results and also pre-

Kinetic parameters for fl -galactosidase sented more deviation for the larger enzyme concentrations. Models B and C accounted for competitive inhibition by galactose but they considered different numbers of parameters. Based on the reduction of the residual sum of squares (Table 1), the addition of the extra parameter did not significantly improve the fitting values and resulted in equal sum of squares. While comparing the first and second models with the F-test, a statistically significant difference was obtained, improving the prediction of experimental results with Models B and C. Similar behaviour was observed with the immobiliT.ed enzyme. The kinetic parameters calculated with the three models for free and immobilized enzyme are shown in Table 2. Figure 2 shows the values determined for the six initial lactose concentrations and the theoretical values of glucose concentration for Model B. Good agreement was obtained between Model B and the experimental data. The accuracy of the initial parameter estimation was an important factor that affected the closeness of the final parameter estimates. Starting

Z. Kinetics parameters for soluble immobilized enzyme (IE) with different models

Table

(SE)

and

Model

k~

K,. (mM)

k+(raM)

kp (raM)

ASE BSE

1"8 2"0 2"0 0"64 0"78 0"73

75"1 43"6 44"0 141"1 137"0 135"8

-51"9 52"1 -234"0 246"8

--10"5 --10"8

CSE

AlE BIE Cm

"Specific activity mM (of glucose)/min/g of protein.

0.4.

1

247

from a poor estimation, the non-linear regression could sometimes satisfy the convergence criterion, but the parameter estimated could be physically unrealistic as negative rate constants, etc. Figure 3 shows the experimental data obtained with immobilized enzyme on chitosan beads and calculated values with the intrinsic kinetic parameters of Model B. For the immobiliT.ed enzyme it was important to ensure that interparticle diffusion did not affect the rates of reaction. The values calculated with the six lactose concentrations for Re between 630 and 1400, Sc 1700 and 3800, and Sh 230 and 264, indicated that external mass transfer was negligible and that the correlations presented could be applied. For comparison, Table 3 shows the calculated values for soluble and immobiliT.ed enzymes at 43°C, the values of enzyme from the same sources in dried cells in Tris-maleato-NaOH buffer published by Kuo-Cheng6 and pure soluble enzyme obtained by Mahoney.4 The Km and ki values obtained for soluble and immobilized enzymes fell between values for pure enzyme and dried cells. Bernal9 has indicated that the purification of enzyme preparations could result in a decrease in Km value if any inhibitor present in these preparations was removed. The greater temperature and ionic concentrations in buffers could also influence enzyme activity and kinetics parameters. Kuo-Cheng6 reported that glucose acted as a non-competitive inhibitor, but this effect was not noted in these and other experiments.4 The intrinsic kinetic parameters of the immobilized enzyme were significantly different to those of the corresponding soluble enzyme. This

1.0

1 . 0

•

/i, j o

f

•

0.8

~

o

I

° 8 o.0-

0 . 2 "

0.41

~o.I.

JJb~"IF

I • so ~.5% I " SO1°'°~

N.--.--'-~ I .so12.5%

0.2-

0.0

1.0

I 2.0

. =~'"

3.0

4.0

0.0,~ 5.0

E° , t (m2.min/mt) Fig. 2,

Glucose concentration

• SO 2.5% o SO 5.o%

i~.

• so 7.5%

//~/v,

o So 10.0% • So 12.5% v SO t5.0%

,,~/

¥

o.o,r . . . . .

~/~,',~' ~//~i'."'

I

I --.o~B

. . . . . . . . . . . 10203040506070 "nrne (min) a) E°: 0.967 mg de pro~n/ml

I

/ o

'°

--

....~''~ • '~r 0.4-

I

I ~, . "TI ," i l

*

'1

•

So 2.5%

o

SO 5,0%

*

SO

7.5%

j o SOlOO%1

o2!//~: /~: I' 0 . 0

•

I

•

so12.5% I

I " SO15.0% I I --M°~ ,

'tO

"

,

-

,

-

,

20 30 40 ~me (min)

-

a ,

50

-

} I

60

b)E°: 0.886 mg de p¢otein/mi

in f u n c t i o n o f t h e p r o d u c t

time-soluble enzyme concentration for different initial lactose concentrations.

Fig. 3. Experimental values and values estimated by Model B for lactose conversion by immobilized enzyme.

C. R. Carrara, A. C. Rubiolo

248

Table 3. Kinetics parameters for soluble and immobilized fl-galactosidase

Enzyme state Soluble a Dried cells b Soluble d Insoluble d

Temperature (*C)

pn

g,, (raM)

k~(raM)

k_~

37 37 43 43

6.60 7.00 6.86 6.86

13.9 62.5 43.6 137.0

27.7 77.0 51"9 234-0

-0.578" 2.0 e 0.78 ~

aMahoney. 4 bKuo-Cheng.6 'Specific activity,/~M (of glucose)/min/mg of dried cells. dDeterminated values in this work. eSpecific activity mM (of glucose)/min/mg of protein.

behaviour is due to the immobilization process that altered some of its properties, but not the kinetics behaviour since the experimental results were predicted with the same model and with similar statistic errors. The larger inhibition constant for the immobilized enzyme indicated that in this case steric effects could be present.

9.

10. 11.

REFERENCES 1. Gekas, V. & Lopez-Leiva, M., Hydrolysis of lactose: a literature review. Process Biochem., 20 (1985) 2. 2. Mahoney, R. & Adamchuk, C., Effect of milk constituents on the hydrolysis of lactose by lactase from Kluyveromyces fragilis. J. Food Sci., 45 (1980) 962-4, 968. 3. Weetall, H. & Noshir, B., The preparation of immobilized lactase and its use in enzymatic hydrolysis of acid whey. Biotechnol. Bioeng., 16 (1974) 295-313. 4. Mahoney, R. & Whitaker, J., Stability and enzymatic properties of fl-galactosidase from Kluyveromyces fragilis. J. Food Biochem., 1 (1977) 327. 5. Wondolowski, M. V. & Woychik, J. H., Characterization and immobilization of E. coli (ATCC-26) fl-galactosidase. Biotechnol. Bioeng., (1974) 1633-44. 6. Kuo-Cheng, C., Jer-Ying, H. & Alvin, C., Product inhibition of the enzymatic hydrolysis of lactose. Enzyme Microb. Technol., 7 (1985). 7. Flaschel, E. & Raetz, E., The kinetics of lactose hydrolysis for fl-galactosidase of Aspergillus niger. Biotechnol. Bioeng., 24 (1982) 2499. 8. Yang, S. & Tang, I., Lactose hydrolysis and oligosaccha-

12.

13. 14. 15. 16.

17. 18. 19.

ride formation catalyzed by fl-galactosidase -- kinetics and mathematical modeling. Ann. N.Y. Acad. Sci., 542 (1988) 417-22. Bernal, V. & Pavel, J., Lactose hydrolysis by Kluyveromyces lactis fl-o-galactosidase in skim milk, whey, permeate and model system. Can. Inst. Food Sci. Technol. J., 18 (1985) 97-9. Roberts, D., Enzyme Kinetics, Cambridge University Press, 1977, pp. 48-82. Halwachs, H., Km and Vm from only one experiment. Biotechnol. Bioeng., 20 (1978) 281-5. Yang, S. & Okos, M., A new graphical method for determining parameters in Michaelis-Menten type kinetics for enzymatic lactose hydrolysis. Biotechnol. Bioeng., 34 (1989) 763-73. Carrara, C. & Rubiolo, A., The immobilization of flgalactosidase on chitosan. Biotechnol. Prog., 10 (1994) 220-4. Braun, J., Le Chanu, P. & Le Goffic, The immobilization of peniciUun G-acylase on chitosan. Biotechnol. Bioeng., 33 (1989) 242. Trinder, P., Enzymatic method for determination of glucose. Ann. Clin. Biochem., 6 (24) (1969). Prass, N. & Hesselink, P., Kinetic aspects of bioconversion of L-tyrosine into L-DOPA by cells of Macuna pruriens L. entrapped in different matrices. Biotechnol. Bioeng., 34 (1989) 214. Perry, R. & Chilton, C., Chemical Engineer's Handbook, 6th edn, MacGraw-Hill, 1984. Weast, R. & Melvin, A., CRC -- Handbook of Chemistry and Physics, 69th edn, 1989, pp. D125-7. Prenosil, J. E, Stuker, E. & Bourne, J. R., Formation of oligosaccharides during enzymatic lactose hydrolysis. Part I: State of art. Biotechnol. Bioeng., 30 (1987) 1019-25.