A new approach for determination of enzyme kinetic constants using response surface methodology

Biochemical Engineering Journal 25 (2005) 55–62

A new approach for determination of enzyme kinetic constants using response surface methodology ˙Ismail Hakkı Boyacı ∗ Department of Food Engineering, Faculty of Engineering, Hacettepe University, Beytepe Campus, TR-06532 Ankara, Turkey Received 2 October 2003; received in revised form 16 March 2005; accepted 10 April 2005

Abstract A statistical approach called response surface methodology (RSM) is used for the prediction of the kinetic constants of glucose oxidase (GOx) as a function of reaction temperature and pH. Lineweaver–Burk transformation of the Michaelis–Menten equation was utilized as the integral part of the RSM algorithm. The effects of variables, namely reciprocal of substrate concentration (0.033–0.5 mM−1 ), reaction temperature (14.9–40.1 ◦ C) and reaction pH (pH 4.4–8.5) on the reciprocal of initial reaction rate were evaluated and a second order polynomial model was fitted by a central composite circumscribed design (CCCD). It was observed that optimum reaction temperature and pH for the GOx reaction depended on the substrate concentration and varied between 27.8 ◦ C and 6.4 pH and 32.7 ◦ C and 6.1 pH in the investigated range of substrate concentration. The maximum reaction rate (Vmax ) and Michaelis–Menten constant (Km ) of GOx were obtained for each reaction parameter by using the model equation. The maximum reaction rate varied between 3.5 ␮mol/min mg enzyme and 29.8 ␮mol/min mg enzyme. Michaelis–Menten constant was determined between 1.9 mM and 16.8 mM in the tested reaction parameters. The kinetic constants of GOx were also determined with the conventional method at six reaction parameters and compared with the results of the proposed method. The correlation coefficients (R2 ) between the results of two methods were determined as 0.940 and 0.869 for Vmax and Km , respectively. © 2005 Elsevier B.V. All rights reserved. Keywords: Enzymes; Kinetic constants; Glucose; Glucose oxidase; Response surface methodology

1. Introduction The characteristics of enzymes such as kinetic constants, activation energy, temperature, pH, water activity profiles, etc. determine the usability and productivity of enzymes. For that reason, these characteristics should be determined for a newly discovered enzyme, an enzyme used in different reaction mediums or an enzyme used in different forms (such as free or immobilized). In developing an enzyme based process, kinetic constants are the most important information which has to be determined. Simple enzyme kinetics is generally described by the Michaelis–Menten kinetics (Eq. (1)) [1]. V =

Vmax S Km + S

(1)

where V is the enzymatic reaction rate (mmol/min mg enzyme), S is the substrate concentration (mM), Vmax is the ∗

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maximum or limiting reaction rate (mmol/min mg enzyme) and Km is the Michaelis–Menten constant (mM). By rearranging Eq. (1), the following equation can be derived: 1 Km 1 1 = + V Vmax Vmax S

(2)

Plotting of Eq. (2) as 1/V versus 1/S (known as a Lineweaver–Burk plot) gives an appropriate linear plot that is commonly used for determination of numerical values of Vmax and Km . Besides of Lineweaver–Burk plot, other plotting methods (Eadie–Hofstee plot, Hanes–Woolf plot) and non-linear regression methods have been also used for determination of kinetic constants [1–3]. Response surface methodology (RSM) is a collection of statistical and mathematical techniques useful for developing, improving, and optimizing processes [4]. RSM defines the effect of the independent variables, alone or in combination, on the process. In addition to analyzing the effects of the independent variables, this experimental methodology

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generates a mathematical model that accurately describes the overall process [5]. It has been successfully applied to optimizing conditions in food, chemical and biological processes [5–9]. In recent years, RSM has been used to determine the kinetic constants of enzymatic reactions as well as for the optimization of reactions [6,8]. Andersson and Adlercreutz [6] used RSM to determine the Vmax and Km values of horse liver alcohol dehydrogenase between pH 4 and pH 8 using a second order polynomial model, obtained from RSM, and the Michaelis–Menten equation. A quite good correlation between the kinetic constants obtained from conventional methods and the ones obtained from RSM was reported. Beg et al. [8] used RSM to determine Vmax , Km , catalytic power of enzyme (Kcat ) and activation energy (Ea ) of alkaline protease from Bacillus mojavensis. The kinetic constants were determined between 45 and 60 ◦ C using a second order polynomial model, obtained from RSM, a Lineweaver–Burk plot and an Arrhenius plot. It was reported that values found were comparable to those obtained with conventional methods and RSM could successfully be adopted for determining kinetic constants for enzyme-catalyzed reactions. In this study, a new approach to determine the kinetic constants of glucose oxidase (GOx) with RSM was studied. In the traditional approach for predicting a polynomial model, the response, which is the enzymatic reaction rate, is defined as a function of substrate concentration and the other independent variables. This method needs a further step such as a non-linear regression method [6] or a Lineweaver–Burk plot [8] for determination of kinetic constants. In this study, Lineweaver–Burk transformation (Eq. (2)) was utilized as a part of the RSM algorithm and a second order polynomial model was obtained for the reciprocal of enzymatic reaction rate (1/V) as a function of reciprocal of substrate concentration (1/S), reaction temperature (T) and pH (pH) using RSM. The polynomial model was used for calculation of Vmax and Km of GOx enzyme at different reaction parameters (T and pH). The accuracy of the calculated values was tested with Vmax and Km values which were obtained in conventional method at six reaction parameters. 2. Experimental 2.1. Chemicals Glucose oxidase (EC 1.1.3.4 from Aspergillus niger), peroxidase (POD) (EC 1.11.1.7 from horseradish) were received from Biozyme Laboratories (Blaenavon, Gwent, UK). Phenol, d-glucose and 4-aminoantipyrine were obtained from Sigma Co. (St. Louis, MO, USA). All other chemicals were commercially available products of reagent grade. 2.2. Equipment A UV–vis spectrophotometer (Shimadzu 1201 PC, Shimadzu Corporation, Japan) was used in the determination of

the enzymatic reaction rate. A circulating water bath (PolyScience 910 Refrigerated Circulator, Polyscience Div. of Preston Ind. Inc., Niles, IL, USA) was used to maintain constant experimental temperature with an accuracy of ±0.1 ◦ C. 2.3. Determination of enzymatic reaction rate The enzymatic reaction rate of GOx enzyme was determined by measuring the hydrogen peroxide production from d-glucose as described elsewhere [10,11]. The reaction was done in 100 ml of a 0.67 mM phosphate buffer solution (pH varies between 4.4 and 8.5 depending on the tested reaction pH) containing 10 mg GOx at defined reaction parameters. The initial substrate concentration was adjusted to the tested concentration by using a stock d-glucose solution (500 mM). The reaction medium was mixed continuously during the incubation period with a magnetic stirrer, and the reaction temperature was kept constant using a water bath. A 2.5 ml aliquot of 0.1 mM phosphate buffer (pH 7.0) containing POD (0.1 mg), 4-aminoantipyrin (0.5 mg) and phenol (2.5 mg) was the assay solution. A 1.0 ml sample of the GOx/d-glucose reaction solution was goended with 4 ml dilute acid solution (sulphuric acid, 0.2%, v/v), and then 100 ␮l of this solution was added to the assay mixture. After incubation at 37 ◦ C for 15 min, hydrogen peroxide produced from glucose was determined spectrophotometrically at 505 nm and the remaining glucose concentration in the reaction medium was calculated stoichiometrically. The graph of glucose concentration versus incubation period was plotted and the initial reaction rate (dC/dt)t = 0 was determined. 2.4. Experimental design with response surface methodology Before arranging an experimental design with response surface methodology, the effects of substrate concentration, reaction temperature and pH on enzymatic reaction rate were tested by varying one factor at a time while keeping the others constant. A three-factor and five-coded level central composite circumscribed design (CCCD) was used to determine the GOx enzyme kinetic constants. Three independent variables (reciprocal of substrate concentration, 1/S; reaction medium temperature, T; reaction medium pH) were varied simultaneously relative to the chosen center point (reciprocal of substrate concentration, 0.267 mM−1 ; reaction medium temperature, 27.5 ◦ C; acidity, pH 6.5), there being six replicates at the center points and single runs for each of the other combinations, 20 runs were done in a totally random order. Duplicate experiments were carried out at all design points. The independent variables and their levels are given in Table 1. The relationship between the natural variables (xi ) and coded variables (Xi ) is Xi =

xi − [xmax + xmin ]/2 [xmax − xmin ]/2

(3)

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Table 1 Decoding values of independent variables used in the experimental design Variables

Coded symbols

Reciprocal of substrate concentration Reaction temperature (◦ C) Reaction pH

(mM−1 )

X1 X2 X3

Decoding value −α

−1

0

1

+α

0.033 14.9 4.4

0.128 20.0 5.3

0.267 27.5 6.5

0.405 35.0 7.7

0.500 40.1 8.5

α = 1.682.

where Xi is a dimensionless coded value of the variable xi , and xmax and xmin are the maximum and minimum values of the natural variable. The response or dependent variable (Y) was the reciprocal of initial GOx reaction rate (min mg enzyme/mM glucose). A second order polynomial model was predicted by a multiple regression procedure. This resulted in an empirical model related to the response by the equation Y = β0 +

3
i=1

βi X i +

3
i=1

βii X12 +

2
3


βij Xi Xj

posed method at six different reaction parameters (18 ◦ C, pH 8; 23 ◦ C, pH 7; 34 ◦ C, pH 6.2; 32 ◦ C, pH 6.4; 30 ◦ C, pH 6; 37 ◦ C, pH 5). Glucose was used as a substrate between the concentration of 2 and 30 mM. The enzymatic consumption of glucose with GOx was measured during the incubation period, and the initial reaction rate of GOx was determined from a plot of glucose concentration versus time. The Vmax and Km values of Michaelis–Menten kinetics were determined using a Lineweaver–Burk plot at defined parameters.

(4)

i=1 j=i+1

where Y is the response, β0 is the constant coefficient, βi s are the linear coefficients, βii s are the quadratic coefficients, βij s are the interaction coefficients, and Xi and Xj are coded independent variables. The predicted polynomial model was analyzed using the response surface regression procedure. The coefficients of the Eq. (4) were determined using an algorithm written in MATLAB® (The Mathworks, Natick, MA). 2.5. Determination of simple enzyme kinetics with the conventional method Kinetic constants of GOx reaction were determined with the conventional method to confirm the accuracy of the pro-

3. Results and discussion 3.1. Determination ranges of the variables The effects of substrate concentration, reaction medium temperature and acidity on enzymatic reaction rate are demonstrated in Fig. 1a, b and c, respectively. It was seen that an increase in glucose concentration caused an increase in enzymatic reaction rate up to 25 mM glucose, above which there was no change. It was indicated that above this concentration GOx activity was the limiting factor for reaction kinetics and glucose did not have any inhibitory effect on GOx activity. By using the first graph, 2 and 30 mM were chosen as a substrate concentration range. In the CCCD, the

Fig. 1. Effects of substrate concentration (a), reaction temperature (b) and reaction pH (c) on enzymatic reaction rate.

˙ Boyacı / Biochemical Engineering Journal 25 (2005) 55–62 I.H.

58 Table 2 Experimental design and results of the CCCD Run order

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a

Independent variablesa

Response

X1

X2

X3

Y

−1 0 1 0 0 −1.682 0 0 0 0 1 −1 −1 0 0 1 1.682 1 −1 0

1 0 −1 1.682 0 0 0 0 0 −1.682 1 −1 −1 0 0 −1 0 1 1 0

−1 0 −1 0 1.682 0 0 0 0 0 −1 1 −1 −1.682 0 1 0 1 1 0

106.77 193.59 429.67 167.78 352.33 62.25 197.94 202.49 197.94 391.48 248.12 202.49 177.95 314.58 193.59 440.42 314.58 314.58 171.04 193.59

Coded symbols and levels of independent variables refer to Table 1.

reciprocal of the substrate concentration (1/S) was used as well as S, and therefore varied from 0.033 to 0.500 mM−1 . An increase in temperature increased the enzymatic reaction rate up to 30 ◦ C, beyond which the reaction rate decreased (Fig. 1b). Therefore, the minimum and maximum temperatures were chosen as 20 and 35 ◦ C, respectively. Similarly, an increase in pH caused an increase in reaction rate up to pH 6 after which the rate decreased (Fig. 1c). The minimum and maximum pH values for experimental design were therefore determined as 5.3 and 7.7, respectively. 3.2. Response surface analysis and determination of kinetic constants The effects of 1/S, T and pH on the reciprocal of the enzymatic reaction rate were investigated by response surface methodology. The experimental design and the results of the CCCD are illustrated in Table 2. Multiple regression coefficients obtained from a least squares analysis used to predict a quadratic polynomial model for the reciprocal of enzymatic reaction rate are summarized in Table 3. All coefficients in the full quadratic model were analyzed with a t-test. The coefficients β11 and β13 were found significant (p < 0.01). Therefore, it can be said that the intercept, linear quadratic and interaction terms of all independent variables except the quadratic effect of 1/S and the interaction effect of 1/S and pH were significant to the model. The coefficient of determination (R2 = 0.990) implies that the model was satisfactory. The F0.05 value for lack of fit (2.97) did not exceed the tabulated value of 5.05 (5, 5, 0.05) indicating that lack of fit was not significant and therefore the fitted model was appropriate for describing of the response surface.

Table 3 Regression coefficients of predicted quadratic polynomial model for response (Y) Term

β0 β1 β2 β3 β11 β22 β33 β12 β13 β23 Lack of fit (F) R2 Adjusted R2

Full quadratic model

Diminished quadratic model

Coefficient estimate

Standard error

Coefficient estimate

Standard error

194.4 142.9 −101.6 23.5 −9.1 82.1 136.0 −58.8 9.5 47.4 3.0 0.990 0.981

5.9 6.5 6.5 6.5 10.7 10.7 10.7 14.4 14.4 14.4

191.8 142.9 −101.6 23.5 – 83.0 136.9 −58.8 – 47.4 2. 5 0.989 0.982

4.8 6.3 6.3 6.3 – 10.3 10.3 13.9 – 13.9

It was expected that the β11 coefficient was not significant because the inverse of the Michaelis–Menten equation (Eq. (2)) does not contain a quadratic term for 1/S. Similarly, predicted quadratic polynomial model does not contain quadratic term for 1/S. Hence, the predicted polynomial model was rearranged by eliminating terms which were not significant in full quadratic model. The coefficients of the diminished quadratic model are also seen in Table 3. In this model, all the coefficients were significant (p < 0.01), the model was satisfactory (R2 = 0.989), and the lack of fit of the model was not significant (F = 2.45 < F7, 5, 0.05 = 4.88). Finally the model for the reciprocal of enzymatic reaction rate was obtained for the coded unit as: Y = 191.8 + 142.9X1 − 101.6X2 + 23.5X3 + 83.0X22 + 136.9X32 − 58.8X1 X2 + 47.4X2 X3

(5)

Afterwards the same data analysis was carried out with the natural variables and Eq. (6) was obtained.   1 1 = 1970 + 1064.6 − 28.8(T ) − 466.8(pH) V S   1 2 2 + 0.3 (T ) + 33.9(pH ) − 15.0 + 1.4(T pH) ST (6) This predicted polynomial model was used to obtain three dimensional response surface graphs for all interactions (Figs. 2–4). The third independent variables were kept at the middle level to obtain these figures. The critical values (optimum) of the T and pH were determined by setting the derivatives of Eq. (5) ∂Y/∂X2 and ∂Y/∂X3 to zero yielding Eqs. (7) and (8). ∂Y = −101.6 + 166X2 + 58.8X1 + 47.4X3 ∂X2

(7)

˙ Boyacı / Biochemical Engineering Journal 25 (2005) 55–62 I.H.

∂Y = 23.5 + 273.8X3 + 47.4X2 ∂X3

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(8)

The system of equations gave a group of solution for the optimum temperature and pH combinations for different reciprocal of substrate concentration (X1 ). Optimum values of temperature and pH combination shifted from T = 0.044pH = −0.093 (un-coded T = 27.8 ◦ C-pH = 6.4) to T = 1.296

Fig. 2. A three-dimensional response surface plot illustrating optimal conditions for reciprocal of enzymatic reaction rate as a function of reaction temperature and reaction pH.

Fig. 3. A three-dimensional response surface plot showing reciprocal of enzymatic reaction rate as a function of reciprocal of substrate concentration and reaction temperature.

Fig. 4. A three-dimensional response surface plot showing reciprocal of enzymatic reaction rate as a function of reciprocal of substrate concentration and reaction pH.

Fig. 5. Effect of reciprocal of the substrate concentration on optimal values of reaction temperature and pH (a) 1/S = 0.033 mM−1 (b) 1/S = 0.267 mM−1 and (c) 1/S = 0.500 mM−1 .

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and pH = −0.310 (un-coded T = 32.7 ◦ C-pH = 6.1) when the reciprocal of substrate concentration varied from −1.68 to 1.68 (un-coded 0.033–0.5 mM−1 ) (Fig. 5a, b and c). These results showed that optimum values of reaction parameters depended on substrate concentration. It should be related with the three dimensional structural changes of enzyme in the presence of different amounts of substrate. About these results further study is needed and these results will be important for industrial application of the enzymes. It was possible to obtain three dimensional response surfaces of the reciprocal of reaction rate as a function of 1/S and T and 1/S and pH (Figs. 3 and 4, respectively). Although the Lineweaver–Burk plot is defined for a set of reaction parameters (T, pH, etc.), Figs. 4 or 5 was valid for a range of reaction parameters (T and pH, respectively) and a constant value of the other parameters (pH and T, respectively). This means that these graphs contain as many Lineweaver–Burk plots as the investigated parameters for the enzymatic reaction, thus providing the opportunity to calculate Vmax and Km values for each reaction parameter. One can evaluate Eq. (6) under the condition 1/S = 0 which immediately leads to Eq. (9). 1 Vmax

= 1970 − 28.8T − 466.8pH + 0.3T 2 + 33.9pH2 + 1.4T pH

(9)

The slope of the Lineweaver–Burk plot gives the Km /Vmax ratio [1]. In order to determine the slopes of Figs. 2 and 3 for each T and pH, Eq. (6) was differentiated (∂(1/V)/∂(1/S)) and Eq. (10) was obtained. Km = 1064.6 − 15.0T Vmax

(10)

It was obvious that calculations of Vmax and Km values could be calculated for each combination of T and pH within the investigated ranges using Eqs. (9) and (10). The changes in Vmax and Km were calculated using these equations within the range of parameters worked in this study. Three-dimensional plots of Vmax and Km were obtained as a function of T and pH (Fig. 6a and b). Vmax and Km values were determined between 3.5 ␮mol/min mg enzyme (14.9 ◦ C-pH

Fig. 6. Three-dimensional plots of the maximum reaction rate (Vmax ) (a) and the Michaelis–Menten constant (Km ) (b) as a function of reaction temperature and reaction pH.

4.5) and 29.8 ␮mol/min mg enzyme (35 ◦ C-pH 6.1) and between 1.9 mM (40.1 ◦ C-pH 8.5) and 16.8 mM (at 33 ◦ C-pH 6.3), respectively. It can be seen in the figures that Vmax and Km had similar trends when the reaction parameters were changed. It can be shown that when the maximum reaction rate increased, the substrate concentration necessary to reach half of the maximum reaction rate (Km ) also increased. The Km /Vmax ratio varied between 839.3 min mg enzyme/l and 388.0 min mg enzyme/l in the ranges of tested parameters (The data was not given). It decreased when T was increased without being affected by changes of pH. Similar results related with Vmax and Km were reported in the literature [12,13].

Table 4 Comparison of Vmax and Km values obtained from RSM and conventional method Temperature (◦ C)

pH

Vmax (mmol/min mg enzyme) Conventional

RSM

Conventional

RSM

18 23 34 32 30 37

8.0 7.0 6.2 6.4 6.0 5.0

0.0059 0.0101 0.0325 0.0273 0.0224 0.0164

0.0055 0.0125 0.0301 0.0282 0.0246 0.0127

5.30 6.25 13.62 11.27 11.03 7.07

4.40 9.02 16.66 16.45 15.08 6.45

R2

0.940

Km (mM)

0.869

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3.3. Determination of Km and Vmax of GOx enzyme with conventional method The kinetic constants of GOx were determined with the conventional method at six different reaction parameters (T and pH). At the same reaction parameters, Vmax and Km were calculated using Eqs. (9) and (10). The results are given in Table 4. The accuracy of the new approach for determining Vmax and Km was tested by comparing the results of these two methods. The correlation coefficients (R2 ) between these two methods were obtained as 0.940 and 0.869 for Vmax and Km , respectively. These results indicate that the proposed approach gave the reasonable results for the determination of Vmax and Km in the range of tested parameters.

4. Conclusions Three dimensional structures of enzymes are easily affected from the reaction parameters and the changes in the structure cause a decrease or increase in the catalytic power of the enzyme. In most of the studies, enzyme kinetics and kinetic constants of the reactions are investigated. Due to the uncontrolled reaction parameters and nature of the enzymes (source and production procedure of enzyme) different kinetic models or constants can be obtained from the reaction of the same enzyme. Typical examples of this situation are the Vmax and Km values of GOx enzyme determined in different studies. Vmax and Km values for the free form of GOx were reported in variations of 1.0 and 66.0 ␮mol/min mg enzyme and 2.9 and 30.0 mM, respectively [11,14,15]. The primary reason is that the kinetic constants were determined at different or uncontrolled reaction parameters. Generally, it is not possible to compare Vmax and Km values of the same enzyme when obtained from different studies. Determination of these parameters using conventional methods for each reaction parameter is not possible due to their time consuming procedures and also their cost. At this point, RSM would be helpful to characterize enzymatic reactions. RSM (recognized in other fields where applicable) can predict overall behavior of the enzyme with limited number of experimental points, without further knowledge of enzyme mechanism and irrespective of the factors that are out of control. In this study Lineweaver–Burk transformation was utilized as integral part of the RSM algorithm and the effects of reaction parameters on kinetic constants were investigated. The main advantage of integration is that it is possible to determine kinetic constants as functions of the tested reaction parameters with minimum experimentation. It is also possible to determine interaction effects of reaction parameters on the kinetic constants and optimize reaction parameters during determination of kinetic constants. The results of this study provide an improvement to the basic enzyme kinetics knowledge. General approach is that, determination of only one kinetic model or constant for an enzyme is enough for characterization of an enzyme. How-

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ever the result of this study showed that the kinetic constants depend on the reaction parameters and should be determined for each reaction parameter. It could be done using RSM. The other one is that the optimum reaction parameters are affected from substrate concentration not only at the extreme values but also in the moderate concentration range. Besides of these results, this study also produced some questions which will be answered with further studies. These are (i) Selection of experimental data points has critical importance in RSM studies and using second order model equation limits the RSM application. “Is it possible to produce a generalized procedure for RSM application in enzyme kinetics study?” (ii) The effect of substrate concentration on optimum reaction parameters were observed but could not explained. “The reason of this change should be explained by doing more experiments using RSM and searching the molecular structure of enzymes.” (iii) There are other methods which could be useable for estimation of response using limited number of experiment data such as genetic algorithm, artificial neural network or so on. “Is there an other method which is more convenient then RSM for the prediction of overall behavior of the enzyme with limited number of experimental study?” (iv) Maybe the most critical is that “Could we find a model equation which is more informatics for the user and covers the characteristics and the behavior of the enzyme towards the reaction parameters?” The answers of these questions will helpful for improvement of the basic enzyme knowledge.

Acknowledgements The author would like to thank to Prof. Dr. H. Brian Halsall, Prof. Dr. Thomas H. Ridgway, Dr. Necati Kaval, Dr. Zoraida P. Aguilar, and Dr. John N. Richardson from Cincinati University, Department of Chemistry, Cincinnati, OH, for their kind helps during preparation of this manuscript.

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