- Email: [email protected]
Modeling and optimization III: Reaction rate estimation using artificial neural network (ANN) without a kinetic model
Journal of Food Engineering 79 (2007) 622–628 www.elsevier.com/locate/jfoodeng
Modeling and optimization III: Reaction rate estimation using artificial neural network (ANN) without a kinetic model _ Deniz Basß, Fahriye Ceyda Dudak, Ismail Hakkı Boyacı
*
Department of Food Engineering, Faculty of Engineering, Hacettepe University, Beytepe, TR-06532 Ankara, Turkey Received 6 October 2005; accepted 13 February 2006 Available online 5 April 2006
Abstract In this study, the usability of artificial neural networks (ANN) for the estimation of enzymatic reaction rate was investigated. The study was performed by following a model reaction, enzymatic hydrolysis of maltose, catalyzed by amyloglucosidase enzyme. The effects of substrate (maltose) and product (glucose) concentration on enzymatic reaction rate were studied. Data obtained from seven time courses were used for training of the ANN and another set of data obtained from eight time courses were used for testing of the trained network. This network was designed as a feed forward neural network with three neurons in the input layer, four neurons in the hidden layer and one neuron in the output layer. The network was trained till the mean square value between the targets and the outputs obtained was 1 · 104. The enzymatic reaction rate for the defined maltose and glucose concentration was estimated using the trained network. The regression coefficient of determination (R2) showed a good correlation between estimated and experimental data sets for both train (0.992) and test data sets (0.965). In further part of the study, estimated data by the ANN was used in a numerical solution of batch reactor modeling equation to obtain time courses data. There are high correlations between experimental and estimated time course curves and that was another proof of the high performance of ANN for estimation of enzyme-catalyzed reaction rate. 2006 Elsevier Ltd. All rights reserved. Keywords: Enzymatic reaction rate; Feed forward; Artificial neural networks; Amyloglucosidase
1. Introduction Enzymes are responsible for biochemical and physiological reactions that occur within all living organisms. In addition, enzymatic reactions are used for different purposes in food, pharmacology, textiles and other related industries. The biggest part of the enzyme is protein, and the protein structure is sensitive to reaction conditions such as temperature, pH, ionic strength and activator or inhibitor concentrations. Since the enzyme is sensitive to environmental conditions, the enzymatic reaction rate is dependent on these conditions. The most important matter for the kinetic study is to find an accurate model equation that has the capability *
Corresponding author. Tel.: +90 312 297 71 00; fax: +90 312 299 21
23. _ E-mail address: [email protected] (I.Hakkı Boyacı). 0260-8774/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2006.02.021
to estimate the reaction rate as similar as the experimental reaction rate. In a kinetic study, the first step is to find the suitable model(s), which defines the experimental data properly. Based on the experimental data, one or more model equations are selected and then kinetic constants (such as KM, Vmax or Ki) are estimated by fitting the model equation to the experimental data. In the last step, the most proper model equation, which could explain the experimental data, is selected to use in further studies. In the studies based on the selected model equation and kinetic constant prediction method, all model equations have an error value and researchers try to minimize this value. The value of the error is generally low for an enzymatic reaction that obeys simple enzyme kinetics. However, it is hard to find a proper model for an enzymatic reaction, which obeys inhibition kinetics. To save time, the affect of some independent parameters are not taken into consideration and this causes an increase in the error value. The
D. Basß et al. / Journal of Food Engineering 79 (2007) 622–628
623
Nomenclature AAD ANN A1 A2 dx i Ki KM MSE P P0
absolute average deviation artificial neural networks constant in Boltzmann function constant in Boltzmann function width of Boltzmann function step indices inhibition constant of enzyme Michaelis–Menten constant (mM) mean square error product (glucose) concentration (mM) initial product concentration (mM)
kinetic model and its constants are only valid for an enzyme with its reaction conditions in which the model and the constant are determined. When the reaction conditions change, the usability of the model equation in those reaction conditions should be investigated and the kinetic constants in the model need to be re-estimated. The aim of this study was to explore an alternative method for estimation of the reaction rate without using a model equation. The alternative method has the following properties: (i) not requiring so much experimental work, (ii) applicable for all type enzymatic reactions without any limitations, (iii) not requiring any assumption about kinetic study, such as ignorance of the effect of some parameters to obtain a simpler mathematical model, (iv) usable as a part of numerical solution for complex engineering problems, similar to the model equation. For this purpose, in this study, artificial neural networks (ANN) were used for the estimation of reaction rates instead of a model equation. Artificial neural networks have been previously applied successfully in modeling of biological systems (Basß & Boyacı, 2006; Geeraerd, Herremans, Cenens, & Van Impe, 1998; Hajmeer, Basheer, & Najjar, 1997; Lou & Nakai, 2001; Sun, Peng, Chen, & Shukla, 2003; Torrecilla, Otero, & Sanz, 2004). Essentially, artificial neural networks are designed in an attempt to mimic the methods of information processing and knowledge acquisition of the human brain (Najjar, Basheer, & Hajmeer, 1997). An ANN is a massively interconnected network structure consisting of many simple processing elements capable of performing parallel computations for data processing. In effect, ANN can handle multiple independent and dependent variables simultaneously without having prior knowledge about the functional relationship. ANN differs from traditional computing methods in many ways. The processing style of ANN is parallel and there is not a strict rule or algorithm to follow, whereas traditional methods are sequential and logical. Moreover, traditional methods can learn by rules while ANN learns by examples and therefore artificial neural networks are known as universal function approximators (Anjum, Tasadduq, & Al-Sultan, 1997). A neural
R2 r S S0 t T t0 Vmax Dt
regression coefficient of determination reaction rate (lmol/min/mg enzyme) substrate (maltose) concentration (mM) initial substrate concentration (mM) reaction time (min) total reaction time (min) constant in Boltzmann function maximal reaction rate of enzyme (lmol/min/mg enzyme) time increments (min)
network derives its computing power through first its massively parallel-distributed structure and second its ability to learn and therefore generalize. Here generalization refers to the neural network producing reasonable outputs for inputs not encountered during training (Haykin, 1994). In this study, we used artificial neural networks (ANN) to estimate the reaction rate of maltose hydrolysis by amyloglucosidase enzyme. The network was trained using time course data obtained from enzymatic reaction. The trained network was used for estimation of the reaction rate and then estimated rate was used to obtain concentration profile of substrate (maltose) and product (glucose) in batch reactor during reaction time for the reaction of amyloglucosidase enzyme. 2. Material and methods 2.1. Chemicals Maltose, glucose, amyloglucosidase (From Rhizopus sp., E.C. 3.2.1.3) and 4-aminoantipyrine were supplied from Sigma Chemical Co. (USA). Glucose oxidase (From Aspergillus niger, E.C. 1.1.3.4) and peroxidase (From horseradish, E.C. 1.11.1.7) were obtained from Biozyme Laboratories Lim. (UK). Phenol, Na2HPO4, and KH2PO4 were obtained from J.T. Baker (The Netherlands). 2.2. Maltose hydrolysis and determination of reaction rate using time course data The enzymatic reaction performed in this study is given below: Amyloglucosidase
Maltose þ H2 O ! 2Glucose Enzymatic hydrolysis of maltose was carried out in 5 ml of reaction solution (phosphate buffer, 100 mM, pH 5.5). Reaction solution was heated up to 37 C using a continuously stirred water bath and after adjustment of initial maltose and glucose concentration in solution, the reaction was started by adding amyloglucosidase enzyme into the solution. The final enzyme concentration in the reaction
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medium was 0.20 mg/ml. Maltose hydrolysis was monitored by measuring the amount of glucose produced during the incubation period. For this purpose, 10 ll sample was taken from reaction medium at different time intervals and immediately mixed with 2 ml of assay solution which was prepared by the method given by Trinder (1969). Time course data was obtained by measuring the amount of glucose produced during the enzymatic reaction. This procedure was replicated for fifteen different initial maltose or glucose concentrations. The initial maltose and glucose concentrations were varied between 2 and 16 mM and between 0 and 10 mM, respectively. In all measurements, glucose concentration was monitored during 75 min incubation period and progress curves were obtained. A model time course is seen in Fig. 1. Maltose concentration in the reaction vessel was calculated using stoichiometric relation between maltose and glucose concentrations. Finally, maltose hydrolysis during incubation time was obtained which is also shown in Fig. 1. After obtaining time course data, enzymatic reaction rate was determined in two steps: (i) Experimental time course data was fitted to the Boltzmann function (Eq. (1)). S¼
A1 A2 þ A2 1 þ eðtt0 Þ=dx
ð1Þ
Boltzmann function (BF) has a special condition. In a nutshell, most of the time course data of the biochemical reactions could be fitted to this equation with a high accuracy. It adequately fits the time course data of glucose production and maltose hydrolysis with a high coefficient of correlation (R2). The equations fitted to the experimental data and their R2 values are seen in Fig. 1.
(ii) Eq. (1) was derived with respect to t and Eq. (2) was obtained. This equation could be used to determine reaction rate (dS/dt) at any reaction time (t).
dS ðA1 A2 Þðeðtt0 Þ=dx Þ ¼ 2 dt dxð1 þ eðtt0 Þ=dx Þ
ð2Þ
By using this procedure, 150 different reaction rate values were determined for 15 time courses.
2.3. Development of artificial neural networks and prediction of reaction rates The artificial neural network was built with The Neural Network Toolbox, MATLAB Release 14 (The Mathworks, Natick, MA). A feed forward neural network, which consists of three neurons in input layer, four neurons in hidden layer and one neuron in output layer, was built. The inputs of the neural network were maltose concentration, glucose concentration and initial glucose concentration, and output of the system was the reaction rate. Transfer function between the input and the hidden layer was ‘tansig’ and the one between the hidden layer and the output layer was ‘logsig’. Training of the network was performed with the function ‘trainlm’, which updates weight and bias values according to Levenberg–Marquardt optimization. The other parameters of the neural network were taken as the defaults of MATLAB. Data of seven different reaction batches was used for training and the remaining (eight reaction batches) data was used for testing the network. Training was performed to minimize the mean square error (MSE) between targets and outputs (Target MSE value was defined as 1 · 104). Trained network was saved as a mat file for further applications. 2.4. Progress curves for maltose hydrolysis obtained from the estimated reaction rates
35 P = 34.875 +
Glucose Maltose
30
-59.579 (1 + e t/31.758 )
Enzymatic reaction rate for defined maltose and glucose concentration was estimated using trained network. In further part of the study, outputs of the ANN were used in a numerical solution of modeling equation obtained for batch reactor. The modeling part of the study and the numerical solution is described below briefly. The change in substrate concentration in a batch reactor with time can be given by Eq. (3).
2
R = 0.996
Concentration (mM)
Predicted with Boltzmann 25
20
15
dS ð3Þ dt where r is the reaction rate. The backward difference approximation for dS/dt gives
r¼ 10
R2 = 0.996
5
S = 1.062 +
29.790 (1 + e t/31.758 )
0 0
10
20
30
40
50
60
70
80
Time (min) Fig. 1. Time courses for hydrolysis of maltose into glucose. Initial maltose and glucose concentrations are 16.0 mM and 5.0 mM, respectively.
dS S i þ S iþ1 ¼ dt Dt
ð4Þ
The central differences approximation for dS/dt gives dS S i1 þ S iþ1 ¼ dt 2Dt
ð5Þ
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625
For time t, step indices (i) are used with increments in time being represented by Dt. i = 1 represents initial value of maltose concentration at t = 0. In the beginning of the reaction initial values of maltose and glucose concentrations were fed to the program as Si=1 = S0 and Pi=1 = P0, respectively. Using input values, enzymatic reaction rate ri=1 was estimated using trained network and then Si=2 was calculated using finite difference equation given in Eq. (6). Then Pi=2 was calculated using Eq. (7). S i¼2 ¼ S i¼1 ri¼1 Dt P i¼2 ¼ P i¼1 þ 2 ri¼1 Dt
ð6Þ ð7Þ
Calculated maltose and glucose concentrations were presented to the network and reaction rate was estimated for further calculation. For i > 2, the central differences approximation was used and Eqs. (8) and (9) were derived for the finite difference equations used in the numerical solution S i¼iþ1 ¼ S i¼i1 2 ri¼i Dt
ð8Þ
P i¼iþ1 ¼ P i¼i1 þ 4 ri¼i Dt
ð9Þ
This procedure was replicated during finite reaction time. In addition, a program which combined ANN with numerical solution of the model equation was coded using MATLAB to obtain maltose and glucose concentrations versus time profile in the batch reactor. The profiles were correlated with the ones obtained experimentally. Performance of numerical solution was studied using test data set, which was not used in training of the network. 3. Results and discussion 3.1. Maltose hydrolysis and determination of reaction rate using time course data Glucose production from maltose by amyloglucosidase enzyme was investigated at fifteen different initial maltose or glucose concentrations. Fifteen time courses were obtained and experimental time courses were fitted to the Boltzmann function. Model equation and its R2 value were obtained for each reaction batch. (A sample time course data and its model equation and R2 values were given in Fig. 1.) The R2 values of the all predicted model equations were quite high (R2 > 0.99) and these values indicated that time courses of the enzyme catalyzed reaction can be properly explained by the BF. An equation, obtained by the derivation of the predicted model equation, was used for determining the enzymatic reaction rate for the period of reaction time. The obtained initial reaction rates (t = 0) were visualized in Fig. 2 by drawing surface and contour plots of the enzymatic reaction rate as a function of the maltose and glucose concentration. As seen in the figure, increasing maltose concentration cause an increase in the reaction rate. In low maltose concentrations, glucose concentration does not affect the reaction rate. However, at high maltose concentrations as
Fig. 2. Surface and contour plots of the initial reaction rate as a function of maltose and glucose concentration.
glucose concentration increases, reaction rate decreases. This result indicates that inhibition kinetics is valid for amyloglucosidase reactions. Similar results were reported by several authors (Li, Kim, & Peeples, 2002; Najafpour & Shan, 2003; Polakovic & Bryjak, 2004). 3.2. Training and testing of artificial neural network Data obtained from seven time courses was used for training of the ANN and the data of other time courses (eight time courses) was used for testing the trained network. The training data set (three input and their corresponding desired responses) was presented to the network and a feed forward algorithm automatically adjusted the weights so that the output response to input values were as close as possible to the desired response. Estimation was made and the results were compared with the corresponding desired value. Then the estimation error (the difference between the estimated and the desired values) was distributed across the network in a manner which allowed the interconnection weights to be modified according to the scheme specified by the learning rule. This process was repeated while the prediction error decreased. After 250 learning epochs, the target was achieved and the learning stage was completed. In order to test the trained network another data set was used and the input test set was presented to the network and the output was obtained. The output of the ANN was compared with the experimental data for the trained and test data sets in Fig. 3(a) and (b), respectively. It is clear that the regression coefficients of determination (R2 = 0.992 and R2 = 0.965) are high between estimated and experimental data for both of the data sets. Since excellent estimation performances were obtained using the trained network, it demonstrates that the trained network was
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D. Basß et al. / Journal of Food Engineering 79 (2007) 622–628
y = 0.997x 1.25
2
R = 0.992
Experimental Rates (μ mol/min/mg enzyme)
1.00
0.75
0.50
0.25
0.00 0.00
0.25
0.50
0.75
1.00
1.25
Predicted Rates (μ mol/min/mg enzyme)
(a)
y = 1.000x
1.25
2
R = 0.965
in this study, r value was estimated by using ANN without a model equation. A small program for numerical solution of differential equation was coded in MATLAB and ANN was integrated to this solution. Algorithm of the program is seen in Fig. 4. The algorithm written for this study will be sent as requested from the address above. The developed program was used for the prediction of time courses in the batch reactor. The initial concentrations of maltose and glucose, and variables were presented to the system. The products (time courses of maltose and glucose concentration) were obtained. Experimental and estimated time courses of maltose concentration are seen in Fig. 5(a). Similarly, those of glucose concentration were given in Fig. 5(b) (Fifteen time courses were predicted using developed program and only five of them were presented in Fig. 5). In both of the graphs, there are high correlations between experimental and estimated curves. It was verified that the developed program has high capacity for estimating time courses of substrate and product concentrations. One critical point is that the performance of the program was directly related with estimation performance of the ANN. This result was another proof of the high performance of ANN for estimation of enzyme-catalyzed reaction rate. Estimation capability of ANN does not
Experimental Rates (μ mol/min/mg enzyme)
1.00
Input initial values and variables 0.75
Estimate the reaction rate based on the initial condition 0.50
0.25
0.00 0.00
(b)
Calculate new maltose concentration, S Calculate new glucose concentration, P Calculate time, t Add the data into the matris 0.25
0.50
0.75
1.00
1.25
Predicted Rates (μ mol/min/mg enzyme)
Fig. 3. Comparison of the estimated reaction rate with experimental data (a) training data set and (b) test data set.
reliable, accurate and hence could be employed in the further part of the study. 3.3. Progress curves for maltose hydrolysis obtained from the estimated reaction rates In conventional numerical solution of the differential equation, constant or variable in the model equation should be known. In Eq. (3), r variable could be explained with an equation such as Michaelis–Menten Equation. Otherwise, one of the more complex inhibition kinetics equations is required. It is necessary to decide which model equation is valid to start the numerical solution. However,
Estimate the new reaction rate, r Calculate new maltose concentration, S Calculate new glucose concentration, P Calculate time, t Add the data into the matris
If t
YES
NO Plot reaction time vs. substrate and product concentration Fig. 4. Algorithm of the program.
D. Basß et al. / Journal of Food Engineering 79 (2007) 622–628
14
Maltose Concentration (mM)
and the trained network was used to estimate the reaction rate at any substrate and product concentration, which is a new approach in this field. In the further part of the study, ANN was successfully employed as a part of the numerical solution of the differential equations that was obtained by modeling the batch reactor. The usage of ANN with time course in reaction kinetics had several advantages. Some of them are listed below.
S0 = 16 mM; P0 = 10 mM S0 = 12 mM; P0 = 0 mM S0 = 8 mM; P0 = 0 mM S0 = 4 mM; P0 = 5 mM S0 = 2 mM; P0 = 5 mM
16
Time courses obtained from developed program
12 10 8 6 4 2 0 0
15
(a)
45
60
75
60
75
Time (min) 35
S0 = 16 mM; P0 = 10 mM S0 = 12 mM; P0 = 0 mM S0 = 8 mM; P0 = 0 mM S0 = 4 mM; P0 = 5 mM S0 = 2 mM; P0 = 5 mM
30
Glucose Concentration (mM)
30
Time courses obtained from developed program
25 20 15 10 5 0
0
(b)
15
30
45
627
Time (min)
Fig. 5. Time courses of (a) maltose and (b) glucose concentrations obtained from estimated reaction rates. (The points represent experimental data while the lines are estimated concentrations obtained from developed program.)
depend on the type of reaction and it is directly related with the training stage of the network. It is obvious that well trained network will have high performance to obtain time courses correctly for both reversible and irreversible reactions. 4. Conclusion The time course of an enzyme-catalyzed reaction contains a wealth of information and the increased speed and capacity of desktop computers allow the better use of reaction time courses (Duggleby, 2001). In conventional methods, kinetic models and kinetic constants are determined by using the initial reaction rate values or kinetic constants are directly determined by utilizing the time course data. In this study, the reaction rate values obtained from time courses were used for the training of the ANN
i. ANN estimates reaction rate without requiring any kinetic model equation. ii. Estimation of reaction rate without a kinetic model eliminates the errors arising from the selection of kinetic model and the estimation of kinetic constants. iii. Usage of ANN reduces the number of the experimental work or provides more information than the conventional method, if same amount of experimental study is employed. iv. In the conventional method, at the beginning of the modeling study, it is required to make assumptions about the kinetic equation and estimate the constant in the equation. This procedure increases paperwork. The first assumption is generally not correct and the researcher has to check more than one kinetic model. However, if researcher uses ANN, he/she does not need any kinetic model. As a result, kinetic work could be performed without any limitation of model equation. v. ANN could be useable in a part of the complex calculation using an improved program. In this study, usability of the ANN for numerical solution of the differential equation obtained by modeling of the batch reaction was demonstrated successfully. The results are very promising and we believe that ANN will become a popular method for estimation of reaction rate without using the kinetic model. However, for different purposes investigation of reaction kinetics and prediction of kinetic constants may be required and researchers may want to determine those constants. In a subsequent paper, we will perform a research on the applicability of ANN for estimation of the reaction kinetics and kinetic constants. Acknowledgements The authors gratefully acknowledge the financial support from State Planning Organization of Republic of Turkey; Project No. 03 K 120 570-4. References Anjum, M. F., Tasadduq, I., & Al-Sultan, K. (1997). Response surface methodology: a neural network approach. European Journal of Operational Research, 101, 65–73. _ Basß, D., Boyacı, I.H. (2006). Modeling and optimization II: comparison of estimation capabilities of response surface methodology with artificial neural networks in a biochemical reaction, Journal of Food Engineering, in press, doi:10.1016/j.jfoodeng.2005.11.025.
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Duggleby, R. G. (2001). Quantitative analysis of the time courses of enzyme-catalyzed reactions. Methods, 24, 168–174. Geeraerd, A. H., Herremans, C. H., Cenens, C., & Van Impe, J. F. (1998). Application of artificial neural networks as a non-linear modular modeling technique to describe bacterial growth in chilled food products. International Journal of Food Microbiology, 44, 49– 68. Hajmeer, M. N., Basheer, I. A., & Najjar, Y. M. (1997). Computational neural networks for predictive microbiology II. Application to microbial growth. International Journal of Food Microbiology, 34, 51–66. Haykin, S. (1994). Neural networks: A comprehensive foundation. New Jersey: Prentice Hall (pp. 696–723). Li, M., Kim, J., & Peeples, T. L. (2002). Kinetic enhancement of starch bioconversion in thermoseparating aqueous two-phase reactor systems. Biochemical Engineering Journal, 11, 25–32. Lou, W., & Nakai, S. (2001). Application of artificial neural networks for predicting the thermal inactivation of bacteria: a combined effect of
temperature, pH and water activity. Food Research International, 34, 573–579. Najafpour, G. D., & Shan, C. P. (2003). Enzymatic hydrolysis of molasses. Bioresource Technology, 86, 91–94. Najjar, Y. M., Basheer, I. A., & Hajmeer, M. N. (1997). Computational neural networks for predictive microbiology: I. Methodology. International Journal of Food Microbiology, 34, 27–49. Polakovic, M., & Bryjak, J. (2004). Modelling of potato starch saccharification by an Aspergillus niger glucoamylase. Biochemical Engineering Journal, 18, 57–63. Sun, Y., Peng, Y., Chen, Y., & Shukla, A. J. (2003). Application of artificial neural networks in the design of controlled release drug delivery systems. Advanced Drug Delivery Reviews, 55, 1201–1215. Torrecilla, J. S., Otero, L., & Sanz, P. D. (2004). A neural network approach for thermal/pressure food processing. Journal of Food Engineering, 62, 89–95. Trinder, P. (1969). Determination of blood glucose using 4-aminophenazone. Journal of Clinical Pathology, 22, 246.
Modeling and optimization III: Reaction rate estimation using artificial neural network (ANN) without a kinetic model _ Deniz Basß, Fahriye Ceyda Dudak, Ismail Hakkı Boyacı
*
Department of Food Engineering, Faculty of Engineering, Hacettepe University, Beytepe, TR-06532 Ankara, Turkey Received 6 October 2005; accepted 13 February 2006 Available online 5 April 2006
Abstract In this study, the usability of artificial neural networks (ANN) for the estimation of enzymatic reaction rate was investigated. The study was performed by following a model reaction, enzymatic hydrolysis of maltose, catalyzed by amyloglucosidase enzyme. The effects of substrate (maltose) and product (glucose) concentration on enzymatic reaction rate were studied. Data obtained from seven time courses were used for training of the ANN and another set of data obtained from eight time courses were used for testing of the trained network. This network was designed as a feed forward neural network with three neurons in the input layer, four neurons in the hidden layer and one neuron in the output layer. The network was trained till the mean square value between the targets and the outputs obtained was 1 · 104. The enzymatic reaction rate for the defined maltose and glucose concentration was estimated using the trained network. The regression coefficient of determination (R2) showed a good correlation between estimated and experimental data sets for both train (0.992) and test data sets (0.965). In further part of the study, estimated data by the ANN was used in a numerical solution of batch reactor modeling equation to obtain time courses data. There are high correlations between experimental and estimated time course curves and that was another proof of the high performance of ANN for estimation of enzyme-catalyzed reaction rate. 2006 Elsevier Ltd. All rights reserved. Keywords: Enzymatic reaction rate; Feed forward; Artificial neural networks; Amyloglucosidase
1. Introduction Enzymes are responsible for biochemical and physiological reactions that occur within all living organisms. In addition, enzymatic reactions are used for different purposes in food, pharmacology, textiles and other related industries. The biggest part of the enzyme is protein, and the protein structure is sensitive to reaction conditions such as temperature, pH, ionic strength and activator or inhibitor concentrations. Since the enzyme is sensitive to environmental conditions, the enzymatic reaction rate is dependent on these conditions. The most important matter for the kinetic study is to find an accurate model equation that has the capability *
Corresponding author. Tel.: +90 312 297 71 00; fax: +90 312 299 21
23. _ E-mail address: [email protected] (I.Hakkı Boyacı). 0260-8774/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2006.02.021
to estimate the reaction rate as similar as the experimental reaction rate. In a kinetic study, the first step is to find the suitable model(s), which defines the experimental data properly. Based on the experimental data, one or more model equations are selected and then kinetic constants (such as KM, Vmax or Ki) are estimated by fitting the model equation to the experimental data. In the last step, the most proper model equation, which could explain the experimental data, is selected to use in further studies. In the studies based on the selected model equation and kinetic constant prediction method, all model equations have an error value and researchers try to minimize this value. The value of the error is generally low for an enzymatic reaction that obeys simple enzyme kinetics. However, it is hard to find a proper model for an enzymatic reaction, which obeys inhibition kinetics. To save time, the affect of some independent parameters are not taken into consideration and this causes an increase in the error value. The
D. Basß et al. / Journal of Food Engineering 79 (2007) 622–628
623
Nomenclature AAD ANN A1 A2 dx i Ki KM MSE P P0
absolute average deviation artificial neural networks constant in Boltzmann function constant in Boltzmann function width of Boltzmann function step indices inhibition constant of enzyme Michaelis–Menten constant (mM) mean square error product (glucose) concentration (mM) initial product concentration (mM)
kinetic model and its constants are only valid for an enzyme with its reaction conditions in which the model and the constant are determined. When the reaction conditions change, the usability of the model equation in those reaction conditions should be investigated and the kinetic constants in the model need to be re-estimated. The aim of this study was to explore an alternative method for estimation of the reaction rate without using a model equation. The alternative method has the following properties: (i) not requiring so much experimental work, (ii) applicable for all type enzymatic reactions without any limitations, (iii) not requiring any assumption about kinetic study, such as ignorance of the effect of some parameters to obtain a simpler mathematical model, (iv) usable as a part of numerical solution for complex engineering problems, similar to the model equation. For this purpose, in this study, artificial neural networks (ANN) were used for the estimation of reaction rates instead of a model equation. Artificial neural networks have been previously applied successfully in modeling of biological systems (Basß & Boyacı, 2006; Geeraerd, Herremans, Cenens, & Van Impe, 1998; Hajmeer, Basheer, & Najjar, 1997; Lou & Nakai, 2001; Sun, Peng, Chen, & Shukla, 2003; Torrecilla, Otero, & Sanz, 2004). Essentially, artificial neural networks are designed in an attempt to mimic the methods of information processing and knowledge acquisition of the human brain (Najjar, Basheer, & Hajmeer, 1997). An ANN is a massively interconnected network structure consisting of many simple processing elements capable of performing parallel computations for data processing. In effect, ANN can handle multiple independent and dependent variables simultaneously without having prior knowledge about the functional relationship. ANN differs from traditional computing methods in many ways. The processing style of ANN is parallel and there is not a strict rule or algorithm to follow, whereas traditional methods are sequential and logical. Moreover, traditional methods can learn by rules while ANN learns by examples and therefore artificial neural networks are known as universal function approximators (Anjum, Tasadduq, & Al-Sultan, 1997). A neural
R2 r S S0 t T t0 Vmax Dt
regression coefficient of determination reaction rate (lmol/min/mg enzyme) substrate (maltose) concentration (mM) initial substrate concentration (mM) reaction time (min) total reaction time (min) constant in Boltzmann function maximal reaction rate of enzyme (lmol/min/mg enzyme) time increments (min)
network derives its computing power through first its massively parallel-distributed structure and second its ability to learn and therefore generalize. Here generalization refers to the neural network producing reasonable outputs for inputs not encountered during training (Haykin, 1994). In this study, we used artificial neural networks (ANN) to estimate the reaction rate of maltose hydrolysis by amyloglucosidase enzyme. The network was trained using time course data obtained from enzymatic reaction. The trained network was used for estimation of the reaction rate and then estimated rate was used to obtain concentration profile of substrate (maltose) and product (glucose) in batch reactor during reaction time for the reaction of amyloglucosidase enzyme. 2. Material and methods 2.1. Chemicals Maltose, glucose, amyloglucosidase (From Rhizopus sp., E.C. 3.2.1.3) and 4-aminoantipyrine were supplied from Sigma Chemical Co. (USA). Glucose oxidase (From Aspergillus niger, E.C. 1.1.3.4) and peroxidase (From horseradish, E.C. 1.11.1.7) were obtained from Biozyme Laboratories Lim. (UK). Phenol, Na2HPO4, and KH2PO4 were obtained from J.T. Baker (The Netherlands). 2.2. Maltose hydrolysis and determination of reaction rate using time course data The enzymatic reaction performed in this study is given below: Amyloglucosidase
Maltose þ H2 O ! 2Glucose Enzymatic hydrolysis of maltose was carried out in 5 ml of reaction solution (phosphate buffer, 100 mM, pH 5.5). Reaction solution was heated up to 37 C using a continuously stirred water bath and after adjustment of initial maltose and glucose concentration in solution, the reaction was started by adding amyloglucosidase enzyme into the solution. The final enzyme concentration in the reaction
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medium was 0.20 mg/ml. Maltose hydrolysis was monitored by measuring the amount of glucose produced during the incubation period. For this purpose, 10 ll sample was taken from reaction medium at different time intervals and immediately mixed with 2 ml of assay solution which was prepared by the method given by Trinder (1969). Time course data was obtained by measuring the amount of glucose produced during the enzymatic reaction. This procedure was replicated for fifteen different initial maltose or glucose concentrations. The initial maltose and glucose concentrations were varied between 2 and 16 mM and between 0 and 10 mM, respectively. In all measurements, glucose concentration was monitored during 75 min incubation period and progress curves were obtained. A model time course is seen in Fig. 1. Maltose concentration in the reaction vessel was calculated using stoichiometric relation between maltose and glucose concentrations. Finally, maltose hydrolysis during incubation time was obtained which is also shown in Fig. 1. After obtaining time course data, enzymatic reaction rate was determined in two steps: (i) Experimental time course data was fitted to the Boltzmann function (Eq. (1)). S¼
A1 A2 þ A2 1 þ eðtt0 Þ=dx
ð1Þ
Boltzmann function (BF) has a special condition. In a nutshell, most of the time course data of the biochemical reactions could be fitted to this equation with a high accuracy. It adequately fits the time course data of glucose production and maltose hydrolysis with a high coefficient of correlation (R2). The equations fitted to the experimental data and their R2 values are seen in Fig. 1.
(ii) Eq. (1) was derived with respect to t and Eq. (2) was obtained. This equation could be used to determine reaction rate (dS/dt) at any reaction time (t).
dS ðA1 A2 Þðeðtt0 Þ=dx Þ ¼ 2 dt dxð1 þ eðtt0 Þ=dx Þ
ð2Þ
By using this procedure, 150 different reaction rate values were determined for 15 time courses.
2.3. Development of artificial neural networks and prediction of reaction rates The artificial neural network was built with The Neural Network Toolbox, MATLAB Release 14 (The Mathworks, Natick, MA). A feed forward neural network, which consists of three neurons in input layer, four neurons in hidden layer and one neuron in output layer, was built. The inputs of the neural network were maltose concentration, glucose concentration and initial glucose concentration, and output of the system was the reaction rate. Transfer function between the input and the hidden layer was ‘tansig’ and the one between the hidden layer and the output layer was ‘logsig’. Training of the network was performed with the function ‘trainlm’, which updates weight and bias values according to Levenberg–Marquardt optimization. The other parameters of the neural network were taken as the defaults of MATLAB. Data of seven different reaction batches was used for training and the remaining (eight reaction batches) data was used for testing the network. Training was performed to minimize the mean square error (MSE) between targets and outputs (Target MSE value was defined as 1 · 104). Trained network was saved as a mat file for further applications. 2.4. Progress curves for maltose hydrolysis obtained from the estimated reaction rates
35 P = 34.875 +
Glucose Maltose
30
-59.579 (1 + e t/31.758 )
Enzymatic reaction rate for defined maltose and glucose concentration was estimated using trained network. In further part of the study, outputs of the ANN were used in a numerical solution of modeling equation obtained for batch reactor. The modeling part of the study and the numerical solution is described below briefly. The change in substrate concentration in a batch reactor with time can be given by Eq. (3).
2
R = 0.996
Concentration (mM)
Predicted with Boltzmann 25
20
15
dS ð3Þ dt where r is the reaction rate. The backward difference approximation for dS/dt gives
r¼ 10
R2 = 0.996
5
S = 1.062 +
29.790 (1 + e t/31.758 )
0 0
10
20
30
40
50
60
70
80
Time (min) Fig. 1. Time courses for hydrolysis of maltose into glucose. Initial maltose and glucose concentrations are 16.0 mM and 5.0 mM, respectively.
dS S i þ S iþ1 ¼ dt Dt
ð4Þ
The central differences approximation for dS/dt gives dS S i1 þ S iþ1 ¼ dt 2Dt
ð5Þ
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For time t, step indices (i) are used with increments in time being represented by Dt. i = 1 represents initial value of maltose concentration at t = 0. In the beginning of the reaction initial values of maltose and glucose concentrations were fed to the program as Si=1 = S0 and Pi=1 = P0, respectively. Using input values, enzymatic reaction rate ri=1 was estimated using trained network and then Si=2 was calculated using finite difference equation given in Eq. (6). Then Pi=2 was calculated using Eq. (7). S i¼2 ¼ S i¼1 ri¼1 Dt P i¼2 ¼ P i¼1 þ 2 ri¼1 Dt
ð6Þ ð7Þ
Calculated maltose and glucose concentrations were presented to the network and reaction rate was estimated for further calculation. For i > 2, the central differences approximation was used and Eqs. (8) and (9) were derived for the finite difference equations used in the numerical solution S i¼iþ1 ¼ S i¼i1 2 ri¼i Dt
ð8Þ
P i¼iþ1 ¼ P i¼i1 þ 4 ri¼i Dt
ð9Þ
This procedure was replicated during finite reaction time. In addition, a program which combined ANN with numerical solution of the model equation was coded using MATLAB to obtain maltose and glucose concentrations versus time profile in the batch reactor. The profiles were correlated with the ones obtained experimentally. Performance of numerical solution was studied using test data set, which was not used in training of the network. 3. Results and discussion 3.1. Maltose hydrolysis and determination of reaction rate using time course data Glucose production from maltose by amyloglucosidase enzyme was investigated at fifteen different initial maltose or glucose concentrations. Fifteen time courses were obtained and experimental time courses were fitted to the Boltzmann function. Model equation and its R2 value were obtained for each reaction batch. (A sample time course data and its model equation and R2 values were given in Fig. 1.) The R2 values of the all predicted model equations were quite high (R2 > 0.99) and these values indicated that time courses of the enzyme catalyzed reaction can be properly explained by the BF. An equation, obtained by the derivation of the predicted model equation, was used for determining the enzymatic reaction rate for the period of reaction time. The obtained initial reaction rates (t = 0) were visualized in Fig. 2 by drawing surface and contour plots of the enzymatic reaction rate as a function of the maltose and glucose concentration. As seen in the figure, increasing maltose concentration cause an increase in the reaction rate. In low maltose concentrations, glucose concentration does not affect the reaction rate. However, at high maltose concentrations as
Fig. 2. Surface and contour plots of the initial reaction rate as a function of maltose and glucose concentration.
glucose concentration increases, reaction rate decreases. This result indicates that inhibition kinetics is valid for amyloglucosidase reactions. Similar results were reported by several authors (Li, Kim, & Peeples, 2002; Najafpour & Shan, 2003; Polakovic & Bryjak, 2004). 3.2. Training and testing of artificial neural network Data obtained from seven time courses was used for training of the ANN and the data of other time courses (eight time courses) was used for testing the trained network. The training data set (three input and their corresponding desired responses) was presented to the network and a feed forward algorithm automatically adjusted the weights so that the output response to input values were as close as possible to the desired response. Estimation was made and the results were compared with the corresponding desired value. Then the estimation error (the difference between the estimated and the desired values) was distributed across the network in a manner which allowed the interconnection weights to be modified according to the scheme specified by the learning rule. This process was repeated while the prediction error decreased. After 250 learning epochs, the target was achieved and the learning stage was completed. In order to test the trained network another data set was used and the input test set was presented to the network and the output was obtained. The output of the ANN was compared with the experimental data for the trained and test data sets in Fig. 3(a) and (b), respectively. It is clear that the regression coefficients of determination (R2 = 0.992 and R2 = 0.965) are high between estimated and experimental data for both of the data sets. Since excellent estimation performances were obtained using the trained network, it demonstrates that the trained network was
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y = 0.997x 1.25
2
R = 0.992
Experimental Rates (μ mol/min/mg enzyme)
1.00
0.75
0.50
0.25
0.00 0.00
0.25
0.50
0.75
1.00
1.25
Predicted Rates (μ mol/min/mg enzyme)
(a)
y = 1.000x
1.25
2
R = 0.965
in this study, r value was estimated by using ANN without a model equation. A small program for numerical solution of differential equation was coded in MATLAB and ANN was integrated to this solution. Algorithm of the program is seen in Fig. 4. The algorithm written for this study will be sent as requested from the address above. The developed program was used for the prediction of time courses in the batch reactor. The initial concentrations of maltose and glucose, and variables were presented to the system. The products (time courses of maltose and glucose concentration) were obtained. Experimental and estimated time courses of maltose concentration are seen in Fig. 5(a). Similarly, those of glucose concentration were given in Fig. 5(b) (Fifteen time courses were predicted using developed program and only five of them were presented in Fig. 5). In both of the graphs, there are high correlations between experimental and estimated curves. It was verified that the developed program has high capacity for estimating time courses of substrate and product concentrations. One critical point is that the performance of the program was directly related with estimation performance of the ANN. This result was another proof of the high performance of ANN for estimation of enzyme-catalyzed reaction rate. Estimation capability of ANN does not
Experimental Rates (μ mol/min/mg enzyme)
1.00
Input initial values and variables 0.75
Estimate the reaction rate based on the initial condition 0.50
0.25
0.00 0.00
(b)
Calculate new maltose concentration, S Calculate new glucose concentration, P Calculate time, t Add the data into the matris 0.25
0.50
0.75
1.00
1.25
Predicted Rates (μ mol/min/mg enzyme)
Fig. 3. Comparison of the estimated reaction rate with experimental data (a) training data set and (b) test data set.
reliable, accurate and hence could be employed in the further part of the study. 3.3. Progress curves for maltose hydrolysis obtained from the estimated reaction rates In conventional numerical solution of the differential equation, constant or variable in the model equation should be known. In Eq. (3), r variable could be explained with an equation such as Michaelis–Menten Equation. Otherwise, one of the more complex inhibition kinetics equations is required. It is necessary to decide which model equation is valid to start the numerical solution. However,
Estimate the new reaction rate, r Calculate new maltose concentration, S Calculate new glucose concentration, P Calculate time, t Add the data into the matris
If t
YES
NO Plot reaction time vs. substrate and product concentration Fig. 4. Algorithm of the program.
D. Basß et al. / Journal of Food Engineering 79 (2007) 622–628
14
Maltose Concentration (mM)
and the trained network was used to estimate the reaction rate at any substrate and product concentration, which is a new approach in this field. In the further part of the study, ANN was successfully employed as a part of the numerical solution of the differential equations that was obtained by modeling the batch reactor. The usage of ANN with time course in reaction kinetics had several advantages. Some of them are listed below.
S0 = 16 mM; P0 = 10 mM S0 = 12 mM; P0 = 0 mM S0 = 8 mM; P0 = 0 mM S0 = 4 mM; P0 = 5 mM S0 = 2 mM; P0 = 5 mM
16
Time courses obtained from developed program
12 10 8 6 4 2 0 0
15
(a)
45
60
75
60
75
Time (min) 35
S0 = 16 mM; P0 = 10 mM S0 = 12 mM; P0 = 0 mM S0 = 8 mM; P0 = 0 mM S0 = 4 mM; P0 = 5 mM S0 = 2 mM; P0 = 5 mM
30
Glucose Concentration (mM)
30
Time courses obtained from developed program
25 20 15 10 5 0
0
(b)
15
30
45
627
Time (min)
Fig. 5. Time courses of (a) maltose and (b) glucose concentrations obtained from estimated reaction rates. (The points represent experimental data while the lines are estimated concentrations obtained from developed program.)
depend on the type of reaction and it is directly related with the training stage of the network. It is obvious that well trained network will have high performance to obtain time courses correctly for both reversible and irreversible reactions. 4. Conclusion The time course of an enzyme-catalyzed reaction contains a wealth of information and the increased speed and capacity of desktop computers allow the better use of reaction time courses (Duggleby, 2001). In conventional methods, kinetic models and kinetic constants are determined by using the initial reaction rate values or kinetic constants are directly determined by utilizing the time course data. In this study, the reaction rate values obtained from time courses were used for the training of the ANN
i. ANN estimates reaction rate without requiring any kinetic model equation. ii. Estimation of reaction rate without a kinetic model eliminates the errors arising from the selection of kinetic model and the estimation of kinetic constants. iii. Usage of ANN reduces the number of the experimental work or provides more information than the conventional method, if same amount of experimental study is employed. iv. In the conventional method, at the beginning of the modeling study, it is required to make assumptions about the kinetic equation and estimate the constant in the equation. This procedure increases paperwork. The first assumption is generally not correct and the researcher has to check more than one kinetic model. However, if researcher uses ANN, he/she does not need any kinetic model. As a result, kinetic work could be performed without any limitation of model equation. v. ANN could be useable in a part of the complex calculation using an improved program. In this study, usability of the ANN for numerical solution of the differential equation obtained by modeling of the batch reaction was demonstrated successfully. The results are very promising and we believe that ANN will become a popular method for estimation of reaction rate without using the kinetic model. However, for different purposes investigation of reaction kinetics and prediction of kinetic constants may be required and researchers may want to determine those constants. In a subsequent paper, we will perform a research on the applicability of ANN for estimation of the reaction kinetics and kinetic constants. Acknowledgements The authors gratefully acknowledge the financial support from State Planning Organization of Republic of Turkey; Project No. 03 K 120 570-4. References Anjum, M. F., Tasadduq, I., & Al-Sultan, K. (1997). Response surface methodology: a neural network approach. European Journal of Operational Research, 101, 65–73. _ Basß, D., Boyacı, I.H. (2006). Modeling and optimization II: comparison of estimation capabilities of response surface methodology with artificial neural networks in a biochemical reaction, Journal of Food Engineering, in press, doi:10.1016/j.jfoodeng.2005.11.025.
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Duggleby, R. G. (2001). Quantitative analysis of the time courses of enzyme-catalyzed reactions. Methods, 24, 168–174. Geeraerd, A. H., Herremans, C. H., Cenens, C., & Van Impe, J. F. (1998). Application of artificial neural networks as a non-linear modular modeling technique to describe bacterial growth in chilled food products. International Journal of Food Microbiology, 44, 49– 68. Hajmeer, M. N., Basheer, I. A., & Najjar, Y. M. (1997). Computational neural networks for predictive microbiology II. Application to microbial growth. International Journal of Food Microbiology, 34, 51–66. Haykin, S. (1994). Neural networks: A comprehensive foundation. New Jersey: Prentice Hall (pp. 696–723). Li, M., Kim, J., & Peeples, T. L. (2002). Kinetic enhancement of starch bioconversion in thermoseparating aqueous two-phase reactor systems. Biochemical Engineering Journal, 11, 25–32. Lou, W., & Nakai, S. (2001). Application of artificial neural networks for predicting the thermal inactivation of bacteria: a combined effect of
temperature, pH and water activity. Food Research International, 34, 573–579. Najafpour, G. D., & Shan, C. P. (2003). Enzymatic hydrolysis of molasses. Bioresource Technology, 86, 91–94. Najjar, Y. M., Basheer, I. A., & Hajmeer, M. N. (1997). Computational neural networks for predictive microbiology: I. Methodology. International Journal of Food Microbiology, 34, 27–49. Polakovic, M., & Bryjak, J. (2004). Modelling of potato starch saccharification by an Aspergillus niger glucoamylase. Biochemical Engineering Journal, 18, 57–63. Sun, Y., Peng, Y., Chen, Y., & Shukla, A. J. (2003). Application of artificial neural networks in the design of controlled release drug delivery systems. Advanced Drug Delivery Reviews, 55, 1201–1215. Torrecilla, J. S., Otero, L., & Sanz, P. D. (2004). A neural network approach for thermal/pressure food processing. Journal of Food Engineering, 62, 89–95. Trinder, P. (1969). Determination of blood glucose using 4-aminophenazone. Journal of Clinical Pathology, 22, 246.