Article pubs.acs.org/jchemeduc
Diagnosis of Enzyme Inhibition Using Excel Solver: A Combined Dry and Wet Laboratory Exercise Albino A. Dias, Paula A. Pinto, Irene Fraga, and Rui M. F. Bezerra* CITABDepartamento de Biologia e Ambiente, Universidade de Trás-os-Montes e Alto Douro, Apartado 1013, 5001-801 Vila Real, Portugal S Supporting Information *
ABSTRACT: In enzyme kinetic studies, linear transformations of the Michaelis−Menten equation, such as the Lineweaver−Burk double-reciprocal transformation, present some constraints. The linear transformation distorts the experimental error and the relationship between x and y axes; consequently, linear regression of transformed data is less accurate when compared with methodologies that use nonlinear regression. However, linear transformations are widely used. Explanations for this are the facility to determine model parameters by hand calculations, and until recently, the use of nonlinear regression was difficult as specialized software was not readily available to most scientists and students. Because utilization of personal computers is widespread, these constraints are no longer applicable. This work describes how to perform nonlinear regression with the Solver supplement of Microsoft Office Excel. It is easy to use and to view the results graphically. The F-test was applied to discriminate between models. These methodologies are important in any biochemistry syllabus and can be used to create an active-learning environment where students discriminate between different kinetic models and explore their own experimental results based on several hypotheses. KEYWORDS: Upper-Division Undergraduate, Biochemistry, Computer-Based Learning, Enzymes, Kinetics
T
There are different iterative techniques and several different algorithms can be used, including the Gauss−Newton, the Marquardt−Levenberg, and the steepest descent methods. Excel Solver, however, employs the generalized reduced gradient algorithm. Nevertheless, with the same data set all of these methods should yield the same parameter values.6 As nonlinear regression is an interactive methodology, the program must start with “initial values” for each parameter attributed by the investigator. Unrealistic “initial values” can result in convergence failure or an unacceptable convergence often associated with impossible parameter values (e.g., negative Km). A poor convergence means that the result (local minimum) is not the lowest possible minimum. To prevent this, it is important to estimate reasonable initial values, for example, by using Lineweaver−Burk parameters, as initial starting points as they are close to the actual values.6 Another important procedure is starting a kinetic analysis with simple models in which the number of parameters is small (e.g., only Vmax and Km). The values obtained for the parameters of the simple model should be used as initial values to more complex models. Several problems associated with nonlinear regression include nonconvergence, slow convergence, redundant variables, and poor convergence. According to Motulsky and Ransnas,2 these problems (and suggested resolution) are
he determination of kinetic parameters is a central point in enzyme characterization. Linear transformations of the Michaelis−Menten equations, such as the Lineweaver−Burk double-reciprocal transformation, are still widely used despite much criticism;1 linear transformations are not appropriate for accurate determination of kinetic parameters because they distort experimental errors.2 Assuming the Michaelis−Menten equation, it was reported that kinetic constants obtained after linear regression of the transformed data may differ by up to 150% when compared with nonlinear regression applied to the nontransformed data.3 Nonlinear regression is a more accurate methodology, but until recently it was difficult task to use. This is no longer a problem as it is now included in commonly used software applications, such as Microsoft Office Excel.4,5 Optimization of nonlinear equation parameters is obtained after minimizing the sum of squares of the distances of the data from the curve (sum of the square error (SSE), SSE = sum[ydata − ycurve)2]). Two different processes of curve fitting can be considered according to the uncertainty of dependent variable data: (i) unweighted, when all values of y have equal uncertainty, and (ii) weighted, when different values of y have different uncertainties.4 In this work, the first approach will be used. Unlike linear regression, nonlinear regression must be solved iteratively where the results of one iteration are used as the starting point for the next iteration. These iterations proceed until no improvement in the goodness of curve fit is obtained.2 © 2014 American Chemical Society and Division of Chemical Education, Inc.
Published: June 6, 2014 1017
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and 60 s was determined by absorbance readings at 405 nm (reaction: 4-nitrophenylphosphate + H2O → p-nitrophenol + phosphate). The spectrophotometer was adjusted to zero with reference cuvettes (i.e., blanks) that contained all reaction compounds except the enzyme. This zero value of absorbance was assumed at the time zero. Reaction was started adding enzyme to the cuvette at time zero.
usually related to (1) data values too large or too small (change the units), (2) the selected equation not fitting the data (change the equation), (3) poor initial values (estimate them from a Lineweaver−Burk double-reciprocal plot), (4) data not spanning a sufficient range (obtain more experimental points), (5) computer calculations with insufficient significant digits (change the value), and (6) too many parameters to be fit (fix one or more as a constant). One goal in the study of enzymology is parameter determination and another goal is to discriminate between different kinetic models (e.g., competitive inhibition or noncompetitive inhibition). Methods, such as extra sum-ofsquare F test (as Mannervik discrimination), allow comparison of models.7−10 It should be pointed out that, in kinetic studies that are beyond the accuracy of statistical measures, it is also important to consider the physical plausibility of the model and to take into account only models with theoretical justification.2 After model discrimination, the uncertainties in parameters can be determined by procedures including Billo’s SolvStat macro,11 de Levie’s SolverAid macro,12 or Ogren’s variancecovariance matrix.13
Data Processing Analysis and Model Discrimination
Initial velocities (Figure 1) were obtained under assay conditions where the substrate depletion never exceeded
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OVERVIEW The major goal of this work is to present an easily implemented methodology to obtain kinetic constants for an enzymecatalyzed reaction by use of nonlinear regression. As Excel is one of the most widely used software packages and offers a friendly user interface with the possibility of instantaneous data graphing,14 it is used in this work. This methodology, in association with the King Altman approach,15 can be used to create an active-learning environment where students discriminate between different kinetic models and explore their own experimental results based on several hypotheses as explained elsewhere.15,16 The exercise contains three sessions of 2 h each: (i) dry lab training where the instructors provide a data sheet containing experimental data previously obtained to apply the nonlinear regression with the Solver and learn to discriminate different models; (ii) wet lab session where the students perform kinetic assays described in this work; and (iii) processing of their own data, with the discrimination of different enzymatic models and concomitant determination of kinetic parameters. This experiment is suitable for a typical undergraduate biochemistry or enzymology course. More than 250 students completed the lab work. All but a few student groups successfully obtained the expect results. The experiment was performed with a typical class size of 16 students where students worked in groups of two.
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Figure 1. Screen capture of a worksheet template to determine the values of Km, Vmax, Kic, and Kiu (MI model). Numbers in columns A and B (from line 7 to 24) are experimental data.16 Numbers in cells B2−B5 are initial values assigned to Km, Vmax, Kic, and Kiu (see the text). Reprinted with permission from Microsoft Corp.
10%. The students filled in the data cells on a blank spreadsheet similar to Figure 1 and then followed the step-by-step instructions. Cells B2−B5 corresponded to model parameters, Km, Vmax, Kic, and Kiu and were assigned initial, arbitrary values of 1. The data were fitted to the Michaelis−Menten equation (Table 1) considering different types of inhibition: without inhibition (WI), competitive inhibition (CI), noncompetitive inhibition (NCI), uncompetitive inhibition (UCI), and mixed inhibition (MI). Because the MI model is the more complex model, it was advantageous to start the simulations with the simpler model (i.e., without inhibition model, WI). Students copied the worksheet in Figure 1 to four additional Excel worksheets and changed the equation of cell C7 in each worksheet to correspond to one of the models from Table 1. Each spreadsheet was modified, considering only the parameters present in the respective model. In the first spreadsheet the equation corresponded to the Michaelis−Menten equation without inhibition (WI model), which is the simplest model. The process was complete when the five spreadsheets corresponded to the equations WI, CI, NCI, UCI (obtained
MATERIAL AND METHODS
Enzyme, Reagents, and Reaction Conditions
The experimental conditions were previously published.16 Briefly, stock solutions of 4-nitrophenylphosphate and 4nitrophenol were prepared at concentrations of 20 mM. Reactions (2.75 mL) were carried out at 37 °C containing 9.5 μg of alkaline phosphatase and eight different concentrations of 4-nitrophenylphosphate disodium salt (25 μM; 35 μM; 45 μM; 90 μM; 200 μM; 500 μM; 1000 μM; 2000 μM) in 0.1 M Tris/HCl buffer, pH 9.0. Another set of similar reaction experiments was also performed, but in the presence of 5.0 μM disodium hydrogen phosphate (inhibitor). The quantity of reaction product (4-nitrophenol) formed at 10, 20, 30, 40, 50, 1018
dx.doi.org/10.1021/ed3006677 | J. Chem. Educ. 2014, 91, 1017−1021
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Table 1. Equations for Different Michaelis−Menten Models Considering Different Types of Inhibition Model Without inhibition (WI) Competitive inhibition (CI)
Noncompetitive inhibition (NCI)
vo =
vo =
vo =
Calculated vo = $B$3*A7/($B$2+A7) Assume: Km = $B$2, Vmax = $B$3
Vmax[S] K m + [S]
Km 1 +
(
[1] K ic
) + [S]
(
[1] K ic
Vmax[S]
) + [S](1 + ) [1] K iu
(Remember:
(
vo =
(
Km 1 + a
[1] K iu
Calculated vo = ($B$3*A7)/($B$2+A7*(1+F7/$B$4)) Assume: Km = $B$2, Vmax = $B$3, Kiu = $B$4
)
Vmax[S] [1] K ic
Calculated vo = ($B$3/(1+F7/$B$4))*A7/($B$2+A7) Assume: Km = $B$2, Vmax = $B$3, Kic = Kiu = $B$4
K ic = K iu)
Vmax[S]
vo =
K m + [S] 1 + Mixed inhibition (MI)
Calculated vo = $B$3*A7/($B$2*(1+F7/$B$4)+A7) Assume: Km = $B$2, Vmax = $B$3, Kic = $B$4
Vmax[S]
Km 1 +
Uncompetitive inhibition (UCI)
Excel Equationsa (cell C7)
Michaelis−Menten Equation
) + [S](1 + ) [1] K iu
Calculated vo = ($B$3*A7)/(($B$2*(1+F7/$B$4))+A7*(1+F7/$B$5)) Assume: Km = $B$2, Vmax = $B$3, Kic = $B$4, Kiu = $B$5
See the text for explanation.
from the simplification of the more complex model, MI), and MI. Students copied cells C7−C24 to obtain the calculated initial velocity. The square of the difference between experimental and calculated velocities was found in column G and was obtained by typing the equation =POWER(B7−C7,2), in cell G7 and copying through cell G24. Then the SSE value was calculated by typing the equation =SUM(G7:G24) in cell G25. The students put the cursor in cell “G25” and chose Solver (to enable Solver in Office Excel, see Supporting Information) from the data menu. In the space entitled “by changing cells” they wrote “$B$2,$B$3” and checked the “Min” checkbox in the “Equal to” selection. Note that the target cell (G25) is the sum of the square error. By selecting Solve, the SSE value was minimized by changing the cells that contained the initial values of parameters (in this case Km and Vmax). The students examined the graph plot (Figure 2) to see what happened. The process was repeated to ensure that the values of SSE and parameters were stabilized. The students then changed to another Excel spreadsheet (e.g., competitive inhibition, CI) and utilized the previous estimated parameter values as “initial values” of Km and Vmax and assigned 1 to the “initial value” of Kic. The same procedure was repeated for the remaining models (in Table 1).
Figure 2. Excel graph plot (A) before and (B) after parameter estimation (Km, Vmax, and Kic) by Solver (competitive inhibition model, CI). Experimental data points were obtained without inhibitor (squares) and in the presence of inhibitor (circles), whereas the solid lines are model simulations.
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RESULTS AND DISCUSSION In this analysis, the main aim is to compare linear mixed inhibition that includes all the common types of reversible linear inhibitions (noncompetitive inhibition, NCI; competitive inhibition, CI; and uncompetitive inhibition, UCI) and even the lack of inhibition (WI), to find the best kinetic model. This methodology has already been explained.16,17 According to Table 1, MI is written as required in the Excel program and can be simplified,16 giving rise to models with fewer constants, such as CI, UCI, NCI, or WI. For example, if Kic approaches ∞, the equation does not contain the term with Kic in the denominator (Table 1), and the model obtained is UCI (uncompetitive linear inhibition). The SSE values obtained with different equations, after nonlinear regression, are given in Table 2 where n is the number of experimental points and p is the number of parameters for each model. Screen captures for all five models after convergence by Solver
Table 2. Sum of Square Error (SSE) Values for Different Michaelis−Menten Models, Number of Parameters (p), and Number of Experimental Points (n) Model Metric
WI
CI
NCI
UCI
MI
SSE p n
0.365 2 17
0.155 3 17
0.272 3 17
0.313 3 17
0.156 4 17
are presented in the Supporting Information. Additionally, it is interesting to see what happens to Michaelis−Menten graph plots (Figure 2) before and after parameter estimation. This is a simple way to visualize the effects of parameter change. 1019
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Table 3. Summary of Discrimination between Models No Inhibition (WI) and Competitive Inhibition by Phosphate (CI) Models A/B
SSEA
SSEB
n
pA
pB
w
pB − pA
n − pB
f 0.95
w − f 0.95
WI/CI
0.365
0.155
17
2
3
19.0
1
14
4.6
14.4
This kinetic investigation consisted of two parts: parameter estimation and discrimination between available models (Tables 2 and 3). As previously explained,7 discrimination is possible between any two models, for example, model A (with pA parameters) and model B (with pB parameters), when the two models are fitted separately to the same experimental points. If model B is an extension of the simpler model A, the significance of improvement obtained by the addition of the new parameters (pB − pA) can be tested by comparing the Fstatistic, F(pB−pA, n − pB), at the desired level of probability (f 0.95) with quotient w: w=
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported by European Union Funds (FEDER/ COMPETEOperational Competitiveness Programme) and by national funds (FCTPortuguese Foundation for Science and Technology) under the Project FCOMP-01-0124-FEDER022692.
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(1)
When the w − f value > 0, the more complex model should be used; otherwise, the simpler model can be applied. The F values were obtained from statistical tables (F distribution, F(pB− pA,n−pB)) at the desired level of probability (f 0.95) knowing pB − pA (number of parameters added) and n − pB (degrees of freedom) for these models. To discriminate between models (Table 2) with the same number of parameters (CI, NCI, and UCI), it is only necessary to consider the SSE value. Thus, the CI model is preferable to UCI and NCI because the SSE value is lower. Also, because the MI model has a SSE value higher than CI, it is not necessary to compare CI with MI because MI has more parameters than CI. Thus, it is only necessary to compare WI and CI (Table 3). Because the w − f value > 0, the CI model (more complex model) should be preferred, which means that competitive inhibition is the best model. For this enzyme-catalyzed reaction, it is now possible to summarize the constants obtained with this model: Km = 62.1 μM, Kic = 8.9 μM, and Vmax = 3.0 μmol min−1 mg−1. Experimental results and fitted curves obtained with these parameters are shown in Figure 2B.
REFERENCES
(1) Martin, R. B. Disadvantages of Double Reciprocal Plots. J. Chem. Educ. 1997, 74, 1238−1240. (2) Motulsky, H. J.; Ransnas, L. A. Fitting Curves to Data Using Nonlinear Regression: A Practical and Nonmathematical Review. FASEB J. 1987, 1, 365−374. (3) Yildirim, N.; Akçay, F.; Okur, H.; Yildirim, D. Parameter Estimation of Nonlinear Models in Biochemistry: A Comparative Study on Optimization Methods. Appl. Math. Comput. 2003, 29−36. (4) Harris, D. C. Nonlinear Least-Squares Curve Fitting with Microsoft Excel Solver. J. Chem. Educ. 1998, 75, 119−121. (5) Silverstein, T. P. Nonlinear and Linear Regression Applied to Concentration versusTime Kinetic Data from Pinhas’s Sanitizer Evaporation Project. J. Chem. Educ. 2011, 88, 1589−1590. (6) Motulsky, H.; Christopoulos, A. Fitting models to biological data using linear and nonlinear regression, a practical guide to curve fitting; GraphPad Software: San Diego, CA, USA, 2003; p 351. (7) Mannervik, B. Regression Analysis, Experimental Error and Statistical Criteria in the Design and Analysis of Experiments for Discrimination between Rival Kinetic Models. Methods Enzymol. 1982, 87, 370−391. (8) Lieb, S. G. Simplex Method of Nonlinear Least-SquaresA Logical Complementary Method to Linear Least-Square Analysis of Data. J. Chem. Educ. 1997, 74, 1008−1011. (9) Masel, R. I. Chemical Kinetics and Catalysis; John Wiley & Sons: New York, 2001; p 968. (10) Zielinski, T. J.; Allendoerfer, R. D. Least Squares Fitting of NonLinear Data in the Undergraduate Laboratory. J. Chem. Educ. 1997, 74, 1001−1007. (11) Billo, E. J. Excel for Chemists; Wiley VCH: New York, 2001; pp 223−233. (12) de Levie, R. Advanced Excel for Scientific Data Analysis, 3rd ed.; Atlantic Academic: Brunswick: ME, USA, 2012; p 646. (13) Ogren, P.; Davis, B.; Guy, N. Curve Fitting, Confidence Intervals and Envelopes, Correlations, and Monte Carlo Visualizations for Multilinear Problems in Chemistry: A General Spreadsheet Approach. J. Chem. Educ. 2001, 78, 827−836. (14) Brown, A. M. A Step-by-Step Guide to Non-linear Regression Analysis of Experimental Data Using a Microsoft Excel Spreadsheet. Comput. Methods Programs Biomed. 2001, 65, 191−200. (15) Sims, P. A. An ″Aufbau″ Approach To Understanding How the King−Altman Method of Deriving Rate Equations for EnzymeCatalyzed Reactions Works. J. Chem. Educ. 2009, 86, 385−389. (16) Bezerra, R. M. F.; Dias, A. A. Utilization of Integrated MichaelisMenten Equation to Determine Kinetic Constants. Biochem. Mol. Biol. Educ. 2007, 35, 145−150. (17) Bezerra, R. M. F.; Fraga, I.; Dias, A. A. Utilization of Integrated Michaelis−Menten Equations for Enzyme Inhibition Diagnosis and
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CONCLUSION Nonlinear regression is a powerful technique for data analysis, and with this methodology it is easy and fast for routine use. No sophisticated knowledge of computers or mathematics is required to use such programs. Nevertheless, for extended applications it is important to have a theoretical knowledge to formulate kinetic equations, without which it is not possible to understand the results. This methodology can also be extended to the analysis of entire progress curves by the utilization of the integrated Michaelis−Menten equation in its classical implicit form16−18 or in its explicit form in which the concentration of the substrate as a function of time can be obtained by approximations to the Lambert W function.19,20
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AUTHOR INFORMATION
Corresponding Author
(SSEA − SSEB)(n − pB ) (pB − pA )SSE B
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ASSOCIATED CONTENT
S Supporting Information *
Screen capture of a worksheet template with all five models after convergence by Solver and instructions on how to add Solver supplement to Microsoft Office Excel 2010 for Windows. This material is available via the Internet at http:// pubs.acs.org. 1020
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Determination of Kinetic Constants Using Solver Supplement of Microsoft Office Excel. Comput. Methods Programs Biomed. 2013, 109, 26−31. (18) Bezerra, R. M. F.; Dias, A. A.; Fraga, I.; Pereira, A. N. Cellulose Hydrolysis by Cellobiohydrolase Cel7A Shows Mixed Hyperbolic Product Inhibition. Appl. Biochem. Biotechnol. 2011, 165, 178−189. (19) Schnell, S.; Mendoza, C. J. Closed Form Solution for TimeDependent Enzyme Kinetics. Theor. Biol. 1997, 187, 207−212. (20) Goličnik, M. Exact and Approximate Solutions for the DecadesOld Michaelis-Menten Equation: Progress-Curve Analysis through Integrated Rate Equations. Biochem. Mol. Biol. Educ. 2011, 39, 117− 125.
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