Laboratory Exercises. Utilization of Integrated Michaelis-Menten Equation to Determine Kinetic Constants

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1 Q 2007 by The International Union of Biochemistry and Molecular Biology BIOCHEMISTRY AND MOLECULAR BIOLOGY EDUCATION Vol. 35, No. 2, pp , 2007 Laboratory Exercises Utilization of Integrated Michaelis-Menten Equation to Determine Kinetic Constants Received for publication, August 4, 2006, and in revised form, November 2, 2006 Rui M. F. Bezerra and Albino A. Dias From the Departamento de Engenharia Biológica e Ambiental, CETAV, Universidade de Trás-os-Montes e Alto Douro, Apartado 1013, Vila Real, Portugal Students of biochemistry and related biosciences are urged to solve problems where kinetic parameters are calculated from initial rates obtained at different substrate concentrations. Troubles begin when they go to the laboratory to perform kinetic experiments and realize that usual laboratory instruments do not measure initial rates but only substrate or product concentrations as a function of reaction time. To overcome this problem we present a methodology which uses the integrated form of Michaelis-Menten equation. The method presented has a theoretical and pedagogic basis which is not as arbitrary as other approaches. Here we present and describe the methodology for analyzing time course data together with some examples of the essential computer procedures to implement these analyses. To simplify the understanding of this methodology the experimental examples are confined to linear inhibitions and experimental points utilized are the same from which the initial rates are determined. Keywords: Enzyme inhibition diagnosis, general kinetic mechanism, integrated Michaelis-Menten equations, linear inhibitions. Characterization of enzyme kinetics today plays an essential role in important areas such as medical diagnostics, proteomics research, or even enzyme inhibitionbased biosensors for environmental monitoring. Alkaline phosphatase is an excellent model enzyme for teaching practical classes ranging from beginner level to more advanced biochemistry courses, in order to demonstrate important principles of kinetic studies and parameter determination (V max, K m, K i ) [1]. Almost any kinetic experiment on an enzyme-catalyzed reaction, exhibits a fundamental incompatibility between the data and the underlying model. The catalytic properties of the enzyme are usually evaluated by measuring and analyzing reactions rates, but experimental measurements of these reactions never determine rates directly. As previously pointed out [2], to avoid this incompatibility either the model must be integrated to give a description of the course of the reaction or the data must be differentiated (determining rates) by measuring tangents to the reaction curves at zero time. However, there is not a simple and accurate procedure to estimate initial velocities and in most of the cases this is done by eye. In addition, when the reaction product is an enzyme inhibitor, determination of initial velocities or kinetic parameters without theoretical errors still presents a considerable challenge. The utilization of integrated equations provides a unique methodology that can surpass such limitations associated with the presence of this type of inhibitor. To whom correspondence should be addressed. bezerra@utad.pt. This paper is available on line at The advantages and disadvantages of initial velocities and progress curve analysis have been recorded by several authors [3, 4]. Analyzing all reaction time-course of progress curve can be advantageous because it contains additional information about the properties of the enzyme. One of the major barriers to the wide application of timecourse studies is that is computationally challenging. This is no longer a problem because the new generation is the so-called laptop student [5]. Another caveat regarding progress curve analysis is related to the fact that enzymes may undergo gradual loss of activity under the assay conditions. Fortunately, there is a simple test to detect this problem [6] and there are also methods to be used for kinetic characterization of an unstable enzyme that allows progress curve analysis [7]. In the methodology presented here the integrated equation is applied with the same points usually utilized to determine initial velocities (conditions that never surpass 10% of substrate depletion). This methodology has been successfully applied to the progress curve in conditions that even surpass 50% of substrate hydrolysis [8 10]. Furthermore, this methodology also has been successfully applied where more complex inhibition models such as parabolic, hyperbolic, and substrate inhibition were compared and discriminated [9], even with more than one inhibitor [10, 11]. Several general design criteria have been described for enzymological studies [12] in order to discriminate between putative kinetic models. A special mention to one of them should be introduced, that is the use of a discrimination function [13] for determining which of two rival kinetic models is the most appropriate. DOI /bambed.32

2 146 BAMBED, Vol. 35, No. 2, pp , 2007 The major goal of this work is to present a step-bystep methodology for obtaining the kinetic constants for an enzyme-catalyzed reaction without prior determination of initial reaction velocities. MATERIALS AND METHODS Enzyme, Reagents, and Reaction Conditions Alkaline phosphatase (E.C ) was obtained from BDH Chemicals Ltd. Other reagents used are readily available from major suppliers. Stock solutions of 4-nitrophenylphosphate (Merck) and 4-nitrophenol were prepared at concentrations of 20 mm. Reactions (2.75 ml) were carried out at 37 8C containing9.5lg ofalkalinephosphatase and eight different amounts of 4-nitrophenylphosphate (25, 35, 45, 90, 200, 500, 1000, and 2000 lm) in 0.1 M Tris/HCl buffer, ph 9.0. The amount of reaction product (4-nitrophenol) formed at several time intervals was determined by absorbance readings at 405 nm on a Spectronic 601, Milton Roy spectrophotometer. Another set of similar reaction experiments were also performed but in the presence of 5.0 lm phosphate (inhibitor). Data Processing and Analysis Raw data (Annex) were treated by two different approaches. The first one was to calculate initial reaction velocities (v) (v ¼ d[p]/dt at t ¼ 0) at several different substrate concentrations with and without inhibitor added. The alternative method consisted of fitting an integrated form of the Michaelis-Menten equation to the progress curves. Both methods (initial velocities and progress curve analysis) used the same data points. All experimental points were obtained during 60 sec assays under conditions in which substrate depletion never exceeded 10%. Kinetic constants were first estimated from initial rates by the Michaelis-Menten equation, after model discrimination by Lineweaver-Burk graphical representation, and the constants were determined by non-linear regression. The second method is the progress curve analysis performed by non-linear regression according to Equations 4 8, utilizing the Manervik methodology [13] to discriminate the models. Both non-linear regressions were performed with Gauss Newton algorithm of SAS software (SAS Institute Inc.). Raw data were analyzed assuming only the values without phosphate and also these values plus the data obtained with inhibition by phosphate (5 lm phosphate). Non-Linear Regression Using SAS Program (Proc nlin procedure) The beginning of the SAS 1 program has the following formulation (Appendix): data; input y x z h w; cards; the input variables are: y ¼ time (sec); x ¼ phosphate concentration (lm); z ¼ 4-nitrophenyl phosphate (lm); h ¼ enzyme (lg/2.75 ml), and w ¼ initial phosphate concentration (lm). The values for each variable are written in the program as explained in Appendix. Values are written into the program as follows (example for CI model from Table II): proc nlin method ¼ Gauss-Newton maxiter ¼ 500 parms k m ¼ 1500, 5000 k a ¼ 10, 100 k al ¼ 50, 500 k3 ¼ 1, 3; model y ¼ ((1 k m /k a )*(x w) þ k m *(z/k a þ w/k a þ 1)*log(z/(z (x w))))*1/(h*k3); bounds k m > 0, k3 > 0, k a > 0; run; The PROC NLIN statement invokes the procedure. The Gauss- Newton iterative method regresses the residuals onto the partial derivatives of the model with respect to the parameters until the estimates converge (it is also possible to use other methods such FIG. 1. Mixed linear (total) inhibition model MI where: E - enzyme, ES - enzyme substrate complex, EIS - enzyme substrate inhibitor complex, EI - enzyme inhibitor complex, P - product, K m - Michaelis-Menten constant, K ic, K iu - inhibition constants and k cat - rate constant. A model without inhibition WI can be obtained by simplification of previous model assuming K ic, K iu, as infinite. Thus it is possible to obtain the following linear (total) inhibition models: competitive inhibition CI (K iu ¼1); noncompetitive inhibition NCI (K ic ¼ K iu ); uncompetitive inhibition UI (K ic ¼1). as Marquardt, Newton, or Descendent). First and second-order derivatives are automatically computed. MAXITER define the maximum number of iterations allowed for maximum likelihood fit. The PARMS statement lists names of parameters and specifies initial values. If the Gauss Newton does not converge immediately with the first parameters, it will be necessary to repeat the process introducing formerly obtained parameters, which now become the initial parameters, until the value of the constants stabilize. The MODEL statement specifies dependent (endogenous) variables and independent (exogenous) variables for the model. The BOUNDS statement restrains the parameter estimates within specified bounds. In each BOUNDS statement, you can specify a series of bounds separated by commas. The program ends with the RUN statement that executes a user-defined program. When initial phosphate does not exist it is necessary to write where w ¼ 0 before the model or rewrite the model. To convert the k3 constant obtained by the SAS software to usual enzyme units (U mg 1 ) it is necessary to remember that k3 units are in lm sec 1 (lg/2.75 ml) 1. Thus it is necessary to multiply by 165 to convert k3 units in lmol min 1 mg 1. RESULTS AND DISCUSSION Theoretical Framework Linear mixed inhibition (MI) includes all of the common types of inhibition and will therefore be taken as a general case (Fig. 1), although some inhibition constants can tend to infinity, meaning that they are irrelevant. This methodology has already been explained in previous works [8 11]. Assuming the general model for mixed linear inhibition (MI) in Fig. 1 and the Michaelis-Menten kinetic equation, we obtain the following rate equation: v ¼ K m 1 þ ½PŠ K ic V½ Š (Eq: 1) þ½ Š 1 þ ½PŠ K iu where K m, Michaelis constant; K ic, inhibitor dissociation constant of enzyme-inhibitor, and K iu, inhibitor dissociation constant of enzyme substrate inhibitor complexes; V, maximum velocity of reaction; v, initial velocity of reaction; [P] concentration of inhibitor; [ ] initial concentration of substrate. Now assuming that the inhibitor P is also formed during the reaction ([P] ¼ [P t ] þ [P 0 ] where [ ] [S t ] ¼ [P t ]) and [P 0 ] is the initial inhibitor concentration:

3 147 v ¼ d½s tš ¼ dt V½ Š K m ð1 þ½p t Š=K ic þ½p 0 Š=K ic Þþ½ Šð1 þ½p t Š=K iu þ½p 0 Š=K iu Þ (Eq: 2) where [S t ] is the concentration of substrate at the time t and the subscript 0 as, P 0, means initial concentration at t ¼ 0. Rearranging and integrating gives Z t 0 Z ½St Š ðk m ð1 þ½p t Š=K ic þ½p 0 Š=K ic Þþ½ Šð1 þ½p t Š=K iu þ½p 0 Š=K iu ÞÞ dt ¼ d½s t Š (Eq: 3) ½S t0 Š V½ Š Carrying out the integration gives t ¼ 1 V K ½ Š m þ ½P 0Š þ1 ln ½S tš K ic K ic ½ Š þ 1 K m þ ½Š þ ½P 0Š K ic K iu K iu ½S ð t Š ½ ŠÞþ 1 ½S t Š 2 ½ Š 2 ðeq: 4Þ 2K iu According to Table I, Equation 4 (MI, mixed linear model) can be simplified, giving rise to models with less constants such as the competitive inhibition (CI), uncompetitive inhibition (UCI), noncompetitive inhibition (NCI), or without inhibiton (WI). The four kinetic equations for WI, CI, NCI, and UCI models (Table I) are simplifications of Equation 4 (MI model). For example, if K ic approaches 1 the equation does not contains the terms where K ic is found in the denominator (Table I) and the model obtained is UCI (uncompetitive linear inhibition). Equations 5 8 also can be obtained from integration of the Michaelis-Menten rate equation assuming different inhibition types. Table II presents the equations from Table I rearranged for use with SAS software. Discrimination and Experimentally Determined Constants This kinetic investigation consists of two parts: discrimination between available models and parameter estimation WI CI NCI UCI TABLE I Integrated Michaelis-Menten equations obtained by simplification of mixed linear inhibition equation Model s name Obtained integrated equation Without inhibition t ¼ 1 V K m ln S t þðs t Þ Competitive inhibition Non competitive inhibition Uncompetitive inhibition t ¼ 1 V t ¼ 1 V t ¼ 1 V K m K m þ P 0 þ 1 K ic K ic þ P 0 þ 1 K iu K iu 1 2K iu ðs 2 t S 2 0 Þ ln S t þ 1 K m K ic ðs t Þ ln S t þ 1 K m þ P 0 þ K iu K iu K iu ðs t Þ ðk ic ¼ K iu Þ K m ln S t þ 1 þ þ P 0 K iu K iu ðs t Þþ 1 2K iu ðs 2 t S 2 0 Þ (Eq: 5) (Eq: 6) (Eq: 7) (Eq: 8) Model WI CI TABLE II Equations for different models arranged in order to use them in SAS software SAS equation y ¼ ððx wþþk m logðz=ðz ðx wþþþþ=ðh k3þ; y ¼ ðð1 k m =k a Þ ðx wþþk m ðz=k a þ w=k a þ 1Þ logðz=ðz ðx wþþþþ 1=ðh k3þ; NCI k a ¼ k b ; a ¼ k m þ k m w=k a þ k m z=k a ; b ¼ k m =k a þ 1 þ w=k b þ z=k b ; c ¼ 1=ð2 k b Þ; s ¼ z ðx wþ; y ¼ ða logðs=zþþb ðs zþ þc ðs 2 z 2ÞÞ ð 1=ðh k3þþ; UCI s ¼ ðz ðx wþþ; y ¼ ðk m logðz=sþ ð1þw=k b þ z=k b Þ ðs zþ þðs 2 z 2Þ=ð2 k b ÞÞ ð1=ðh k3þþ MI a ¼ k m þ k m w=k a þ k m z=k a ; b ¼ k m =k a þ 1 þ w=k b þ z=k b ; c ¼ 1=ð2 k b Þ; s ¼ z ðx wþ; y ¼ ða logðs=zþþb ðs zþ þ c ðs 2 z 2ÞÞ ð 1=ðh k3þþ

4 148 BAMBED, Vol. 35, No. 2, pp , 2007 TABLE III Summary of the obtained constants and statistical parameters in assays without phosphate initially present (w 5 0) WI CI NCI UCI MI K m (lm) 52.6 (61.5) 44.3 (62.4) Failed to converge 52.6 (61.5) 46.1 (61.6) K ic (mm) 6.9 (62.1) 9.2 (60.9) K iu (mm) k 3 (lm seg 1 (lg/2.75 ml) 1 ) ( ) ( ) ( ) ( ) SSE MSE r p n The models are explained in Fig. 1, see also the text. TABLE IV Summary of the obtained constants and statistical parameters in assays with phosphate initially present WI CI NCI UCI MI K m (lm) 58.6 (62.4) 45.5 (61.8) Failed to converge 66.7 (64.2) 45 (61.8) K ic (mm) 8.4 (60.9) 8.4 (60.9) K iu (mm) 62.0 (64.2) k 3 (lm seg 1 (lg/2.75 ml) 1 ) ( ) ( ) ( ) ( ) SSE MSE r p n The models are explained in Fig. 1, see also the text. TABLE V Sum of square errors (SSE) values for different models determined with or without initial phosphate added Conditions WI CI NCI UCI MI (only product of reaction) f.c f.c (also initially present) f.c. failed to converge. (Tables III and IV). As previously demonstrated [13] when two models, for example, A and B with pa and pb parameters are fitted (separately) to the same data set with n experimental points, the model giving the lowest SSE (sum of squares error) value should be regarded as giving the best fit and is a measure of the goodness of fit of the mathematical model to the data set. w ¼ ðsse A SSE B Þðn p B Þ ðp B p A ÞSSE B When w f-value > 0 the more complex model should be preferred, otherwise a simpler model can be applied. The f-values are obtained from statistical tables (F distribution, f(p B p A, n p B )) at the desired level of probability (f 0.95 ) knowing p B p A (number of parameters added) and n p B (degrees of freedom) for these models (Table VI). Inhibition by In spite of the fact that two different assays were used [without phosphate initially present (Table III) and with phosphate (Table IV)] the results obtained are very close. This is one of the principal advantages of this approach, i.e. to obtain kinetic constants of product inhibition without the initial addition of the product. This is of special mention when the product of reaction is a labile molecule. By studying the values presented in Tables III and IV we can conclude that CI, NCI, and UCI models have the same parameters and so it is enough to compare the SSE values (Table V). The comparison reveals that the CI model is better than any other with the same number of parameters (CI exhibit the lowest SSE value). When we compare CI versus MI model since the CI model has fewer parameters and a lower SEE, it is indubitably the best model. Thus, it is only necessary to compare w f 0.95 value to the WI versus CI model (Table VI). The w f 0.95 comparison between models (Table VI) shows that CI should be preferred when compared with the WI. The results show that the more complex model (CI model) should be preferred (w f-value > 0). This last result was expected by analysis of SSE values, but as there were different parameters in the models WI and CI it is convenient to apply this test. The constants obtained by this methodology (progress curve analysis) and by initial rates (Table VII and Fig. 2) are compared (Table VIII). The results showed lowest values for inhibition constant K ic and K m when determined with integrated Michaelis-Menten equation (Table VIII). The presence of initial product (when it is also an inhibitor) causes an evident decrease in initial velocity determinations. The decrease in the value of the parameters can be explained by theoretical errors connected with determination of initial velocities when the reaction product is an inhibitor. These errors are related to the occurrence of reaction product inhibition that prevents accurate determination of initial veloc-

5 149 TABLE VI Summary of the w f 0.95 values in assays without initial phosphate (first line) and with initial phosphate (second line) Models A/B SSE A SSE B n p A p B w p B p A n p B f 0.95 w f 0.95 WI/CI a c WI/CI b c The models are explained in Fig 1, see also the text. a Compare models without initial phosphate present. b Compare models with initial phosphate present. c These values point out that w is larger than the f-value. The f-value is f(p B p A, n p B ) at the desired level of probability (f 0.95 ). TABLE VII Initial rates determined by tangent of experimental points [S] (lm) I ¼ 0.00 lm, vo (U mg 1 ) I ¼ lm, vo (U mg 1 ) TABLE VIII Comparison of kinetic constants as estimated by two methods (integrated Michaelis-Menten equation, competitive Michaelis-Menten equation) using non linear regression Inhibitors Method a K m (lm) K ic (lm) (only product of reaction) (initially present and also product of reaction) V max (lmol min 1 mg 1 ) IMME 44.3 (62.4) 6.9 (62.1) 2.9 (60.1) CMME 61.2 (65.1) b 3.0 (60.1) IMME 45.5 (61.8) 8.4 (60.9) 3.0 (60.1) CMME 62.1 (65.6) 8.9 (62.7) 3.0 (60.1) a Methods: IMME Integrated Michaelis-Menten equation; CMME competive Michaelis-Menten equation (initial rates). b Without initial phosphate is not possible to determine this value. FIG. 2. A Lineweaver-Burk plot of alkaline phosphatase kinetic catalyzed at 37 8C in 0.1 M Tris HCl buffer, ph 9.0. D effect of phosphate (5 lm); & without inhibitor. ities by the tangent to the progress curve. Thus initial velocities estimated by a tangent to the progress curve give an inferior value than the real velocity. CONCLUSIONS Even if it is true that initial velocity methods are preferred in common laboratory work, the understanding of the integrated Michaelis-Menten equation by the students should not be considered a useless effort. Actually, this integrated equation is useful to obtain more accurate kinetic parameters. Moreover, this is a potent methodology which can be applied in more advanced studies to overcome problems of kinetic parameters determination. Acknowledgment The authors thank anonymous reviewers for valuable comments and suggestions on an earlier version of the manuscript. REFERENCES [1] N. C. Price, B. L. Newman (2000) Demonstration of the principles of enzyme-catalysed reactions using alkaline phosphatase, Biochem. Mol. Biol. Educ. 28, [2] R. G. Duggleby (2001) Quantitative analysis of the time courses of enzyme-catalyzed reactions, Methods 24, [3] B. A. Orsi, K. F. Tipton (1979) Kinetic analysis of progress curves, Methods Enzymol. 63, [4] R. G. Duggleby (1995) Analysis of enzyme progress curves by nonlinear regression, Methods Enzymol. 249, [5] J. Moreno (1985) The use of the integrated Michaelis-Menten equation in the determination of kinetic parameters, Biochem. Educ. 13, [6] M. J. Selwyn (1965) A simple test for inactivation of an enzyme during assay, Biochim. Biophys. Acta 105, [7] R. G. Duggleby (1994) Analysis of progress curves for enzymecatalyzed reactions: Application to unstable enzymes, coupled reactions and transient-state kinetics, Biochim. Biophys. Acta 1205, [8] R. M. F. Bezerra, A. A. Dias (2004) Discrimination among eight modified Michaelis-Menten kinetics models of cellulose hydrolysis with a large range of substrate/enzyme ratios: Inhibition by cellobiose, Appl. Biochem. Biotechnol. 112, [9] R. M. F. Bezerra, A. A. Dias (2005) Enzymatic kinetic of cellulose hydrolysis, Inhibition by ethanol and cellobiose, Appl. Biochem. Biotechnol. 126, [10] R. M. F. Bezerra, A. A. Dias, I. Fraga, A. N. Pereira (2006) Simultaneous ethanol and cellobiose inhibition of cellulose hydrolysis studied with integrated equations assuming constant or variable substrate concentration, Appl. Biochem. Biotechnol. 134, [11] R. M. F. Bezerra (1999) A simple process to identify the type of kinetics and estimate the kinetic parameters in the presence of two different inhibitors, J. Med. Biochem. 3, [12] M. A. Pitt, I. J. Myung (2002) When a good fit can be bad, Trends Cogn. Sci. 6, [13] B. Mannervik (1982) Regression analysis, experimental error and statistical criteria in the design and analysis of experiments for discrimination between rival kinetic models, Methods Enzymol. 87,

6 150 BAMBED, Vol. 35, No. 2, pp , 2007 APPENDIX SAS Program to Model CI with Experimental Points data; input y x z w h; cards; proc nlin method ¼ gauss newton maxiter ¼ 500; parms k m ¼1500, k a ¼13, k3 ¼ 2; model y ¼ ((1 k m /k a )*(x w) þ k m *(z/k a þ w/k a þ 1)*log(z/(z (x w))))*1/(h*k3); bounds k m > 0, k a > 0, K3 > 0; run;