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Novel methods to estimate the enantiomeric ratio and the kinetic parameters of enantiospecific enzymatic reactions
Enzyme and Microbial Technology 28 (2001) 322–328
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Novel methods to estimate the enantiomeric ratio and the kinetic parameters of enantiospecific enzymatic reactions Georgia D.C. Machado, Lucia M.C. Paiva*, Gerson F. Pinto, Enrique G. Oestreicher Departamento de Bioquı´mica, Instituto de Quı´mica, Universidade Federal do Rio de Janeiro, 5° Andar Bloco A, Centro de Tecnologia, Ilha do Funda˜o, Rio de Janeiro 21945-900, Brazil Received 7 July 2000; received in revised form 11 August 2000; accepted 11 September 2000
Abstract The Enantiomeric Ratio (E) of the enzyme, acting as specific catalysts in resolution of enantiomers, is an important parameter in the quantitative description of these chiral resolution processes. In the present work, two novel methods hereby called Method I and II, for estimating E and the kinetic parameters Km and Vm of enantiomers were developed. These methods are based upon initial rate (v) measurements using different concentrations of enantiomeric mixtures (C) with several molar fractions of the substrate (x). Both methods were tested using simulated “experimental data” and actual experimental data. Method I is easier to use than Method II but requires that one of the enantiomers is available in pure form. Method II, besides not requiring the enantiomers in pure form shown better results, as indicated by the magnitude of the standard errors of estimates. The theoretical predictions were experimentally confirmed by using the oxidation of 2-butanol and 2-pentanol catalyzed by Thermoanaerobium brockii alcohol dehydrogenase as reaction models. The parameters E, Km and Vm were estimated by Methods I and II with precision and were not significantly different from those obtained experimentally by direct estimation of E from the kinetic parameters of each enantiomer available in pure form. © 2001 Elsevier Science Inc. All rights reserved. Keywords: Enantiomeric ratio; Enzyme kinetic; Kinetic resolution; Thermoanaerobium brockii alcohol dehydrogenase
1. Introduction Chiral technology has progressed a long way. It has found applications in biochemicals, pesticides, flavor chemAbbreviations: A - Faster reacting enantiomer (Eq.1a); B - Slower reacting enantiomer (Eq.1b); C - Total concentration of substrate; x - Molar fraction; HRGC - High resolution gas chromatography; C.V.–Variation coefficient; E - Enzyme; E - Enantiomeric ratio; Eo - Total free enzyme concentration; Error (%) - Relative error; kcat- Catalytic constant; kcatACatalytic constant for A; kcatB- Catalytic constant for B; Kia - Dissociation constant of TBADH-NADP complex; KmA - Michaelis constant for A; Kmapp - Apparent value of Michaelis constant; KmB - Michaelis constant for B; KNADP - Michaelis constant for NADP; KR - Michaelis constant for enantiomer R; KS - Michaelis constant for enantiomer S; Kx –Apparent Michaelis constant when the substrate is an enantiomeric mixture of total concentration C with constant molar fraction x; TBADH - Alcohol dehydrogenase from Thermanaerobium brockii; vi- Initial velocity; vxA - Initial velocity for A at a constant molar fraction x; vA - Initial velocity for A; vB - Initial velocity for B; Vm- Maximum velocity; VmA - Maximum velocity for A; Vmapp - Apparent value of the maximum velocity; VmB - Maximum velocity for B; VmR - Maximum velocity for R; VmS - Maximum velocity for S; Vmx - Apparent value of maximum velocity for an enantiomeric mixture with a constant molar fraction x. * Corresponding author. Fax: ⫹55-21-537-3022. E-mail address: [email protected] (L.M.C. Paiva).
icals, pigments, liquids crystals, nonlinear optical materials and polymers. This is justified because enantiomers are often readily distinguished by biologic systems, and may have different pharmacological or toxicological effects [1]. This fact induced the US Food and Drug Administration (FDA) to require since 1992, some specifications regarding chiral drug products. One of the efficient methodologies to achieve the synthesis of chiral building blocks suitable for production of pharmaceuticals and other important chiral industrial compounds is based on the use of enzymes [2]. One of the best enzymatic approaches to carry out the production of enantioenriched or enantiopure compounds is based on the kinetic resolution of racemates [2]. Chen et al. [3] conveniently described the enantiospecificity by introducing the dimensionless number, E, as the enantiomeric ratio. It was expressed by E ⫽ (kcatA/KmA)/(kcatB /KmB,) where A is defined as the fast-reacting enantiomer. Since then, several methods are available in literature for estimating the E-value of enantiospecific enzymatic reactions [3–7]. However, those methods have important limitations that restrict their use. One of them is the elegant method described by Jongejan et al. [7] based upon initial rate (v) measurements, using several molar fractions of enantio-
0141-0229/01/$ – see front matter © 2001 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 1 - 0 2 2 9 ( 0 1 ) 0 0 3 6 0 - 4
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meric mixtures with a fixed concentration of substrate. The main drawback of this method is the high standard error verified for the estimates of E. In the present work two advantageous improvements over this method are presented. 2. Theoretical aspects and modeling The model considered in the present paper is based upon the assumption that the resolution is operated in a homogeneous batch system (Eqs. 1.a and 1.b). Consequently, the enantiomeric ratio can be expressed by E ⫽ (VmA/KmA)/ (VmB./KmB). This system was described [7] by Eq. 2 k2 k1 A⫹EL | ; EA O ¡ PA ⫹ E k ⫺1
(1.a)
k2 k1 B⫹EL | ; EB O ¡ PB ⫹ E k⬘⫺1
(1.b)
共kcatA /KmA 兲.x.C ⫹ 共kcatB /KmB 兲.共1 ⫺ x兲.C x ⫽ 䡠 E0 1 ⫹ x.C/KmA ⫹ C.共1 ⫺ x兲/KmB (2) where the initial reactions rates (vx) are functions of the total enzyme concentration (E0), chiral substrate concentration (C) and its molar fraction (x). Parameters kcatA, kcatB, KmA and KmB represent the kinetic parameters of the homochiral enantiomers. In this approach, the value of E is obtained by one of the two derived equations (Eqs. 3 and 4) where C is fixed. Eq. 3 should be used when the fast-reacting pure enantiomer is available. On the other hand, Eq. 4 requires the slow-reacting homochiral enantiomer. (3) 共1 ⫺ x兲 䡠 共A ⫺ B 兲 A ⫺ x ⫽ B B E䡠 ⫹ 共1 ⫺ x兲 䡠 1 ⫺ E 䡠 A A
冉冊
x ⫺ B ⫽
再
x 䡠 E 䡠 共 A ⫺ B兲 A A/ B ⫹ E ⫺ 䡠x B
冉
冊
冉 冊冎
(4)
2.1. Method I Method I is also constrained by the availability of one homochiral enantiomer, but is able to estimate the parameters Vm and Km of the other enantiomer. Eqs. 3 and 4 were rewritten for vCx, vCA and vCB indicating that these values were obtained at a fixed concentration C. In order to estimate Vm and Km of the non available enantiomer, initial rate values for each pure enantiomers (vCA and vCB) in these equations were substituted by the Michaelis-Menten expression
⫽ Vm 䡠 C/共C ⫹ Km兲
(5)
323
giving equations 6 and 7
CA ⫺ Cx ⫽ 共1 ⫺ x兲 䡠
冢 冣
再
冎
VmB 䡠 C VmA 䡠 C ⫺ KmA ⫹ C KmB ⫹ C
冦 冢 冣冧
VmB 䡠 C VmB 䡠 C Km ⫹ C Km B B⫹C E䡠 ⫹ 共1 ⫺ x兲 䡠 1 ⫺ E 䡠 VmA 䡠 C VmA 䡠 C KmA ⫹ C KmA ⫹ C
(6) x䡠E䡠
Cx ⫺ CB ⫽
冉
Vm B 䡠 C Vm A 䡠 C ⫺ Km A ⫹ C Km B ⫹ C
冊
冢 冣 关 冢 冣兴 Vm A 䡠 C Vm A 䡠 C Km A ⫹ C Km A ⫹ C ⫹ E⫺ Vm B 䡠 C Vm B 䡠 C Km B ⫹ C Km B ⫹ C
䡠x
(7)
where Vm and Km of one enantiomer must be known and vx and vA (Eq. 6) or vx and vB (Eq. 7) are determined as a function of x and C. Each one of these equations describing Method I can be fitted to the experimental data by using nonlinear regression to determine the kinetic parameters and, therefore, allow E estimation. 2.2. Method II Method II was developed to be used when none of the enantiomers is available in pure form and when it is necessary to estimate the kinetic parameters of both of them. Rewriting Eq. 2 by setting kcat.E0 ⫽ Vm and vx ⫽ Vmx.C/(C ⫹ Kx) where Kx is defined as the apparent Michaelis constant when the substrate is an enantiomeric mixture of total concentration C with constant molar fraction x and after a mathematical rationalization we get Eq. 8: C 1 Kx ⫹ ⫽ Vmx Vmx VmA VmB 䡠x⫹ 䡠 共1 ⫺ x兲 KmA KmB
冉
冊
x 1⫺x ⫹ 䡠C KmA KmB ⫹ VmB VmA 䡠x⫹ 共1 ⫺ x兲 KmA KmB
(8)
The development of this method should be separated in two main steps. 2.2.1. KmA and KmB estimation It is worth to point out that both sides of this equality (Eq. 8) are formed by summation of concentration depen-
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dent with concentration independent terms. In other words, two novel equations (Eqs. 9 and 10) arise from Eq. 8 and were expressed as: 1 1 䡠 Kx ⫽ Vm B Vm x Vm A 䡠x⫹ 䡠 共1 ⫺ x兲 Km A Km B 1⫺x x ⫹ l Km A Km B ⫽ Vm x Vm A Vm B 䡠x⫹ 䡠 共1 ⫺ x兲 Km A Km B
(9)
(10)
Substitution of 1/Vmx in the Eq. 9 for 1/Vmx expression in the Eq. 10, leads to the appropriate form of the first equation of Method II.
冉
冊
1 1 Km B ⫺ Km A ⫽ 䡠x⫹ Kx Km A 䡠 Km B Km B
(11)
When 1/Kx is plotted versus x, 1/KmB and consequently KmB can be estimated from the linear coefficient of the straight line, whereas the value of KmA can be determined from the slope of the plot and the KmB value previously calculated 2.2.2. E, VmA and VmB estimation After inversion and some mathematical manipulation of Eq. 9, a new equation can be obtained:
冉
冊
Vm B Vm A Vm B Vm x ⫽ ⫺ 䡠x⫹ Kx Km A Km B Km B
(12)
that can be rewritten by setting VmA/KmA ⫽ E.VmB /KmB. The result of this procedure is expressed by Eq. 13: Vm x Vm B Vm B ⫽ 䡠 共E ⫺ 1兲 䡠 x ⫹ Kx Km B Km B
(13)
where the E-value is obtained from the slope of the straight line arising by plotting Vmx/Kx versus x. Alternatively, least-squares regression analysis of Eq. 13 can be applied to estimate E. The VmA/KmA can then be calculated by setting x ⫽ 1 and the VmB /KmB and E values previously estimates. Vmx /Kx and 1/Kx must be obtained from fitting the Michaelis-Menten equation to some groups of initial rate experimental data for fixed values of molar fraction varying the total concentration C of the substrate.
EC 1.1.1.2) and NADP sodium salt were obtained from Sigma Chemical Co. 2-butanol and 2-pentanol (pure enantiomers) were purchased from Aldrich Chemical Co. All other chemicals used were of analytical grade. 3.2. Analyses The enantiomeric excess (ee) of all chiral compounds was determined by GC (Hewlett Packard, model HP-5890), using a Duran-50 capillary column (20 ⫻ 0.3 mm). This is -cyclodextrin column with a 2,3-di-O-methyl-6-O-t-butyldimethylsilane coating. True values of concentrations and molar fractions of enantiomeric mixtures were calculated from the ee-values. 3.3. Assay of TBADH activity TBADH activity assayed in the direction of 2-butanol oxidation was measured by following NADPH absorption at 340 nm at pH 7.8 and 37°C essentially as described by Pereira et al. [8]. 3.4. Simulated experimental data Sets of ‘perfect’(i.e. error free) initial velocity data for each variable molar fraction and/or total substrate concentration were formed by setting the values of parameters of rate equations (5, 6, 7, 11 and 13) which were utilized to calculate v. Sets of simulated data (i.e. containing error in v) for each variable were constructed from the respective “perfect” set by multiplying each value of v by a series of normally distributed pseudo-random numbers with mean of 1 and standard deviation of 0.05. Three series of 18 “pseudo-experimental” values with normally distributed error with coefficient of variation equals to 5% were computed for each equation analyzed [9]. In the special case of Method II, values of Vmx /Kx and 1/Kx were previously determined. In order to overtake this goal, three error-free values of Vmx /Kx were obtained from VmB /KB and E of Eq. 13 for three different x. The same procedure was used to produce three error-free values of 1/Kx from KmA and KmB of Eq. 11. Then, the MichaelisMenten equation was fitted [10] to the simulated error-free values of Vmx /Kx for six different values of C. In this way 18 simulated “experimental” values of v were obtained . 3.5. Actual experimental data
3. Materials and methods 3.1. Materials Thermoanaerobium brockii alcohol dehydrogenase (purified-lyophilized powder, alcohol: NADP oxidoreductase
When 2-pentanol was used as substrate, five stock solutions containing a total concentration of 20 mM of this alcohol were made up varying the molar fraction of S(⫹)-2-pentanol in the following range: 0.0; 0.5; 0.6; 0.8 and 1.0. From each one of these solutions, adequate dilutions were made in order to obtain the total concentration (C) of 2-pentanol indicated in the abscissa of Fig. 1 B. The initial
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325
were used to fit Eq. 5 thus obtaining estimates of Vmx and Kmx. Five pairs of Vmx and Kmx were in this way obtained and Eqs. 11 and 13, respectively, were fitted to these values as described in Theoretical Aspects and Modeling allowing estimates of all kinetic parameters of those equations. 3.6. Data processing Estimates of parameters and of their standard error (SE) were obtained by fitting the appropriate rate equations to data using nonlinear least-squares computer programs, as indicated in the legends of the figures and tables.
4. Results and discussion 4.1. Results with simulated data
Fig. 1. (A)–Velocity curves for oxidation of 2-butanol catalyzed by TBADH in a fixed concentration of NADP and different molar fractions of (R)-(-)-2-butanol xS: ({) 0.04; (䊐) 0.5; (F) 0.6; (Œ) 0.9; (f) 1.0. (B) -Velocity curves for oxidation of 2-pentanol catalyzed by TBADH in a fixed concentration of NADP and different molar fractions of (S)-(⫹)-2pentanol xS: (⌬) 0.0; (䊐) 0.5; (F) 0.6; () 0.8; (f) 1.0. The values (symbols) are the mean of experimentally determined initial velocities and the curves are the best fitting curves for Michaelis-Menten rate equation to the experimental data.
velocity (vx) was then determined as a function of the total concentration of 2-pentanol at a constant molar fraction (x). This procedure was repeated for each solution of different constant molar fraction. The same experimental design was followed when 2-butanol was the substrate but in this case, the molar fraction of R-(-)-2-butanol was varied in the following range: 0.04; 0.5; 0.6; 0.9 and 1.0. For utilization of Method I experimental data obtained for xA ⫽ 0.0 gave vB, i.e. the initial velocity for R-(-)-2butanol and fitting of Eq. 5 to these data gave estimates of VmB and KmB. Fitting of Eq. 7 to the values of (vx - vB) gave estimates VmA, KmA and E. For 2-pentanol, experimental data obtained for xA ⫽ 1.0 gave vA (S-(⫹)-2-pentanol) and fitting of Eq. 5 to these data gave estimates of VmA and KmA. Fitting of Eq. 6 to the values of (vA - vx) gave estimates of VmB , KmB and E. For utilization of Method II the initial velocity data obtained for each value of x and total concentration C
Since the standard errors of parameters of the equations are estimated by the nonlinear regression computer programs used [10], the precision of the procedures tested could be evaluated by the magnitude of the C.V.(%). These results are shown in Table 1 where it can be noted the low magnitude of C.V.(%) arising from utilization of Methods I and II. It must be pointed out that both methods outlined in the present paper were able to estimate values of Vm and Km and of the enantiomeric ratio with high precision and accuracy using simulated data. Comparison between the theoretical values of the kinetic parameters used to obtain the simulated values of v with the ones estimates by nonlinear-square regression, allowed determination of the relative errors (Errors (%)) for each parameter of each equation used. This initial analysis reveals that a good accuracy is present in the determination of E by both methods proposed since the relative errors obtained were of the same order of magnitude than that of the simulated experimental data (Table 1). 4.2. Experimental data Since the error structure and magnitude it is not known when experimentally obtained kinetic data are used, a reference value of the kinetic parameter and of E was needed in order to evaluate the performance of Methods I and II. For this purpose E for 2-pentanol was estimated in the traditional way from the Vm and Km values estimated for both enantiomers available in pure form by using two different kinetic approaches: Y Oxidation of 2-pentanol catalyzed by TBADH: initial velocity experiments were performed by using each pure enantiomers of this alcohol as the variable substrate and a single fixed concentration of NADP (0.5 mM). Under these experimental conditions only apparent values of the kinetic parameters Vm and Km
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Table 1 Precision and accuracy of Methods I and II evaluated with simulated data Parameter
E
Vm A
Km A
Vm B
Km B
Method/equation
Arbitrated value
Estimated value
C.V. (%)*
Error (%)*
I/Eq. 6 I/Eq. 7 II I/Eq. 6 I/Eq. 7 II I/Eq. 6 I/Eq.7 II I/Eq. 6 I/Eq.7 II I/Eq. 6 I/Eq. 7 II
3.61 2.79 8.0 0.0649 0.100 15.0 0.187 1.700 0.00240 0.0624 0.08 10.00 0.649 3.8 0.013
3.71 2.78 8.4 — 0.102 15.7 — 1.752 0.00239 0.0601 — 9.94 0.643 — 0.0128
11.7 6.6 ⬍5.0 — 11.7 7.0 — 9.1 2.5 14.2 — 6.5 8.1 — 0.06
2.8 0.4 5.0 — 2.0 4.5 — 3.1 0.1 3.7 — 0.6 0.2 — 1.5
* The Relative error (Error) and variation coefficient (CV) were calculated from simulated data of initial rate obtained as described under Material and Methods by using Methods I and II using computer programs described in Pinto, et al., 1991 [10] and the Prism™ program respectively.
could be determined by fitting the Michaelis-Menten rate equation to these data by using a nonlinear leastsquares regression method [10]. Y Oxidation of 2-pentanol catalyzed by TBADH: initial velocity experiments were now performed using a fixed ratio between the concentration of NADP and each pure enantiomers of the alcohol. Estimates of parameters were obtained by fitting steady-state sequential ordered BiBi initial rate equation [8] by using a computer program based on the Hooke-Jeeves direct search method with acceleration in distance [11]. Since the Hooke-Jeeves method [11] was used to fit the steady-state sequential ordered BiBi initial rate equation and this method does not give the asymptotic standard errors of parameters estimates, the 95% confidence limits of the estimates of parameters of the Michaelis-Menten equation were calculated according to the Student’s t-test adapted for nonlinear regression [12]. As shown in Table 2 the 95% confidence limits for Kmapp and Vmapp include the values of
the respective parameter obtained on fitting the former rate equation. This analysis reveals that both sets of kinetic parameters estimates are not significantly different and that both experimental designs used to determine the kinetic parameters, gave essentially the same results, i.e. the fixed NADP concentration used (0.5 mM) could be considered practically saturating. The apparent value of Km and Vm for R-(-)-2-butanol oxidation were also estimated (Table 2). This experimental observation was used to choose sets of initial velocity data at each molar fraction by varying the alcohol concentration at this fixed concentration of NADP (0.5 mM) and Methods I and II were tested with these sets of experimentally obtained data.
4. 3.Utilization and comparison of Methods I and II Fig. 1 shows the velocity curves obtained for oxidation of 2-butanol and 2-pentanol respectively catalyzed by
Table 2 Values of kinetic parameters estimated from the application of the reference methods by using two experimental design Parameter
(R)-(-)-2-butanol value ⫾ SE (95% confidence limits)
(R)-(-)-2-butanol value ⫾ SE (95% confidence limits)
(S)-(⫹)-2-pentanol value ⫾ SE (95% confidence limits)
Vm (mM/min) Vm app (mM/min)
— 0.067 ⫾ 0.001 (0.065 ⫺ 0.070) — 0.218 ⫾ 0.011 (0.195 ⫺ 0.240) — — —
0.09 0.080 ⫾ 0.002 (0.07 ⫺ 0.091) 3.919 3.826 ⫾ 0.227 (2.151 ⫺ 4.321) 0.010 0.045
0.127 0.113 ⫾ 0.007 (0.098 ⫺ 0.129) 1.687 1.682 ⫾ 0.217 (1.223 ⫺ 2.141) 0.034 0.056
Km (mM) Km app (mM) K NADP (mM) Kia (mM) E
3.236 ⫾ 0.512
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Table 3 Estimates of kinetic parameters derived from the application of the Method I and II on the oxidation of the 2-butanol and the 2-pentanol catalyzed by TBADH Methods
Substrate
Vm R (mM/min) value ⫾ SE
KR (mM) value ⫾ SE
Vm S (mM/min) value ⫾ SE
KS (mM) value ⫾ SE
E value ⫾ SE
[7]
2-butanol 2-pentanol 2-butanol 2-pentanol 2-butanol 2-pentanol
— — — — 0.0554 ⫾ 0.0137 0.080 ⫾ 0.011
— — — — 0.219 ⫾ 0.0175 3.687 ⫾ 0.181
— — 0.0647 ⫾ 0.0126 0.1034 ⫾ 0.0092 0.0607 ⫾ 0.0094 0.108 ⫾ 0.022
— — 0.697 ⫾ 0.123 1.435 ⫾ 0.106 0.571 ⫾ 0.0509 1.621 ⫾ 0.083
5.590 ⫾ 0.972 2.912 ⫾ 0.627 3.338 ⫾ 0.286 3.465 ⫾ 0.447 2.864 ⫾ 0.405 3.050 ⫾ 0.406
I II
TBADH obtained by using 0.5 mM of NADP and different molar fraction of each alcohol as indicated in the legends of these figures. A temperature of 37°C was chosen for performing all the experimental work described in the present article since at this temperature, TBADH presents a good enzymatic activity and its enantiospecificity is not significantly decreased [13–14]. As a matter of fact this temperature has been used to carry out most of reduction of carbonyl compounds catalyzed by this enzyme [13]. These initial velocity data used to test the performance of both novel methods described in this article were obtained with standard deviations varying between 5 and 12% of the mean value. Since, as shown with contrived data of known magnitude and error structure in Table 1, the C.V. of the kinetic parameters were of the same order of magnitude as that of these data, this fact is indicative of a good precision of these estimates. The same conclusion can be obtained with the result summarized in Table 3 where again standard errors of estimates of parameters, expressed as percent of mean value, are of the same order of magnitude as the C.V. of the actual experimental data. It is also demonstrated by the results shown in Fig. 1 that R-(-)-2-butanol and S-(⫹)-2pentanol are better substrates than their respective enantiomers and consequently were considered as the A enantiomer of the equation that express the enantiomeric ratio (Eq. 1). The dashed region pointed out with an arrow in Fig. 1 represents the set of experimental data used for the determination of E based on the method previously described [7]. Because the intrinsic limitations of this latter method, already mentioned under Theoretical Aspects and Modeling, total substrate concentration (C) has to be fixed and constant. As a consequence of this limitation, very few experimental points are available to be used in E determination. As a result, Table 3 shows that the kinetic parameters and E values were estimated by Methods I and II with a much higher precision, as indicated by the magnitude of the SE of these parameters, than by the method described in the literature [7]. It must be emphasized here that for enzymatic systems with small E values an increase of the SE of parameters is expected. This occurs because the lower the E value is, the smaller will be the difference of velocity for each molar fraction used. Kinetic parameters and enan-
tiomeric ratios estimated by using Method I with the experimental data (Eqs. 6 and 7) are shown in Table 3. As expected and confirmed by the simulation study shown in Table 1, the utilization of two variables, x and C, which represents one of the major advantages of Methods I and II, allowed higher precision in the determination of these parameters than with the method previously described by Jongejan et al. [7]. Although Method I also requires that one of the enantiomers is in the pure form, its precision (lower SE) for estimating kinetic parameters was clearly higher than that obtained with the method described in the literature [7]. Besides this significant advantage over that method [7], Method I allows, in addition, the precise determination of kinetic parameters Vm and Km. For utilization of Method II, Vmx and Kx were obtained by fitting the Michaelis-Menten equation to the initial velocity data, shown in Fig. 1 as described under Theoretical Aspects and Modeling. Fitting of Equations 11 and 13 to the values of 1/Kx and (Vmx /Kx), respectively, were used for application of Method II. The results obtained are also shown in Table 3 and they clearly suggest that Method II is more general than Method I and the one described by Jongejan et al. [7], since it does not need a pure enantiomer and the estimates of E and of kinetic parameters have a high precision. As a matter of fact, there is no significant difference in precision between both novel methods for estimating E but only Method II is efficient in estimating the kinetic parameters Vm and Km of both enantiomers. On comparing the estimates of Vm and Km for both enantiomers of 2-pentanol and for R-(-)-2butanol obtained by using Method II (Table 3) with those obtained with the traditional method considered in the present work as the reference method (Table 2), it can be seen that no statistically significant difference exists between both sets of estimates.
Acknowledgments Financial support by CNPq, FINEP, FAPERJ, CAPES and FUJB/UFRJ. We also wish to express our gratitude to
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Professor Maria da Conceic¸a˜o K.V. Ramos for running chiral GC analysis.
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[8] Pereira DA, Pinto GF, Oestreicher EG. Kinetic mechanism of the oxidation of 2-propanol catalyzed by Thermoanaerobium brockii alcohol dehydrogenase. J Biotechnology 1996;46:23–31. [9] Pinto JC, Loba˜o MW, Monteiro JL. Sequential experimental design for parameter estimation: analysis of relative deviations. Chem Eng Sci 1991;46:3129 –38. [10] Pinto GF, Oestreicher EG. A microcomputer program for fitting two-substrate enzyme rate equations. Comput Biol Med 1988;18: 135– 44. [11] Hooke R, Jeeves A. A direct search solution of numerical and statistical problems in optimization theory and practice. J Assoc Comput Mach 1961;8:212–29, In: Gordon SG, Beveridge RS, editors. Optimization: Theory and Practice. Tokyo: McGraw Hill Kogakusha, 1970. pp. 384 –9. [12] Metzler CM. Statistical Properties of Kinetic Estimates. In: Kinetic Data Analysis: Design and Analysis of Enzyme and Pharmacokinetic Experiments, L. Endrenyi (Ed.), Plenum Press, N.U., 1981:25–37. [13] Keinan E, Seth KK, Lamed R. Synthetic Applications of AlcoholDehydrogenase from Thermoanaerobium brockii. Ann NY Acad Sci 1987;501:130 – 49. [14] Yang H, Jo¨nsson Å, Wehtje E, Adlerereutz P, Mattiasson B. The Enantiomeric Purity of Alcohols Formed by Enzymatic Reduction of Ketones can be Improved by Optimisation of the Temperature and by Using a High Co-substrate Concentration. Biochim Biophys Acta 1997;1336:51– 8.
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Novel methods to estimate the enantiomeric ratio and the kinetic parameters of enantiospecific enzymatic reactions Georgia D.C. Machado, Lucia M.C. Paiva*, Gerson F. Pinto, Enrique G. Oestreicher Departamento de Bioquı´mica, Instituto de Quı´mica, Universidade Federal do Rio de Janeiro, 5° Andar Bloco A, Centro de Tecnologia, Ilha do Funda˜o, Rio de Janeiro 21945-900, Brazil Received 7 July 2000; received in revised form 11 August 2000; accepted 11 September 2000
Abstract The Enantiomeric Ratio (E) of the enzyme, acting as specific catalysts in resolution of enantiomers, is an important parameter in the quantitative description of these chiral resolution processes. In the present work, two novel methods hereby called Method I and II, for estimating E and the kinetic parameters Km and Vm of enantiomers were developed. These methods are based upon initial rate (v) measurements using different concentrations of enantiomeric mixtures (C) with several molar fractions of the substrate (x). Both methods were tested using simulated “experimental data” and actual experimental data. Method I is easier to use than Method II but requires that one of the enantiomers is available in pure form. Method II, besides not requiring the enantiomers in pure form shown better results, as indicated by the magnitude of the standard errors of estimates. The theoretical predictions were experimentally confirmed by using the oxidation of 2-butanol and 2-pentanol catalyzed by Thermoanaerobium brockii alcohol dehydrogenase as reaction models. The parameters E, Km and Vm were estimated by Methods I and II with precision and were not significantly different from those obtained experimentally by direct estimation of E from the kinetic parameters of each enantiomer available in pure form. © 2001 Elsevier Science Inc. All rights reserved. Keywords: Enantiomeric ratio; Enzyme kinetic; Kinetic resolution; Thermoanaerobium brockii alcohol dehydrogenase
1. Introduction Chiral technology has progressed a long way. It has found applications in biochemicals, pesticides, flavor chemAbbreviations: A - Faster reacting enantiomer (Eq.1a); B - Slower reacting enantiomer (Eq.1b); C - Total concentration of substrate; x - Molar fraction; HRGC - High resolution gas chromatography; C.V.–Variation coefficient; E - Enzyme; E - Enantiomeric ratio; Eo - Total free enzyme concentration; Error (%) - Relative error; kcat- Catalytic constant; kcatACatalytic constant for A; kcatB- Catalytic constant for B; Kia - Dissociation constant of TBADH-NADP complex; KmA - Michaelis constant for A; Kmapp - Apparent value of Michaelis constant; KmB - Michaelis constant for B; KNADP - Michaelis constant for NADP; KR - Michaelis constant for enantiomer R; KS - Michaelis constant for enantiomer S; Kx –Apparent Michaelis constant when the substrate is an enantiomeric mixture of total concentration C with constant molar fraction x; TBADH - Alcohol dehydrogenase from Thermanaerobium brockii; vi- Initial velocity; vxA - Initial velocity for A at a constant molar fraction x; vA - Initial velocity for A; vB - Initial velocity for B; Vm- Maximum velocity; VmA - Maximum velocity for A; Vmapp - Apparent value of the maximum velocity; VmB - Maximum velocity for B; VmR - Maximum velocity for R; VmS - Maximum velocity for S; Vmx - Apparent value of maximum velocity for an enantiomeric mixture with a constant molar fraction x. * Corresponding author. Fax: ⫹55-21-537-3022. E-mail address: [email protected] (L.M.C. Paiva).
icals, pigments, liquids crystals, nonlinear optical materials and polymers. This is justified because enantiomers are often readily distinguished by biologic systems, and may have different pharmacological or toxicological effects [1]. This fact induced the US Food and Drug Administration (FDA) to require since 1992, some specifications regarding chiral drug products. One of the efficient methodologies to achieve the synthesis of chiral building blocks suitable for production of pharmaceuticals and other important chiral industrial compounds is based on the use of enzymes [2]. One of the best enzymatic approaches to carry out the production of enantioenriched or enantiopure compounds is based on the kinetic resolution of racemates [2]. Chen et al. [3] conveniently described the enantiospecificity by introducing the dimensionless number, E, as the enantiomeric ratio. It was expressed by E ⫽ (kcatA/KmA)/(kcatB /KmB,) where A is defined as the fast-reacting enantiomer. Since then, several methods are available in literature for estimating the E-value of enantiospecific enzymatic reactions [3–7]. However, those methods have important limitations that restrict their use. One of them is the elegant method described by Jongejan et al. [7] based upon initial rate (v) measurements, using several molar fractions of enantio-
0141-0229/01/$ – see front matter © 2001 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 1 - 0 2 2 9 ( 0 1 ) 0 0 3 6 0 - 4
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meric mixtures with a fixed concentration of substrate. The main drawback of this method is the high standard error verified for the estimates of E. In the present work two advantageous improvements over this method are presented. 2. Theoretical aspects and modeling The model considered in the present paper is based upon the assumption that the resolution is operated in a homogeneous batch system (Eqs. 1.a and 1.b). Consequently, the enantiomeric ratio can be expressed by E ⫽ (VmA/KmA)/ (VmB./KmB). This system was described [7] by Eq. 2 k2 k1 A⫹EL | ; EA O ¡ PA ⫹ E k ⫺1
(1.a)
k2 k1 B⫹EL | ; EB O ¡ PB ⫹ E k⬘⫺1
(1.b)
共kcatA /KmA 兲.x.C ⫹ 共kcatB /KmB 兲.共1 ⫺ x兲.C x ⫽ 䡠 E0 1 ⫹ x.C/KmA ⫹ C.共1 ⫺ x兲/KmB (2) where the initial reactions rates (vx) are functions of the total enzyme concentration (E0), chiral substrate concentration (C) and its molar fraction (x). Parameters kcatA, kcatB, KmA and KmB represent the kinetic parameters of the homochiral enantiomers. In this approach, the value of E is obtained by one of the two derived equations (Eqs. 3 and 4) where C is fixed. Eq. 3 should be used when the fast-reacting pure enantiomer is available. On the other hand, Eq. 4 requires the slow-reacting homochiral enantiomer. (3) 共1 ⫺ x兲 䡠 共A ⫺ B 兲 A ⫺ x ⫽ B B E䡠 ⫹ 共1 ⫺ x兲 䡠 1 ⫺ E 䡠 A A
冉冊
x ⫺ B ⫽
再
x 䡠 E 䡠 共 A ⫺ B兲 A A/ B ⫹ E ⫺ 䡠x B
冉
冊
冉 冊冎
(4)
2.1. Method I Method I is also constrained by the availability of one homochiral enantiomer, but is able to estimate the parameters Vm and Km of the other enantiomer. Eqs. 3 and 4 were rewritten for vCx, vCA and vCB indicating that these values were obtained at a fixed concentration C. In order to estimate Vm and Km of the non available enantiomer, initial rate values for each pure enantiomers (vCA and vCB) in these equations were substituted by the Michaelis-Menten expression
⫽ Vm 䡠 C/共C ⫹ Km兲
(5)
323
giving equations 6 and 7
CA ⫺ Cx ⫽ 共1 ⫺ x兲 䡠
冢 冣
再
冎
VmB 䡠 C VmA 䡠 C ⫺ KmA ⫹ C KmB ⫹ C
冦 冢 冣冧
VmB 䡠 C VmB 䡠 C Km ⫹ C Km B B⫹C E䡠 ⫹ 共1 ⫺ x兲 䡠 1 ⫺ E 䡠 VmA 䡠 C VmA 䡠 C KmA ⫹ C KmA ⫹ C
(6) x䡠E䡠
Cx ⫺ CB ⫽
冉
Vm B 䡠 C Vm A 䡠 C ⫺ Km A ⫹ C Km B ⫹ C
冊
冢 冣 关 冢 冣兴 Vm A 䡠 C Vm A 䡠 C Km A ⫹ C Km A ⫹ C ⫹ E⫺ Vm B 䡠 C Vm B 䡠 C Km B ⫹ C Km B ⫹ C
䡠x
(7)
where Vm and Km of one enantiomer must be known and vx and vA (Eq. 6) or vx and vB (Eq. 7) are determined as a function of x and C. Each one of these equations describing Method I can be fitted to the experimental data by using nonlinear regression to determine the kinetic parameters and, therefore, allow E estimation. 2.2. Method II Method II was developed to be used when none of the enantiomers is available in pure form and when it is necessary to estimate the kinetic parameters of both of them. Rewriting Eq. 2 by setting kcat.E0 ⫽ Vm and vx ⫽ Vmx.C/(C ⫹ Kx) where Kx is defined as the apparent Michaelis constant when the substrate is an enantiomeric mixture of total concentration C with constant molar fraction x and after a mathematical rationalization we get Eq. 8: C 1 Kx ⫹ ⫽ Vmx Vmx VmA VmB 䡠x⫹ 䡠 共1 ⫺ x兲 KmA KmB
冉
冊
x 1⫺x ⫹ 䡠C KmA KmB ⫹ VmB VmA 䡠x⫹ 共1 ⫺ x兲 KmA KmB
(8)
The development of this method should be separated in two main steps. 2.2.1. KmA and KmB estimation It is worth to point out that both sides of this equality (Eq. 8) are formed by summation of concentration depen-
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dent with concentration independent terms. In other words, two novel equations (Eqs. 9 and 10) arise from Eq. 8 and were expressed as: 1 1 䡠 Kx ⫽ Vm B Vm x Vm A 䡠x⫹ 䡠 共1 ⫺ x兲 Km A Km B 1⫺x x ⫹ l Km A Km B ⫽ Vm x Vm A Vm B 䡠x⫹ 䡠 共1 ⫺ x兲 Km A Km B
(9)
(10)
Substitution of 1/Vmx in the Eq. 9 for 1/Vmx expression in the Eq. 10, leads to the appropriate form of the first equation of Method II.
冉
冊
1 1 Km B ⫺ Km A ⫽ 䡠x⫹ Kx Km A 䡠 Km B Km B
(11)
When 1/Kx is plotted versus x, 1/KmB and consequently KmB can be estimated from the linear coefficient of the straight line, whereas the value of KmA can be determined from the slope of the plot and the KmB value previously calculated 2.2.2. E, VmA and VmB estimation After inversion and some mathematical manipulation of Eq. 9, a new equation can be obtained:
冉
冊
Vm B Vm A Vm B Vm x ⫽ ⫺ 䡠x⫹ Kx Km A Km B Km B
(12)
that can be rewritten by setting VmA/KmA ⫽ E.VmB /KmB. The result of this procedure is expressed by Eq. 13: Vm x Vm B Vm B ⫽ 䡠 共E ⫺ 1兲 䡠 x ⫹ Kx Km B Km B
(13)
where the E-value is obtained from the slope of the straight line arising by plotting Vmx/Kx versus x. Alternatively, least-squares regression analysis of Eq. 13 can be applied to estimate E. The VmA/KmA can then be calculated by setting x ⫽ 1 and the VmB /KmB and E values previously estimates. Vmx /Kx and 1/Kx must be obtained from fitting the Michaelis-Menten equation to some groups of initial rate experimental data for fixed values of molar fraction varying the total concentration C of the substrate.
EC 1.1.1.2) and NADP sodium salt were obtained from Sigma Chemical Co. 2-butanol and 2-pentanol (pure enantiomers) were purchased from Aldrich Chemical Co. All other chemicals used were of analytical grade. 3.2. Analyses The enantiomeric excess (ee) of all chiral compounds was determined by GC (Hewlett Packard, model HP-5890), using a Duran-50 capillary column (20 ⫻ 0.3 mm). This is -cyclodextrin column with a 2,3-di-O-methyl-6-O-t-butyldimethylsilane coating. True values of concentrations and molar fractions of enantiomeric mixtures were calculated from the ee-values. 3.3. Assay of TBADH activity TBADH activity assayed in the direction of 2-butanol oxidation was measured by following NADPH absorption at 340 nm at pH 7.8 and 37°C essentially as described by Pereira et al. [8]. 3.4. Simulated experimental data Sets of ‘perfect’(i.e. error free) initial velocity data for each variable molar fraction and/or total substrate concentration were formed by setting the values of parameters of rate equations (5, 6, 7, 11 and 13) which were utilized to calculate v. Sets of simulated data (i.e. containing error in v) for each variable were constructed from the respective “perfect” set by multiplying each value of v by a series of normally distributed pseudo-random numbers with mean of 1 and standard deviation of 0.05. Three series of 18 “pseudo-experimental” values with normally distributed error with coefficient of variation equals to 5% were computed for each equation analyzed [9]. In the special case of Method II, values of Vmx /Kx and 1/Kx were previously determined. In order to overtake this goal, three error-free values of Vmx /Kx were obtained from VmB /KB and E of Eq. 13 for three different x. The same procedure was used to produce three error-free values of 1/Kx from KmA and KmB of Eq. 11. Then, the MichaelisMenten equation was fitted [10] to the simulated error-free values of Vmx /Kx for six different values of C. In this way 18 simulated “experimental” values of v were obtained . 3.5. Actual experimental data
3. Materials and methods 3.1. Materials Thermoanaerobium brockii alcohol dehydrogenase (purified-lyophilized powder, alcohol: NADP oxidoreductase
When 2-pentanol was used as substrate, five stock solutions containing a total concentration of 20 mM of this alcohol were made up varying the molar fraction of S(⫹)-2-pentanol in the following range: 0.0; 0.5; 0.6; 0.8 and 1.0. From each one of these solutions, adequate dilutions were made in order to obtain the total concentration (C) of 2-pentanol indicated in the abscissa of Fig. 1 B. The initial
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325
were used to fit Eq. 5 thus obtaining estimates of Vmx and Kmx. Five pairs of Vmx and Kmx were in this way obtained and Eqs. 11 and 13, respectively, were fitted to these values as described in Theoretical Aspects and Modeling allowing estimates of all kinetic parameters of those equations. 3.6. Data processing Estimates of parameters and of their standard error (SE) were obtained by fitting the appropriate rate equations to data using nonlinear least-squares computer programs, as indicated in the legends of the figures and tables.
4. Results and discussion 4.1. Results with simulated data
Fig. 1. (A)–Velocity curves for oxidation of 2-butanol catalyzed by TBADH in a fixed concentration of NADP and different molar fractions of (R)-(-)-2-butanol xS: ({) 0.04; (䊐) 0.5; (F) 0.6; (Œ) 0.9; (f) 1.0. (B) -Velocity curves for oxidation of 2-pentanol catalyzed by TBADH in a fixed concentration of NADP and different molar fractions of (S)-(⫹)-2pentanol xS: (⌬) 0.0; (䊐) 0.5; (F) 0.6; () 0.8; (f) 1.0. The values (symbols) are the mean of experimentally determined initial velocities and the curves are the best fitting curves for Michaelis-Menten rate equation to the experimental data.
velocity (vx) was then determined as a function of the total concentration of 2-pentanol at a constant molar fraction (x). This procedure was repeated for each solution of different constant molar fraction. The same experimental design was followed when 2-butanol was the substrate but in this case, the molar fraction of R-(-)-2-butanol was varied in the following range: 0.04; 0.5; 0.6; 0.9 and 1.0. For utilization of Method I experimental data obtained for xA ⫽ 0.0 gave vB, i.e. the initial velocity for R-(-)-2butanol and fitting of Eq. 5 to these data gave estimates of VmB and KmB. Fitting of Eq. 7 to the values of (vx - vB) gave estimates VmA, KmA and E. For 2-pentanol, experimental data obtained for xA ⫽ 1.0 gave vA (S-(⫹)-2-pentanol) and fitting of Eq. 5 to these data gave estimates of VmA and KmA. Fitting of Eq. 6 to the values of (vA - vx) gave estimates of VmB , KmB and E. For utilization of Method II the initial velocity data obtained for each value of x and total concentration C
Since the standard errors of parameters of the equations are estimated by the nonlinear regression computer programs used [10], the precision of the procedures tested could be evaluated by the magnitude of the C.V.(%). These results are shown in Table 1 where it can be noted the low magnitude of C.V.(%) arising from utilization of Methods I and II. It must be pointed out that both methods outlined in the present paper were able to estimate values of Vm and Km and of the enantiomeric ratio with high precision and accuracy using simulated data. Comparison between the theoretical values of the kinetic parameters used to obtain the simulated values of v with the ones estimates by nonlinear-square regression, allowed determination of the relative errors (Errors (%)) for each parameter of each equation used. This initial analysis reveals that a good accuracy is present in the determination of E by both methods proposed since the relative errors obtained were of the same order of magnitude than that of the simulated experimental data (Table 1). 4.2. Experimental data Since the error structure and magnitude it is not known when experimentally obtained kinetic data are used, a reference value of the kinetic parameter and of E was needed in order to evaluate the performance of Methods I and II. For this purpose E for 2-pentanol was estimated in the traditional way from the Vm and Km values estimated for both enantiomers available in pure form by using two different kinetic approaches: Y Oxidation of 2-pentanol catalyzed by TBADH: initial velocity experiments were performed by using each pure enantiomers of this alcohol as the variable substrate and a single fixed concentration of NADP (0.5 mM). Under these experimental conditions only apparent values of the kinetic parameters Vm and Km
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Table 1 Precision and accuracy of Methods I and II evaluated with simulated data Parameter
E
Vm A
Km A
Vm B
Km B
Method/equation
Arbitrated value
Estimated value
C.V. (%)*
Error (%)*
I/Eq. 6 I/Eq. 7 II I/Eq. 6 I/Eq. 7 II I/Eq. 6 I/Eq.7 II I/Eq. 6 I/Eq.7 II I/Eq. 6 I/Eq. 7 II
3.61 2.79 8.0 0.0649 0.100 15.0 0.187 1.700 0.00240 0.0624 0.08 10.00 0.649 3.8 0.013
3.71 2.78 8.4 — 0.102 15.7 — 1.752 0.00239 0.0601 — 9.94 0.643 — 0.0128
11.7 6.6 ⬍5.0 — 11.7 7.0 — 9.1 2.5 14.2 — 6.5 8.1 — 0.06
2.8 0.4 5.0 — 2.0 4.5 — 3.1 0.1 3.7 — 0.6 0.2 — 1.5
* The Relative error (Error) and variation coefficient (CV) were calculated from simulated data of initial rate obtained as described under Material and Methods by using Methods I and II using computer programs described in Pinto, et al., 1991 [10] and the Prism™ program respectively.
could be determined by fitting the Michaelis-Menten rate equation to these data by using a nonlinear leastsquares regression method [10]. Y Oxidation of 2-pentanol catalyzed by TBADH: initial velocity experiments were now performed using a fixed ratio between the concentration of NADP and each pure enantiomers of the alcohol. Estimates of parameters were obtained by fitting steady-state sequential ordered BiBi initial rate equation [8] by using a computer program based on the Hooke-Jeeves direct search method with acceleration in distance [11]. Since the Hooke-Jeeves method [11] was used to fit the steady-state sequential ordered BiBi initial rate equation and this method does not give the asymptotic standard errors of parameters estimates, the 95% confidence limits of the estimates of parameters of the Michaelis-Menten equation were calculated according to the Student’s t-test adapted for nonlinear regression [12]. As shown in Table 2 the 95% confidence limits for Kmapp and Vmapp include the values of
the respective parameter obtained on fitting the former rate equation. This analysis reveals that both sets of kinetic parameters estimates are not significantly different and that both experimental designs used to determine the kinetic parameters, gave essentially the same results, i.e. the fixed NADP concentration used (0.5 mM) could be considered practically saturating. The apparent value of Km and Vm for R-(-)-2-butanol oxidation were also estimated (Table 2). This experimental observation was used to choose sets of initial velocity data at each molar fraction by varying the alcohol concentration at this fixed concentration of NADP (0.5 mM) and Methods I and II were tested with these sets of experimentally obtained data.
4. 3.Utilization and comparison of Methods I and II Fig. 1 shows the velocity curves obtained for oxidation of 2-butanol and 2-pentanol respectively catalyzed by
Table 2 Values of kinetic parameters estimated from the application of the reference methods by using two experimental design Parameter
(R)-(-)-2-butanol value ⫾ SE (95% confidence limits)
(R)-(-)-2-butanol value ⫾ SE (95% confidence limits)
(S)-(⫹)-2-pentanol value ⫾ SE (95% confidence limits)
Vm (mM/min) Vm app (mM/min)
— 0.067 ⫾ 0.001 (0.065 ⫺ 0.070) — 0.218 ⫾ 0.011 (0.195 ⫺ 0.240) — — —
0.09 0.080 ⫾ 0.002 (0.07 ⫺ 0.091) 3.919 3.826 ⫾ 0.227 (2.151 ⫺ 4.321) 0.010 0.045
0.127 0.113 ⫾ 0.007 (0.098 ⫺ 0.129) 1.687 1.682 ⫾ 0.217 (1.223 ⫺ 2.141) 0.034 0.056
Km (mM) Km app (mM) K NADP (mM) Kia (mM) E
3.236 ⫾ 0.512
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327
Table 3 Estimates of kinetic parameters derived from the application of the Method I and II on the oxidation of the 2-butanol and the 2-pentanol catalyzed by TBADH Methods
Substrate
Vm R (mM/min) value ⫾ SE
KR (mM) value ⫾ SE
Vm S (mM/min) value ⫾ SE
KS (mM) value ⫾ SE
E value ⫾ SE
[7]
2-butanol 2-pentanol 2-butanol 2-pentanol 2-butanol 2-pentanol
— — — — 0.0554 ⫾ 0.0137 0.080 ⫾ 0.011
— — — — 0.219 ⫾ 0.0175 3.687 ⫾ 0.181
— — 0.0647 ⫾ 0.0126 0.1034 ⫾ 0.0092 0.0607 ⫾ 0.0094 0.108 ⫾ 0.022
— — 0.697 ⫾ 0.123 1.435 ⫾ 0.106 0.571 ⫾ 0.0509 1.621 ⫾ 0.083
5.590 ⫾ 0.972 2.912 ⫾ 0.627 3.338 ⫾ 0.286 3.465 ⫾ 0.447 2.864 ⫾ 0.405 3.050 ⫾ 0.406
I II
TBADH obtained by using 0.5 mM of NADP and different molar fraction of each alcohol as indicated in the legends of these figures. A temperature of 37°C was chosen for performing all the experimental work described in the present article since at this temperature, TBADH presents a good enzymatic activity and its enantiospecificity is not significantly decreased [13–14]. As a matter of fact this temperature has been used to carry out most of reduction of carbonyl compounds catalyzed by this enzyme [13]. These initial velocity data used to test the performance of both novel methods described in this article were obtained with standard deviations varying between 5 and 12% of the mean value. Since, as shown with contrived data of known magnitude and error structure in Table 1, the C.V. of the kinetic parameters were of the same order of magnitude as that of these data, this fact is indicative of a good precision of these estimates. The same conclusion can be obtained with the result summarized in Table 3 where again standard errors of estimates of parameters, expressed as percent of mean value, are of the same order of magnitude as the C.V. of the actual experimental data. It is also demonstrated by the results shown in Fig. 1 that R-(-)-2-butanol and S-(⫹)-2pentanol are better substrates than their respective enantiomers and consequently were considered as the A enantiomer of the equation that express the enantiomeric ratio (Eq. 1). The dashed region pointed out with an arrow in Fig. 1 represents the set of experimental data used for the determination of E based on the method previously described [7]. Because the intrinsic limitations of this latter method, already mentioned under Theoretical Aspects and Modeling, total substrate concentration (C) has to be fixed and constant. As a consequence of this limitation, very few experimental points are available to be used in E determination. As a result, Table 3 shows that the kinetic parameters and E values were estimated by Methods I and II with a much higher precision, as indicated by the magnitude of the SE of these parameters, than by the method described in the literature [7]. It must be emphasized here that for enzymatic systems with small E values an increase of the SE of parameters is expected. This occurs because the lower the E value is, the smaller will be the difference of velocity for each molar fraction used. Kinetic parameters and enan-
tiomeric ratios estimated by using Method I with the experimental data (Eqs. 6 and 7) are shown in Table 3. As expected and confirmed by the simulation study shown in Table 1, the utilization of two variables, x and C, which represents one of the major advantages of Methods I and II, allowed higher precision in the determination of these parameters than with the method previously described by Jongejan et al. [7]. Although Method I also requires that one of the enantiomers is in the pure form, its precision (lower SE) for estimating kinetic parameters was clearly higher than that obtained with the method described in the literature [7]. Besides this significant advantage over that method [7], Method I allows, in addition, the precise determination of kinetic parameters Vm and Km. For utilization of Method II, Vmx and Kx were obtained by fitting the Michaelis-Menten equation to the initial velocity data, shown in Fig. 1 as described under Theoretical Aspects and Modeling. Fitting of Equations 11 and 13 to the values of 1/Kx and (Vmx /Kx), respectively, were used for application of Method II. The results obtained are also shown in Table 3 and they clearly suggest that Method II is more general than Method I and the one described by Jongejan et al. [7], since it does not need a pure enantiomer and the estimates of E and of kinetic parameters have a high precision. As a matter of fact, there is no significant difference in precision between both novel methods for estimating E but only Method II is efficient in estimating the kinetic parameters Vm and Km of both enantiomers. On comparing the estimates of Vm and Km for both enantiomers of 2-pentanol and for R-(-)-2butanol obtained by using Method II (Table 3) with those obtained with the traditional method considered in the present work as the reference method (Table 2), it can be seen that no statistically significant difference exists between both sets of estimates.
Acknowledgments Financial support by CNPq, FINEP, FAPERJ, CAPES and FUJB/UFRJ. We also wish to express our gratitude to
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Professor Maria da Conceic¸a˜o K.V. Ramos for running chiral GC analysis.
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