Determination of the Kinetic Constants of Glucose-6-phosphate l-epimerase by Non-Linear Optimization

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1 Eur. J. Biochem. 50,19-2 (1975) Determination of the Kinetic Constants of Glucose-6-phosphate l-epimerase by Non-Linear Optimization Edwin M. CHANCE, Benno HESS, Theodor PLESSER, and Bernd WURSTER Biochemistry Department, University College London, and Max-Planck-Institut fur Ernahrungsphysiologie, Dortmund (Received July 15, 197) 1. The overall kinetic constants of the reversible anomerisation of D-glucopyranose 6-phosphate from c( to 0 non-enzymatically as well as catalysed by glucose-6-phosphate l-epimerase are determined by application of a novel computerized non-linear optimization technique. 2. The non-enzymic rate constants for the anomerisation of D-glucopyranose 6-phosphate from CI to p and reverse are and s-l, respectively. The Michaelis constants of the enzymic reaction are K; 1 pm and K i 55.5 pm with the turnover numbers of 1950 s-l and 6 s-' for the conversion of D-glucopyranose 6-phosphate from a to p and reverse, respectively. Glucose-6-phosphate 1-epimerase, an enzyme catalysing the reversible anomerization of D-glucopyranose 6-phosphate from a to p, was discovered in bakers' yeast [I] and subsequently purified and characterized [2]. Similar enzymes were recently described to occur in Escherichia coli, Rhodotorula gracilis and potato tubers [3]. The test systems described for the determination of the anomerization catalysed by glucose-6-phosphate 1 -epimerase of D-glucopyranose 6-phosphate from a to [l] are limited to small concentrations of the substrate D-glucose 6-phosphate, because of the interference of non-enzymic anomerization of the two substrates. Therefore, only pseudo first-order reaction velocity constants (activity constants) which correspond to the ratio V/K,,,, but neither the maximum velocities (turnover numbers) nor the Michaelis constants could be evaluated [2]. The determination of these enzyme parameters from the progress curves of the reaction system has been achieved by application of computerized optimization techniques [], the results of which are reported in this paper. Abbreviation. SD,,, standard deviation of natural logarithm of the optimized prameter. Enzvmes. Glucose-6-phosphate dehydrogenase (EC ) ; gluconolactonase (EC ). MATERIALS AND METHODS Materials All chemicals used were p.a. grade and purchased from E. Merck AG (Darmstadt). D-Glucose 6-phosphate, NADP' and glucose-6-phosphate dehydrogenase from yeast were bought from Boehringer Mannheim GmbH (Mannheim, Germany). Methods Glucose-6-phosphate l-epimerase from bakers' yeast was purified according to [2]. From a partially purified preparation of glucose-6-phosphate 1 -epimerase, the enzyme 6-phosphogluconolactonase [5,6] was separated by isoelectric focusing using an LKB 101 column (110 ml) with a 2% ampholine ph 3-7 gradient stabilized by sucrose ( "C and 500 V). Since the isoelectric point of glucosed-phosphate l-epimerase is 5. [l] and the isoelectric point of 6-phosphogluconolactonase is about.7 (B. Wurster and B. Hess, unpublished experiments) a clear separation of the two enzymes was obtained. Enzymes were freed from ammonium sulphate by dialysis. The determination of substrate concentrations and enzyme activities was performed in the following buffer : 50 mm imidazoie-hc1,50 mm KCl,

2 20 Non-Linear Optimization of Glucose-6-phosphate I-Epimerase mm MgSO,, ph 7.6 at 25"C, using an Eppendorf photometer connected to a recorder (A = 366 nm). The activity of glucose-6-phosphate dehydrogenase was determined as described in [7]. The activity constant of 6-phosphogluconolactonase was determined in the test system described by Horecker and Smyrniotis []. The spontaneous anomerization of D-glucopyranose 6-phosphate from a to fi, and that catalysed by glucose-6-phosphate 1 -epimerase, was followed in test system 2 described in [l] in the presence of 6-phosphogluconolactonase : a - D - Glucopyranose 6 - phosphate L spontaneous and /or glucose - 6- phosphate 1 - epimerase formation of P-D-glucopyranose 6-phosphate from the a form. At ph 7.6 and 25 "C the apparent equilibrium constant of the reaction catalysed by glucose-6-phosphate dehydrogenase is only about 32, with respect to /?-D-glucopyranose 6-phosphate [7]. The quasiirreversible treatment of this reaction is justified at high concentration of NADP' and, on addition of 6-phosphogluconolactonase, which accelerates the hydrolysis of one product of the glucose-6-phosphate dehydrogenase reaction. The spontaneous and enzymatically catalysed anomerization of the a and /I form of D-glucopyranose 6-phosphate can be expressed by the following set of chemical equations, which were supplied to the computer: - D - Glucopyranose 6 - phosphate 1 NADP' glucose - 6- phosphate dehydrogenase NADPH A+E 2 C b 6- Phosphogluconic acid 6 - lactone t spontaneous and 6 - Phosphogluconate Initial Conditions 6- p hosphogluconolactonase Approximately 57 pm or 230 pm D-glucose 6-phosphate, 2 mm NADP', glucose-6-phosphate 1 -epimerase varied, 30 U/ml glucose-6-phosphate dehydrogenase, activity constant of 6-phosphogluconolactonase 0.6 s-'. Actual initial conditions of D-glucose 6-phosphate were measured optically. The reactions are initiated by addition of equilibrated D-glucose 6-phosphate and recorded until the reaction is complete. Since glucose- 6-phosphate dehydrogenase selectively reacts with fi-d-glucopyranose 6-phosphate (for reference see [9]), -D-glucopyranose 6-phosphate is oxidized at- high activities of this enzyme in a fast initial reaction; the succeeding slower reaction represents the rate of where A is a-d-glucopyranose 6-phosphate, B is j?-d-glucopyranose 6-phosphate, E is glucose-6-phosphate 1-epimerase, C is Michaelis-Menten complex of glucose-6-phosphate 1-epimerase, N is NADP+, NH is NADPH, P is 6-phosphogluconate. The first equation describes the spontaneous anomerization, and the second and third equation describe the enzymatically catalysed anomerization of D-glucopyranose 6-phosphate from M. to p. The fourth equation describes the oxidation of fi-d-glucopyranose 6-phosphate in the glucose-6-phosphate dehydrogenase reaction, simplified as reaction of second order and lumped together with the reaction describing the hydrolysis of 6-phosphogluconic acid d-lactone occurring spontaneously and catalysed by 6-phosphogluconolactonase, a treatment which was justified as described below. The K, as well as V for the reaction in the forward (a) and reversed (/3) direction are defined as follows: V" = k, x [E,], VB= kx[e,].

3 E. M. Chance, B. Hess, T. Plesser, and B. Wurster 2 1 The following equilibrium relationships must be observed : where Glcp-6-P is glucopyranose 6-phosphate; K"' is equilibrium constant of the glucose-6-phosphate dehydrogenase reaction, and Kw is equilibrium constant of the 6-phosphogluconolactonase reaction. The concentration of glucose-6-phosphate l-epimerase (E) was computed from the specific activity constant of min-' x mg-' [2] and from the molecular weight of this enzyme of [I]. The equilibrium constant K"' = 32 (ph 7.6,25 "C [7]) and K" = lop3 [lo]. Experimental data were taken from the progress curves recorded with the Eppendorf recorder using an absorbance scale of 1 unit = 20 cm with a time basis of 60 cm per min. Computer Techniques Simulation and optimizations reported in this paper were carried out by similar methods to those reported by Curtis and Chance [ll]. Since that time the computer program has been modified to calculate the nonlinear covariance of the systems of rate constants to be optimized. Furthermore, the optimization algorithm has been replaced by one that produces rapid convergence of the sum of squares for non-linear systems of equations [12]. Because the rate constants need to be varied over a large range of values, the natural log of the unknown constant is varied and as a consequence, the standard deviation of the natural log of the unknown constant (SD,,) is calculated from the non-linear covariance. This parameter is a quantitative measure of how accurately the constant has been determined by optimization. If the SD,, is less than 0.12, the constant has a well determined minimum in multidimensional parameter space. Because of the linearity of logarithms less than 0.1, a constant whose SDI, lies below this value has a standard deviation of & SD,,. In the range of , due to divergence from linearity, the constant has a standard deviation of approximately 15%. For values above 0.12, the standard deviation increases in value exponentially, so that the constants which are not accurately determined by the experimental conditions can be easily identified. If the correlation coefficient between two unknown constants is greater than 0.9, the ratio of these constants may be accurately determined. If the correlation Table 1. initial concentration of glucose 6-phosphate and glucose-6-phosphate I-epimerase Experiment Total glucose Enzyme no. 6-phosphate PM nm coefficient is less than the product may be well determined. If, for example, the correlation coefficient between k, and k, + e.g. the forward and backward rate constants for a reaction, is greater than 0.9 and although the SD1, for k, and k,+, individually is greater than 0.12, the values for k, and k,+, are poorly determined but their ratio (k,/k, +,) may be correctly estimated. The SDI, of the ratio is calculated by holding either k, or k, + constant and recalculating the covariance. If k,+, were held constant and the covariance recalculated, the SDI, of k, would be the measure of the accuracy of the value of the equilibrium constant of the reaction k,/k, + Computer facilities used were IBM 3601, , RESULTS AND DISCUSSION The first optimization problem was done with the non-enzymic reaction with the data from Expts 1,, 7. The initial conditions from all experiments are given in Table 1. The results of this computation (see Table 2A) show that kl and k2 as well as the equilibrium constant are well determined. The value of k, and k2 determine the initial distribution of the a and B form of glucose 6-phosphate and were treated as unknowns in the non-enzymic problem. The rate equation solved was : L'= -- [B1 - kl [A]- k2 [B]- k7 [B] [N] dt (5) The values of k, and k2 were held constant in the subsequent optimization problems in which the curves

4 d[b1 22 Non-Linear Optimization of Glucose-6-phosphate 1-Epimerase Table 2. Results of the optimutizution problems (5 = standard deviation of the overall fit; n = the number of the degree of freedom equation was substituted into Eqn (6) yielding the following rate equation: Parameter Value SDI" A. Non-enzymic epimerisation kl s-' k s-' K kllkz = ( n 3 B. Enzymic epimerisution using u linear rute law kjk; 11.5 pm-' x s-l D n 2 C. Enzymic epimerisution using the Michaelis-Menten rute law k s-' % 172 pm k,lr 13. pm-' x s-' n 3 D. Enzymic epimerisation using differential equations derived from the chemical reuction mechanism k pm-' x s-l kik pm k s-' ( n obtained in the presence of two epimerase concentrations were analyzed (Expts 2, 3, 5,6, - 1 l), and thus, the influence of the non-enzymic anomerization on the enzymatically catalysed anomerization can be accurately accounted for. The next optimization problem was carried out with the enzymic data obtained in the presence of two enzyme concentrations and various initial conditions, using different equations for the rate of production of the form of glucose 6-phosphate. First, a pseudo first-order relationship was tested and the following rate equation solved : t'= ~ = kl [A] - k2 [B] dt k5 [EO1 [B] Rl As shown in Table 2B the ratio k5/k;r is well determined but the overall standard deviation (n) is larger than in problem A. In order to analyze the validity of the first model, the Michaelis-Menten If both mechanisms [Eqns (6) and (7)] are equally valid, then the value for Q should be about the same, if the number of degrees of freedom is the same. As shown in the results given in Table2C k5 and K; as well as the ratio k5/k; are well determined, and the value for Q is approximately half of the value obtained by optimization of the Eqn (6). A comparison of the results of the analysis of both mechanisms showed that K: and I/ are best estimated from the model of Eqn (7), although Eqn (6) gives a reasonable order of magnitude as indicated in the difference between the values in both problems. In order to determine whether the reverse reaction of epimerase is significant for the determination of the kinetic constants, a reversible reaction mechanism was used as a model. The relationship describing the reaction velocity was: u=-- [B1 - kl [A] - k2 [B] + k5 [C] dt - k6 PI [El - k7 PI "I + 3.2x k7 (7) [PI [NH]. Optimization results in the data summarized in Table 2D demonstrating a good determination for all constants given. The data allow the calculation of K;. resulting in a value of 1 pm, slightly lower than the value obtained by using Eqn (7). A comparison between the turnover numbers in both models (see Table 2C and D) show significant differences. The evidence, that the back reaction is significant, although small, comes from the observation that both the turnover numbers with an Q of are significantly different. Also, since the number of degrees of freedom is taken into account when calculating 0, the lower value of 0 in problem D is again important, indicating a small contribution of the back reaction. Examination of the individual curves and groups of curves in problems C and D yields constants whose values were ill-determined (SD,, > 0.15). A reasonable determination of the constants was achieved only when the complete system of eight experimental curves was optimized. The distribution

~ equilibrium E. M. Chance, B. Hess, T. Plesser, and B. Wurster 23 Table 3. Individual rate constants for the kinetic mechanism Con- Data Dimensions Remarks stants 0.065 0.039 16.6 6 1950 3.2 1 13.

5 ~ equilibrium E. M. Chance, B. Hess, T. Plesser, and B. Wurster 23 Table 3. Individual rate constants for the kinetic mechanism Con- Data Dimensions Remarks stants Expts 1., 7 calculated from k/k3 calculated from the constant calculated from k3, k and ks Expts 2, 5,, 10 Expts 3,6,9, 11 calculated from k, ks, k6 Table. Turnover numbers and K, of various aldose l-epimerases and ~-glucose-6-phosphate I-epimerase Source f Turnover Km Refernumber ence "C s-l mm E. coli ~ lo [I31 Bovine kidney x lo [1] Green pepper x lo [15] Bakers' yeast x lo3 0.1 this Paper of residuals of the individual curves is at present under investigation in order to determine the nature of any deviation from a Gaussian distribution of experimental error. Preliminary analysis however, showed that the magnitude of the reported parameters are not grossly affected by the errors implicit in the experimental curves. The optimization yielded the values for the individual rate constants of the anomerization catalysed by glucose-6-phosphate 1-epimerase as well as V" and K; as summarized in Table 3. Using the Haldane equation the VB and K& can be computed as shown in the table. Using the equilibrium constant K', k6 can also be obtained from the equation k3 x k5 k, = 1.69 x k as shown in Table 3. A comparison of the turnover number and K,,, of glucose-6-phosphate 1-epimerase from bakers' yeast obtained by computerized optimization with the graphically determined kinetic constants of related enzymes, namely aldose 1-epimerases (mutarotases) indicates differences as summarized in Table. In general, the enzymes seem to operate with a turnover number roughly one magnitude higher than the enzyme described here. In addition, the K,,, of the three enzymes given in Table, are roughly two orders of magnitude higher. With the lower accuracy of the graphical first-order linearisation technique applied, the difference between the Michaelis constants is out of the range, which could simply be explained by deviation from non-linearity in the first-order treatment (see above). Thus, the difference in the Michaelis constants should be due to different substrate affinities of the various enzymes, and it is suggested that the high substrate specificity of glucose- 6-phosphate 1-epimerase towards a and forms of D-glucopyranose 6-phosphate is reflected by a relatively low K, in comparison to the substrate specificity of the aldose 1-epimerases given in Table. It is interesting to note that under physiological conditions of gluconeogenesis the levels of D-glucose 6-phosphate in yeast are in the range of 353 pm and about twice the concentration of the Michaelis constant [16]. Earlier [2,9] we suggested that the biological function of glucose-6-phosphate 1 -epimerase seems to be restricted to physiological situations where glucose-6-phosphate isomerase is inhibited. Indeed, under aerobic conditions this situation is given in the starving glycogen-catabolising yeast cell, where low concentrations of % pm D-glucose 6-phosphate corresponding to rc pm a-d-glucopyranose 6-phosphate are found experimentally [2]. The upper level would be approximately one third of Michaelis constant and allow a rapid reaction to any change of the level of X-Dglucose 6-phosphate. The authors are indebted to the Atlas Computer Laboratory and the University College London Computer Centre for Computer facilities. Furthermore, a guest research fellowship to one of us (E. M.C.) of the Max-Planck-Gesellschaft is gratefully acknowledged. REFERENCES 1. Wurster, B. & Hess, B. (1972) FEBS Lett. 23, Wurster, B. & Hess, B. (197) Hoppe-Seyler's Z. Phjsiol. ~~ Chem. 355, Wurster. B. & Hess. B. (1973)., FEBS Lett Hess, B.; Chance, E: M., Busse, H. & Wurster, B. (1972) in Analysis and Simulation of Biochemical Systems (Hemker, H. C. & Hess, B., eds) vol. 25, pp , North-Holland Amsterdam/American Elsevier, New York. 5. Brodie, A. F. & Lipman, F. (1955) J. Biol. Chem. 212,

6 2 E. M. Chance, B. Hess, T. Plesser, and B. Wurster : Non-Linear Optimization of Glucose-6-phosphate 1-Epimerdse 6. Kawada, M., Kagawa, Y., Takiguchi, H. & Shimazono, N. (1962) Biochim. Biophys. Acta, 57, Wurster, B. & Hess, B. (2973) Hoppe-Seyler s Z. Physiol. Chem. 35,01-20,. Horecker, B. L. & Smyrniotis, P. Z. (1953) Biochem. Biophys. Acts, 12, Wurster,B. & Hess,B. (197) FEBS Lett. 0S, Mahler. H. R. & Cordes, E. H. (1961) in Biological Chemistry, p. 50, Harper & Row, New York. 11. Curtis, A. R. & Chance, E. M. (1972) in Analysis and Simulation of Biochemical Systems (Hemker, H. C. & Hess, B., eds) vol. 25, pp , North-Holland Amsterdam/American Elsevier, New York. 12. Powell, M. J. D. (1965) Comput. J. 7, Hucho, F. & Wallenfels, K. (1971) Eur. J. Biochem. 23, Bailey, J. M., Fishman, P. H. & Pentchev, P. G. (1969) J. Biol. Cham. 2, Bailey, J. M., Fishman, P. H. & Pentchev, P. G. (1967) J. Biol. Chem. 22, Barwell, C. J. & Hess, B. (1971) FEBS Lett. 19, 1-. E. M. Chance, Department of Biochemistry, University College London, Gower Street, London, Great Britain WCI 6BT B. Hess, T. Plesser, and B. Wurster, Max-Planck-Institut fur Ernahrungsphysiologie, D-600 Dortmund, Rheinlanddamm 201, Federal Republic of Germany

A Kinetic Study of Glucose-6-phosphate Dehydrogenase

A Kinetic Study of Glucose-6-phosphate Dehydrogenase (Received for publication, September 10, 1975) MOHAMMED. KANJ, MYRON L. TOEWS, AND W. ROBERT CARPER* From the Department of Chemistry, Wichita State