A New Method to Estimate Intrinsic Parameters in the Ping-pong Bisustrate Kinetic: Application to the Oxipolymerization of Phenol

A new method for determining the intrinsic parameters of reaction in processes involving a high initial rate has been developed. The usefulness of this alternative, which consists of determining several sets of apparent parameters at different times and then extrapolating these to time zero, is demonstrated proved by the linear dependence obtained between the apparent parameters and the reaction time. The method permitted the values of the intrinsic parameters (enzyme specific activity and Michaelis-Menten constants of both substrates) to be obtained for the system under study and was checked with experimental reaction rate data for the soybean peroxidase/phenol/hydrogen peroxide system.


INTRODUCTION
Usually, the intrinsic parameters of enzymatic systems with soluble enzymes are determined from experimental data of the reaction rate in the first instants of the process. Under these conditions, the rate varies lineally with time and the initial value can be determined by extrapolating the values of the reaction rate to zero time. This is possible with most of the systems, regardless of their mechanism.
However, the reaction rates of some systems show no such lineal variation in the first instants of the reaction, which it impossible to apply this procedure. Among such systems is the oxipolymerization reaction of the phenol with hydrogen peroxide in the presence of soybean peroxidase. This process is very important, since it constitutes an alternative for the elimination of phenol from wastewaters.
Phenolic compounds are present in widely varying concentrations in the wastewaters of oil refineries and numerous other industries, including the plastics, resins, textiles, iron, steel and forestry industries [1][2][3][4][5] . So that they can be eliminated from the medium, these compounds must previously undergo some treatment that will reduce their concentrations to the limits established by environmental legislation. Until now no such treatment has shown itself to be better than any other, although the most studied has been peroxidasecatalysed oxidation by hydrogen peroxide.
In the bibliography numerous studies exist which propose the kinetic mechanisms for peroxidase/phenolic compounds systems. Most have been developed by Nicell et al. [6][7][8][9][10] and consist of complex equations which include enzymatic inhibition and de-activation terms.
Other authors [11][12][13][14] have described the behavior of the enzymatic reaction that takes place during the elimination of phenolic compounds, such as phenol, 4chlorophenol and 2-chlorophenol, in processes in which horseradish peroxidase and hydrogen peroxide intervene, and some authors [15][16][17] have proposed a bisubstrate kinetic that follows a ping-pong mechanism.
This study forms part of a project "Biotreatment of contaminating phenolic effluents" being carried out by the University of Murcia, in which two phenolic substrates (phenol and 4-chorophenol) and two enzymes (soybean and horseradish peroxidase), both in soluble and immobilized form, have been studied [18][19][20] .
The previous studies carried out with soybean peroxidase (enzyme) and phenol (substrate) enabled us to establish the optimal conditions of pH, temperature and the influence of the reagent concentrations used in the reactor on the elimination of phenol, as a previous step to the industrial application of the process. The object of the present work was to study the kinetic of the SBP/phenol system in a discontinuous reactor, decreasing the buffer load and without adding polyethylene glycol, in an attempt to confirm whether or not the above system follows a bisubstrate ping-pong kinetic.
In such enzymatic systems, the reaction rate in the first minutes is very high and its variation is not linear with the time, which makes it impossible to obtain reliable data for the values of the kinetic constants. Here, we develop an alternative method which permits us to obtain the intrinsic parameters from three sets of parameters calculated from the mean rates at 5, 10 and 15 min.

THEORY:
Because the proposed new method will be checked with the soybean peroxidase / phenol / hydrogen peroxide system, the theoretical background will be developed in terms of this system. In agreement with the bibliography [15][16][17] , a bi-substrate ping-pong mechanism is assumed for the reaction of phenol with hydrogen peroxide in the presence of soybean peroxidase. The initial rate of such a system would be given by: Where, r 0 is the initial rate of the reaction, V max is the maximum rate, [Ph]

Kinetic parameters determinations: usual method:
For a kinetic of this type, the bibliography recommends a study based on an analysis of particular situations, as described follows.

Determination of V max :
When the initial concentrations of both substrates are equal, Eq. (1) reduces to Eq. (2): This equation can be linearised, as in Eq. (3): and the value of V max is obtained from the inverse of the intercept. Also, we can obtain, from de slope, the value of the sum not the individual values of both Michaelis constants.

Determination of
From the slope of the Eq. (4) and taken into account the obtained value for V max , This equation reveals the limitations of this type of representation for the system under study. Thus, for low concentrations of hydrogen peroxide, large variations in the inverse of the initial rate are obtained. However, in these conditions, the hydrogen peroxide becomes the limiting reagent and, given the high initial rate of this system, it is consumed within the first few moments, hindering the correct measurement of the initial rate. For the same reason, the results are no better when the hydrogen peroxide concentration is much higher than that of phenol; furthermore, in this situation, the inverse values of the initial rate give rise to a cloud of very close points in the region of the small values given by the inverse of the hydrogen peroxide concentration.
For these reasons, we propose another form of linear representation, (Eq. 6), which has proved to be less sensitive to errors in measuring the initial rate: For the same reasons as mentioned above, Eq. (7) is proposed to obtain the were purchased from Sigma-Aldrich Fine Chemicals. An Eppendorf (MiniSpin) centrifuge was used to separate the precipitates in the samples and a Shimadzu UV-160 spectrophotometer was used for all absorbance measurements.
Experimental system: The experimental system used in the different assays consisted of a sleeved reactor tank of 50 cm 3 maximum capacity. The reaction mixture was continuously stirred using magnetic stirrers and Teflon-coated stir bars. Aliquots were taken at different time intervals and were mixed with catalase (1 cm 3 of sample + 1cm 3 of catalase). Each sample was treated with 0.2 cm 3 of 40 g dm 3 of (AlK(SO 4 ) 2 12H 2 O) and then centrifuged for 30 min at 10,000 rpm. The supernatant was analyzed to determine residual phenol concentration by the colorimetric method described above.
Experimental planning: The following experimental conditions, established in previous assays [20] , were maintained in the reactor: pH 7.0, temperature 30 ºC and a reaction time of 90 min. The enzyme concentrations was also established in previous works as 4·10 2  Analytical procedure: The phenol concentration was determined using a colorimetric assay as recommended by Standard Methods [21] , in which the phenolic compounds within a sample react with 2.08 mM AAP in the presence of 8.34 mM potassium ferricyanide reagent. The reaction product absorbs light at a wavelength of 505 nm with an extinction coefficient of 10.235 mM 1 cm 1 . The assay mixture consisted of 0.3 cm 3 of ferricyanide solution, 0.3 cm 3 of 4aminoantipyrine solution and 2.4 cm 3 of phenol sample.

RESULTS AND DISCUSSION
To check the validity of the proposed mechanism, the initial rate of the enzymatic reaction was represented as in Eqs. (3), (6) and (7). Because the system studied does not permit reliable extrapolation to time zero, the mean rate in the first five min of the reaction was taken an approximation of the initial rate With these approximations, the initial rate values for the three series of experiments were obtained. For each individual assay duplicate runs were made and triplicate when necessary. The experimental reaction rate values used for checking the method are the mean values; the estimated maximum error was 3.25 % and the average error was 1.12 %.
By fitting these values to Eq. (3), (6) and (7), Figures 1, 2 and 3, and the following lineal relationship was obtained, with the indicated values of R 2 :

Intrinsic kinetic parameters:
The above results confirmed that the process under study follows a bisubstrate ping-pong mechanism. We obtained approximate values for the kinetic constants, calculated from the mean rate of the reaction during the first five min. The definitive values of these constants, known as the intrinsic kinetic parameters, will depend on the availability of rate data at time zero, which is not possible in the present system because the reaction rate is not linear at the outset and cannot, therefore, be extrapolated to time zero.
As an alternative, we chose to determine three sets of parameters calculated from the mean rates at 5, 10 and 15 min, which were then extrapolated to time zero. We used the experimental data from series 1, in which the phenol and hydrogen peroxide concentrations were the same at the outset. The respective representations of the inverse of the rate versus the inverse of the initial concentration, for the mean rates at 5, 10 and 15 min, are depicted in Fig. 4.
As can be seen from Fig. 4, a good linear fit is obtained. This result provides the three pairs of kinetic parameters depicted in Table 1, where the coefficients  As regards the intrinsic values of the Michaelis constants, the linearity of the sum of both with respect to time permits that the following equations ( ) ( ) ( ) 5