where [S] = [PCP] = [H2O2]. Fitting Equation 17.15 to the data shown in Figure 17.3 provides a simple way of estimating the four constants.

The best-fit values of the four constants were estimated by minimizing the error between experimental data and model calculations. The minimization algorithm is based on a genetic algorithm, which is a stochastic optimization technique patterned after the natural selection process taking place during biological evolution. It explores all regions of the solution space using a population of individuals. Each individual represents a set of the parameters to be optimized. Initially, a population of individuals is formed randomly. The fitness of each individual is evaluated using an objective function. Upon completion of the fitness evaluation, genetic operations such as mutation and cross-

0.008

0.002

FIGURE 17.3 Experimental data (symbols) showing the variation of reaction rate V with equimolar substrate concentration ([H2O2] = [PCP]) for different initial enzyme concentrations [£0]. Also shown are the theoretical curves (lines) calculated according to Equation 17.15 with the constants of set A as given in Table 17.1.

over are applied to individuals selected according to their fitness to produce the next generation of individuals for fitness evaluation. This process continues until a near-optimum solution is found.

Because the genetic algorithm searches the entire input space in parallel, it is more robust than traditional deterministic methods and is more likely to converge to a unique global minimization. As with any artificial intelligence technique, the performance of a genetic algorithm is affected by a number of design parameters such as the initial population size, parent selection, crossover rate, mutation rate, and the number of generations. Some initial tests indicate that the genetic algorithm used in this work is robust to parameter variations, with the population size and the number of generations having the largest effect on performance. Using a population of 100, the solution successfully converged to the optimum values after 2000 to 3000 generations. All computations were conducted using the software package Matlab®. An excellent description of the implementation of genetic algorithms and their use as a problem-solving and optimization technique can be found in the book by Goldberg.42

Repetitive optimization runs reveal the existence of two distinct sets of best-fit values within the search space of 0 to 500 for each constant. These best-fit values are listed in Table 17.1. A comparison between the reaction rate profiles calculated from the two sets of constants (lines) and experimental data (symbols) is shown in Figure 17.3 and Figure 17.4. It is clear that there is generally good agreement, although at the highest [E0] examined the two theoretical curves underestimate the middle part of the experimentally measured reaction rate data. It is further observed that both sets of constants give congruent theoretical profiles. It can therefore be concluded that unique parameter estimates cannot be obtained for the simplified nonlinear model (Equation 17.15), because more than one combination of parameters can describe the same data set. In addition, the value of KPmCP

0.008

0.002

o |
[£0] = 0.13 |M | |||

• |
[£„] = 0.148 |M | |||

□ |
[£0] = 0.295 |M | |||

- m/ |
- • |
u n-iD-f • • • ( |
□ |
[£0] = 0.34 |M |

// ^ | ||||

ro |
TABLE 17.1 Best-Fits Values of Kcat, Km2°2, Kmcp, and K FIGURE 17.4 Experimental data (symbols) showing the variation of reaction rate V with equimolar substrate concentration ([H2O2] = [PCP]) for different initial enzyme concentrations [£0]. Also shown are the theoretical curves (lines) calculated according to Equation 17.15 with the constants of set B as given in Table 17.1. FIGURE 17.4 Experimental data (symbols) showing the variation of reaction rate V with equimolar substrate concentration ([H2O2] = [PCP]) for different initial enzyme concentrations [£0]. Also shown are the theoretical curves (lines) calculated according to Equation 17.15 with the constants of set B as given in Table 17.1. identified by this multiparameter estimation routine is either zero or very close to zero, indicating that it is not a significant parameter. Simultaneous retrieval of unique estimates of the four constants may require fitting the original model equation (Equation 17.14) to data obtained from experiments with different combinations of [£0], [PCP], and [H2O2]. As the fitted parameters are able to capture the general trends of the experimental data, as shown in Figure 17.3 and Figure 17.4, the best-fit constants of set A are used in the simulation studies reported in Section 17.5. FIGURE 17.5 Experimental data (symbols) showing reaction rate V as a function of [H2O2]. The initial enzyme and PCP concentrations are [£0] = 0.72 ||M and [PCP] = 1.5 mM, respectively. The theoretical curve (line) calculated from Equation 17.14 with the constants of set A (Table 17.1) is shown for comparison. FIGURE 17.5 Experimental data (symbols) showing reaction rate V as a function of [H2O2]. The initial enzyme and PCP concentrations are [£0] = 0.72 ||M and [PCP] = 1.5 mM, respectively. The theoretical curve (line) calculated from Equation 17.14 with the constants of set A (Table 17.1) is shown for comparison. ## 17.4.3 Model ValidationBecause the constants identified by the parameter estimation approach described above are not unique, it is important to assess the predictive capability of the model equation before the model is used for simulation studies. The predictive capability of Equation 17.14 can be assessed by comparing its predictions with data obtained from experiments conducted at conditions that are different from those used to generate data for parameter estimation. A set of such data (symbols) is shown in Figure 17.5, together with the theoretical curve (line) calculated from Equation 17.14 with the constants of set A. As can be seen in Figure 17.5, although the simulation does not capture the measured reaction rate data accurately, it does predict the trend very well. Given that the experimentally measured data show some scatter, for all practical purposes the agreement achieved using the rate constants of set A is quite satisfactory. The results presented in Figure 17.5 clearly show that the reaction rate is inhibited when the H2O2 concentration is higher than ~3 to 4 mM. Having developed confidence in the theoretical model after matching the simulation results with the experimental observations, the model is used to examine the inhibitory effect of H2O2 in greater detail in the next section. 17.5 MODEL SIMULATION 17.5.1 Dependence of the Reaction Rate on PCP Concentration We first examine the dependence of the reaction rate V on PCP concentration. Figure 17.6 and Figure 17.7 show the effect of PCP concentration on reaction rate for different initial enzyme and H2O2 concentrations, respectively. The reaction rate profiles shown in these two figures are calculated from Equation 17.14 with the constants of set A. Both figures show highly rectangular reaction rate profiles, indicating that the reaction rate reaches its maximum value at very low PCP concentrations (~0.1 to 0.2 mM) for a given initial enzyme or H2O2 concentration. The plateau of the profiles gives the maximum rate of reaction. The profiles shown in Figure 17.6 indicate that the maximum reaction rate increases if more enzyme is added. This is of course a typical feature of enzyme kinetics. On the other hand, the profiles in Figure 17.7 do not show a monotonic rise in maximum reaction rate with FIGURE 17.6 Theoretical profiles showing the variation of reaction rate V with [PCP] for different initial enzyme concentrations [£0]. [H2O2] = 2mM. FIGURE 17.6 Theoretical profiles showing the variation of reaction rate V with [PCP] for different initial enzyme concentrations [£0]. [H2O2] = 2mM. H2O2 concentration. The simulation results suggest that the maximum reaction rate at first increases and then decreases with increasing H2O2 concentration, reflecting the inhibitory effect of H2O2. 17.5.2 Dependence of the Reaction Rate on H2O2 Concentration In this section, we describe the dependence of the reaction rate V on H2O2 concentration, as the reaction rate has been shown to be suppressed at high H2O2 concentrations (see Figure 17.5). Representative theoretical curves calculated according to Equation 17.14 with the constants of set A 0.008 0.006 0.004 0.002 0.006 0.004 0.002 FIGURE 17.8 Theoretical profiles showing the variation of reaction rate V with [H2O2] for different initial enzyme concentrations [E0]. [PCP] = 2mM. The solid circles indicate the location of the maximum reaction rate. FIGURE 17.8 Theoretical profiles showing the variation of reaction rate V with [H2O2] for different initial enzyme concentrations [E0]. [PCP] = 2mM. The solid circles indicate the location of the maximum reaction rate. FIGURE 17.9 Theoretical profiles showing the variation of reaction rate V with [H2O2] for different [PCP]. The initial enzyme concentration is [£0] = 0.5 |M. The solid circles indicate the location of the maximum reaction rate. FIGURE 17.9 Theoretical profiles showing the variation of reaction rate V with [H2O2] for different [PCP]. The initial enzyme concentration is [£0] = 0.5 |M. The solid circles indicate the location of the maximum reaction rate. are shown in Figure 17.8 and Figure 17.9. As expected, the figures show that the reaction rate increases with increasing initial enzyme or PCP concentration at a given H2O2 concentration. Also, the reaction rate profiles in both figures have almost the same shape, although they differ in absolute values; they initially increase with increasing H2O2 concentration, reaching a maximum value before declining with a further increase in H2O2 concentration. This type of curve is commonly observed for systems in which the substrate is inhibiting. However, two major differences between the two sets of profiles are observed. First, the variation in the shape of the profiles in Figure 17.8 is directly proportional to the enzyme concentration, but although the variation in Figure 17.9 is quite pronounced in the low PCP concentration region, at higher values of PCP concentration this is no longer true. The curves lie quite close together when the PCP concentration is >0.5 mM. Second, Figure 17.8 shows that the maximum reaction rate for each profile, as indicated by the solid circles, does not vary with H2O2 concentration and it appears to occur at a H2O2 concentration of ~3 mM. By contrast, Figure 17.9 shows that the occurrence of the maximum reaction rate is governed by the H2O2 concentration when the PCP concentration is varied from 0.01 to 5 mM. The simulation results presented in Figure 17.9 suggest that for a given PCP concentration an optimum H2O2 concentration exists that gives the maximum reaction rate. As it is desirable to run the enzymatic reaction at the maximum possible reaction rate, knowledge of the relationship between the optimum H2O2 concentration and PCP concentration is of great practical interest. The next section describes how this relationship may be derived from the model equation. ## 17.5.3 Determination of Optimum H2O2 ConcentrationThe relationship between the optimum H2O2 concentration and PCP concentration may be derived from Equation 17.14. Differentiating V with respect to the H2O2 concentration and setting the derivative to zero (dV/d[H2O2] = 0) yields the following equation: where [H2O2]opt is the optimum H2O2 concentration. The solid line in Figure 17.10 is a plot of Equation 17.16 using the constants of set A. This curve gives the optimum H2O2 concentration at which the maximum reaction rate occurs for a given PCP concentration. As can be seen in Figure 17.10, when the PCP concentration is <0.5 mM, the optimum H2O2 concentration increases nonlin-early with increasing PCP concentration. When the PCP concentration is >0.5 mM the optimum H2O2 concentration approaches an asymptote and becomes independent of the PCP concentration. Figure 17.10 therefore serves as a useful guide for selecting combinations of H2O2 and PCP concentrations that would avoid enzyme inactivation by H2O2. For example, the noninactivation zone is designated by the area below the solid line, and the area above the solid line depicts the inactivation zone where the inactivated form of the enzyme, Eiii, is formed in the presence of excess H2O2, leading to reduced reaction rates. The low operational stability of HRP as a result of inactivation by H2O2 seriously impedes commercial applications of the enzyme in detoxification of waste streams and industrial organic synthesis. One approach to improving the operational stability of HRP is to maintain the concentration of H2O2 at a low level. This can be achieved using an appropriate method of H2O2 addition or generation. Examples of these methods include the stepwise addition of H2O2, feed-on-demand addition of H2O2, and in situ generation of H2O2.43 A different curve is observed when Equation 17.16 is plotted using the constants of set B. The optimum H2O2 concentration curve goes straight up from the origin to a certain H2O2 concentration and then extends horizontally from that point, as depicted by the broken line in Figure 17.10. This curve overestimates the optimum H2O2 concentration when the PCP concentration is <0.5 mM. Such a limiting form is a consequence of being set to 0. As a result, we can see from Equation 17.16 that the optimum H2O2 concentration becomes independent of the PCP concentration. Nevertheless, from Figure 17.10 it is clear that both curves predict a limiting optimum H2O2 concentration of ~2.9 mM, even though the values of K^02 and Ki used in generating the two curves are quite different, as may be seen from Table 17.1. This is evidently due to the fact that for high values of PCP concentration Equation 17.16 will approach the asymptote yKm202 Ki . The practically important conclusion from this analysis is that the effective use of mathematical models for simulation studies requires the development of sound methodologies to identify the key model parameters. It is essential to know whether the measured data are sufficient for identifying the unknown parameters and the conditions under which they are identifiable. The development of robust parameter estimation methodologies is beyond the scope of this chapter. NOMENCLATURE iii HRP ki k-i k2 k3 Kcat Ki r^H2O2 Km KPCP Km PCP PPCHD [H2O2]opt Reducing substrate Free enzyme Enzyme-substrate complex Compound I ## Enzyme-substrate complex Compound IIEnzyme-substrate complex Compound III Horseradish peroxidase Forward rate constant, L/mol-s Reverse rate constant, s-1 Rate constant, s-1 Forward rate constant, L/mol-s Reverse rate constant, s-1 Rate constant, s-1 Forward rate constant, L/mol-s Reverse rate constant, s-1 Rate constant, s-1 Rate constant, s-1 Inactivation constant, mM Constant, mM Constant, mM Pentachlorophenol 2,3,4,5,6-Pentachloro-4-pentachlorophenoxy-2,5-cyclohexadienone Reaction rate, mol/L-s Radical product of AH2 Radical product of PCP Concentration of substrate, mM Concentration of substrate, mM Concentration of free enzyme, mM Initial concentration of enzyme, mM Concentration of enzyme-substrate complex, mM Concentration of compound I, mM Concentration of enzyme-substrate complex, mM Concentration of compound II, mM Concentration of enzyme-substrate complex, mM Concentration of compound III, mM Concentration of hydrogen peroxide, mM Optimum concentration of hydrogen peroxide, mM Concentration of pentachlorophenol, mM Concentration of substrate, mM ## APPENDIXThis appendix illustrates the steps involved in deriving the reaction rate equation (Equation 17.11) from the reaction scheme given in Section 17.3.2 using the King and Altman method.41 This schematic method allows the derivation of a rate equation for essentially any enzyme mechanism in terms of the individual rate constants of the various steps in biocatalysis. Step 1. An enzymatic reaction is considered as a cyclic process that displays all the interconversions among the various enzyme forms involved. For each step in the reaction a rate constant is defined in terms of the product of the actual rate constant for that step and the concentration of free substrate involved in the step. Hence, the cyclic form of the reaction scheme given in Equations 17.6, 17.7, and 17.8 is represented by ka[PCP] ka[PCP] k5[PCP] Because the enzyme serves as a catalyst and is not consumed, the conservation equation on the enzyme yields [£0] = [E ] + [ E *] + [EJ + [ E*] + [EJ + [ E* ] (Am) Step 2. Every reaction pathway in the reaction scheme involving five arrows, by which a particular enzyme species might be formed, is constructed. The concentration of a particular enzyme species is given by the sum of the rate constant products for that enzyme form. Consideration of the above cyclic reaction scheme yields the relationships given in Table A17.1. Step 3. Equation 17.11 can now be derived from the overall reaction rate equation, Equation 17.10, using the expressions derived in Step 2 for the concentrations of the six enzyme species. Dividing Equation 17.10 by [E0] gives Substituting the enzyme conservation Equation A17.1 in the left-hand side of Equation A17.2 yields Substituting the expressions derived in Step 2 for the six enzyme species into Equation A17.3 gives [E0] _ 1 j (k-1 + k2)k6 + + (k-3 + k4 )k6 + ^ + k-5 + k6 + ^ TABLE A17.1 King-Altman Relationships for the Various Enzyme Species Enzyme Form Pathways to Form Sum of Rate Constant Products ypcp] k k3[PCP] k5[PCP] k1[H202] k3[PCP] Ei ki[H202] ki[H202] ki[H202] ypcp] k3[PCP] k_3kjk2k3k6[H2O2][PCP] k1k2k4k5k6[H2O2][PCP] k5[PCP] k2k3k4k5k6[PCP]2 Ei k TABLE A17.1 (continued) Enzyme Form Pathways to Form Sum of Rate Constant Products kJHpj k3[PCP] yH2O2] k3[PCP] ^k^fcJHAHPCP] k1k2k3k4k6[H2O2][PCP] k3[PCP] yPCP] Solving Equation A17.4 for V we find kç[Eo][H2?2][PCP] [PCP] + <(k_3 k4)k6 + k-5 + K ^[h2o2] + <1 + ^ + + ^ +1 ^[H2o2][PCP] k2 k4 Rearranging the right-hand side of Equation A17.5 we obtain k2 k4 k6 k2 k4 k6 The above equation is formally identical to Equation 17.11. |

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