Enzyme numerical solution

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

It should be noted that this solution procedure requires the knowledge of elementary rate constants, klt k2, and k3. The elementary rate constants can be measured by the experimental techniques such as pre-steady-state kinetics and relaxation methods (Bailey and Ollis, pp. 111 -113, 1986), which are much more complicated compared to the methods to determine KM and rmax. Furthermore, the initial molar concentration of an enzyme should be known, which is also difficult to measure as explained earlier. However, a numerical solution with the elementary rate constants can provide a more precise picture of what is occurring during the enzyme reaction, as illustrated in the following example problem. 

The linear approach described here is expandable to multienzyme electrodes as well as multilayer electrodes. At least for the stationary case, multilayer models of bienzyme electrodes may be easily treated, too. The whole system is readily adaptable to potentiometric electrodes (Carr and Bowers, 1980). It must be noted, however, that the superiority over purely numerical solution procedures decreases with increasing number of enzyme species and in the multilayer model. The advantage in calculation speed using the sum formulas described (e.g., in Section 2.5.2) amounts to about two orders of magnitude. With multilayer electrodes and formulas containing double and triple sums it is reduced to one order of magnitude. 

An important parameter for suicide inactivators is the partition ratio - the number of catalytic events which are necessary for one active site to be inactivated. If the inactivator is stable, it can be estimated by incubating excess enzyme with a limited amount of inactivator if it is not, and particularly if the partition ratio is low but > 1, numerical solutions to the differential equations involved are needed. ... 

Mell and Maloy [18] postulated a numerical approach to simulate the steady-state amperometric measurements for an enzyme electrode [18]. Since then, the reaction-diffusion problems describing biochemical processes are often solved numerically [6,7,19]. The analytical solutions are often applied to validation of the corresponding numerical solutions. 

In the first reaction glucose reacts with ATP to produce ADP and PEP the enzyme for this first step is hexokinase the notation is similar for the remaining reaction steps, with the enzymes as indicated. The rate of influx of glucose into the system is constant. Due to the feedback mechanisms in both the PFK and PK reactions chemical oscillations of some species may occur, see Fig. 16.4. These oscillations have been observed [1] and are also obtained from numerical solutions of the deterministic mass action rate equations of the model in Fig. 16.1 for given glucose inflow conditions, see Fig. 16.4. 

The efficiency of solution-phase (two aqueous phase) enzymatic reaction in microreactor was demonstrated by laccase-catalyzed l-DOPA oxidation in an oxygen-saturated water solution, and analyzed in a Y-shaped microreactor at different residence times (Figure 10.24) [142]. Up to 87% conversions of l-DOPA were achieved at residence times below 2 min. A two-dimensional mathematical model composed of convection, diffusion, and enzyme reaction terms was developed. Enzyme kinetics was described with the double substrate Michaelis-Menten equation, where kinetic parameters from previously performed batch experiments were used. Model simulations, obtained by a nonequidistant finite differences numerical solution of a complex equation system, were proved and verified in a set of experiments performed in a microreactor. Based on the developed model, further microreactor design and process optimization are feasible. 

The development of kinetic schemes for sequences of enzyme reactions contributes to the resolution of two problems. The first of these, the more complex one, concerns the study of the control of metabolic pathways and has been of major interest to biochemists for some time. Two related approaches to the problem have been developed for the interpretation of the behaviour of large assemblies of coupled enzyme reactions. Models can be made which contain the differential equations for the progress of the reactions for all the enzymes of a system. The numerical solutions of this set of equations can be compared with the experimental data for the concentrations of intermediates and their rates of change. Iterative improvements of the model can then be made. Alternatively, if data are only available for a... 

Numerical solutions of equations of this type are included in the programme of complex reactions reported by Sel6gny et al. [68] giving the concentration of species in solutions in contact with multi-enzyme membranes. The evolution of substrate-concentration-profiles with symmetrical boundary conditions in an originally void membrane and reversible reactions has been calculated according to the same technique in cooperation between Thomas and Kernevez and is reported in different contexts by both of them in their respective theses [44, 45]. At a given time interestingly two minima separated by a maximum in this profile are predicted. 

Numerical solution on computer of the partial differential equations of these diffusion-reactions has shown that in such an enzyme membrane a small inverse shift of the pH on the limits of the enzyme layer (about 0.1 pH unit up on one side and about the same value simultaneously down on the other side) will rapidly generate a progressively self increasing activity for a different enzyme on each side of the membrane. A space oscillaiory type profile appears rapidly through the membrane for the hydro-... 

Duggleby, R. G. (1986). Progress-curve analysis in enzyme kinetics. Numerical solution of integrated rate equations. Biochem. J. 235, 613-615. 

The most numerous cases of homogeneous catalysis are by certain ions or metal coordination compounds in aqueous solution and in biochemistry, where enzymes function catalyticaUy. Many ionic effects are known. The hydronium ion and the hydroxyl ion OH" cat-... 

Free Enzymes in Flow Reactors. Substitute t = zju into the DDEs of Example 12.5. They then apply to a steady-state PFR that is fed with freely suspended, pristine enzyme. There is an initial distance down the reactor before the quasisteady equilibrium is achieved between S in solution and S that is adsorbed on the enzyme. Under normal operating conditions, this distance will be short. Except for the loss of catalyst at the end of the reactor, the PFR will behave identically to the confined-enzyme case of Example 12.4. Unusual behavior will occur if kfis small or if the substrate is very dilute so Sj Ej . Then, the full equations in Example 12.5 should be (numerically) integrated. 

P-Gal has a molecular weight of 540,000 and is composed of four identical subunits of MW 135,000, each with an independent active site (Melchers and Messer, 1973). The enzyme has divalent metals as cofactors, with chelated Mg+2 ions required to maintain active site conformation. The presence of NaCl or dilute solutions (5 percent) of low-molecular-weight alcohols (methanol, ethanol, etc.) causes enhanced substrate turnover. P-Gal contains numerous sulfhy-dryl groups and is glycosylated. 

The structural integrity of enzymes in aqueous solution is often compromised by the addition of small quantities of water-miscible organic solvents. However, there are numerous examples, particularly using extremophiles, where enzymes have been successfully employed in organic solvent-aqueous mixtures.A good example is the savinase-catalysed kinetic resolution of an activated racemic lactam precursor to abacavir in 1 1 THF/water (Scheme 1.39). The organic solvent is beneficial as it retards the rate of the unselective background hydrolysis. 

In addition to the many enzyme systems available, there are with each a series of chromogenic substrate solutions that can be used to create different colors and locations of reaction products. For the peroxidase system, there are numerous oxidizable compounds that precipitate as a permanent color. The most common and still widely used is 3,3 diaminobenzidine tetrahydro-chloride (DAB). This compound precipitates to a golden brown color when in solution with peroxidase and hydrogen peroxide. This brown color has many subtleties and readily stands out in a tissue section. With practice, it is possible to differentiate specific from nonspecific staining patterns just by examining the characteristics of the precipitated pigment. This material is also insoluble in alcohol and xylene, and therefore the tissue may be routinely dehydrated and cleared without loss of chromogen. 

---