Autocatalytic enzyme reactions

Fig. 2.10. Scheme of an autocatalytic enzyme reaction. The allosteric enzyme catalysing the reaction is formed by several subunits (not shown), which exist in the conformational states R and T. These states differ in their affinity for the substrate and/or in their catalytic activity. In the model by Monod et al. (1965), the transition between the two conformational states is concerted for all subunits. Here the product is a positive effector, i.e. an activator, as it binds in an exclusive or preferential manner to the most active, R state, of the enzyme. The system is open as the substrate S is injected at a constant rate while the product P disappears as a result of its utilization in a subsequent enzyme reaction. 

Fig. 7.2 Model for the coupling of two autocatalytic enzyme reactions. The system admits three simultaneously stable periodic regimes for appropriate values of the parameters.

Autocatalytic enzyme reactions are frequent, and in many or all of those known, the effect is due to production of an acid, such as phosphoric, lactic, pyruvic, or oleic acid. Here the rate of reaction changes with pH, since ionization and destruction of the enzyme at different pH values affect the amount of the active form of the enzyme. Hence no general rules can be set up, and it is not surprising that reaction rates do not follow changes in reactant concentration. For example, the degree of buffering of the reaction mixture will affect the pH changes produced by the autocatalytic acid, and hence the rate of reaction. 

A coating bearing one enzyme (papain) is produced on the surface of a glass pH electrode by the method previously introduced (co-crosslinking). The papain reaction decreases the pH, and the pH-activity variation gives an autocatalytic effect for pH values greater than the optimum under zero-order kinetics for the substrate (benzoyl arginine ethyl ester) the pH inside the membrane is studied as a function of the pH in the bulk solution in which the electrode is immersed. A hysteresis effect is observed and the enzyme reaction rate depends not only on the metabolite concentrations, but also on the history of the system. 

The coupling in series of two enzyme reactions with autocatalytic regulation (fig. 4.1) permits the construction of a three-variable biochemical prototype containing two instability-generating mechanisms (Decroly, 1987a,b Decroly Goldbeter, 1982). As in the model for glycolytic oscillations, the substrate S of the first enzyme is introduced at a constant rate into the system this substrate is transformed by enzyme Ej into product Pi, which serves as substrate for a second enzyme E2 that transforms Pj into P2. The two allosteric enzymes are both activated by their reaction product Pj and P2 are thus positive effectors of enzymes Ej and E2, respectively. 

Fig. 4.1. Three-variable biochemical model for the coupling in series of two enzyme reactions with autocatalytic regulation (Decroly Goldbeter, 1982, 1987).

The peroxidase reaction provides another prototype for periodic behaviour and chaos in an enzyme reaction. As noted by Steinmetz et al. (1993), in view of its mechanism based on free radical intermediates, this reaction represents an important bridge between chemical oscillations of the Belousov-Zhabotinsky type, and biological oscillators. In view of the above discussion, it is noteworthy that the model proposed by Olsen (1983), and further analysed by Steinmetz et al. (1993), also contains two parallel routes for the autocatalytic production of a key intermediate species in the reaction mechanism. As shown by experiments and accounted for by theoretical studies, the peroxidase reaction possesses a particularly rich repertoire of dynamic behaviour (Barter et al, 1993) ranging from bistability (Degn, 1968 Degn et al, 1979) to periodic oscillations (Yamazaki et al, 1965 Nakamura et al, 1969 ... 

The specific turnover rate for the autocatalytic biomass formation itself rx is classically referred to as. Results on the biomass level give insight into the performance of a given strain and its physiological behavior under the specific conditions in the reactor. The fluxes are driven by light and substrate availabihty and coupled by enzyme reactions. The cells can adapt to the environment on a control level, where enzyme activities are regulated or the cell composition changes with respect to functional macromolecules. 

Initial pre-life reproducing cellular systems emerge as a membrane-bonded vesicle incorporating autocatalytic enzyme-type catalysts. There is an ongoing metabolic process that lead to cell growth and cell multiplication. Evolutionary adaptive stematics leads to cell growth and cell death, depending on the relative value of different reaction parameters. Computational models of such processes are discussed in Sections 9.3 and 9.4. 

The cubic rate-law model presented here affords a prototype for many systems of practical interest, including solution-phase kinetics and enzyme reactions. It cannot be stressed too strongly that steps 1(a) and (b) do not have to be regarded as elementary steps. They may be combinations of such steps. Thus the reaction between arsenite and iodate ions, which is autocatalytic [9,10] in the production of r, is well approximated at constant pH by the rate expression... 

Now we will return briefly to Sections 3.8-3.11 and 4.6-4.8 where we considered the general problem of multiple flows, here of H, C, N, O, S and P. We observe immediately that all the products are from the same small molecule environmental sources and are required to be formed in relatively fixed amounts using the same source of energy and a series of intermediates. Controlling all the processes to bring about optimum cellular production are feedbacks between them and linked with the code which generates proteins, and here we note particularly enzymes, i.e. catalysts. The catalysts are made from the amino acids, the synthesis of which they themselves manage, while the amino acids control the catalysts so as to maintain a restricted balanced set of reaction pathways in an autocatalytic assembly. It is also the feedback controls on both the DNA (RNA) from the same units used in the... 

This type of autocatalytic reaction is a simplification of many biological reactions such as fermentation, where the reaction produces products (species B in the previous example), which accelerates the rate. In fermentation, yeast cells in the solution produce enzymes that catalyze the decomposition of sugar to produce ethanol as a byproduct of yeast reproduction. Since the yeast population increases as the reaction proceeds, the enzyme concentration increases, and the process appears to be autocatalytic. A highly simplified description of fermentation might be... 

However, the RTR is an efficient way to introduce a small amount of backmix into a reactor. This is particularly useful for an autocatalytic reactor where the rate is proportional to the concentration of a product. In the enzyme-catalyzed reactions discussed previously and modeled as A B, r = kCACs, need to seed the reactor with enzyme, and this is easily accomplished using recycle of a small amount of product back into the feed. 

There are still other causes of nonlinearities than (apparent or real) higher-order transformation kinetics. In Section 12.3 we discussed catalyzed reactions, especially the enzyme kinetics of the Michaelis-Menten type (see Box 12.2). We may also be interested in the modeling of chemicals which are produced by a nonlinear autocatalytic reaction, that is, by a production rate function, p(Q, which depends on the product concentration, C,. Such a production rate can be combined with an elimination rate function, r(C,), which may be linear or nonlinear and include different processes such as flushing and chemical transformations. Then the model equation has the general form ... 

Dopa can also undergo autocatalytic oxygenation with Fem, but Cu11 inhibits the reaction. This difference in reactivity reflects different coordination modes (see Section 20.2.2.2.8).82 The enzyme superoxide dismutase (vide infra) catalyzes disproportionation of the superoxide ion. The same catalysis is exhibited by Cun-L-His solutions, the reactive species, from the pH dependence, appearing to be [Cu(L-HisO)(L-His)]+. 

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