Autocatalysis mathematics

To quantify this idea in mathematical terms, we can recognize that we are really talking about the partial derivative quantities d(da/dt)/da and d(db/dt)/db. Stability has been associated in some sense with these two quantities being negative (i.e. da/dt decreases as a increases, so d(da/dt)/da < 0), instability with these being positive. In most normal chemical systems, e.g. those with deceleratory kinetics, the two partial derivatives will be negative. It is a characteristic of autocatalysis, however, that at least one of these may become positive — at least over some ranges of composition and experimental conditions. 

The CSTR is, in many ways, the easier to set up and operate, and to analyse theoretically. Figure 6.1 shows a typical CSTR, appropriate for solution-phase reactions. In the next three chapters we will look at the wide range of behaviour which chemical systems can show when operated in this type of reactor. In this chapter we concentrate on stationary-state aspects of isothermal autocatalytic reactions similar to those introduced in chapter 2. In chapter 7, we turn to non-isothermal systems similar to the model of chapter 4. There we also draw on a mathematical technique known as singularity theory to explain the many similarities (and some differences) between chemical autocatalysis and thermal feedback. Non-stationary aspects such as oscillations appear in chapter 8. 

Reaction 8.114 is genuine autocatalysis because the product P acts as catalyst. It is a strange reaction, though, because the product could never come into being if it were not present at start in at least a trace quantity (or could arise in some other fashion). The simple, single-step mechanism does not correspond to any known reaction. Its mathematical behavior reflects the most typical symptoms of autocatalysis, but is of little quantitative value. 

The unrealistic "cubic autocatalysis" form 14.4 was chosen here for simplicity s sake (a "quadratic" form X+Y— 2Y does not produce instability). Gray and Scott [34] discuss this case in much greater detail. They include reverse steps as well as an uncatalyzed, parallel step X— Y the latter keeps the mathematics from... 

Tyson [38] classified destabilizing processes in chemical reaction systems according to mathematical relations among the JMEs. He distinguished direct autocatalysis,... 

From the methodological point of view the hypercycle model is the most impressive example of the possibility of unifying detailed biochemical examinations and mathematical models of chemical reactions. Both the prebiotic relevance of the model and its mathematical structure were studied, sometimes critically (e.g. King, 1981 Miiller-Herold, 1983 Szathmary, 1984). It was suggested that the hypercyclic evolution is not as effective as conventional autocatalysis, and the model should be defined by multiple time singular perturbation theory. 

A key feature of this system, and of most chemical systems that exhibit oscillations, is autocatalysis, which means that the rate of growth of a species, whether animal or chemical, increases with the population or concentration of that species. Even autocatalytic systems can reach a steady state in which the net rate of increase of all relevant species is zero— for example, the rate of reproduction of rabbits is exactly balanced by that species consumption by lynxes, and lynxes die at the same rate that baby lynxes are born. Mathematically, we find such a state by setting all the time derivatives equal to zero and solving the resulting algebraic equations for the populations. As we shall see later, a steady state is not necessarily stable-, that is, the small perturbations or fluctuations that always exist in a real system may grow, causing the system to evolve away from the steady state. 

An alternative form of positive feedback is indirect autocatalysis. Here, although / and Jjj are both negative, the combined effects of species i on species j and of j on i are such that a rise in one of these concentrations ultimately leads to an increase in how rapidly it is produced. Mathematically, we require that JyJji > 0. There are two possibilities competition, in which i inhibits y, and j inhibits i, so that both Jacobian elements are negative and symbiosis, in which the two species activate each other, resulting in positive Jy and /y,. A simple example of symbiosis arises from introducing an intermediate Y and breaking down reaction (5.19) into a pair of steps ... 

The tools that we need to help the modelling of complex reaction systems have to fill the gap between chemical and mathematical modelling. They also should allow the chemist to gain practical and effecient access to the required mathematical knowledge. Finally, those tools should be able to take into account the specific features and properties of chemical reaction equations, and, at the same time, to do this using a chemical language to describe the expected behaviour and dynamical structure of the model, for instance, in terms of chemical network, reaction processes, autocatalysis, activation or inhibition. We are far from that, which indicates that we are still lacking theoretical methods to handle those problems. Moreover, even well established mathematical theories are still not usually implemented in effecient practical procedures. 

Consider next a system in which explosion occurs primarly because of chemical kinetics. Real world systems of this type involve multiple steps and competition between various pathways, many of which contain autocatalytic or inhibitory effects associated with the appearance of chain reactions and free radicals [4]. Instead of developping the analysis of such a system however, we present hereafter a mathematical model which captures the essence of the phenomenon while still allowing a fairly complete mathematical treatment. The specific example we choose is the autocatalytic mechanism suggested by Schlbgl [6]. Further comments on the role of autocatalysis are to be found in the Chapters by P. Gray and S.K. Scott and by I. Epstein. 

In 1953, Frank developed a mathematical model showing that spontaneous asymmetric synthesis is theoretically possible (21). If the chiral product of a catalytic reaction would act as a catalyst for its own formation and at the same time suppress the formation of its enantiomer, a basically enantiopirre product could be formed from near-racemic starling materials. About forty years later, Soai and coworkers provided the first experimental proof for this concept of asymmetric autocatalysis with the alkylation of pyrimidyl aldehydes with diallgrlzinc reagents (Figure 1) (22). 

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