Autocatalysis linear

Useful reviews of the kinetics of autocatalytic reactions have recently been published by Mata-Perez and Perez-Benito (22) and by Schwartz Q2) For an autocatalytic reaction a plot of r/c, where r is the rate of reaction and c the concentration of reactant, as a function of c should be linear with a negative slope (22). When this analysis is applied to the possibility of autocatalysis of liquids production by LOG, no dependence of r/c on c was found. In fact, least uares "correlation coefficients were in the range 0.01 - 0.03. Although the initid hypothesis of autocatalysis by thiols is shown to be untenable, an alternative is the possibility of autocatalysis by H2S. 

Autocatalysis will play a central role in driving the oscillations and other non-linear phenomena of interest in this book. Usually, an autocatalytic process will be combined in a larger mechanism with other steps. Before considering such systems, however, we investigate the properties and behaviour of autocatalysis on its own—in particular how the concentrations and rate vary with time and with respect to each other. We start with quadratic autocatalysis, and then look at the cubic form. 

This chapter and chapter 5 study the prototypical thermokinetic oscillator. Thermal feedback replaces autocatalysis, and the Arrhenius temperature dependence of rate coefficients supplies non-linearity in the scheme P - A - B + heat. After careful study of this chapter the reader should be able to ... 

In earlier publications we have shown that the principle of selection can be deduced from the premises of a self-replicating system as an extremum principle. It states that inherent linear autocatalysis causes the relative population numbers to take on values that correspond to the highest reproductive efficiency of the system as a whole. The distribution of relative concentrations in the stationary population is, after a short induction period, independent of changes of the system as a whole. The population consists of a uniquely defined wild-type (or several equivalent,... 

A positive non-linear effect is also necessary for amplifying asymmetric autocatalysis... 

The dimer model was anticipated by Kagan in his analysis of non-linear effects. In simple catalytic cases it is much less common than the reservoir model where the monomeric species is the active catalyst. Figure 7 shows how both of these work in practice for autocatalyic systems, and demonstrate how the two pathways can be distinguished when the catalyst structure is understood. A further consequence of autocatalysis is that (counter-intuitively) the smaller the initial proportion of catalyst, the greater the enhancement of ee. 

The key NMR observations (i) that the proportion of homo- and heterochiral dimers is near-equal, and (ii) that their interconversion by a dissociative process is rapid compared to catalytic turnover, preclude the possibility of a monomer autocatalyst. In Kagan s classification, monomer catalysis with a positive NLE may only arise when there is an unequal concentration of homo- and heterochiral oligomers, in favour of the heterochiral form, which acts as a reservoir for the deficient enantiomer. NMR results show that the resting state for Soai s autocatalysis is an equal mixture of homo-and heterochiral species, predominantly dimeric. The lack of ground-state stereo-discrimination requires that the number of chiral entities in the resting state must be less than or equal to the number in the enantioselectivity-determining transition state, else there is no possibility of the vital non-linear effect. Even after the publication of these results in late 2004, their consequences are not always applied. For recent discussions where a monomeric catalyst for Soai s system is permitted or promoted, see [91-93]. 

As for theoretical research on the chiral symmetry breaking, Frank was the first to show that a linear autocatalysis with an antagonistic nonlinear chemical reaction can lead to homochirality [16]. His formulation with rate equations corresponds to the mean-field analysis of the phase transition in a nonequilibrium situation [17], and other variants have been proposed [6, 18-23]. All these analyses have been carried out only for open systems where... 

Since the spontaneous production (Eq. 1) yields only a racemic mixture of two enantiomers, one has to assume some autocatalytic processes. The simplest is a linear autocatalysis with a reaction coefficient k ... 

Beyond the linear autocatalyses, nonlinear effects such as a quadratic autocatalysis have been considered [6,24,31] ... 

If the erroneous linear catalysis (Eq. 3) is at work in addition to the linear autocatalysis (Eq. 2), rate equations are modified ... 

Another model for the chiral asymmetry amplification is a variant of the Frank model in a closed system [34]. In its simplest form, the model consists of reaction schemes of spontaneous production (Eq. 1), linear autocatalysis (Eq. 2), and heterodimer formation (Eq. 31). Then, the rate equations for monomers are the same as those of the original Frank model [ 16], but they are supplemented with the heterodimer formation ... 

When A is positive, as in the case of Fig. 6c, the coefficient of the cubic termB is also positive, and the velocity 4>i vanishes at three values of 0i in the range of - 1 < 0i < 1. This is possible if a strong quadratic autocatalysis > k 2 exists together with a linear recycling X > 0, or if a linear autocatalysis k and a nonlinear recycling x > 0 coexist. By following the direction indicated by the arrows for positive order parameter ends up at a definite value ... 

As for the second example, we consider the case with a quadratic autocatalysis k2 > 0 and a linear recycling A > 0. Because the linear recycling takes place whenever there is nonzero concentrations of enantiomers, a few achiral substrate always remain, a t = oo) 0. Thus, the diagonal line q = r + s = c is no longer a fixed line. Instead, there appear fixed points in general. In the present case with a finite A. > 0, a total of seven fixed points appear three stable (0, Si,2) and four unstable ones (1/1-4), as shown in Fig. 7b. As the reaction proceeds, the system approaches a state associated with one of the stable fixed points. The origin 0 is not interesting, since all the chiral products are recycled back to the achiral substrate. Also its influence extends only in a small... 

Fig. 7 Flow diagrams with a linear autocatalysis and nonlinear recycling (ki = /z > 0, k(1 = k2 =X = 0) similar to the Frank model, and b quadratic autocatalysis and linear recycling (k2c2 = /. > 0, fc, = ki = [ = 0). O origin, S stable fixed points, U instable fixed points...

In all the cases considered, auto catalytic processes must be present, whether linear or nonlinear. To understand the actual mechanism of autocatalysis for the Soai reaction, identification of the process at a molecular level is necessary, but is out of scope of the present review. 

There is a theoretical study on the asymptotic shape of probability distribution for nonautocatalytic and linearly autocatalytic systems with a specific initial condition of no chiral enantiomers [35,36]. Even though no ee amplification is expected in these cases, the probability distribution with a linear autocatalysis has symmetric double peaks at 0 = 1 when ko is far smaller than k -,kototal number of all reactive chemical species, A, R, and S. This can be explained by the single-mother scenario for the realization of homo chirality, as follows From a completely achiral state, one of the chiral molecules, say R, is produced spontaneously and randomly after an average time l/2koN. Then, the second R is produced by the autocatalytic process, whereas for the production of the first S molecule the... 

We now turn to the selection of reactions exhibiting autocatalysis. Chemical reactions with abrupt, non-linear changes of concentrations have been known for some time. They are the spectacular clock reactions, first described by Landolt in the nineteenth century41. The key to understanding the sudden and predictable (you could set your watch by their occurrence, hence the name clock ) color changes was provided over 60 years ago by Eggert and Schamow42. They analyzed the Landolt reaction (IV) (in the presence of excess iodate)... 

Nevertheless, the RNA World hypothesis would seem to answer most of the questions raised above how can an essentially linear molecule be autocatalytic, how can it synthesize proteins, and how can it replicate Autocatalysis can occur because RNA can adopt a wide range of secondary and tertiary structures that position RNA monomers into a preorganized sequence and link them together, it can apply the same flexibility to bind other small molecules and catalyse their polymerization, and it could form weakly interacting sense and antisense duplexes. The main problem is... 

Autocatalysis is a distinctive phenomenon while in ordinary catalysis the catalyst re-appears from the reaction apparently untouched, additional amounts of catalyst are actively produced in an autocatalytic cycle. As atoms are not interconverted during chemical reactions, this requires (all) the (elementary or otherwise essential) components of autocatalysts to be extracted from some external reservoir. After all this matter was extracted, some share of it is not introduced in and released as a product but rather retained, thereafter supporting and speeding up the reaction(s) steadily as amounts and possibly also concentrations of autocatalysts increase. At first glance, such a system may appear doomed to undergo runaway dynamics ( explosion ), but, apart from the limited speeds and rates of autocatalyst resupply from the environment there are also other mechanisms which usually limit kinetics even though non-linear behavior (bistability, oscillations) may not be precluded ... 

Fig. 5.1. Variation of reaction rate R with extent of reaction (a) linear relationship for first-order reaction (b) non-linear deceleratory reactions of overall order n (c) reactions showing chemical feedback in the form of autocatalysis (d) comparison of chemical and...

The next milestone appeared in the 1950s in the context of the development of asymmetric reactions. Various stereochemical reactions induced by facial discrimination of the carbonyl group have always been pivotal in this field. Cram s rule inspired an explosion of studies on diastereoselective reactions followed by enan-tioselective versions. The recent outstanding progress in the non-linear effect of chirality or asymmetric autocatalysis heavily relies on the carbonyl addition reactions. Thanks to these achievements, natural products chemistry has enjoyed extensive advancement in the synthesis of complex molecules. It is no exaggeration to say that we are now in a position to be able to make any molecules in as highly selective a manner as we want. 

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