Mathematical Analysis of a Prototypical Autocatalytic Reaction Network.

Ekaterina V. Skorb,Sergey N. Semenov

doi:10.3390/life9020042

Ekaterina V. Skorb, Sergey N. Semenov

Open Access

https://doi.org/10.3390/life9020042

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Abstract

Network autocatalysis, which is autocatalysis whereby a catalyst is not directly produced in a catalytic cycle, is likely to be more common in chemistry than direct autocatalysis is. Nevertheless, the kinetics of autocatalytic networks often does not exactly follow simple quadratic or cubic rate laws and largely depends on the structure of the network. In this article, we analyzed one of the simplest and most chemically plausible autocatalytic networks where a catalytic cycle is coupled to an ancillary reaction that produces the catalyst. We analytically analyzed deviations in the kinetics of this network from its exponential growth and numerically studied the competition between two networks for common substrates. Our results showed that when quasi-steady-state approximation is applicable for at least one of the components, the deviation from the exponential growth is small. Numerical simulations showed that competition between networks results in the mutual exclusion of autocatalysts; however, the presence of a substantial noncatalytic conversion of substrates will create broad regions where autocatalysts can coexist. Thus, we should avoid the accumulation of intermediates and the noncatalytic conversion of the substrate when designing experimental systems that need autocatalysis as a source of positive feedback or as a source of evolutionary pressure.

Highlights

Autocatalytic reactions are of profound importance for at least three problems in the chemical sciences: (i) chemical evolution and the origin of life, [1,2,3,4,5,6] (ii) dissipative chemical systems [7,8,9], and (iii) chemical signaling and information processing systems [10]
We can separate a catalytic cycle of cysteamine (CSH), which catalyzed the acylation of cystamine (CSSC) by a thioester; this is followed by converting one of its byproducts, ethanethiol, into CSH (Figure 1a) [37]
The first term in Equation (7) has a positive exponent (λ1 ) and has the higher coefficient in front of the exponent other than the second term, and it will dominate from the beginning. This equation means that if intermediates of the catalytic cycle do not accumulate in significant amounts, the reaction will behave as exponential autocatalysis from the beginning of the experiment and any deviations from exponential growth will decay over time

Summary

Introduction

Autocatalytic reactions are of profound importance for at least three problems in the chemical sciences: (i) chemical evolution and the origin of life, [1,2,3,4,5,6] (ii) dissipative chemical systems [7,8,9], and (iii) chemical signaling and information processing systems [10]. The catalytic cycle of the copper-catalyzed azide-alkyne cycloaddition reactions is followed by the formation of a catalytically active complex from triazole derivatives (Figure 1b) [38]. Because of the abundance of catalytic reactions in Examples of autocatalysts are cysteamine (CSH) and a copper complex (Cu(I)TATZ).

Results

Conclusion