Abstract
The emergence of autocatalytic sets of molecules seems to have played an important role in the origin of life context. Although the possibility to reproduce this emergence in laboratory has received considerable attention, this is still far from being achieved.
In order to unravel some key properties enabling the emergence of structures potentially able to sustain their own existence and growth, in this work we investigate the probability to observe them in ensembles of random catalytic reaction networks characterized by different structural properties.
From the point of view of network topology, an autocatalytic set have been defined either in term of strongly connected components (SCCs) or as reflexively autocatalytic and food-generated sets (RAFs).
We observe that the average level of catalysis differently affects the probability to observe a SCC or a RAF, highlighting the existence of a region where the former can be observed, whereas the latter cannot. This parameter also affects the composition of the RAF, which can be further characterized into linear structures, autocatalysis or SCCs.
Interestingly, we show that the different network topology (uniform as opposed to power-law catalysis systems) does not have a significantly divergent impact on SCCs and RAFs appearance, whereas the proportion between cleavages and condensations seems instead to play a role.
A major factor that limits the probability of RAF appearance and that may explain some of the difficulties encountered in laboratory seems to be the presence of molecules which can accumulate without being substrate or catalyst of any reaction.
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Notes
- 1.
The only exception is presented in the final part of the results section.
- 2.
Forward and backward reactions, during the creation of the reactions graph, are in principle handled as two separated reactions.
- 3.
Since different \(M\) will be assessed, in this work we prefer to adopt the average level of catalysis \(\langle c \rangle \), as in [16], instead of the standard reaction probability \(p\), as we adopted in our previous works [10, 12], and Kauffman [21] and others [2, 25] in important works on this topic. Nevertheless, according to \(|S|\), it is always possible to move from \(\langle c \rangle \) to \(p\) and vice-versa.
- 4.
We obtain the power law distribution by slightly modifying the algorithm proposed by Barabási and Albert [3]. We increase \(p^i_k\) as a function of the already catalyzed reactions, i.e., the probability to catalyze a reaction is weighted with the number of reactions already catalyzed so that \(p^i_k=\#r_i/\sum _{z=1}^{|S|} \#r_z\), where \(\#r_i\) and \(\#r_z\) indicate the number of reactions already catalyzed by the \(i\)-th and the \(z\)-th species respectively. In such a way, we obtain a power-law distribution in the number of reactions each chemical species can catalyze.
- 5.
Since the analysis are static and only statistic structural properties of the networks will be assessed, the choice on which reaction is the direct one and which is the reverse one is only implementative and does not affect the analysis.
- 6.
It is important to notice that in [17] the transition is found at \(1.25\). Such a difference is basically related to a different way to count forward and reverse reactions. Hordijk [17] consider both forward and backward reactions as a unique reaction whereas in our work, though it is clear that they account for the same reaction scheme, we consider them as two different reactions.
- 7.
In [17], the authors show the so-called RA sets—which are reflexively autocatalytic but not necessarily food-generated—whose transition and nature correspond to that of SCC, where the transitions happen on the same zones of scale-free topologies in case of non catalytic activity of the foodset molecular species, i.e., “\(BUF_2\)” scenario in this work.
- 8.
It is worth stressing that an autocatalysis is a strongly connected component, nevertheless we decided to deal with them separately.
- 9.
At high catalysis levels almost each RAF owns at least one SCC and one autocatalysis, so with our measurements it is not possible to observe the exact RAFs’ structure. This aspect will be analyzed in further works.
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Acknowledgements
The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007–2013) under grant agreement n. 284625 and from “INSITE - The Innovation Society, Sustainability, and ICT” Pr.ref. 271574, under the 7th FWP - FET programme.
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Filisetti, A. et al. (2014). On RAF Sets and Autocatalytic Cycles in Random Reaction Networks. In: Pizzuti, C., Spezzano, G. (eds) Advances in Artificial Life and Evolutionary Computation. WIVACE 2014. Communications in Computer and Information Science, vol 445. Springer, Cham. https://doi.org/10.1007/978-3-319-12745-3_10
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