A stochastic model of the emergence of autocatalytic cycles

Autocatalytic cycles are rather common in biological systems and they might have played a major role in the transition from non-living to living systems. Several theoretical models have been proposed to address the experimentalists during the investigation of this issue and most of them describe a phase transition depending upon the level of heterogeneity of the chemical soup. Nevertheless, it is well known that reproducing the emergence of autocatalytic sets in wet laboratories is a hard task. Understanding the rationale at the basis of such a mismatch between theoretical predictions and experimental observations is therefore of fundamental importance. We here introduce a novel stochastic model of catalytic reaction networks, in order to investigate the emergence of autocatalytic cycles, sensibly considering the importance of noise, of small-number effects and the possible growth of the number of different elements in the system. Furthermore, the introduction of a temporal threshold that defines how long a specific reaction is kept in the reaction graph allows to univocally define cycles also within an asynchronous framework. The foremost analyses have been focused on the study of the variation of the composition of the incoming flux. It was possible to show that the activity of the system is enhanced, with particular regard to the emergence of autocatalytic sets, if a larger number of different elements is present in the incoming flux, while the specific length of the species seems to entail minor effects on the overall dynamics. Introduction The investigation of the generic properties of catalytic reaction networks, with specific regard to the emergence of the so-called autocatalytic cycles or sets (referred to as ACS hereafter), is one of the key issues in systems chemistry and, particularly, in the research concerning the origin of life. The debate on the transition from non-living to living is still open and compelling [1-4] and several different theoretical frameworks have been proposed, such as the metabolic-first hypothesis [5-10], the protein first scenario [11-13], the compartmentalization [14], the compositional approach [15,16] and the well-known gene first hypothesis embodied in the RNA world theory [4,17-22]. Despite the differences, the underlining principle beyond these theories is the necessity for the existence of robust reaction pathways for the production of the molecular species involved in the transition. Some authors hypothesised the existence of linear chemical processes capable of produce substantial amount of the species under plausible prebiotic conditions [23] or at energy-rich site such as hydrothermal vents [24]. Alternatively, great emphasis has been put on chemical cycles embodied by autocatalytic network [25] as a means to produce molecular species and achieve self-sustenance. Several models have been developed in the course of time to describe this kind of systems, including the works by Dyson [26], Eigen and Schuster [27-30], Kauffman [7], Jain and Khrishna [31], Lancet [15,16] and Kaneko [32], while recently the formalism introduced by Steel et al. [33-35] has contributed in setting a standard theoretical framework for the algorithmic detection of autocatalytic networks within the original model designed by Kauffman [7]. Following [7] an ACS is a subset of chemicals which production is catalysed by at least one other member of the subset. In spite of their important differences, most * Correspondence: [email protected] European Centre for Living Technology, S.Marco 2940, 30124, Venice, Italy Full list of author information is available at the end of the article Filisetti et al. Journal of Systems Chemistry 2011, 2:2 http://www.jsystchem.com/content/2/1/2 © 2011 Filisetti et al; licensee Chemistry Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. models describe a phase transition such that, once certain conditions are satisfied, ACSs spontaneously emerge. It is important to notice that the presence of autocatalytic cycles may lead to a significant departure of the concentrations of the elements belonging to the cycle from the expected one. These out-of-equilibrium concentrations are in principle amenable to experimental analysis. In particular, Kauffman, through the analysis of the structural properties of the reaction graph [7] claims that, increasing the maximum length of the molecules present in the system, the number of possible reactions increases faster than the number of molecules: therefore, whatever is the probability for a reaction to be catalysed, it will be sufficient to feed the system with the right number of different molecules to observe the emergence of an autocatalytic set. The kinetics was explicitly introduced within the model, through a set of ordinary differential equations, by Farmer and coworkers [36,37]. They observed that once that the average connectivity of the reactions graph is greater than 1, i.e. supra-critical, cycles of all length begin to form. These results were confirmed also in the model by Bagley [38,39] in which all the molecules characterised by concentrations that are below a certain threshold are removed from the graph and new molecules are introduced within the system according to the reaction scheme. Nevertheless, despite the theoretical predictions, observing ACSs in wet experiments is an indeed difficult task. It is possible that the simplifying assumptions at the basis of the in-silico experiments are unrealistic with respect to the actual features of the real world or, on the contrary, that the experiments lack the correct features indicated by the theory and hence the necessary conditions for the emergence of ACSs are not met. As progress beyond the state of the art, in this work we introduce a novel model for the study of the emergence of autocatalytic cycles that is explicitly intended to fill the gap between theoretical predictions and experimental findings and provide useful insight for further experimentations. In our model two kind of reactions are taken into account, as firstly proposed by Kauffman [7]: namely cleavage and condensation, whereby one polymer is divided into two short polymers in the former case and two polymers are concatenated forming a longer polymer in the latter case. Each reaction must be catalysed by another species in the system to occur and one of the assumptions is that any chemical species has an independent probability to catalyse a randomly chosen reaction. With respect to the original model, the main novelties are the following. First, the model is stochastic and is aimed to specifically consider the influence of the molecules present with few copies and the importance of noise and random fluctuations on the overall dynamics. Second, the model is capable to increase in complexity as new entities can be introduced in the system by either condensation or cleavage of the existing elements. Third, we introduce a temporal threshold which allows to exclude those reaction that do not occur at least twice within the specific time frame when analysing the dynamical evolution of the system: in this way it is possible to clearly detect cycles in a asynchronous stochastic system and to neglect the structural influence of very rare reactions. Finally, particular attention is devoted to those cases in which the system is close to instability points (i.e. critical systems) and those in which the total amount of some species is particularly low. In this work we are not interested in investigating the nature of the specific chemicals represented in the system, nor in the particular interactions among them. We are rather interested in the description and characterisation of the dynamical behaviour that emerges from the interaction of a whole set of interacting chemicals and in the detection of possible generic properties of this kind of systems. Description of the model The basic entities of the model are linear chains, species from now on, oriented from left to right, composed of symbols or letters from a finite alphabet (e.g. A, B, C...; G, A, T, C). We refer to species composed of single symbols also as “monomers” and to species composed of more than one letter as “polymers”. Following the formalism introduced by Steel [40]X stands for the entire set of species, while each single species will be denoted by xi, i = 1... N. The amount (quantity) of species xi in the reaction vessel (equal to its concentration times the reactor volume) will be denoted by x̂i. In the rest of the work we will make use either of amount or concentration in accordance with the specific context; nevertheless since the two quantities are proportional the relation is straightforward. The dynamics of the system is ruled, as in the original Kauffman model, by two different reactions, namely condensation and cleavage. By means of the former two species are concatenated together forming a longer species (e.g. AB + BA ® ABBA) whereas by means of the latter a longer species is cut to create two shorter species (e.g. ABBB ® A + BBB). Notice that, with regard to the condensation, there could be two equal products starting from two different substrates (e.g. AA + B ® AAB and A + AB ® AAB), and on the other hand, with regard to the cleavage, there could be two (or more) different products starting from the same substrate (e.g. BAAB ® BAA + B and BAAB ® BA + AB). Filisetti et al. Journal of Systems Chemistry 2011, 2:2 http://www.jsystchem.com/content/2/1/2 Page 2 of 10