Autocatalytic Replication of Polymers Revisited

A simple computational model for the emergence of autocatalytic sets as described in (Farmer et al., 1986) is reimplemented. Results are found to generally agree with the major theme in the original work: increasing the initial polymer variety in a toy chemical soup scenario increases the likelihood that a complex autocatalytic set will suddenly bootstrap itself into existence. Quantitatively, however, critical probabilities derived from this careful re-implementation are very much higher than those reported in the original work. A full resolution is not reached, but a theoretical argument supports the simulation results gained in this instance. Introduction The principle of an autocatalytic set, a set of molecules which collectively catalyses its own production, holds intuitive interest. There exists obvious relations to primitive metabolic systems, and contemporary minimal definitions of life such as autopoiesis (McMullin, 1999). By achieving catalytic closure, a set of relatively inert molecules can organise into a self-sustaining identity, a persistent presence in a chemical soup. Different questions can be asked of autocatalytic sets. In general, one might be interested in (1) how a set came to be and the preconditions necessary for it’s emergence, (2) which critical molecular species the set consists of, or (3) how the set chemically operates in real physical space and time. Original work on autocatalytic sets (Kauffman, 1986; Farmer et al., 1986) pursued the first question as the main point of interest, although all questions are inter-related to some extent. Question 2 has recently been given a deep formal treatment (Hordijk and Steel, 2004). Question 3 is of considerable depth and of most contemporary interest, involving concepts such as dynamics, spatial compartmentalisation, reaction kinetics and concentrations in particular physical autocatalytic instantiations (see, for example (Ono and Ikegami, 2000) for application in a spatial abstract cell model). This paper describes a careful re-implementation of the original (graph theoretic) model investigating the inevitable emergence of complex autocatalytic sets (Farmer et al., 1986). Section 1 recaps the motivations and assumptions of the original model. Section 2 describes in detail the reimplementation carried out. Remaining sections present and discuss the results, which generally follow the same qualitative pattern as original results, but differ by a factor some 100 in quantitative predictions of the critical probability of autocatalysis. 1 Original Work The original work on autocatalytic sets by Stuart Kauffman is concerned with making a tentative link to the grand problem of the origin of life itself (Kauffman, 1986, 1993) and levelling a respectable argument against entrenched explanations of template based replication. In the original model (Farmer et al., 1986), the emergence of autocatalytic sets is investigated as a connectivity feature of directed graphs. A reaction graph captures the core chemical relationships in a system of polymers, expressing the reaction possibilities in that system. Operational details such as space, time and quantity are not represented in this canonical description. The chemical system is assumed to exist in a wellstirred overflowing reactor environment. The central idea is built upon the phase transition phenomena in connectivity problems. As systems become increasingly connected a critical limit is reached when, very suddenly, each component of the system is connected directly or indirectly to every other. A large component crystallises from a mass of independent sub-systems. By the same logic, when a reaction network is expressed as a reaction graph, there must exist some critical catalytic connectivity beyond which each polymer will directly or indirectly catalyse every other at which point the existence of a complex autocatalytic set can be inferred with almost certainty. The original model focuses on finding this critical connectivity. A basic reaction system where polymers consist of directional strings of characters is successively grown from an original ‘firing disk’ (food set). In this scenario, Artificial Life XI 2008 56 the reaction system is always autocatalytic in a strict sense, but criticality is judged when the rate of change of polymer species becomes exponential (autocatalytic networks which continually create large complex proteins were of prime interest to the authors). Significant assumptions of the model include the prerequisite of flow reactor conditions and the assumption that the distribution of catalytic capacities in peptide space can be modelled by a fixed probability P that any one polymer will catalyse any other. Farmer et al. find that that the critical value of P required for an autocatalytic set decreases as the initial polymer variation in the system increases, lending support to their general autocatalytic account for the origin of life from a sufficiently diverse pre-biotic chemical soup. 2 Re-implementation Original Graph Growth Algorithm For clarity, the original graph growth algorithm is presented below. Square braces represent cross-references to the more detailed implementation to follow. Our rule for random assignment of reactions is implemented as follows: For a given starting list of molecular species, we compute the maximum number of allowed condensation and cleavage reactions by counting the number of distinguishable combinations of string concatenations and string cleavages [see Note 2]. The number of reactions that we actually assign is obtained by multiplying the number of allowed reactions of each type by a probability P . To assign condensation reactions, we chose two molecules at random [Box 1], while for cleavage reactions we chose a molecule and a cleavage point at random [Box 2]. In both cases enzymes are chosen at random from the set of species currently present. Assignment of reactions can be viewed as a dynamical process. We initialise the system by choosing a starting list, called the “firing disk”, typically chosen to be all possible strings shorter than a given length L [Iteration 0, Step 1]. Reactions within the firing disk are assigned as described above [Iteration 0, Step 2]. Condensation reactions may generate new species outside the firing disk, thereby expanding the list [Iteration 0, Steps 3 and 4]. The introduction of new species creates new reaction possibilities; to take these into account, on the next time step we count the number of combinatorial possibilities involving the new species [Iterations 1 to 1000, Steps 2 and 3]. Multiplying by P gives the number of new reactions [Iterations 1 to 1000, Steps 6,7,8,9]. This process is repeated on subsequent time steps. As long as new species are created on each step the graph continues to grow; otherwise growth stops. (Farmer et al., 1986), p. 54 Graph Growth Algorithm as Implemented Definitions S Set of all distinct polymer species currently in the system. Initially empty. N Set of distinct polymer species, new on the current iteration. Initially empty. sn Size of set S. Number of distinct polymer species in the system. nn Size of set N . Number of new distinct polymer species on current iteration. B Alphabet size of polymers. M Order of initial firing disk (or ’maximum sized polymer’ in firing disk, also referred to as Lf elsewhere in this paper). P Probability that a random polymer catalyses an arbitrary reaction.