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Mathematical Modeling Reveals Spontaneous Emergence of Self

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Mathematical Modeling Reveals Spontaneous Emergence of Self ARTICLE cro Author’s Choice Mathematical modeling reveals spontaneous emergence of self-replication in chemical reaction systems Received for publication, May 10, 2018, and in revised form, September 29, 2018 Published, Papers in Press, October 3, 2018, DOI 10.1074/jbc.RA118.003795 X Yu Liu1 and David J. T. Sumpter From the Department of Mathematics, Uppsala University, 75105 Uppsala, Sweden Edited by John M. Denu Explaining the origin of life requires us to elucidate how self- sor life does not have nucleic acids but consists of a self-repli- replication arises. To be specific, how can a self-replicating cating (or “autocatalytic,” in his words) metabolic network (6). entity develop spontaneously from a chemical reaction system Another proposal, by Szathma´ry (7–9), is that life evolved from in which no reaction is self-replicating? Previously proposed “holistic limited hereditary replicators” such as the formose mathematical models either supply an explicit framework for a reaction—in which sugar is replicated from formaldehyde—to Downloaded from minimal living system or consider only catalyzed reactions, and “modular unlimited hereditary replicators” such as RNA and thus fail to provide a comprehensive theory. Here, we set up a today’s DNA. general mathematical model for chemical reaction systems that There are lots of biological examples of self-replicating sys- properly accounts for energetics, kinetics, and the conservation tems that do not rely on a single complex template molecule (as law. We found that 1) some systems are collectively catalytic, a is the case for DNA and RNA molecules). These include the mode whereby reactants are transformed into end products malic acid cycle (a metabolic path of some bacteria and plants http://www.jbc.org/ with the assistance of intermediates (as in the citric acid cycle), for synthesis of malates), the Calvin cycle in photosynthesis whereas some others are self-replicating, that is, different parts (10), the reductive citric acid cycle for a certain group of che- replicate each other and the system self-replicates as a whole (as moautotrophs (11, 12), the artificially designed myokinase– in the formose reaction, in which sugar is replicated from form- pyruvate kinase cycling system (13), the metabolic pathways of ϩ aldehyde); 2) side reactions do not always inhibit such systems; ATP (as well as some other coenzymes such as NAD and CoA) at Uppsala Universitetsbibliotek on March 4, 2019 3) randomly chosen chemical universes (namely random artifi- found in many different living organisms (14), and the whole cial chemistries) often contain one or more such systems; 4) it is metabolic reaction network of Escherichia coli (15). Self-repli- possible to construct a self-replicating system in which the cation has also been identified in nonliving systems, such as the entropy of some parts spontaneously decreases, in a manner formose reaction (10, 16), and experiments in laboratories (17– similar to that discussed by Schro¨dinger; and 5) complex self- 19), including self-replication of nucleotide-based oligomers replicating molecules can emerge spontaneously and relatively (20). easily from simple chemical reaction systems through a se- Many models have been put forward to explain the origin of quence of transitions. Together, these results start to explain the self-replication in terms of a system of coupled chemical origins of prebiotic evolution. reactions (21). Some of these can be categorized as artificial chemistry models, as reviewed by Dittrich et al. (22) and by Banzhaf and Yamamoto (23). For example, the chemoton Self-replication is one of the central properties of life (1), and model describes a system composed of three coupled quasi- to explain life’s origins, we need to explain how self-replication self-replicating subsystems (metabolism, membrane, and tem- arose. It is widely accepted that before DNA, life replicated plate), which as a whole is able to self-replicate (10). The che- through RNA molecules (2). However, it remains unclear how moton can be considered as a model of a minimal living system, the building blocks of RNA, such as nucleotides, became avail- but cannot explain how this system spontaneously develops able on the primitive Earth, and even if these building blocks from a soup of simple molecules. The metabolically coupled were abundant, it is unclear how they were assembled into the replicator system is another type of model in this direction (24, first RNA (3, 4). It is plausible that self-replication did not orig- 25). But it mainly focuses on the information heredity and eco- inate from a single complex independent self-replicating mol- logical stability, that is, how to maintain coexistence of different ecule. In the early stage of evolution, the “precursor life” could replicators and constrain parasitism, rather than investigating be very different from what we see today (3, 5). For example, why self-replication can occur in the first place. The most influ- in Wa¨chtersha¨user’s iron-sulfur world hypothesis, the precur- ential model in this direction is the reflexively autocatalytic and food-generated (RAF)2 theory (26–28), extended from Kauff- man’s autocatalytic sets theory (29). A set of chemical reactions The authors declare that they have no conflicts of interest with the contents of this article. is RAF if 1) every reaction in this set is catalyzed by at least one Author’s Choice—Final version open access under the terms of the Creative molecule involved in this set and 2) every molecule involved in Commons CC-BY license. this set can be produced from a small food molecule set. RAF This article contains sections S1–S10. 1 Supported by the Erasmus Mundus Action 2 program (Lotus III Scholarship), funded by the European Commission. To whom correspondence should 2 The abbreviations used are: RAF, reflexively autocatalytic and food-gener- be addressed. Tel.: 46-76-5635-008; E-mail: [email protected]. ated; ODE, ordinary differential equation. 18854 J. Biol. Chem. (2018) 293(49) 18854–18863 © 2018 Liu and Sumpter. Published by The American Society for Biochemistry and Molecular Biology, Inc. Modeling spontaneous self-replication sets are shown to be able to readily emerge from a set of chem- In this paper, we set up an artificial chemistry model, in the ical reactions and always consist of smaller RAF sets, demon- form of a general mathematical framework for chemical reac- strating the capability to evolve (30, 31). Other similar models tion systems, that properly accounts for energetics, kinetics, also contributed to this theory (32–36), including the chemical and the conservation law. Catalysts are not explicitly included organization theory (37) and the graded autocatalysis replica- in the model, but we later find that catalysis can emerge, along tion domain model (38). The former is closely related to RAF with self-replication and potentially complex molecules, in our theory (see Hordijk et al. (39) for a detailed comparison), system. whereas the latter suffers from lacking evolvability (see Vasas et In what follows, we give an overview of our model. The spec- al. (40, 41) for detailed critical analyses). In addition, many of ifications are given under “Theory” at the end of this paper. We the biological observations mentioned in the previous para- would encourage the reader, on first reading, to watch a short graph can be put into the framework of RAF theory (15, 17–20). movie included in the supporting information that illustrates Although RAF theory is influential, it has limitations as an how our model works, and see “Theory” directly afterward. explanation of self-replication. First, the theory stipulates that We model a well-mixed soup of molecules, each of which is every reaction in the set is catalyzed. Even though catalyzed defined by its integer mass, i. A molecule’s type is thus denoted reactions are very common in living systems, not every reaction ı¯. Only synthesis reactions and decomposition reactions are possible. Only reactions that conserve mass can occur (i.e. the involves a catalyst (e.g. condensation reactions (8, 42)). In Downloaded from the early stages of biological evolution, probably no reaction total mass on the reactant side adds up to that on the product required sophisticated biotic catalysts (e.g. enzymes) (11, 15), so side). Each type of molecule ı¯has its own standard Gibbs energy ° there is a strong motivation not to include these enzymes (20, of formation Gi, that decides whether a reaction is spontaneous. 43, 44) or even catalysts as given in a model of the origins of Moreover, a reaction has to overcome an energy barrier self-replication. Uncatalyzed reactions have significant effects (namely Gibbs energy of activation) to occur. It is called low- http://www.jbc.org/ on the dynamics of the whole system (e.g. “innovating” new barrier if its energy barrier is low and its reaction rate is thus species of molecules and then triggering other RAF sets) (45, high or high-barrier otherwise. To define a specific chemical reaction system, we show a list of low-barrier (and spontane- 46). The second concern with RAF theory is that it is a purely ous) reactions only (e.g. Scheme 1). graph-theoretic approach. As a result, there is no constraint on how chemical reactions are constructed and coupled (although 5 3 1 ϩ 4 at Uppsala Universitetsbibliotek on March 4, 2019 it gives the theory much freedom, the construction of reaction 3 ϩ systems is too arbitrary to investigate the model systematically). Ά 6 1 5 ϩ 3 As another result, extra assumptions need to be made about 2 4 6 chemical kinetics to investigate the dynamics of how popula- SCHEME 1 tions of molecules change over time (47). Here, we see another reason to relax the assumption of studying only catalyzed reac- Many other reaction systems can also be constructed.
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