Autocatalytic Sets Arising in a Combinatorial Model of Chemical Evolution.

The idea that chemical evolution led to the origin of life is not new, but still leaves open the question of how exactly it could have led to a coherent and self-reproducing collective of molecules. One possible answer to this question was proposed in the form of the emergence of an autocatalytic set: a collection of molecules that mutually catalyze each other's formation and that is self-sustaining given some basic "food" source. Building on previous work, here we investigate in more detail when and how autocatalytic sets can arise in a simple model of chemical evolution based on the idea of combinatorial innovation with random catalysis assignments. We derive theoretical results, and compare them with computer simulations. These results could suggest a possible step towards the (or an) origin of life.

Autocatalytic chemical networks at the origin of metabolism.

Modern cells embody metabolic networks containing thousands of elements and form autocatalytic sets of molecules that produce copies of themselves. How the first self-sustaining metabolic networks arose at life's origin is a major open question. Autocatalytic sets smaller than metabolic networks were proposed as transitory intermediates at the origin of life, but evidence for their role in prebiotic evolution is lacking. Here, we identify reflexively autocatalytic food-generated networks (RAFs)-self-sustaining networks that collectively catalyse all their reactions-embedded within microbial metabolism. RAFs in the metabolism of ancient anaerobic autotrophs that live from H2 and CO2 provided with small-molecule catalysts generate acetyl-CoA as well as amino acids and bases, the monomeric components of protein and RNA, but amino acids and bases without organic catalysts do not generate metabolic RAFs. This suggests that RAFs identify attributes of biochemical origins conserved in metabolic networks. RAFs are consistent with an autotrophic origin of metabolism and furthermore indicate that autocatalytic chemical networks preceded proteins and RNA in evolution. RAFs uncover intermediate stages in the emergence of metabolic networks, narrowing the gaps between early Earth chemistry and life.

Dynamics of a birth-death process based on combinatorial innovation.

A feature of human creativity is the ability to take a subset of existing items (e.g. objects, ideas, or techniques) and combine them in various ways to give rise to new items, which, in turn, fuel further growth. Occasionally, some of these items may also disappear (extinction). We model this process by a simple stochastic birth-death model, with non-linear combinatorial terms in the growth coefficients to capture the propensity of subsets of items to give rise to new items. In its simplest form, this model involves just two parameters (P, α). This process exhibits a characteristic 'hockey-stick' behaviour: a long period of relatively little growth followed by a relatively sudden 'explosive' increase. We provide exact expressions for the mean and variance of this time to explosion and compare the results with simulations. We then generalise our results to allow for more general parameter assignments, and consider possible applications to data involving human productivity and creativity.

Developmental structuring of phenotypic variation: A case study with a cellular automata model of ontogeny.

Developmental mechanisms not only produce an organismal phenotype, but they also structure the way genetic variation maps to phenotypic variation. Here, we revisit a computational model for the evolution of ontogeny based on cellular automata, in which evolution regularly discovered two alternative mechanisms for achieving a selected phenotype, one showing high modularity, the other showing morphological integration. We measure a primary variational property of the systems, their distribution of fitness effects of mutation. We find that the modular ontogeny shows the evolution of mutational robustness and ontogenic simplification, while the integrated ontogeny does not. We discuss the wider use of this methodology on other computational models of development as well as real organisms.

Average Fitness Differences on NK Landscapes.

The average fitness difference between adjacent sites in a fitness landscape is an important descriptor that impacts in particular the dynamics of selection/mutation processes on the landscape. Of particular interest is its connection to the error threshold phenomenon. We show here that this parameter is intimately tied to the ruggedness through the landscape's amplitude spectrum. For the NK model, a surprisingly simple analytical estimate explains simulation data with high precision.

Autocatalytic networks in biology: structural theory and algorithms.

Self-sustaining autocatalytic networks play a central role in living systems, from metabolism at the origin of life, simple RNA networks and the modern cell, to ecology and cognition. A collectively autocatalytic network that can be sustained from an ambient food set is also referred to more formally as a 'reflexively autocatalytic food-generated' (RAF) set. In this paper, we first investigate a simplified setting for studying RAFs, which is nevertheless relevant to real biochemistry and which allows an exact mathematical analysis based on graph-theoretic concepts. This, in turn, allows for the development of efficient (polynomial-time) algorithms for questions that are computationally intractable (NP-hard) in the general RAF setting. We then show how this simplified setting for RAF systems leads naturally to a more general notion of RAFs that are 'generative' (they can be built up from simpler RAFs) and for which efficient algorithms carry over to this more general setting. Finally, we show how classical RAF theory can be extended to deal with ensembles of catalysts as well as the assignment of rates to reactions according to which catalysts (or combinations of catalysts) are available.

Molecular Diversity Required for the Formation of Autocatalytic Sets.

Systems chemistry deals with the design and study of complex chemical systems. However, such systems are often difficult to investigate experimentally. We provide an example of how theoretical and simulation-based studies can provide useful insights into the properties and dynamics of complex chemical systems, in particular of autocatalytic sets. We investigate the issue of the required molecular diversity for autocatalytic sets to exist in random polymer libraries. Given a fixed probability that an arbitrary polymer catalyzes the formation of other polymers, we calculate this required molecular diversity theoretically for two particular models of chemical reaction systems, and then verify these calculations by computer simulations. We also argue that these results could be relevant to an origin of life scenario proposed recently by Damer and Deamer.

Autocatalytic Networks at the Basis of Life's Origin and Organization.

Life is more than the sum of its constituent molecules. Living systems depend on a particular chemical organization, i.e., the ways in which their constituent molecules interact and cooperate with each other through catalyzed chemical reactions. Several abstract models of minimal life, based on this idea of chemical organization and also in the context of the origin of life, were developed independently in the 1960s and 1970s. These models include hypercycles, chemotons, autopoietic systems, (M,R)-systems, and autocatalytic sets. We briefly compare these various models, and then focus more specifically on the concept of autocatalytic sets and their mathematical formalization, RAF theory. We argue that autocatalytic sets are a necessary (although not sufficient) condition for life-like behavior. We then elaborate on the suggestion that simple inorganic molecules like metals and minerals may have been the earliest catalysts in the formation of prebiotic autocatalytic sets, and how RAF theory may also be applied to systems beyond chemistry, such as ecology, economics, and cognition.

Population Dynamics of Autocatalytic Sets in a Compartmentalized Spatial World.

Autocatalytic sets are self-sustaining and collectively catalytic chemical reaction networks which are believed to have played an important role in the origin of life. Simulation studies have shown that autocatalytic sets are, in principle, evolvable if multiple autocatalytic subsets can exist in different combinations within compartments, i.e., so-called protocells. However, these previous studies have so far not explicitly modeled the emergence and dynamics of autocatalytic sets in populations of compartments in a spatial environment. Here, we use a recently developed software tool to simulate exactly this scenario, as an important first step towards more realistic simulations and experiments on autocatalytic sets in protocells.

Niche emergence as an autocatalytic process in the evolution of ecosystems.

The utilisation of the ecospace and the change in diversity through time has been suggested to be due to the effect of niche partitioning, as a global long-term pattern in the fossil record. However, niche partitioning, as a way to coexist, could be a limited means to share the environmental resources and condition during evolutionary time. In fact, a physical limit impedes a high partitioning without a high restriction of the niche's variables. Here, we propose that niche emergence, rather than niche partitioning, is what mostly drives ecological diversity. In particular, we view ecosystems in terms of autocatalytic sets: catalytically closed and self-sustaining reaction (or interaction) networks. We provide some examples of such ecological autocatalytic networks, how this can give rise to an expanding process of niche emergence (both in time and space), and how these networks have evolved over time (so-called evoRAFs). Furthermore, we use the autocatalytic set formalism to show that it can be expected to observe a power-law in the size distribution of extinction events in ecosystems. In short, we elaborate on our earlier argument that new species create new niches, and that biodiversity is therefore an autocatalytic process.

The Influence of Modularity, Seeding, and Product Inhibition on Peptide Autocatalytic Network Dynamics.

Chemical networks often exhibit emergent, systems-level properties that cannot be simply derived from the linear sum of the individual components. The design and analysis of increasingly complex chemical networks thus constitute a major area of research in Systems Chemistry. In particular, much research is focused on the emergence of functional properties in prebiotic chemical networks relevant to the origin and early evolution of life. Here, we apply a formal framework known as RAF theory to study the dynamics of a complex network of mutually catalytic peptides. We investigate in detail the influence of network modularity, initial template seeding, and product inhibition on the network dynamics. We show that these results can be useful for designing new experiments, and further argue how they are relevant to origin of life studies.

Autocatalytic confusion clarified.

There is frequent confusion about the terms autocatalytic reaction, autocatalytic cycle, and autocatalytic set. As the use of the same adjective implies, these three systems do indeed share common properties, in particular their potential for exponential growth. However, the ways in which they achieve this potential are different, giving rise to different internal network structures and dynamics. Therefore, care should be taken which term is used in which context. Here, we explain and discuss the similarities and differences between the three systems in detail, in an effort to avoid any further confusion. We then also discuss the relevance of these autocatalytic systems for possible origin of life scenarios, with an emphasis on how autocatalytic sets may have played an important role in this.

Autocatalytic Sets and RNA Secondary Structure.

The dominant paradigm in origin of life research is that of an RNA world. However, despite experimental progress towards the spontaneous formation of RNA, the RNA world hypothesis still has its problems. Here, we introduce a novel computational model of chemical reaction networks based on RNA secondary structure and analyze the existence of autocatalytic sub-networks in random instances of this model, by combining two well-established computational tools. Our main results are that (i) autocatalytic sets are highly likely to exist, even for very small reaction networks and short RNA sequences, and (ii) sequence diversity seems to be a more important factor in the formation of autocatalytic sets than sequence length. These findings could shed new light on the probability of the spontaneous emergence of an RNA world as a network of mutually collaborative ribozymes.

Chasing the tail: The emergence of autocatalytic networks.

A ubiquitous feature of all living systems is their ability to sustain a biochemistry in which all reactions are coordinated by catalysts, and all reactants (along with the catalysts) are either produced by the system itself or are available from the environment. This led to the hypothesis that 'autocatalytic networks' play a key role in both the origin and the organization of life, which was first proposed in the early 1970s, and has been enriched in recent years by a combination of experimental studies and the application of mathematical and computational techniques. The latter have allowed a formalization and detailed analysis of such networks, by means of RAF theory. In this review, we describe the development of these ideas, from pioneering early work of Stuart Kauffman through to more recent theoretical and experimental studies. We conclude with some suggestions for future work.

Evolution of Autocatalytic Sets in Computational Models of Chemical Reaction Networks.

Several computational models of chemical reaction networks have been presented in the literature in the past, showing the appearance and (potential) evolution of autocatalytic sets. However, the notion of autocatalytic sets has been defined differently in different modeling contexts, each one having some shortcoming or limitation. Here, we review four such models and definitions, and then formally describe and analyze them in the context of a mathematical framework for studying autocatalytic sets known as RAF theory. The main results are that: (1) RAF theory can capture the various previous definitions of autocatalytic sets and is therefore more complete and general, (2) the formal framework can be used to efficiently detect and analyze autocatalytic sets in all of these different computational models, (3) autocatalytic (RAF) sets are indeed likely to appear and evolve in such models, and (4) this could have important implications for a possible metabolism-first scenario for the origin of life.

Prebiotic network evolution: six key parameters.

The origins of life likely required the cooperation among a set of molecular species interacting in a network. If so, then the earliest modes of evolutionary change would have been governed by the manners and mechanisms by which networks change their compositions over time. For molecular events, especially those in a pre-biological setting, these mechanisms have rarely been considered. We are only recently learning to apply the results of mathematical analyses of network dynamics to prebiotic events. Here, we attempt to forge connections between such analyses and the current state of knowledge in prebiotic chemistry. Of the many possible influences that could direct primordial network, six parameters emerge as the most influential when one considers the molecular characteristics of the best candidates for the emergence of biological information: polypeptides, RNA-like polymers, and lipids. These parameters are viable cores, connectivity kinetics, information control, scalability, resource availability, and compartmentalization. These parameters, both individually and jointly, guide the aggregate evolution of collectively autocatalytic sets. We are now in a position to translate these conclusions into a laboratory setting and test empirically the dynamics of prebiotic network evolution.

Autocatalytic sets in E. coli metabolism.

BACKGROUND: A central unsolved problem in early evolution concerns self-organization towards higher complexity in chemical reaction networks. In theory, autocatalytic sets have useful properties to help model such transitions. Autocatalytic sets are chemical reaction systems in which molecules belonging to the set catalyze the synthesis of other members of the set. Given an external supply of starting molecules - the food set - and the conditions that (i) all reactions are catalyzed by at least one molecule, and (ii) each molecule can be constructed from the food set by a sequence of reactions, the system becomes a reflexively autocatalytic food-generated network (RAF set). Autocatalytic networks and RAFs have been studied extensively as mathematical models for understanding the properties and parameters that influence self-organizational tendencies. However, despite their appeal, the relevance of RAFs for real biochemical networks that exist in nature has, so far, remained virtually unexplored. RESULTS: Here we investigate the best-studied metabolic network, that of Escherichia coli, for the existence of RAFs. We find that the largest RAF encompasses almost the entire E. coli cytosolic reaction network. We systematically study its structure by considering the impact of removing catalysts or reactions. We show that, without biological knowledge, finding the minimum food set that maintains a given RAF is NP-complete. We apply a randomized algorithm to find (approximately) smallest subsets of the food set that suffice to sustain the original RAF. CONCLUSIONS: The existence of RAF sets within a microbial metabolic network indicates that RAFs capture properties germane to biological organization at the level of single cells. Moreover, the interdependency between the different metabolic modules, especially concerning cofactor biosynthesis, points to the important role of spontaneous (non-enzymatic) reactions in the context of early evolution. Graphical AbstractE. coli metabolic network in the context of autocatalytic sets.

Algorithms for detecting and analysing autocatalytic sets.

BACKGROUND: Autocatalytic sets are considered to be fundamental to the origin of life. Prior theoretical and computational work on the existence and properties of these sets has relied on a fast algorithm for detectingself-sustaining autocatalytic sets in chemical reaction systems. Here, we introduce and apply a modified version and several extensions of the basic algorithm: (i) a modification aimed at reducing the number of calls to the computationally most expensive part of the algorithm, (ii) the application of a previously introduced extension of the basic algorithm to sample the smallest possible autocatalytic sets within a reaction network, and the application of a statistical test which provides a probable lower bound on the number of such smallest sets, (iii) the introduction and application of another extension of the basic algorithm to detect autocatalytic sets in a reaction system where molecules can also inhibit (as well as catalyse) reactions, (iv) a further, more abstract, extension of the theory behind searching for autocatalytic sets. RESULTS: (i) The modified algorithm outperforms the original one in the number of calls to the computationally most expensive procedure, which, in some cases also leads to a significant improvement in overall running time, (ii) our statistical test provides strong support for the existence of very large numbers (even millions) of minimal autocatalytic sets in a well-studied polymer model, where these minimal sets share about half of their reactions on average, (iii) "uninhibited" autocatalytic sets can be found in reaction systems that allow inhibition, but their number and sizes depend on the level of inhibition relative to the level of catalysis. CONCLUSIONS: (i) Improvements in the overall running time when searching for autocatalytic sets can potentially be obtained by using a modified version of the algorithm, (ii) the existence of large numbers of minimal autocatalytic sets can have important consequences for the possible evolvability of autocatalytic sets, (iii) inhibition can be efficiently dealt with as long as the total number of inhibitors is small.

Autocatalytic sets and boundaries.

Autopoietic systems, chemotons, and autogens are models that aim to explain (the emergence of) life as a functionally closed and self-sustaining system. An essential element in these models is the notion of a boundary containing, maintaining, and being generated by an internal reaction network. The more general concept of collectively autocatalytic sets, formalized as RAF theory, does not explicitly include this notion of a boundary. Here, we argue that (1) the notion of a boundary can also be incorporated in the formal RAF framework, (2) this provides a mechanism for the emergence of higher-level autocatalytic sets, (3) this satisfies a necessary condition for the evolvability of autocatalytic sets, and (4) this enables the RAF framework to formally represent and analyze (at least in part) the other models. We suggest that RAF theory might thus provide a basis for a unifying formal framework for the further development and study of such models. Graphical abstractThe emergence of an autocatalytic (super)set of autocatalytic (sub)sets.