Autocatalytic networks in biology: structural theory and algorithms

Self-generating autocatalytic networks: structural results, algorithms and their relevance to early biochemistry

The concept of an autocatalytic network of reactions that can form and persist, starting from just an available food source, has been formalized by the notion of a reflexively autocatalytic and food-generated (RAF) set. The theory and algorithmic results concerning RAFs have been applied to a range of settings, from metabolic questions arising at the origin of life, to ecological networks, and cognitive models in cultural evolution. In this article, we present new structural and algorithmic results concerning RAF sets, by studying more complex modes of catalysis that allow certain reactions to require multiple catalysts (or to not require catalysis at all), and discuss the differing ways catalysis has been viewed in the literature. We also focus on the structure and analysis of minimal RAFs and derive structural results and polynomial-time algorithms. We then apply these new methods to a large metabolic network to gain insights into possible biochemical scenarios near the origin of life.

Interior Operators and Their Relationship to Autocatalytic Networks

The emergence of an autocatalytic network from an available set of elements is a fundamental step in early evolutionary processes, such as the origin of metabolism. Given the set of elements, the reactions between them (chemical or otherwise), and with various elements catalysing certain reactions, a Reflexively Autocatalytic F-generated (RAF) set is a subset R[Formula: see text] of reactions that is self-generating from a given food set, and with each reaction in R[Formula: see text] being catalysed from within R[Formula: see text]. RAF theory has been applied to various phenomena in theoretical biology, and a key feature of the approach is that it is possible to efficiently identify and classify RAFs within large systems. This is possible because RAFs can be described as the (nonempty) subsets of the reactions that are the fixed points of an (efficiently computable) interior map that operates on subsets of reactions. Although the main generic results concerning RAFs can be derived using just this property, we show that for systems with at least 12 reactions there are generic results concerning RAFs that cannot be proven using the interior operator property alone.Kindly check and confirm the edit made in the title.I confirm that the edit is fine.

An algebraic characterization of self-generating chemical reaction networks using semigroup models

The ability of a chemical reaction network to generate itself by catalyzed reactions from constantly present environmental food sources is considered a fundamental property in origin-of-life research. Based on Kaufmann's autocatalytic sets, Hordijk and Steel have constructed the versatile formalism of catalytic reaction systems (CRS) to model and to analyze such self-generating networks, which they named reflexively autocatalytic and food-generated. Recently, it was established that the subsequent and simultaenous catalytic functions of the chemicals of a CRS give rise to an algebraic structure, termed a semigroup model. The semigroup model allows to naturally consider the function of any subset of chemicals on the whole CRS. This gives rise to a generative dynamics by iteratively applying the function of a subset to the externally supplied food set. The fixed point of this dynamics yields the maximal self-generating set of chemicals. Moreover, the set of all functionally closed self-generating sets of chemicals is discussed and a structure theorem for this set is proven. It is also shown that a CRS which contains self-generating sets of chemicals cannot have a nilpotent semigroup model and thus a useful link to the combinatorial theory of finite semigroups is established. The main technical tool introduced and utilized in this work is the representation of the semigroup elements as decorated rooted trees, allowing to translate the generation of chemicals from a given set of resources into the semigroup language.

The hierarchical organization of autocatalytic reaction networks and its relevance to the origin of life

Prior work on abiogenesis, the emergence of life from non-life, suggests that it requires chemical reaction networks that contain self-amplifying motifs, namely, autocatalytic cores. However, little is known about how the presence of multiple autocatalytic cores might allow for the gradual accretion of complexity on the path to life. To explore this problem, we develop the concept of a seed-dependent autocatalytic system (SDAS), which is a subnetwork that can autocatalytically self-maintain given a flux of food, but cannot be initiated by food alone. Rather, initiation of SDASs requires the transient introduction of chemical "seeds." We show that, depending on the topological relationship of SDASs in a chemical reaction network, a food-driven system can accrete complexity in a historically contingent manner, governed by rare seeding events. We develop new algorithms for detecting and analyzing SDASs in chemical reaction databases and describe parallels between multi-SDAS networks and biological ecosystems. Applying our algorithms to both an abiotic reaction network and a biochemical one, each driven by a set of simple food chemicals, we detect SDASs that are organized as trophic tiers, of which the higher tier can be seeded by relatively simple chemicals if the lower tier is already activated. This indicates that sequential activation of trophically organized SDASs by seed chemicals that are not much more complex than what already exist could be a mechanism of gradual complexification from relatively simple abiotic reactions to more complex life-like systems. Interestingly, in both reaction networks, higher-tier SDASs include chemicals that might alter emergent features of chemical systems and could serve as early targets of selection. Our analysis provides computational tools for analyzing very large chemical/biochemical reaction networks and suggests new approaches to studying abiogenesis in the lab.

A Dynamic Autocatalytic Network Model of Therapeutic Change

Psychotherapy involves the modification of a client's worldview to reduce distress and enhance well-being. We take a human dynamical systems approach to modeling this process, using Reflexively Autocatalytic foodset-derived (RAF) networks. RAFs have been used to model the self-organization of adaptive networks associated with the origin and early evolution of both biological life, as well as the evolution and development of the kind of cognitive structure necessary for cultural evolution. The RAF approach is applicable in these seemingly disparate cases because it provides a theoretical framework for formally describing under what conditions systems composed of elements that interact and 'catalyze' the formation of new elements collectively become integrated wholes. In our application, the elements are mental representations, and the whole is a conceptual network. The initial components-referred to as foodset items-are mental representations that are innate, or were acquired through social learning or individual learning (of pre-existing information). The new elements-referred to as foodset-derived items-are mental representations that result from creative thought (resulting in new information). In clinical psychology, a client's distress may be due to, or exacerbated by, one or more beliefs that diminish self-esteem. Such beliefs may be formed and sustained through distorted thinking, and the tendency to interpret ambiguous events as confirmation of these beliefs. We view psychotherapy as a creative collaborative process between therapist and client, in which the output is not an artwork or invention but a more well-adapted worldview and approach to life on the part of the client. In this paper, we model a hypothetical albeit representative example of the formation and dissolution of such beliefs over the course of a therapist-client interaction using RAF networks. We show how the therapist is able to elicit this worldview from the client and create a conceptualization of the client's concerns. We then formally demonstrate four distinct ways in which the therapist is able to facilitate change in the client's worldview: (1) challenging the client's negative interpretations of events, (2) providing direct evidence that runs contrary to and counteracts the client's distressing beliefs, (3) using self-disclosure to provide examples of strategies one can use to diffuse a negative conclusion, and (4) reinforcing the client's attempts to assimilate such strategies into their own ways of thinking. We then discuss the implications of such an approach to expanding our knowledge of the development of mental health concerns and the trajectory of the therapeutic change.

Modeling Discontinuous Cultural Evolution: The Impact of Cross-Domain Transfer

This paper uses autocatalytic networks to model discontinuous cultural transitions involving cross-domain transfer, using as an illustrative example, artworks inspired by the oldest-known uncontested example of figurative art: the carving of the Hohlenstein-Stadel Löwenmensch, or lion-human. Autocatalytic networks provide a general modeling setting in which nodes are not just passive transmitters of activation; they actively galvanize, or "catalyze" the synthesis of novel ("foodset-derived") nodes from existing ones (the "foodset.") This makes them uniquely suited to model how new structure grows out of earlier structure, i.e., cumulative, generative network growth. They have been used to model the origin and early evolution of biological life, and the emergence of cognitive structures capable of undergoing cultural evolution. We conducted a study in which six individual creators and one group generated music, prose, poetry, and visual art inspired by the Hohlenstein-Stadel Löwenmensch, and answered questions about the process. The data revealed four through-lines by which they expressed the Löwenmensch in an alternative art form: (1) lion-human hybrid, (2) subtracting from the whole to reveal the form within, (3) deterioration, and (4) waiting to be found with a story to tell. Autocatalytic networks were used to model how these four spontaneously derived through-lines form a cultural lineage from Löwenmensch to artist to audience. We used the resulting data from three creators to model the cross-domain transfer from inspirational source (sculpted figurine) to creative product (music, poetry, prose, visual art). These four spontaneously-generated threads of cultural continuity formed the backbone of this Löwenmensch-inspired cultural lineage, enabling culture to evolve even in the face of discontinuity at the level conventional categories or domains. We know of no other theory of cultural evolution that accommodates cross-domain transfer or other forms of discontinuity. The approach paves the way for a broad scientific framework for the origins of evolutionary processes.

An Autocatalytic Network Model of Conceptual Change

In reflexively autocatalytic foodset (RAF)-generated networks, nodes are not only passive transmitters of activation, but they also actively galvanize, or "catalyze" the synthesis of novel ("foodset-derived") nodes from existing ones (the "foodset"). Thus, RAFs are uniquely suited to modeling how new structure grows out of currently available structure, and analyzing phase transitions in potentially very large networks. RAFs have been used to model the origins of evolutionary processes, both biological (the origin of life) and cultural (the origin of cumulative innovation), and may potentially provide an overarching framework that integrates evolutionary and developmental approaches to cognition. Applied to cognition, the foodset consists of information obtained through social learning or individual learning of pre-existing information, and foodset-derived items arise through mental operations resulting in new information. Thus, mental representations are not only propagators of spreading activation, but they also trigger the derivation of new mental representations. To illustrate the application of RAF networks in cognitive science, we develop a step-by-step process model of conceptual change (i.e., the process by which a child becomes an active participant in cultural evolution), focusing on childrens' mental models of the shape of the Earth. Using results from (Vosniadou & Brewer, 1992), we model different trajectories from the flat Earth model to the spherical Earth model, as well as the impact of other factors, such as pretend play, on cognitive development. As RAFs increase in size and number, they begin to merge, bridging previously compartmentalized knowledge, and get subsumed by a giant RAF (the maxRAF) that constrains and enables the scaffolding of new conceptual structure. At this point, the cognitive network becomes self-sustaining and self-organizing. The child can reliably frame new knowledge and experiences in terms of previous ones, and engage in recursive representational redescription and abstract thought. We suggest that individual differences in the reactivity of mental representations, that is, their proclivity to trigger conceptual change, culminate in different cognitive networks and concomitant learning trajectories.

Automated Exploration of Prebiotic Chemical Reaction Space: Progress and Perspectives

Prebiotic chemistry often involves the study of complex systems of chemical reactions that form large networks with a large number of diverse species. Such complex systems may have given rise to emergent phenomena that ultimately led to the origin of life on Earth. The environmental conditions and processes involved in this emergence may not be fully recapitulable, making it difficult for experimentalists to study prebiotic systems in laboratory simulations. Computational chemistry offers efficient ways to study such chemical systems and identify the ones most likely to display complex properties associated with life. Here, we review tools and techniques for modelling prebiotic chemical reaction networks and outline possible ways to identify self-replicating features that are central to many origin-of-life models.

Modeling a Cognitive Transition at the Origin of Cultural Evolution Using Autocatalytic Networks

Autocatalytic networks have been used to model the emergence of self-organizing structure capable of sustaining life and undergoing biological evolution. Here, we model the emergence of cognitive structure capable of undergoing cultural evolution. Mental representations (MRs) of knowledge and experiences play the role of catalytic molecules, and interactions among them (e.g., the forging of new associations) play the role of reactions and result in representational redescription. The approach tags MRs with their source, that is, whether they were acquired through social learning, individual learning (of pre-existing information), or creative thought (resulting in the generation of new information). This makes it possible to model how cognitive structure emerges and to trace lineages of cumulative culture step by step. We develop a formal representation of the cultural transition from Oldowan to Acheulean tool technology using Reflexively Autocatalytic and Food set generated (RAF) networks. Unlike more primitive Oldowan stone tools, the Acheulean hand axe required not only the capacity to envision and bring into being something that did not yet exist, but hierarchically structured thought and action, and the generation of new MRs: the concepts EDGING, THINNING, SHAPING, and a meta-concept, HAND AXE. We show how this constituted a key transition toward the emergence of semantic networks that were self-organizing, self-sustaining, and autocatalytic, and we discuss how such networks replicated through social interaction. The model provides a promising approach to unraveling one of the greatest anthropological mysteries: that of why development of the Acheulean hand axe was followed by over a million years of cultural stasis.

An evolutionary process without variation and selection

Natural selection successfully explains how organisms accumulate adaptive change despite that traits acquired over a lifetime are eliminated at the end of each generation. However, in some domains that exhibit cumulative, adaptive change-e.g. cultural evolution, and earliest life-acquired traits are retained; these domains do not face the problem that Darwin's theory was designed to solve. Lack of transmission of acquired traits occurs when germ cells are protected from environmental change, due to a self-assembly code used in two distinct ways: (i) actively interpreted during development to generate a soma, and (ii) passively copied without interpretation during reproduction to generate germ cells. Early life and cultural evolution appear not to involve a self-assembly code used in these two ways. We suggest that cumulative, adaptive change in these domains is due to a lower-fidelity evolutionary process, and model it using reflexively autocatalytic and foodset-generated networks. We refer to this more primitive evolutionary process as self-other reorganization (SOR) because it involves internal self-organizing and self-maintaining processes within entities, as well as interaction between entities. SOR encompasses learning but in general operates across groups. We discuss the relationship between SOR and Lamarckism, and illustrate a special case of SOR without variation.

'Whole Organism', Systems Biology, and Top-Down Criteria for Evaluating Scenarios for the Origin of Life

While most advances in the study of the origin of life on Earth (OoLoE) are piecemeal, tested against the laws of chemistry and physics, ultimately the goal is to develop an overall scenario for life's origin(s). However, the dimensionality of non-equilibrium chemical systems, from the range of possible boundary conditions and chemical interactions, renders the application of chemical and physical laws difficult. Here we outline a set of simple criteria for evaluating OoLoE scenarios. These include the need for containment, steady energy and material flows, and structured spatial heterogeneity from the outset. The Principle of Continuity, the fact that all life today was derived from first life, suggests favoring scenarios with fewer non-analog (not seen in life today) to analog (seen in life today) transitions in the inferred first biochemical pathways. Top-down data also indicate that a complex metabolism predated ribozymes and enzymes, and that full cellular autonomy and motility occurred post-LUCA. Using these criteria, we find the alkaline hydrothermal vent microchamber complex scenario with a late evolving exploitation of the natural occurring pH (or Na⁺ gradient) by ATP synthase the most compelling. However, there are as yet so many unknowns, we also advocate for the continued development of as many plausible scenarios as possible.

The Expected Number of Viable Autocatalytic Sets in Chemical Reaction Systems

The emergence of self-sustaining autocatalytic networks in chemical reaction systems has been studied as a possible mechanism for modeling how living systems first arose. It has been known for several decades that such networks will form within systems of polymers (under cleavage and ligation reactions) under a simple process of random catalysis, and this process has since been mathematically analyzed. In this paper, we provide an exact expression for the expected number of self-sustaining autocatalytic networks that will form in a general chemical reaction system, and the expected number of these networks that will also be uninhibited (by some molecule produced by the system). Using these equations, we are able to describe the patterns of catalysis and inhibition that maximize or minimize the expected number of such networks. We apply our results to derive a general theorem concerning the trade-off between catalysis and inhibition, and to provide some insight into the extent to which the expected number of self-sustaining autocatalytic networks coincides with the probability that at least one such system is present.

A model of the transition to behavioural and cognitive modernity using reflexively autocatalytic networks

This paper proposes a model of the cognitive mechanisms underlying the transition to behavioural and cognitive modernity in the Upper Palaeolithic using autocatalytic networks. These networks have been used to model life's origins. More recently, they have been applied to the emergence of cognitive structure capable of undergoing cultural evolution. Mental representations of knowledge and experiences play the role of catalytic molecules, the interactions among them (e.g. the forging of new associations or affordances) play the role of reactions, and thought processes are modelled as chains of these interactions. We posit that one or more genetic mutations may have allowed thought to be spontaneously tailored to the situation by modulating the degree of (i) divergence (versus convergence), (ii) abstractness (versus concreteness), and (iii) context specificity. This culminated in persistent, unified autocatalytic semantic networks that bridged previously compartmentalized knowledge and experience. We explain the model using one of the oldest-known uncontested examples of figurative art: the carving of the Hohlenstein-Stadel Löwenmensch, or lion man. The approach keeps track of where in a cultural lineage each innovation appears, and models cumulative change step by step. It paves the way for a broad scientific framework for the origins of both biological and cultural evolutionary processes.

The structure of autocatalytic networks, with application to early biochemistry

Metabolism across all known living systems combines two key features. First, all of the molecules that are required are either available in the environment or can be built up from available resources via other reactions within the system. Second, the reactions proceed in a fast and synchronized fashion via catalysts that are also produced within the system. Building on early work by Stuart Kauffman, a precise mathematical model for describing such self-sustaining autocatalytic systems (RAF theory) has been developed to explore the origins and organization of living systems within a general formal framework. In this paper, we develop this theory further by establishing new relationships between classes of RAFs and related classes of networks, and developing new algorithms to investigate and visualize RAF structures in detail. We illustrate our results by showing how it reveals further details into the structure of archaeal and bacterial metabolism near the origin of life, and provide techniques to study and visualize the core aspects of primitive biochemistry.

On the definition of a self-sustaining chemical reaction system and its role in heredity

Background

The ability to self-sustain is one of the essential properties of life. However, a consistent and satisfying definition of self-sustainability is still missing. Currently, self-sustainability refers to either “no-intervention by a higher entity” or “regeneration of all the system’s components”. How to connect self-sustainability with heredity, another essential of life, is another problem, as they are often considered to be independent of each other. Last but not least, current definitions of self-sustainability failed to provide a practical method to empirically discern whether a chemical system is self-sustaining or not.

Results

Here I propose a definition of self-sustainability. It takes into account the chemical reaction network itself and the external environment which is simplified as a continuous-flow stirred tank reactor. One distinct property of self-sustaining systems is that the system can only proceed if molecular triggers (or called, seeds) are present initially. The molecular triggers are able to establish the whole system, indicating that they carry the preliminary heredity of the system. Consequently, life and a large group of fires (and other dissipative systems) can be distinguished. Besides, the general properties and various real-life examples of self-sustaining systems discussed here together indicate that self-sustaining systems are not uncommon.

Conclusions

The definition I proposed here naturally connects self-sustainability with heredity. As this definition involves the continuous-flow stirred tank reactor, it gives a simple way to empirically test whether a system is self-sustaining or not. Moreover, the general properties and various real-life examples of self-sustaining systems discussed here provide practical guidance on how to construct and detect such systems in real biology and chemistry.

Reviewers

This article was reviewed by Wentao Ma and David Baum.

The ambivalent role of water at the origins of life

Life as we know it would not exist without water. However, water molecules not only serve as a solvent and reactant but can also promote hydrolysis, which counteracts the formation of essential organic molecules. This conundrum constitutes one of the central issues in origin of life. Hydrolysis is an important part of energy metabolism for all living organisms but only because, inside cells, it is a controlled reaction. How could hydrolysis have been regulated under prebiotic settings? Lower water activities possibly provide an answer: geochemical sites with less free and more bound water can supply the necessary conditions for protometabolic reactions. Such conditions occur in serpentinising systems, hydrothermal sites that synthesise hydrogen gas via rock-water interactions. Here, we summarise the parallels between biotic and abiotic means of controlling hydrolysis in order to narrow the gap between biochemical and geochemical reactions and briefly outline how hydrolysis could even have played a constructive role at the origin of molecular self-organisation.

Complexity results for autocatalytic network models

A key step in the origin of life is the emergence of a primitive metabolism. This requires the formation of a subset of chemical reactions that is both self-sustaining and collectively autocatalytic. A generic approach to study such processes ('RAF theory') has provided a precise and computationally effective way to address these questions, both on simulated data and in laboratory studies. In this paper, we solve some questions posed in more recent papers concerning the computational complexity of some key questions in RAF theory. In particular, although there is a fast algorithm to determine whether or not a catalytic reaction network contains a subset that is both self-sustaining and autocatalytic (and, if so, find one), determining whether or not sets exist that satisfy certain additional constraints turns out to be NP-hard.

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.

Catalysts, autocatalysis and the origin of metabolism

If life on Earth started out in geochemical environments like hydrothermal vents, then it started out from gasses like CO₂, N₂ and H₂. Anaerobic autotrophs still live from these gasses today, and they still inhabit the Earth's crust. In the search for connections between abiotic processes in ancient geological systems and biotic processes in biological systems, it becomes evident that chemical activation (catalysis) of these gasses and a constant source of energy are key. The H₂–CO₂ redox reaction provides a constant source of energy and anabolic inputs, because the equilibrium lies on the side of reduced carbon compounds. Identifying geochemical catalysts that activate these gasses en route to nitrogenous organic compounds and small autocatalytic networks will be an important step towards understanding prebiotic chemistry that operates only on the basis of chemical energy, without input from solar radiation. So, if life arose in the dark depths of hydrothermal vents, then understanding reactions and catalysts that operate under such conditions is crucial for understanding origins.

Sustainable Growth and Synchronization in Protocell Models

The growth of a population of protocells requires that the two key processes of replication of the protogenetic material and reproduction of the whole protocell take place at the same rate. While in many ODE-based models such synchronization spontaneously develops, this does not happen in the important case of quadratic growth terms. Here we show that spontaneous synchronization can be recovered (i) by requiring that the transmembrane diffusion of precursors takes place at a finite rate, or (ii) by introducing a finite lifetime of the molecular complexes. We then consider reaction networks that grow by the addition of newly synthesized chemicals in a binary polymer model, and analyze their behaviors in growing and dividing protocells, thereby confirming the importance of (i) and (ii) for synchronization. We describe some interesting phenomena (like long-term oscillations of duplication times) and show that the presence of food-generated autocatalytic cycles is not sufficient to guarantee synchronization: in the case of cycles with a complex structure, it is often observed that only some subcycles survive and synchronize, while others die out. This shows the importance of truly dynamic models that can uncover effects that cannot be detected by static graph theoretical analyses.

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.