(M,R)-systems and RAF sets: common ideas, tools and projections (original) (raw)
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Background: In previous work, RAF theory has been developed as a tool for making theoretical progress on the origin of life question, providing insight into the structure and occurrence of self-sustaining and collectively autocatalytic sets within catalytic polymer networks. We present here an extension in which there are two "independent" polymer sets, where catalysis occurs within and between the sets, but there are no reactions combining polymers from both sets. Such an extension reflects the interaction between nucleic acids and peptides observed in modern cells and proposed forms of early life.
AUTOPOIESIS AND (M,R) SYSTEMS IN METABOLIC NETWORKS
Metabolism-repair systems ((M,R)) were introduced by Robert Rosen as an abstract representation of cell metabolic activity. The representation was obtained in the context of Relational Biology, which means that organization prevails over the physico- chemical structure of the components involved. This fact was determinant for algebraically formalizing (M,R) systems using the theory of categories. Two elements are considered in the construction of (M,R) systems: the metabolic activity (M) and the repair functions (R) acting on the unities of the metabolic process. The metabolic system M is considered as an input-output system. In the categorical representation, inputs and outputs are the objects of the category and the processes connecting these elements are represented by the arrows of the category. Autopoiesis is a concept developed by Humberto Maturana and Francisco Varela in order to analyze the nature of living systems. It takes into account the circular organization of metaboli...
Metabolic closure in (M,R) systems
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The work of Robert Rosen, related to metabolic networks called (MR) systems is reviewed and clarified. We study the algebraic formulation of (M,R) systems particularly the mapping , which encapsulates Rosen's solution to the problem of metabolic closure and infinite regress. We construct an arithmetical example of an (MR) system and also an (MR) system based on a three-step minimal metabolism.
Tractable models of self-sustaining autocatalytic networks
2018
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 F-generated' (RAF) set. In this paper, we first investigate a simplified setting for studying RAFs, which are nevertheless relevant to real biochemistry and allows for a more 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 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 ...
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 synchronised 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 organisation 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 visualise 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...
Chasing the tail: The emergence of autocatalytic networks
Bio Systems, 2017
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.
Autocatalytic networks in biology: structural theory and algorithms
Journal of The Royal Society Interface
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...
Organizational invariance and metabolic closure: Analysis in terms of ( M , R ) systems
Journal of Theoretical Biology, 2006
This article analyses the work of Robert Rosen on an interpretation of metabolic networks that he called ðM; RÞ systems. His main contribution was an attempt to prove that metabolic closure (or metabolic circularity) could be explained in purely formal terms, but his work remains very obscure and we try to clarify his line of thought. In particular, we clarify the algebraic formulation of ðM; RÞ systems in terms of mappings and sets of mappings, which is grounded in the metaphor of metabolism as a mathematical mapping. We define Rosen's central result as the mathematical expression in which metabolism appears as a mapping f that is the solution to a fixed-point functional equation. Crucially, our analysis reveals the nature of the mapping, and shows that to have a solution the set of admissible functions representing a metabolism must be drastically smaller than Rosen's own analysis suggested that it needed to be. For the first time, we provide a mathematical example of an ðM; RÞ system with organizational invariance, and we analyse a minimal (three-step) autocatalytic set in the context of ðM; RÞ systems. In addition, by extending Rosen's construction, we show how one might generate self-referential objects f with the remarkable property f ðf Þ ¼ f , where f acts in turn as function, argument and result. We conclude that Rosen's insight, although not yet in an easily workable form, represents a valuable tool for understanding metabolic networks. r
PLOS Computational Biology
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 an...