The nature of life has been a topic of interest from the earliest of times, and efforts to explain it in mechanistic terms date at least from the 18th century. However, the impressive development of molecular biology since the 1950s has tended to have the question put on one side while biologists explore mechanisms in greater and greater detail, with the result that studies of life as such have been confined to a rather small group of researchers who have ignored one another’s work almost completely, often using quite different terminology to present very similar ideas. Central among these ideas is that of closure, which implies that all of the catalysts needed for an organism to stay alive must be produced by the organism itself, relying on nothing apart from food (and hence chemical energy) from outside. The theories that embody this idea to a greater or less degree are known by a variety of names, including (M, R) systems, autopoiesis, the chemoton, the hypercycle, symbiosis, autocatalytic sets, sysers and RAF sets. These are not all the same, but they are not completely different either, and in this review we examine their similarities and differences, with the aim of working towards the formulation of a unified theory of life. – Highlights: There have been many isolated attempts to define the essentials of life, A major unifying feature is metabolic closure, Metabolic closure requires some molecules to fulfill more than one function, There can be no hierarchy in the overall organization of a living system.

Key words: Origin of life, (M, R) systems, Autopoiesis, Chemoton, Self-organization

This article analyses the work of Robert Rosen on an interpretation of metabolic networks that he called (M, R) 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 (M, R) 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 (M, R) system with organizational invariance, and we analyse a minimal (three-step) autocatalytic set in the context of (M, R) systems. In addition, by extending Rosen’s construction, we show how one might generate self-referential objects f with the remarkable property f(f)=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.

Key words: (M, R) systems, Metabolic network, Metabolic closure, Infinite regress, Systems biology