# Key word "metabolic circularity"

Cárdenas M. L. C., Letelier J.-C., Gutierrez C., Cornish-Bowden A. & Soto-Andrade J. (2010) Closure to efficient causation, computability and artificial life. Journal of Theoretical Biology 263(1): 79–92. https://cepa.info/3631

Cárdenas M. L. C., Letelier J.-C., Gutierrez C., Cornish-Bowden A. & Soto-Andrade J.
(

2010)

Closure to efficient causation, computability and artificial life.
Journal of Theoretical Biology 263(1): 79–92.
Fulltext at https://cepa.info/3631
The major insight in Robert Rosen’s view of a living organism as an (M, R)-system was the realization that an organism must be “closed to efficient causation”, which means that the catalysts needed for its operation must be generated internally. This aspect is not controversial, but there has been confusion and misunderstanding about the logic Rosen used to achieve this closure. In addition, his corollary that an organism is not a mechanism and cannot have simulable models has led to much argument, most of it mathematical in nature and difficult to appreciate. Here we examine some of the mathematical arguments and clarify the conditions for closure.

Letelier J.-C., Soto-Andrade J., Guíñez Abarzúa F., Cornish-Bowden A. & Luz Cárdenas M. (2006) Organizational invariance and metabolic closure: analysis in terms of (M, R) systems. Journal of Theoretical Biology 238(4): 949–961. https://cepa.info/3628

Letelier J.-C., Soto-Andrade J., Guíñez Abarzúa F., Cornish-Bowden A. & Luz Cárdenas M.
(

2006)

Organizational invariance and metabolic closure: analysis in terms of (M, R) systems.
Journal of Theoretical Biology 238(4): 949–961.
Fulltext at https://cepa.info/3628
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.

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