Organizational invariance and metabolic closure: Analysis in terms of M;R systems
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2005-08-24Metadata
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Letelier Parga, Juan
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Organizational invariance and metabolic closure: Analysis in terms of M;R systems
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Abstract
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 ðfÞ 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.
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This work was supported by Fondecyt 1030371 (JCL),
Fondecyt 1040444 (JSA) and the CNRS (AC-B, MLC).
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Journal of Theoretical Biology 238 (2006) 949–961
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