Organizational invariance and metabolic closure: Analysis in terms of systems

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 Rosenʹs 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 Rosenʹs 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 Rosenʹs 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 Rosenʹs insight, although not yet in an easily workable form, represents a valuable tool for understanding metabolic networks.