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Organizational invariance and metabolic closure:
analysis in terms of (M,R) systems

This page is the first of a series containing most of the article Organizational invariance and metabolic closure: analysis in terms of (M,R) systems by Juan-Carlos Letelier, Jorge Soto-Andrade, Flavio Guíñez Abarzúa, Athel Cornish-Bowden and María Luz Cárdenas, published in the Journal of Theoretical Biology 238, 949–961 (2006). A PDF file is also available.


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.

1. Introduction

1A measure of this admiration can be seen in a recent paper with the unexpected title of Robert Rosen (1934-1998): A snapshot of biology’s Newton (Mikulecky, 2001). More realistically, the mathematician John Casti recently said The work of Rosen will keep scholars busy for decades (Casti, 2002).

2Rosen summarized his work in his opaque but important book Life Itself: a Comprehensive Inquiry into the Nature, Origin and Fabrication of Life (Rosen, 1991).

Mikulecky, D., 2001. Robert Rosen (1934-1998): a snapshot of biology’s Newton. Comput. Chem. 25, 317–327.

Casti, J., 2002. Biologizing control theory: How to make a control system come alive. Complexity 7, 10–13.

The massive, extended and often cryptic scientific output of Robert Rosen poses a scientific dilemma. While the great majority of biologists are unaware of his work, a few regard it as being of the kind to be expected once in a thousand years1, and a few others are trying to bring understanding of it to the point where its importance for biology can be objectively assessed2. An essential step in gauging its relevance is to understand the core of Rosen’s thinking, which concerns a formal theory of metabolic networks and the notion of circularity or metabolic closure. For this reason we shall concentrate here on his investigations of metabolic systems and his definition of living systems; we shall not touch on some other aspects of his work, such as his epistemological research and his definition of complex systems.

Rosen, R., 1958a. A relational theory of biological systems. Bull. Math. Biophys. 20, 245–341.

Rosen, R., 1958b. The representation of biological systems from standpoint of the theory of categories. Bull. Math. Biophys. 20, 317–341.

Rosen, R., 1959. A relational theory of biological systems II. Bull. Math. Biophys. 21, 109–128.

Rosen, R., 1966a. Abstract biological systems as sequential machines III: some algebraic aspects. Bull. Math. Biophys. 28, 141–148.

Rosen, R., 1967. Further comments on replication in (M,R) systems. Bull. Math. Biophys. 29, 91–94.

Rosen, R., 1971. Some realizations of (M,R) systems and their interpretation. Bull. Math. Biophys. 33, 303–319.

Rosen, R., 1972. Some relational cell models: the metabolism-repair system. In Rosen, R., ed., Foundations of mathematical biology. Academic Press, New York.

Rosen, R., 1991. Life Itself. Columbia University Press, New York.

Rosen, R., 2000. Essays on Life Itself. Columbia University Press, New York.

The central motto of Rosen’s research programme (Organisms are closed to efficient causes) (Rosen, 1991) can be traced to an insight on the nature of cellular metabolic networks (Rosen, 1958ab, 1959); this consists of a semi-formal method to explain how the network of biochemical processes that constitutes metabolism bootstraps itself without the help of external agents generated outside the network, thus keeping cell organization invariant in spite of continuous structural change. He based a large part of the development of his ideas on a branch of mathematics known as Category Theory, and in mathematical terms his major insight appears as a peculiar result about sets and admissible transformations between them. Rosen appears to have been aware of the rather peculiar nature of this result and he gave several slightly different proofs with slightly different interpretations (Rosen, 1958b, 1959, 1966a, 1967, 1971, 1972, 1991, 2000); maddeningly, however, he never provided any concrete examples, not even mathematical ones, let alone ones that would be intelligible to biologists. This result, here called Rosen’s Central Result, is, without doubt, the core of his view of theoretical biology, as well as the unavoidable starting point for analysing his views on complexity.

Casti, J., 2002. Biologizing control theory: How to make a control system come alive. Complexity 7, 10–13.

Landauer, C., Bellman, K., 2002. Theoretical biology: organisms and mechanisms. AIP Conference Proceedings 627, 59–70.

Nomura, T., 2002. Formal description of autopoiesis for analytical models of life and social systems. In Standish, R., Bedau, M., Abbass, H., editors, Artificial Life VIII, pages 15–18. MIT Press, Cambridge, Massachusetts.

Wolkenhauer, O., 2001. Systems biology: the reincarnation of systems theory applied to biology? Briefings Bioinf. 2, 258–270.

Wolkenhauer, O., 2002. Mathematical modelling in the post-genome era: understanding genome expression and regulation—a systems theoretic approach. BioSystems 65, 1–18.

The central result has been used recently in attempts to expand his ideas to other areas, such as bioinformatics (Wolkenhauer, 2001, 2002), control theory (Casti, 2002), and even sociology (Nomura, 2002). Although these recent publications seem to imply increasing acceptance of Rosen’s ideas, we must point out that recently his entire approach has been called into question and even declared false (Landauer and Bellman, 2002). Because of this, and because it is such a special result, with important implications for theoretical biology and computer science, we have found it necessary to revisit and clarify it, trying to give some examples and connect it to other theoretical ideas. In doing so, we have changed somewhat the nomenclature and notations that he used. Renaming established terms is not, of course, something to be done lightly, but we believe it to be unavoidable in this instance because Rosen used a number of words, such as replication, in ways that can only confuse readers familiar with their usual meanings in biology. We shall, however, take care to be explicit whenever we alter one of his terms, and will note why we regard it as unsatisfactory.

Joslyn, C., 1993. Review of Life Itself. Int. J. Gen. Systems 21, 394–402.

Letelier, J.-C., Soto-Andrade, J., Guíñez Abarzúa, F., Cornish-Bowden, A., Cárdenas, M. L., 2004. Metabolic closure in (M,R) systems. In Pollack, J., Bedau, M., Husbands, P., Ikegami, T., Watson, R. A., editors, Artificial Life IX, pages 450–455. MIT Press, Cambridge, Massachusetts.

Pierce, B., 1991. Basic category theory for computer scientists. MIT Press, Cambridge, Massachusetts.

The structure of this paper is as follows: In Sections 2 and 3 we introduce (M,R) systems, in Section 4 we enunciate the central result, and in Section 5 we set out its mathematical context. In Section 6 we show two examples of (M,R) systems, and in Section 7 we introduce what we consider a possible generalization of Rosen’s result about infinite regress and closure. Finally, in Section 8, we discuss the many implications of Rosen’s ideas for the study of metabolic networks. Although category theory was central to Rosen’s thinking, we do not use it here because is this that gives much of his writing its abstract character, making it opaque to most biologists. Nonetheless, a full understanding of his work requires some appreciation of what categories are and how they are relevant to his analysis of metabolism (Pierce, 1991; Joslyn, 1993). Some of these ideas have been briefly addressed in a previous article (Letelier et al., 2004).