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.

Córdova F., Doggenweiler C., Maturana H. R., Mpodozis J., Letelier J. C. & Moyano A. (1993) Alternativas de automatización para el guiado autónomo de vehículos cargadores frontales en una mina subterránea. Automática e Innovación 2: 67–63.

Letelier J. C. (2002) The scientific routes of Francisco Varela (1946–2001). In: Roy R., Köppen M., Ovaska S., Furuhashi T. & Hoffmann F. (eds.) Soft computing and industry. Springer, London: xix–xxvii. Fulltext at https://cepa.info/2760

Francisco Varela’s life as a scientist was not an ordinary experience… some of the most interesting problems addressed in Biology during the last century referred to him. His scientific work developed in such “classical” areas as the electronic microscopy of the eye of the honeybee, as well as in more advanced areas, which in his case were many and varied: the nature of living organization, the neurobiology of mind phenomena, the vertebrate vision, immunology.

Letelier J. C., Marín G. & Mpodozis J. (2002) Computing with autopoietic systems. In: Roy R., Köppen M., Ovaska S., Furuhashi T. & Hoffmann F. (eds.) Soft computing and industry: Recent applications.. Springer, London: 67–80. Fulltext at https://cepa.info/2475

In 1973, in the middle of rather unfortunate political events, two Chilean biologists, Humberto Maturana and Francisco Varela, introduced the concept of Autopoietic systems (“auto”= self and ”poiesis” = generating or producing) as a theoretical construct on the nature of living systems centering on two main notions: the circular organization of metabolism and a redefinition of the systemic concepts of structure and organization. This theoretical contruct has found an important place in theoretical biology, but it can also be used as a foundation for a new type of authentically “soft” computing. To understand the main point of our exposition, how Autopoietic systems can be used to compute, it is first necessary to give a brief summary of Autopoietic theory along with the notion of structural coupling.

Letelier J. C., Marin G. & Mpodozis J. (2003) Autopoietic and (M, R) systems. Journal of Theoretical Biology 222(2): 261–272. Fulltext at https://cepa.info/3627

From the many attempts to produce a conceptual framework for the organization of living systems, the notions of (M, R) systems and Autopoiesis stand out for their rigor, their presupposition of the circularity of metabolism, and the new epistemologies that they imply. From their inceptions, these two notions have been essentially disconnected because each has defined its own language and tools. Here we demonstrate the existence of a deep conceptual link between (M, R) systems and Autopoietic systems. This relationship permits us to posit that Autopoietic systems, which have been advanced as capturing the central aspects of living systems, are a subset of (M, R) systems. This result, in conjunction with previous theorems proved by Rosen, can be used to outline a demonstration that the operation of Autopoietic systems cannot be simulated by Turing machines. This powerful result shows the potential of linking these two models. Finally, we suggest that the formalism of (M, R) systems could be used to model the circularity of metabolism.

Letelier J. C., Marin G., Fredes F., Sentis E., Vargas S., Maturana H. R. & Mpodozis J. (2005) Travelling waves of visually induced very fast oscillations in the optic tectum of the pigeon. Journal of Physiology 565P: C115.

Letelier J. C., Marin G., Mpodozis J. & Andrade J. S. (2002) Anticipatory computing with autopoietic and (M, R) systems. In: Soft Computing Systems: Desing, Management and Applications. IOS Press, Amsterdam: 205–211.

Letelier J.-C., Cárdenas M. L. C. & Cornish-Bowden A. (2011) From L’Homme Machine to metabolic closure: Steps towards understanding life. Journal of Theoretical Biology 286: 100–113.

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.

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.