Calenbuhr V., Bersini H., Stewart J. & Varela F. J. (1995) Natural tolerance in a simple immune network. Journal of Theoretical Biology\>Journal of Theoretical Biology 177: 199–213. Fulltext at https://cepa.info/1998

The following basic question is studied here: In the relatively stable molecular environment of a vertebrate body, can a dynamic idiotypic immune network develop a natural tolerance to endogenous components? The approach is based on stability analyses and computer simulation using a model that takes into account the dynamics of two agents of the immune system, namely B-lymphocytes and antibodies. The study investigates the behavior of simple immune networks in interaction with an antigen whose concentration is held constant as a function of the symmetry properties of the connectivity matrix of the network. Current idiotypic network models typically become unstable in the presence of this type of antigen. It is shown that idiotypic networks of a particular connectivity show tolerance towards auto-antigen without the need for ad hoc mechanisms that prevent an immune response. These tolerant network structures are characterized by aperiodic behavior in the absence of auto-antigen. When coupled to an auto-antigen, the chaotic attractor degenerates into one of several periodic ones, and at least one of them is stable. The connectivity structure needed for this behavior allows the system to adopt particular dynamic concentration patterns which do not lead to an unbounded immune response. Possible implications for the understanding of autoimmune disease and its treatment are discussed.

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\>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.

Detours V., Bersini H., Stewart J. & Varela F. J. (1994) Development of idiotypic network in shape space. Journal of Theoretical Biology\>Journal of Theoretical Biology 170: 401–414.

Based upon the shape-space formalism, a model of an idiotypic network including both bound and free immunoglobulins is simulated. Our point of interest is the network development in the context of self antigens. The investigations are organized around simulations initiated by various spatial configurations of antigens; the behavior of the system with respect to antigens is analyzed in terms of morphogenetic processes occurring in the shape space. For certain values of the parameters, the network expands by traveling waves. The resulting spatial pattern is a partition of the shape space into zones where introduction of an antigen entails an infinite growth of the clones binding to it, and into zones where, on the contrary, the anti-antigen idiotypes decrease. Among the parameter combinations tested, some produce a partition that remains static whereas others produce a partition that changes in time. For other values of the parameters, the patterns generated do not partition shape space into zones; in these cases, it is observed that the system systematically explodes when an antigen is present.

Heschl A. (1990) L = C. A simple Equation with astonishing consequences. Journal of Theoretical Biology\>Journal of Theoretical Biology 185: 13–40. Fulltext at https://cepa.info/3050

At first sight, life and cognition only seem to deal with each other in an indirect way, the former as is perhaps necessary as a precondition for the mere possibility of the latter. However, looking at the question more closely and especially when we include the central problem of the emergence of life from inanimate pre-stages, we arrive at a reasonable conclusion of complete identity between life process and cognitive act: through the spontaneous formation of living systems, i.e. of exceptionally stable processual structures far from the thermodynamic equilibrium, external influence – which only now can be opposed to internal correlations – is transformed into an adaptive integration or, in cognitive terms, into a meaningful interpretation by a (within limited conditions of stability surviving) living system. Thereby the purely quantitative notion of “information” has to be subjected to a decisive relativization, since it is not before the system formation itself that it makes sense to speak of information. Far-reaching conceptual consequences follow from the possibility of conclusively demonstrating the fundamental equation of life and cognition.

Letelier J. C., Marin G. & Mpodozis J. (2003) Autopoietic and (M, R) systems. Journal of Theoretical Biology\>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., Cárdenas M. L. C. & Cornish-Bowden A. (2011) From L’Homme Machine to metabolic closure: Steps towards understanding life. Journal of Theoretical Biology\>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\>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.

Montévil M. & Mossio M. (2015) Biological organisation as closure of constraints. Journal of Theoretical Biology\>Journal of Theoretical Biology 372: 179–191. Fulltext at https://cepa.info/3629

We propose a conceptual and formal characterisation of biological organisation as a closure of constraints. We first establish a distinction between two causal regimes at work in biological systems: processes, which refer to the whole set of changes occurring in non-equilibrium open thermodynamic conditions, and constraints, those entities which, while acting upon the processes, exhibit some form of conservation (symmetry) at the relevant time scales. We then argue that, in biological systems, constraints realise closure, i.e. mutual dependence such that they both depend on and contribute to maintaining each other. With this characterisation in hand, we discuss how organisational closure can provide an operational tool for marking the boundaries between interacting biological systems. We conclude by focusing on the original conception of the relationship between stability and variation which emerges from this framework. – Highlights:Biological systems realise both organisational closure and thermodynamic openness, Organisational closure is a closure of constraints, Constraints exhibit conservation (symmetry) at the relevant time scales, Closure draws the boundaries between interacting biological systems, Closure is a principle of biological stabilisation.

Mossio M., Longo G. & Stewart J. (2009) A computable expression of closure to efficient causation. Journal of Theoretical Biology\>Journal of Theoretical Biology 257(3): 489–498. Fulltext at https://cepa.info/3630

In this paper, we propose a mathematical expression of closure to efficient causation in terms of λ-calculus, we argue that this opens up the perspective of developing principled computer simulations of systems closed to efficient causation in an appropriate programming language. An important implication of our formulation is that, by exhibiting an expression in λ-calculus, which is a paradigmatic formalism for computability and programming, we show that there are no conceptual or principled problems in realizing a computer simulation or model of closure to efficient causation. We conclude with a brief discussion of the question whether closure to efficient causation captures all relevant properties of living systems. We suggest that it might not be the case, and that more complex definitions could indeed create crucial some obstacles to computability.

Sjölander S. (1997) On the evolution of reality: Some biological prerequisites and evolutionary stages. Journal of Theoretical Biology\>Journal of Theoretical Biology 187: 595–600. Fulltext at https://cepa.info/4132

The discussion of the evolutionary origins of consciousness has largely been concentrated to the human mind, and it is only in recent years that a comparative ethological view has come into play. Even here, a tendency has been to look mainly at the primates. There is a vast literature that discusses the differences between human consciousness and cognition, compared with that of the other primates, but much less attention has been given to the fact that evolutionary gaps – fulgurations, emergences, new systems – have occurred at many stages in the evolution of cognition. More especially, the complexity of rather simple cognitive systems in lower animals has been underestimated, as well as the necessary prerequisites for a cognition worthy of the name to exist. Of particular interest in the discussion has been the views from evolutionary epistemology and radical constructivism, since they support the ethologically founded view that mind representations do not depict reality, but are adaptations for a successful way of behaving in the physical world, that reality in this sense is in the mind, that there are many realities, varying for different species – rich or poor in complexity – but all of them basically of the same nature. Even such human achievements as mathematics or logic may thus be seen as specific cognitive adaptions in our species, not as independent aspects of the physical world.