Narrow Results

This Review concerns the Algebraic Relational Theory of living systems as developed by Leguizamón and co-workers since 1973 and based on the categorical approach championed by Robert Rosen and his (ℳ, ℛ) systems. The Algebraic Relational Theory expanded, by using new mathematical developments such as lattice theory, the relational notions found in (ℳ, ℛ) systems with concepts such as material physical nature and extrinsic energy that reflect aspects of physical reality not captured by standard physical concepts like mass, energy or time. The Algebraic Relational Theory is another effort in the direction of developing a theory about the notion of biological organization.

The Review contains the foundations of the theory, the principal mathematical developments, and applications of the theoretical results to different biological problems. They are the substantial base from which new ideas could be developed. It also includes a synthesis of initial experiments connected with this area of research.

Carlos Leguizamón died in 1998. We had worked together for 20 years.

This paper is the first one of a series devoted to the analysis of metabolic networks. Its aim is to establish the theoretical framework for this analysis.

Two different lines of research are considered: the one about metabolism-repair systems ((ℳ, ℛ)), introduced by Robert Rosen as an abstract representation of cell metabolic activity, and the concept of autopoiesis developed by Humberto Maturana and Francisco Varela.

Both concepts have been recently connected by Letelier et al., determining that the set of autopoietic systems is a subset of the set of general abstract (ℳ, ℛ) systems. In fact, every specific (ℳ, ℛ) system is an autopoietic one, being the boundary, which specifies each system as a unity, the main element of autopoietic systems which is not formalized in Rosen's representation.

This paper introduces the definition of boundary — a physical boundary and a functional one — for (ℳ, ℛ) systems in the context of a representation using category theory.

The concept of complete (ℳ, ℛ) system is also introduced by means of a process of completion in categories which is functorial, natural and universal.

An algebraic relational theory is being developed in order to represent biological systems. As a result, it is possible to explain, in terms of qualitative relationships, the behaviors of such systems. This paper deals with the periodic continuous responses of a new state derived from the interaction between low energies and matter. This effect was predicted by categoric developments of the algebraic relational theory.

The property of muscle movement P_m is a central functional property for the qualitative interpretations of behaviors of biological systems. In this paper, based on the example of the muscle, it is shown how relational properties can be identified by responses derived from the functional organization. At a certain level of organization, the manifestation of the property P_m is analyzed in terms of qualitative relationships between the myosin and actin fibres of muscles. A relatively pseudo complemented lattice of nine elements shows algebraic relations in connection with the interaction of the fibres. The response coming from the lattice is in correspondence with the quantitative one expressed by Hill’s equation for muscles.

It is shown how the biological reality correlates with an algebraic modification from a pseudo-Boolean structure for watering processes in normal cells up to a non-modular structure assigned to water interactions in malignant cells. A set of mathematical propositions suggests how to deviate this type of cancer process to new structures mainly maintaining those water structures resulting from the cooperativity between water molecules generated by a surface. A set of disquisitions is made: about the meaning of the change of algebra; on the dual Heyting arrow operations acting for the algebraic triggering of the cancer process; on the loss of energy in the cancer process and about the enhanced value of energy as becoming from the new structures to deviate cancer.

An approach which has the purpose to catch what characterizes the specificity of a living system, pointing out what makes it different with respect to physical and artificial systems, needs to find a new point of view — new descriptive modalities. In particular it needs to be able to describe not only the single processes which can be observed in an organism, but what integrates them in a unitary system. In order to do so, it is necessary to consider a higher level of description which takes into consideration the relations between these processes, that is the organization rather than the structure of the system. Once on this level of analysis we can focus on an abstract relational order that does not belong to the individual components and does not show itself as a pattern, but is realized and maintained in the continuous flux of processes of transformation of the constituents. Using Tibor Ganti's words we call it "Order in the Nothing". In order to explain this approach we analyse the historical path that generated the distinction between organization and structure and produced its most mature theoretical expression in the autopoietic biology of Humberto Maturana and Francisco Varela. We then briefly analyse Robert Rosen's (M,R)-Systems, a formal model conceptually built with the aim to catch the organization of living beings, and which can be considered coherent with the autopoietic theory. In conclusion we will propose some remarks on these relational descriptions, pointing out their limits and their possible developments with respect to the structural thermodynamical description.