PlanetPhysics/Abstract Relational Biology ARB

This is a contributed topic on abstract relational biology and its close connections to the theory of categories, especially the concrete and abstract categories of sets, $$Set$$.

Introduction
Abstract relational biology (ARB) is an area of mathematical or theoretical biology in which networks of linked physiological and biochemical functions of living cells and multi-cellular organisms are defined over sets $$S_i$$ and their elements $$e_j \in S_i$$ representing for example specific metabolic products or other biomolecules; such organismic sets  $$S_i$$ are then assembled in set-related mathematical constructions-such as categories of sets- by making abstraction of the underlying, physical and (bio) molecular structures that implement the cellular or physiological functions of a living organism. Thus, the early formulations of ARB by Nicolas Rashevsky were based on set theory, molecular set theory, and the classical logic of predicates and logical propositions. Therefore, a natural foundation for ARB would currently be the modern relation theory. The relational structure of an organism is strongly emphasized over the anatomical structure and the molecular structure of cells and other components of the organism such as the chromosomes, genome, mitochondria, endoplasmic reticulum, membranes, and so on. However, the relational structure is defined in a general, mathematical sense, such as the relational structure of a category; the initial choice made by Robert Rosen was that of the category of sets, $$\mathbf{Set}$$ (or $$Ens$$). The relational structure was thus limited, both in terms of the objects' relational structure as well as in terms of the relational structures definable by morphisms that were restricted to set-theoretical mappings, or maps between sets. Subsequent developments (beginning in 1968) extended ARB to categories with structure, thus not limiting ARB objects to being sets, or the morphisms to being maps between sets. Furthermore, classes were introduced instead of sets, thus not limiting the categorical framework to small categories.

Since 1952 there have been two major, set-based theories in abstract relational biology that are concisely outlined next.

\section{Organismic Set Theory and Abstract--Relational, \\ Metabolic--Replication, $$(M,R)$$--Systems}

Brief history
Two major proponents were Nicolas Rashevsky (up to 1973) who is one of the founders of mathematical biophysics and mathematical biology, and Robert Rosen, his former PhD student at the University of Chicago. Nicolas Rashevsky formulated the mathematical theory of organismic sets ($$OS$$) that are organized beginning with the genetic level, continuing to the cellular level, and then to higher levels of multi-cellular organization, activities and products; his theory was similarly formulated for societies organized at such different levels. Subsequently, it was shown that Rashevsky's organismic sets can be represented in terms of categories of algebraic theories \cite {ICB70}.

Robert Rosen introduced (metabolic--repair) models, or $$(M,R)$$-systems in 1957 ; such systems will be here abbreviated as $$MR$$-systems, (or simply $$MR$$'s). Rosen, then represented the $$MR$$'s in terms of categories of sets, deliberately selected without any structure other than the discrete topology of sets. He also considered biocomplexity to be an emergent, defining feature of organisms which is not reducible in terms of the molecular structures (or molecular components) of the organism and their physicochemical interactions.

Basic ARB concepts
The simplest $$MR$$-system represents a relational model of the primordial organism which is defined by the following \htmladdnormallink{categorical sequence {http://planetphysics.us/encyclopedia/HomologicalSequence2.html} (or diagram) of sets and set-theoretical mappings}: $$f: A \rightarrow B, \phi: B \rightarrow Hom_{MR}(A,B)$$, where $$A$$ is the set of inputs to the $$MR$$-system, $$B$$ is the set of its outputs, and $$\phi$$ is defined as the `repair map', or $$R$$-component, of the $$MR$$-system which associates to a certain product, or output $$b$$, the metabolic component (such as an enzyme, E, for example) represented by the set-theoretical mapping $$f$$. Then, $$Hom_{MR}(A,B)$$ is defined as the set of all such metabolic components represented by set-theoretical) mappings $$f$$. (occasionally written incorrectly as $$\left\{f\right\}$$)

A general $$(M,R)$$-system was defined by Rosen (1958a,b) as the category of the metabolic and repair components (that were specified above in Definition 0.1 ), which are networked in a complex, abstract `organism' defined by all the abstract relations and connecting maps between the sets specifying all the metabolic and repair components of such a general, abstract model of the biological organism. The morphisms of the $$(M,R)$$-system category are the metabolic and repair set-theoretical mappings, such as $$f$$ and $$\phi$$, and its objects are the sets $$A_i, B_i$$, whereas $$f \in Hom_{MR_i}(A_i,B_i)$$ and $$\phi \in Hom_{MR_i}[B, Hom_{MR_i}(A_i,B_i)]$$, with $$i \in I$$, and $$I$$ being a finite index set, or directed set.

Remarks

With a few, additional notational changes it can be shown that the $$(M,R)$$-system category is a subcategory of the category of automata (or sequential machines; ). However, in his last published book in 1997 on "Essays on Life Itself", Robert Rosen finally accepted the need for representing organisms in terms of categories with structure that entail biological functions, both metabolic and repair ones. Note also that, unlike Rashevsky in his theory of organismic sets, Rosen did not attempt to extend the $$MR$$s to modeling societies, even though with appropriate modifications, such as the introduction of Rosetta biogroupoid structures in generalized $$(M,R)$$-system categories with structure, this is feasible and yields meaningful mathematical and sociological results.