PlanetPhysics/R Systems Category of M

Metabolic-Replication Systems

[[../RobertRosen/|Robert Rosen]] introduced metabolic--repair models , or $(M,R)$-systems in mathematical biology ([[../SystemsBiology/|abstract relational biology]]) in 1957 (); such [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|systems]] will be here abbreviated as $MR$-systems, (or simply $MR$'s). Rosen, then represented the $MR$'s in terms of [[../Cod/|categories]] of sets, deliberately selected without any structure other than the discrete topology of sets .

Theoreticians of life's origins postulate that Life on Earth has begun with the simplest possible organism, called the primordial . Mathematicians interested in biology and this important question of the minimal living organism have attempted to define the functional [[../Bijective/|relations]] that would have made life possible in a such a minimal system-- a grandad and granma of all living organisms on Earth.

The simplest $MR$-system is a relational model of the primordial organism which is defined by the following \htmladdnormallink{categorical sequence {http://planetphysics.us/encyclopedia/HomologicalSequence2.html} (or [[../TrivialGroupoid/|diagram]]) of sets and set-theoretical mappings}: $f:A\to B,\varphi :B\to Ho{m}_{MR}(A,B)$, where $A$ is the set of inputs to the $MR$-system, $B$ is the set of its outputs, and $\varphi$ is 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, $Ho{m}_{MR}(A,B)$ is defined as the set of all such metabolic (set-theoretical) mappings (occasionally written incorrectly by some authors as $\left\{f\right\}$).

A general $(M,R)$-system was defined by Rosen (1958a,b) as the network or [[../Cod/|graph]] of the metabolic and repair components that were specified above in Definition 0.1 ; such components 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 mappings bettwen $(M,R)$-systems are defined as the the metabolic and repair set-theoretical mappings, such as $f$ and $\varphi$ (specified in Definition 0.1 ); moreover, there is also a finite number of sets (just like those that are defined as in Definition 0.1 ): $A_i,B_i$, whereas $f\in Ho{m}_{M{R}_i}(A_i,B_i)$ and $\varphi \in Ho{m}_{M{R}_i}[B,Ho{m}_{M{R}_i}(A_i,B_i)]$, with $i\in I$, and $I$ being a finite index set, or directed set, with $(f,\varphi )$ being a finite number of distinct metabolic and repair components pairs. Alternatively, one may think of a a general $MR$-system as being `made of' a finite number $N$ of interconnected $M{R}_i$, metabolic-repair [[../RModule/|modules]] with input sets $A_i$ and output sets $B_i$. To sum up: a general MR-system can be defined as a family of interconnected quartets : ${\{(A_i,B_i,f_i,{\varphi }_i)\}}_{i\in I}$, where $I$ is an index set of integers $i=1,2,...,n$.

Category of (M,R)--systems

A category of $(M,R)$-system quartet modules , ${\{(A_i,B_i,f_i,{\varphi }_i)\}}_{i\in I}$, with I being an index set of integers $i=1,2,...,n$, is a [[../Cod/|small category]] of sets with set-theoretical mappings defined by the MR-morphisms between the quarted modules ${\{(A_i,B_i,f_i,{\varphi }_i)\}}_{i\in I}$, and also with repair components defined as ${\varphi }_i\in Ho{m}_{M{R}_i}[B,Ho{m}_{M{R}_i}(A_i,B_i)]$, with the $(M,R)$-morphism [[../Cod/|composition]] defined by the usual composition of [[../Bijective/|functions]] between sets.

With a few, additional notational changes it can be shown that the category of $(M,R)$-systems is a subcategory of the [[../AAT/|category of automata]] (or [[../AAT/|sequential machines]]), ${𝒮}_{[M,A]}$ ().

For over two decades, [[../RobertRosen/|Robert Rosen]] developed with several coworkers the MR-systems theory and its applications to life sciences, medicine and general systems theory. He also considered biocomplexity to be an `emergent', defining feature of organisms which is not reducible in terms of the [[../FCS3/|molecular structures]] (or molecular components) of the organism and their physicochemical interactions. However, in his last written book in 1997 on "Essays on Life Itself", published posthumously in 2000" , Robert Rosen finally accepted the need for representing organisms in terms of [[../TrivialGroupoid/|categories with structure]] that entail biological functions, both metabolic and repair ones. Note also that, unlike Rashevsky in his [[../TheoryOfOrganismicSets/|theory of organismic sets]], Rosen did not attempt to extend the $MR$s to modeling societies, even though with appropriate modifications of generalized $(M,R)$-system categories with structure (), this is feasible and yields meaningful mathematical and sociological results. Thus, subsequent publications have generalized MR-system (GMRs) and have studied the fundamental, mathematical properties of [[../CategoryOfLogicAlgebras/|algebraic categories]] of GMRs that were constructed functorially based on the {Yoneda-Grothendieck Lemma} and construction. Then it was shown that such algebraic categories of GMRs are Cartesian closed . Several molecular biology realizations of GMRs in terms of [[../FCS3/|DNA]], RNAs, enzymes, $RNA\to DNA$-reverse trancriptases, and other biomolecular components were subsequently introduced and discussed in ref. in terms of non-linear \htmladdnormallink{genetic network {http://planetphysics.us/encyclopedia/GeneNetDigraph.html} models} in many-valued, $L{M}_n$ logic algebras (or \htmladdnormallink{algebraic category $ℒℳ$ of $L{M}_n$ logic algebras}{http://planetphysics.us/encyclopedia/AlgebraicCategoryOfLMnLogicAlgebras.html}).

If simple $(M,R)$-systems are considered as sequential machines or automata the category of $(M,R)$-systems and $(M,R)$-system [[../TrivialGroupoid/|homomorphisms]] is a subcategory of the automata category. However, when $(M,R)$-systems are considered together with their [[../NewtonianMechanics/|dynamic]] [[../CategoricalGroupRepresentation/|representations]] the category of dynamic $(M,R)$-systems is no longer a subcategory of the category of automata.