AbstractRelationalBiologyARB 2009-01-15 20:31:10 2009-01-15 20:31:10 Topic % this is the default PlanetPhysics preamble. as your knowledge % there are many more packages, add them here as you need them % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@ }}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \section{Abstract relational biology (ARB):} The area of mathematical or theoretical biology in which the physiological and biochemical functions of living cells and multi-cellular organisms are considered in sets of elements, or in set-related mathematical constructions-such as \PMlinkname{categories of sets}{CategoryOfSets}- by \emph{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 were based on set theory and molecular set 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. 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) expanded \emph{ARB} to categories with structure and also `large' categories, thus not limiting ARB objects to being sets, or the morphisms to being maps between sets. Since 1952 there have been two major, \emph{\PMlinkname{set-based}{SetTheory} theories in abstract relational biology} that are concisely outlined next. \textbf{Two major theories: organismic set theory and metabolic--replication systems theory defined in the} \PMlinkname{Categories of Sets}{CategoryOfSets}. 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 {\em algebraic theories} \cite {ICB70}. Robert Rosen introduced Metabolic--Repair models, or $(M,R)$-systems in 1957 (\cite{RRosen1, RRosen2}); 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 {\em 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. \begin{definition} The simplest $MR$-system represents a relational model of the primordial organism which is defined by the following \emph{categorical sequence (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 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' (set-theoretical) mappings $f$ \end{definition} (occasionally written incorrectly as $\left\{f\right\}$) \begin{definition} A {\em general $(M,R)$-system} was defined by Rosen (1958a,b) as the category of the metabolic and repair components (that were specified above in \textbf{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 \emph{$(M,R)$-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. \end{definition} \textbf{Remark:} With a few, additional notational changes it can be shown that the \PMlinkname{$(M,R)$-system category}{CategoryOfMRSystems3} is a subcategory of the category of automata (or sequential machines) (\cite{ICB73, ICBM74}). However, in his last published book in 1997 on \emph{``Essays on Life Itself''}, Robert Rosen finally accepted the need for representing organisms in terms of {\em 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 (\cite{BBGG2k6,BBG2k7}) in generalized $(M,R)$-system categories with structure (\cite{ICB73,ICBM74,ICB87}), this is feasible and yields meaningful mathematical and sociological results. \begin{thebibliography}{99} \bibitem{Rashevsky1-yr1965} Rashevsky, N.: 1965, The Representation of Organisms in Terms of Predicates, {\em Bulletin of Mathematical Biophysics} \textbf{27}: 477-491. \bibitem{Rashevsky2-1969} Rashevsky, N.: 1969, Outline of a Unified Approach to Physics, Biology and Sociology., {\em Bulletin of Mathematical Biophysics} \textbf{31}: 159--198. \bibitem{ICB70} Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems.,{\em Bulletin of Mathematical Biophysics}, \textbf{32}: 539-561. \bibitem{ICB73} Baianu, I.C.: 1973, Some Algebraic Properties of \emph{\textbf{(M,R)}} -- Systems.,{\em Bulletin of Mathematical Biophysics} \textbf{35}, 213-217. \bibitem{ICBM74} Baianu, I.C. and M. Marinescu: 1974, On A Functorial Construction of \textbf{(M,R)}-- Systems., {\em Revue Roumaine de Math\'ematiques Pures et Appliqu\'ees} \textbf{19}: 388-391. \bibitem{ICB6} Baianu, I.C.: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory., {\em Bulletin of Mathematical Biology}, \textbf{39}: 249-258. \bibitem{ICB2} Baianu, I.C.: 1980, Natural Transformations of Organismic Structures., {\em Bulletin of Mathematical Biology},\textbf{42}: 431-446. \bibitem{ICB87} Baianu, I.C.: 1987. {\em Mathematical Models in Medicine}, vol. 7., Ch.11 Pergamon Press, New York, 1513 -1577; URLs: {\em CERN Preprint No. EXT-2004-072:}, \PMlinkexternal{available here as PDF}{http://doc.cern.ch//archive/electronic/other/ext/ext-2004-072.pdf}, or \PMlinkexternal{as as an archived html document}{http://en.scientificcommons.org/1857371}. \bibitem{BBGG2k6} Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and \L ukasiewicz--Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., {\em Axiomathes}, \textbf{16} Nos. 1--2: 65--122. \bibitem{BBG2k7} Baianu, I.C., R. Brown and J.F. Glazebrook. : 2007, Categorical Ontology of Complex Spacetime Structures: The Emergence of Life and Human Consciousness, {\em Axiomathes}, \textbf{17}: 35-168. \bibitem{Rosenbook} Rosen, R.: 1985, {\em Anticipatory Systems}, Pergamon Press: New York. \bibitem{RRosen1} Rosen, R.: 1958a, A Relational Theory of Biological Systems., {\em Bulletin of Mathematical Biophysics} \textbf{20}: 245-260. \bibitem{RRosen2} Rosen, R.: 1958b, The Representation of Biological Systems from the Standpoint of the Theory of Categories., {\em Bulletin of Mathematical Biophysics} \textbf{20}: 317-341. \bibitem{RRosen3} Rosen, R.: 1987, On Complex Systems, {\em European Journal of Operational Research} \textbf{30}:129--134. \end{thebibliography}