GeneNetsPhysicalAndMathemaicalModels 2009-01-25 17:56:09 2009-01-25 19:46:13 Topic gene net Bayesian model genetic network N-state net models two-state models Boolean models category of gene nets gene network topology gene net digraph gene biogroupoid gene nets genetic networks random networks gene net digraphs organismic sets metabolic-replication systems categories of LMn- logic algebras \subsection{Introduction} \emph{Genetic `nets', or networks}, $GN$ -- that form a living organism's genome --are mathematical models of functional genes linked through their non-linear, dynamic interactions. A simple genetic (or gene) network $GN_s$ may be thus represented by a \PMlinkexternal{directed graph}{http://planetmath.org/encyclopedia/OutDegree2.html} $G_D$ -the gene net digraph- whose nodes (or vertices) are the genes $g_i$ of a cell or a multicellular organism and whose edges (arcs) are arrows representing the actions of a gene $a_g^i$ on a linked gene or genes; such a directed graph representing a gene network has a canonically associated biogroupoid $\mathcal{G}_B$ which is generated or directly computed from the directed graph $G_D$. \subsection{Boolean vs. N-state models of genetic networks in LMn- logic algebras} The simplest, Boolean, or two-state models of genomes represented by such directed graphs of gene networks form a proper subcategory of the category of n-state genetic networks, $\textbf{GN}_{\L{}M_n}$ that operate on the basis of a \L{}ukasiewicz-Moisil n-valued logic algebra $LM_n$. Then, the category of genetic networks, $\textbf{GN}_{\L{}M_n}$ was shown in ref. \cite{ICBetal2k6} to form a subcategory of the \PMlinkname{algebraic category of \L{}ukasiewicz algebras}{AlgebraicCategoryOfLMnLogicAlgebras}, $\mathcal{LM}$ \cite{ICB77,ICBetal2k6}. There are several published, extensive computer simulations of Boolean two-state models of both genetic and neuronal networks (for a recent summary of such computations see, for example, ref. \cite{ICBetal2k6}. Most, but not all, such mathematical models are Bayesian, and therefore involve computations for random networks that may have limited biological relevance as the topology of genomes, defined as their connectivity, is far from being random. The category of automata (or sequential machines based on Chrysippean or Boolean logic) and the category of $(M,R)$-systems (which can be realized as concrete metabolic-repair biosystems of enzymes, genes, and so on) are subcategories of the category of gene nets $\textbf{GN}_{\L{}M_n}$. The latter corresponds to organismic sets of zero-th order $S_0$ in the simpler, Rashevsky's theory of organismic sets. \begin{thebibliography}{9} \bibitem{ICB71} References \PMlinkname{[14] to [34] in the ``bibliography of category theory and algebraic topology''}{CategoricalOntologyABibliographyOfCategoryTheory} \bibitem{BGB06} I. C. Baianu, J. F. Glazebrook, R. Brown and G. Georgescu.: Complex Nonlinear Biodynamics in Categories, Higher dimensional Algebra and \L ukasiewicz-Moisil Topos: Transformation of Neural, Genetic and Neoplastic Networks, Axiomathes,16: 65--122(2006). \bibitem{ICBm2} Baianu, I.C. and M. Marinescu: 1974, A Functorial Construction of \emph{\textbf{(M,R)}}-- Systems. \emph{Revue Roumaine de Mathematiques Pures et Appliquees} \textbf{19}: 388-391. \bibitem{ICB6} Baianu, I.C.: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. \emph{Bulletin of Mathematical Biophysics}, \textbf{39}: 249-258. \bibitem{ICB80} Baianu, I.C.: 1980, Natural Transformations of Organismic Structures. \emph{Bulletin of Mathematical Biophysics} \textbf{42}: 431-446 \bibitem{ICB87a} Baianu, I. C.: 1987a, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.), \emph{Mathematical Models in Medicine}, vol. 7., Pergamon Press, New York, 1513-1577; \PMlinkexternal{CERN Preprint No. EXT-2004-072}{http://doe.cern.ch//archive/electronic/other/ext/ext-2004-072.pdf} \bibitem{ICBetal2k6} Baianu, I.C., Brown, R., Georgescu, G., Glazebrook, J.F. (2006). Complex nonlinear biodynamics in categories, higher dimensional algebra and \L{}ukasiewicz-Moisil topos: transformations of neuronal, genetic and neoplastic networks. {\em Axiomathes} \textbf{16}(1-2):65-122. \bibitem{ICBetal2k8} Baianu, I.C., J. Glazebrook, G. Georgescu and R.Brown. (2009). A Novel Approach to Complex Systems Biology based on Categories, Higher Dimensional Algebra and \L{}ukasiewicz Topos. {\em Manuscript in preparation}, 16 pp. \bibitem{GGCV70} Georgescu, G. and C. Vraciu (1970). On the Characterization of \L{}ukasiewicz Algebras., \emph{J. Algebra}, \textbf{16} (4), 486-495. \end{thebibliography}