Introduction

Genetic `nets', or networks, – 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 may be thus represented by a directed graph -the gene net digraph- whose nodes (or vertices) are the genes of a cell or a multicellular organism and whose edges (arcs) are arrows representing the actions of a gene on a linked gene or genes; such a directed graph representing a gene network has a canonically associated biogroupoid which is generated or directly computed from the directed graph .

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, that operate on the basis of a Łukasiewicz-Moisil n-valued logic algebra . Then, the category of genetic networks, was shown in ref. [7] to form a subcategory of the algebraic category of Łukasiewicz algebras, [,7]. 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. [7]. 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 -systems (which can be realized as concrete metabolic-repair biosystems of enzymes, genes, and so on) are subcategories of the category of gene nets . The latter corresponds to organismic sets of zero-th order in the simpler, Rashevsky's theory of organismic sets.

Bibliography

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I. C. Baianu, J. F. Glazebrook, R. Brown and G. Georgescu.: Complex Nonlinear Biodynamics in Categories, Higher dimensional Algebra and Łukasiewicz-Moisil Topos: Transformation of Neural, Genetic and Neoplastic Networks, Axiomathes,16: 65–122(2006).

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Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biophysics, 39: 249-258.

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Baianu, I. C.: 1987a, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.), Mathematical Models in Medicine, vol. 7., Pergamon Press, New York, 1513-1577; CERN Preprint No. EXT-2004-072

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Baianu, I.C., Brown, R., Georgescu, G., Glazebrook, J.F. (2006). Complex nonlinear biodynamics in categories, higher dimensional algebra and Łukasiewicz-Moisil topos: transformations of neuronal, genetic and neoplastic networks. Axiomathes 16(1-2):65-122.

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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 Łukasiewicz Topos. Manuscript in preparation, 16 pp.

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Georgescu, G. and C. Vraciu (1970). On the Characterization of Łukasiewicz Algebras., J. Algebra, 16 (4), 486-495.