0.1 Introduction
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 GNs may be thus represented by a directed graph GD -the
gene net digraph- whose nodes (or vertices) are the genes gi of a cell or a multicellular organism
and whose edges (arcs) are arrows representing the actions of a gene agi on a linked
gene or genes; such a directed graph representing a gene network has a canonically
associated biogroupoid 𝒢B which is generated or directly computed from the directed graph
GD.
0.2 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, GNŁMn that
operate on the basis of a Łukasiewicz-Moisil n-valued logic algebra LMn. Then, the category of
genetic networks, GNŁMn was shown in ref. [7] to form a subcategory of the algebraic category of
Łukasiewicz algebras, ℒℳ[?, 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 (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 GNŁMn. The latter
corresponds to organismic sets of zero-th order S0 in the simpler, Rashevsky’s theory of organismic
sets.
References
[1] References [14] to [34] in the “bibliography of category theory and algebraic topology”
[2] 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).
[3] Baianu, I.C. and M. Marinescu: 1974, A Functorial Construction of (M,R)– Systems.
Revue Roumaine de Mathematiques Pures et Appliquees 19: 388-391.
[4] Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras:
The Non-linear Theory. Bulletin of Mathematical Biophysics, 39: 249-258.
[5] Baianu, I.C.: 1980, Natural Transformations of Organismic Structures. Bulletin of
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[6] 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
[7] 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.
[8] 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.
[9] Georgescu, G. and C. Vraciu (1970). On the Characterization of Łukasiewicz
Algebras., J. Algebra, 16 (4), 486-495.
