genetic nets

0.1 Introduction

Genetic ‘nets’, or networks, $GN$ – that represent 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 directed graph $G_{D}$ 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 $G_{D}$.

0.2 Boolean vs. N-state models of genetic networks in $LM_{n}$- 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 Łukasiewicz-Moisil n-valued logic algebra $LM_{n}$. Then, the category of genetic networks, $\textbf{GN}_{\L{}M_{n}}$ was shown in ref. [2] to form a subcategory of the algebraic category of Łukasiewicz algebras (http://planetmath.org/AlgebraicCategoryOfLMnLogicAlgebras), $\mathcal{LM}$ [1, 2]. 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. [2]. 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.

References

• 1 Baianu, I.C. (1977). A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory., Bulletin of Mathematical Biology, 39:249-258.
• 2 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.
• 3 Baianu, I.C., J. Glazebrook, G. Georgescu and R.Brown. (2008). A Novel Approach to Complex Systems Biology based on Categories, Higher Dimensional Algebra and Łukasiewicz Topos. Manuscript in preparation, 16 pp.
• 4 Georgescu, G. and C. Vraciu (1970). On the Characterization of Łukasiewicz Algebras., J. Algebra, 16 (4), 486-495.
 Title genetic nets Canonical name GeneticNets Date of creation 2013-03-22 18:11:28 Last modified on 2013-03-22 18:11:28 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 50 Author bci1 (20947) Entry type Topic Classification msc 55U99 Classification msc 92D15 Classification msc 03B50 Classification msc 92B20 Classification msc 92B05 Synonym genome network Synonym genome Synonym entity of all interacting genes in a living organism Related topic DirectedGraph Related topic AlgebraicCategoryOfLMnLogicAlgebras Related topic OrganismicSets3 Related topic OrganismicSets2 Related topic JanLukasiewicz Related topic SupercategoriesOfComplexSystems Related topic MolecularSetTheory Related topic CategoryTheory Related topic OrganismicSetTheory Related topic FunctionalBiology Defines gene net Defines Bayesian model Defines genetic network Defines N-state net models Defines two-state models Defines genome Boolean models Defines category of genetic nets