PlanetPhysics/Gene Nets Physical and Mathemaical Models

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 $GN_{s}$ may be thus represented by a directed graph $G_{D}$ -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 $G_{D}$.

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, Failed to parse (unknown function "\L"): {\displaystyle '''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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle '''GN''' _{\L{}M_n}} was shown in ref. to form a subcategory of the \htmladdnormallink{algebraic category of \L{}ukasiewicz algebras}{http://planetphysics.us/encyclopedia/AlgebraicCategoryOfLMnLogicAlgebras.html}, ${\mathcal {LM}}$ . 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. . 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 Failed to parse (unknown function "\L"): {\displaystyle '''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.