Mathematical modeling in genetic networks: relationships between the genetic expression and both chromosomic breakage and positive circuits

Abstract

The human genome with its 23 pairs of chromosomes, is the result of evolution. This evolution has been ruled by the mutation process and also by the physiological and pathological reorganization of the genomic material inside or between the chromosomes, which are conditioning the genomic variability. This reorganization is starting at singular points on the short or long chromosomic arms, called crossing-over, or translocations, insertions, break points. In this paper, we will show that these points, also called weak points or hot spots of the genome are correlated, independently of their origin. In addition, we will give some properties of the genetic interaction matrices in terms of attractors of the genetic expression dynamics.

On the Number of Attractors of Positive and Negative Boolean Automata Circuits

“…Indeed, as it can easily be checked, id and neg can both be expressed as threshold functions and thus, so can all local transition functions using identical interaction weights and threshold values for all arcs and nodes (see [5] for more details on this subject).…”

“…In [5] and [27], positive circuits are characterised this way and indeed, as we will see in the next section, negative circuits never have any fixed points. From this characterisation, the authors of these articles also infer some results concerning more general networks.…”

“…In [5] and [27], the authors proven a result stating that arbitrary networks containing only negative circuits have no fixed points.…”

“…One of the main motivations of many of them was to better understand those emergent dynamical behaviours that networks display and that cannot be explained or predicted by a simple analysis of the local interactions existing between the components of the networks. Amongst the studies published in this context since the end of the 1990's, one may cite [4,5,6,7,8]. Earlier on, Hopfield [9,10] emphasised the notions of memory and learning and Goles et al revealed in [11,12,13] interesting dynamical properties of some particular networks.…”

“…He formulated conjectures concerning the role of positive (i.e., with an even number of inhibitions) and negative (i.e., with an odd number of inhibitions) circuits in the dynamics of regulation networks. Besides the fact that they are known to be decisive patterns for the dynamics of arbitrary biological networks, circuits are also relevant because they may be regarded specifically as internal layers of feedforward networks 5 . Identifying the dynamics of circuits is thus a first step in the process of understanding and formalising the dynamics of such networks.…”