Models of Genetic Networks with Given Properties

Abstract

A multi-parameter system of ordinary differential equations, modelling genetic networks, is considered. Attractors of this system correspond to future states of a network. Sufficient conditions for the non-existence of stable critical points are given. Due to the special structure of the system, attractors must exist. Therefore the existence of more complicated attractors was expected. Several examples are considered, confirming this conclusion.

Comparative Analysis of Models of Gene and Neural Networks

“…The coefficients of regulatory matrix and parameters are the same. The initial conditions are (6). There is no chaotic solution.…”

“…There is no chaotic solution. The nullclines of the system (7) with the regulatory matrix (5) and the solution of the system (7) with the regulatory matrix (5) and the initial conditions (6) are shown in Figure 13 and Figure 14. The graphs of x i (t), i = 1, 2, 3 of the system (7) with the regulatory matrix (5) are depicted in Figure 15.…”

“…The main approaches to the description of gene networks and modeling their dynamics are a logical description; a description of the gene network using a system of nonlinear differential equations [3]; stochastic gene network models; graph theory [1], Boolean modeling [1]. Nonlinear ordinary differential equations are the most-widespread formalism for modeling genetic regulatory networks [4][5][6][7].…”

Genetic engineering – construction of a network of arbitrary dimension with periodic attractor

“…In fact, we have provided an example of the 3D system without stable critical points. More examples of this kind were obtained in the paper [11].…”

On attractors in systems of ordinary differential equations arising in models of genetic networks

“…Stable equilibria are simple attractors. There are systems with a critical point, which is not attractive [5]. A stable periodic solution appears instead.…”