Dynamical complexity of discrete-time regulatory networks

Abstract

Genetic regulatory networks are usually modelled by systems of coupled differential equations, and more particularly by systems of piecewise affine differential equations. Finite state models, better known as logical networks, are also used. In this paper we present a class of models of regulatory networks which may be situated in the middle of the spectrum; they present both discrete and continuous aspects. They consist of a network of units, whose states are quantified by a continuous real variable. The state of each unit in the network evolves according to a contractive transformation chosen from a finite collection of possible transformations. The particular transformation chosen at each time step depends on the state of the neighbouring units. In this way we obtain a network of coupled contractions. As a first approximation to the complete description of the dynamics we shall focus on a global invariant: the dynamical complexity of the system. This is a well-studied notion in the framework of the theory of dynamical systems, and it is related to the proliferation of distinguishable temporal behaviour. The main motivation of this work is to find explicit relations between the topological structure of the regulatory network and the growth rate of the dynamical complexity. In this paper we derive general upper bounds for the dynamical complexity for networks of arbitrary size, and we exhibit specific instances of constraints imposed on the complexity growth by the structure of the underlying network.

Stability and chaos in coupled two-dimensional maps on gene regulatory network of bacterium E. coli

“…In this way, the dynamics of gene regulations is modeled at various levels [13] and using different mathematical techniques [9]. The simplest approaches include boolean networks [10] and discretetime maps [11], while more detailed analysis requires 1D [14] or 2D [12] continuous-time ODE, that can also involve a time-delayed action [15]. Analytical studies of GRN models involving small networks revealed the complexity of their dynamical patterns [14], with the system of two interacting genes solved in detail [11,12].…”

“…The simplest approaches include boolean networks [10] and discretetime maps [11], while more detailed analysis requires 1D [14] or 2D [12] continuous-time ODE, that can also involve a time-delayed action [15]. Analytical studies of GRN models involving small networks revealed the complexity of their dynamical patterns [14], with the system of two interacting genes solved in detail [11,12]. General relationship between structure and function of large GRN was examined, with particular emphasis on their collective dynamics [16], information processing [17], flexibility [10], and functional organization [7].…”

Mean square exponential and robust stability of stochastic discrete-time genetic regulatory networks with uncertainties

“…As a special case, genetic regulatory networks (GRNs) consisting of DNA, RNA, proteins, small molecules and their mutual regulatory interactions, have become an important new area of research in the biological and biomedical sciences and received widely attention recently (Becskei and Serrano 2000;Bolouri and Davidson 2002;Weaver et al 1999;De Jong 2002;Smolen et al 2000). Several models have been developed to investigate the behaviors of the GRNs, for example, Boolean models (Weaver et al 1999), the differential equation models (De Jong 2002;Smolen et al 2000), the Petri net models (Chaouiya 2007) and discrete time piecewise affine model (Lima and Ugalde 2006;Coutinho et al 2006). Among them, GRNs in the form of differential equation models have been well studied in He and Cao (2008), Ren and Cao (2008), Ribeiro et al (2006) and Cao and Ren (2008).…”

“…Hence, it is necessary to consider stochastic delay effects in GRNs. In addition, as pointed out in Lima and Ugalde (2006), Coutinho et al (2006) and Cao and Ren (2008), some GRN models are discrete-time dynamical systems which can be viewed as an extension of discrete-time delay systems and are more important than their continuous-time counterpart in a sense. These kinds of discrete-time models are directly inspired by the systems of differential equations mentioned above, though they do not correspond to a time discretization of the differential equations but rather to a natural discrete-time version of them.…”

Dominant vertices in regulatory networks dynamics

“…We will say that the orbit or controlled trajectory {x t } ∞ t=0 is uniformly separated 1 In previous works [2,11] it has been considered less general interaction functions defined as follows. For each interaction (u, v) ∈ A there are associated a sign σuv ∈ {−1, 1}, and a threshold Tuv ∈ (0, 1).…”