Mathematical Modeling of Gene Networks

Unconditional global exponential stability in Lagrange sense of genetic regulatory networks with SUM regulatory logic

“…In (1), the parameters a i and c i are the decay rates of mRNA and protein, respectively, d i is a constant, which is the translational rate. b i is the regulation function of the ith gene, and it is generally a nonlinear function of the variables (p 1 (t), p 2 (t), …, p n (t)) but has a form of monotone with each variable (Jong 2002;Smolen et al 2000).…”

“…A genetic regulatory network is a dynamic system to describe highly complex interactions among two main species of gene product: mRNAs and proteins, in the interactive transcription and translation processes. Several computational models have been applied to investigate the behaviors of genetic regulatory networks: Bayesian network models (Friedman et al 2000;Hartemink et al 2002), Petri net models (Chaouiya et al 2008;Hardy and Robillard 2004), the Boolean models (Somogyi and Sniegoski 1996;Weaver et al 1999), and the differential equation models (Bolouri and Davidson 2002;Chen and Aihara 2002;Jong 2002;Smolen et al 2000), etc. In the differential model, the variables describe the concentrations of gene product, such as mRNA and proteins, as continuous value of the genetic regulatory systems.…”

Robust filtering for stochastic genetic regulatory networks with time-varying delay

“…It is worth mentioning that the robust filtering circuit design problems and H ∞ stabilization design problems have been considered for the nonlinear genetic regulatory networks in [4,5] without timedelays, and pioneering results have been obtained in the area. As discussed in the introduction, time delays may play an important role in the dynamics of genetic networks, and mathematical models without addressing the delay effects may even have wrong predictions of the mRNA and protein concentrations [30,31]. In this paper, we extend the model (2) further by incorporating time delays in both the translation process and feedback regulation process.…”

“…It has been shown in [25], by mathematically modelling recent data, that the observed oscillatory expression and activity of three proteins is most likely to be driven by transcriptional delays, and time delay is often inevitable when analyzing the dynamical behaviors of GRNs [7,30,31]. On the other hand, the stochastic fluctuations in real-world gene expression data stem from either the probabilistic chemical reactions or random variation of one or more of the externally set control parameters [8,18,26,33], and therefore state-dependent stochastic noise should be recognized as a characteristic that has to be taken into account when modeling GRNs.…”

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).…”