Dynamics in high-dimensional model gene networks

Kappler, K. N.; Edwards, Roderick; Glass, Leon

doi:10.1016/s0165-1684(02)00479-6

Cited by 34 publications

(25 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, the detection and analysis of chaos in systems of piecewise linear differential equations is still largely an open issue, although it was addressed in several, mostly numerical, studies (Mestl et al, 1997;Glass and Hill, 1998;Kappler et al, 2003).…”

Section: P W Imentioning

confidence: 99%

Biologically meaningful update rules increase the critical connectivity of generalized Kauffman networks

Wittmann¹,

Marr²,

Theis³

2010

Journal of Theoretical Biology

View full text Add to dashboard Cite

We generalize random Boolean networks by softening the hard binary discretization into multiple discrete states. These multistate networks are generic models of gene regulatory networks, where each gene is known to assume a finite number of functionally different expression levels. We analytically determine the critical connectivity that separates the biologically unfavorable frozen and chaotic regimes. This connectivity is inversely proportional to a parameter which measures the heterogeneity of the update rules. Interestingly, the latter does not necessarily increase with the mean number of discrete states per node. Still, allowing for multiple states decreases the critical connectivity as compared to random Boolean networks, and thus leads to biologically unrealistic situations. Therefore, we study two approaches to increase the critical connectivity.First, we demonstrate that each network can be kept in its frozen regime by sufficiently biasing the update rules. Second, we restrict the randomly chosen update rules to a subclass of biologically more meaningful functions. These functions are characterized based on a thermodynamic model of gene regu- Preprint submitted to Journal of Theoretical BiologyMay 27, 2010 lation. We analytically show that their usage indeed increases the critical connectivity. From a general point of view, our thermodynamic considerations link discrete and continuous models of gene regulatory networks.

show abstract

Section: P W Imentioning

confidence: 99%

Biologically meaningful update rules increase the critical connectivity of generalized Kauffman networks

Wittmann¹,

Marr²,

Theis³

2010

Journal of Theoretical Biology

View full text Add to dashboard Cite

show abstract

“…The true largest domain on which a global mapping is well-defined, along with all its iterates, is not simple at all, as discussed below eq. (26). Moreover, M is continuous on the domain D, since it is essentially a Dom(M a ) with all discontinuity points removed.…”

Section: Extension To the Whole State Spacementioning

confidence: 99%

“…The main contribution of this paper lies in the fact that working hypotheses are lightened as compared to previous studies, such as [10,16,17,[19][20][21][22]26,30,32]. Namely, several thresholds are allowed for each variable in the system all through this work, and degradation rates are not supposed uniform for most of our results.…”

Section: Introductionmentioning

confidence: 98%

“…This includes switching networks like gene transcriptional regulation networks [6,7,18,19,26], neuron networks [17,16,30], as well as metabolic and chemical pathways [20], which all are currently studied examples in mathematical models of biological dynamics. Neural, metabolic and gene networks models can generally be put in the form of a system of piecewise affine differential equations, with a diagonal matrix as proper linear term.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Geometric properties of a class of piecewise affine biological network models

Farcot

2005

J. Math. Biol.

View full text Add to dashboard Cite

Abstract. The purpose of this report is to investigate some dynamical properties common to several biological systems. A model is chosen, which consists of a system of piecewise affine differential equations. Such a model has been previously studied in the context of gene regulation and neural networks, as well as biochemical kinetics. Unlike most of these studies, nonuniform decay rates and several thresholds per variable are assumed, thus considering a more realistic model. This model is investigated with the aid of a geometric formalism. We first provide an analysis of a continuous-space, discrete-time dynamical system equivalent to the initial one, by the way of a transition map. This is similar to former studies. Especially, the analysis of periodic trajectories is carried out in the case of multiple thresholds, thus extending previous results, which all concerned the restricted case of binary systems.The piecewise affine structure of such models is then used to provide a partition of the phase space, in terms of explicit cells. Allowed transitions between these cells define a language on a finite alphabet. Some words are proved to be forbidden in this language, thus improving the knowledge on such systems in terms of symbolic dynamics. More precisely, we show that taking these forbidden words into account leads to a dynamical system with strictly lower topological entropy. This holds for a class of systems, characterized by the presence of a splitting box, with additional conditions. We conclude after an illustrative three-dimensional example.

show abstract

“…This kind of approach has been used to define critical modular topologies within networks [23][24][25], for the theoretical determination of dynamical network properties [13,26,27], as well as to investigate the evolution of evolvability [28] and other properties of complex networks.…”

Section: Introductionmentioning

confidence: 99%