Inflation of the edge of chaos in a simple model of gene interaction networks

Dejan Stokić, Rudolf Hanel, and Stefan Thurner

Phys. Rev. E 77, 061917 – Published 18 June 2008

Abstract

We study a set of linearized catalytic reactions to model gene and protein interactions. The model is based on experimentally motivated interaction network topologies and is designed to capture some key properties of gene expression statistics. We impose a nonlinearity to the system by enforcing a boundary condition which guarantees non-negative concentrations of chemical substances. System stability is quantified by maximum Lyapunov exponents. We find that the non-negativity constraint leads to a drastic inflation of those regions in parameter space where the Lyapunov exponent exactly vanishes. Within the model this finding can be fully explained as a result of a symmetry breaking mechanism induced by the positivity constraint. The robustness of this finding with respect to network topologies and the role of intrinsic molecular and external noise is discussed. We argue that systems with inflated “edges of chaos” could be much more easily favored by natural selection than systems where the Lyapunov exponent vanishes only on a parameter set of measure zero.

Received 29 November 2007

DOI:https://doi.org/10.1103/PhysRevE.77.061917

Authors & Affiliations

Dejan Stokić¹, Rudolf Hanel¹, and Stefan Thurner^1,2,*

¹Complex Systems Research Group, HNO, Medical University of Vienna, Währinger Gürtel 18-20, A-1090 Vienna, Austria
²Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA

^*thurner@univie.ac.at

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Vol. 77, Iss. 6 — June 2008

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Images

Figure 1
(Color online) (a) Ten randomly picked gene-expression trajectories of yeast (S. cerevisiae) over two cell cycles [38], $N = 6220$ . Fixed point values $x^{0}$ are defined as the time-averages of gene expression levels (horizontal lines). (b) Distribution of stationary state values $x^{0}$ ; inset: cumulative distribution. (c) Cumulative distribution of gene expressions increments for the same data (circles) for the numerical simulation of the evolution of gene expression data with only multiplicative noise only (boxes) and pure additive Gaussian noise (triangles). Data is well-fitted by a $q$ Gaussian (broken line).Reuse & Permissions
Figure 2
(Color online) Time series of five randomly selected trajectories [numerical solutions of Eq. (4)] for the regions (a) $λ < 0$ , (b) at the $λ = 0$ plateau, and (c) $λ > 0$ . $N = 500$ , $σ = \bar{σ} = 0.1$ , and $D = 4$ . These trajectories have been artificially perturbed by an additive shift $x^{p}$ (depicted with the arrows for one trajectory).Reuse & Permissions
Figure 3
(Color online) Maximum Lyapunov exponents $λ$ as a function of average degree $⟨ k ⟩$ , averaged over 50 realizations for ER networks. Simulations are shown without positivity condition for noise $σ = \bar{σ} = 0.1$ (squares) and without noise $σ = \bar{σ} = 0$ (circles). The influence of the positivity condition on forming a plateau is demonstrated with noise $σ = \bar{σ} = 0.1$ (triangles). The solid and broken lines are Eqs. (9, 15). Inset: plateau for the case $x_{j}^{0} = 0$ , with noise $σ = \bar{σ} = 0.1$ . In all cases $N = 500$ , $D = 4$ .Reuse & Permissions
Figure 4
(Color online) Lyapunov exponents for the same parameters as in Fig. 3 for (a) different network types and sizes $N = 200, 500, 1000$ , (b) noise effects for multiplicative noise ( $\bar{σ} = 0.1$ and $σ = 0$ ), additive noise ( $\bar{σ} = 0$ and $σ = 0.1$ ), compared to the deterministic process $(\bar{σ} = σ = 0)$ without the positivity condition. (c) $λ$ compared to the number of inactive nodes as a function of connectivity. ER, $N = 1000$ , $D = 4$ , and $σ = \bar{σ} = 0.1$ .Reuse & Permissions
Figure 5
(Color online) Lyapunov exponents for four different values of $D$ . The dependence of the critical region and the width of the plateau is seen. Lines correspond to Eqs. (9, 15).Reuse & Permissions

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