Emergent Network Structure, evolvable Robustness and non-linear Effects of Point Mutations in an Artificial Genome Model

Genetic regulation is a key component in development, but a clear understanding of the structure and dynamics of genetic networks is not yet at hand. In this paper we investigate these properties within an artificial genome model originally introduced by Reil (1999). We analyze statistical properties of randomly generated genomes both on the sequence- and network level, and show that this model correctly predicts the frequency of genes in genomes as found in experimental data. Using an evolutionary algorithm based on stabilizing selection for a phenotype, we show that dynamical robustness against single base mutations, as well as against random changes in initial states of regulatory dynamics that mimic stochastic fluctuations in environmental conditions, can emerge in parallel. Point mutations at the sequence level have strongly non-linear effects on network wiring, including as well structurally neutral mutations and simultaneous rewiring of multiple connections, which occasionally lead to strong reorganization of the attractor landscape and metastability of evolutionary dynamics. Evolved genomes exhibit characteristic patterns on both sequence and network level.

Morphogenesis by coupled regulatory networks: Reliable control of positional information and proportion regulation

Based on a non-equilibrium mechanism for spatial pattern formation we study how position information can be controlled by locally coupled discrete dynamical networks, similar to gene regulation networks of cells in a developing multicellular organism. As an example we study the developmental problems of domain formation and proportion regulation in the presence of noise, as well as in the presence of cell flow. We find that networks that solve this task exhibit a hierarchical structure of information processing and are of similar complexity as developmental circuits of living cells. Proportion regulation is scalable with system size and leads to sharp, precisely localized boundaries of gene expression domains, even for large numbers of cells. A detailed analysis of noise-induced dynamics, using a mean-field approximation, shows that noise in gene expression states stabilizes (rather than disrupts) the spatial pattern in the presence of cell movements, both for stationary as well as growing systems. Finally, we discuss how this mechanism could be realized in the highly dynamic environment of growing tissues in multi-cellular organisms.

Self-organized criticality and adaptation in discrete dynamical networks

It has been proposed that adaptation in complex systems is optimized at the critical boundary between ordered and disordered dynamical regimes. Here, we review models of evolving dynamical networks that lead to self-organization of network topology based on a local coupling between a dynamical order parameter and rewiring of network connectivity, with convergence towards criticality in the limit of large network size $N$. In particular, two adaptive schemes are discussed and compared in the context of Boolean Networks and Threshold Networks: 1) Active nodes loose links, frozen nodes aquire new links, 2) Nodes with correlated activity connect, de-correlated nodes disconnect. These simple local adaptive rules lead to co-evolution of network topology and -dynamics. Adaptive networks are strikingly different from random networks: They evolve inhomogeneous topologies and broad plateaus of homeostatic regulation, dynamical activity exhibits $1/f$ noise and attractor periods obey a scale-free distribution. The proposed co-evolutionary mechanism of topological self-organization is robust against noise and does not depend on the details of dynamical transition rules. Using finite-size scaling, it is shown that networks converge to a self-organized critical state in the thermodynamic limit. Finally, we discuss open questions and directions for future research, and outline possible applications of these models to adaptive systems in diverse areas.

Assessing Random Dynamical Network Architectures for Nanoelectronics

Independent of the technology, it is generally expected that future nanoscale devices will be built from vast numbers of densely arranged devices that exhibit high failure rates. Other than that, there is little consensus on what type of technology and computing architecture holds most promises to go far beyond todays top-down engineered silicon devices. Cellular automata (CA) have been proposed in the past as a possible class of architectures to the von Neumann computing architecture, which is not generally well suited for future parallel and fine-grained nanoscale electronics. While the top-down engineered semi-conducting technology favors regular and locally interconnected structures, future bottom-up self-assembled devices tend to have irregular structures because of the current lack precise control over these processes. In this paper, we will assess random dynamical networks, namely Random Boolean Networks (RBNs) and Random Threshold Networks (RTNs), as alternative computing architectures and models for future information processing devices. We will illustrate that--from a theoretical perspective--they offer superior properties over classical CA-based architectures, such as inherent robustness as the system scales up, more efficient information processing capabilities, and manufacturing benefits for bottom-up designed devices, which motivates this investigation. We will present recent results on the dynamic behavior and robustness of such random dynamical networks while also including manufacturing issues in the assessment.

Network Structure and Dynamics, and Emergence of Robustness by Stabilizing Selection in an Artificial Genome

Genetic regulation is a key component in development, but a clear understanding of the structure and dynamics of genetic networks is not yet at hand. In this work we investigate these properties within an artificial genome model originally introduced by Reil. We analyze statistical properties of randomly generated genomes both on the sequence- and network level, and show that this model correctly predicts the frequency of genes in genomes as found in experimental data. Using an evolutionary algorithm based on stabilizing selection for a phenotype, we show that robustness against single base mutations, as well as against random changes in initial network states that mimic stochastic fluctuations in environmental conditions, can emerge in parallel. Evolved genomes exhibit characteristic patterns on both sequence and network level.

When correlations matter - response of dynamical networks to small perturbations

We systematically study and compare damage spreading for random Boolean and threshold networks under small external perturbations (damage), a problem which is relevant to many biological networks. We identify a new characteristic connectivity $K_s$, at which the average number of damaged nodes after a large number of dynamical updates is independent of the total number of nodes $N$. We estimate the critical connectivity for finite $N$ and show that it systematically deviates from the annealed approximation. Extending the approach followed in a previous study, we present new results indicating that internal dynamical correlations tend to increase not only the probability for small, but also for very large damage events, leading to a broad, fat-tailed distribution of damage sizes. These findings indicate that the descriptive and predictive value of averaged order parameters for finite size networks - even for biologically highly relevant sizes up to several thousand nodes - is limited.

Critical Line in Random Threshold Networks with Inhomogeneous Thresholds

We calculate analytically the critical connectivity $K_c$ of Random Threshold Networks (RTN) for homogeneous and inhomogeneous thresholds, and confirm the results by numerical simulations. We find a super-linear increase of $K_c$ with the (average) absolute threshold $|h|$, which approaches $K_c(|h|) sim h^2/(2ln{|h|})$ for large $|h|$, and show that this asymptotic scaling is universal for RTN with Poissonian distributed connectivity and threshold distributions with a variance that grows slower than $h^2$. Interestingly, we find that inhomogeneous distribution of thresholds leads to increased propagation of perturbations for sparsely connected networks, while for densely connected networks damage is reduced; the cross-over point yields a novel, characteristic connectivity $K_d$, that has no counterpart in Boolean networks. Last, local correlations between node thresholds and in-degree are introduced. Here, numerical simulations show that even weak (anti-)correlations can lead to a transition from ordered to chaotic dynamics, and vice versa. It is shown that the naive mean-field assumption typical for the annealed approximation leads to false predictions in this case, since correlations between thresholds and out-degree that emerge as a side-effect strongly modify damage propagation behavior.