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.Comment: 4 pages, 4 figures. Accepted for the "Workshop on Computational Systems Biology", Leipzig 200

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.Comment: 7 pages, 7 figures. Submitted to the "8th German Workshop on Artificial Life (GWAL 8)

Learning, Generalization, and Functional Entropy in Random Automata Networks

It has been shown \citep{broeck90:physicalreview,patarnello87:europhys} that feedforward Boolean networks can learn to perform specific simple tasks and generalize well if only a subset of the learning examples is provided for learning. Here, we extend this body of work and show experimentally that random Boolean networks (RBNs), where both the interconnections and the Boolean transfer functions are chosen at random initially, can be evolved by using a state-topology evolution to solve simple tasks. We measure the learning and generalization performance, investigate the influence of the average node connectivity $K$, the system size $N$, and introduce a new measure that allows to better describe the network's learning and generalization behavior. We show that the connectivity of the maximum entropy networks scales as a power-law of the system size $N$. Our results show that networks with higher average connectivity $K$ (supercritical) achieve higher memorization and partial generalization. However, near critical connectivity, the networks show a higher perfect generalization on the even-odd task

Damage Spreading and Criticality in Finite Random Dynamical Networks

We systematically study and compare damage spreading at the sparse percolation (SP) limit for random boolean and threshold networks with perturbations that are independent of the network size $N$. This limit is relevant to information and damage propagation in many technological and natural networks. Using finite size scaling, we identify a new characteristic connectivity $K_s$, at which the average number of damaged nodes $\bar d$, after a large number of dynamical updates, is independent of $N$. Based on marginal damage spreading, we determine the critical connectivity $K_c^{sparse}(N)$ for finite $N$ at the SP limit and show that it systematically deviates from $K_c$, established by the annealed approximation, even for large system sizes. Our findings can potentially explain the results recently obtained for gene regulatory networks and have important implications for the evolution of dynamical networks that solve specific computational or functional tasks.Comment: 4 pages, 4 eps figure

Statistical physics of dynamical networks and morphogenesis

Complex networks of many interacting units occur in diverse areas as, for example, gene regulation, neural networks, economic interactions, and the organization of the internet. Many of these networks exhibit complex, non-Hamiltonian dynamics that strongly depends on network topology. In addition, ensembles of dynamical networks communicating with each other often show emergent behavior - for example, in development (morphogenesis) of multicellular organisms cell-cell communication can coordinate the dynamics of gene regulatory networks, leading to complex spatial patterns. The purpose of this thesis is threefold: first, to gain insight into the interdependence between network dynamics and -topology; second, to explore how this interdependence, together with local rewiring events, can contribute to evolution of network topologies with properties as, e.g., observed for gene regulatory networks; third, to investigate how local interactions between coupled networks can lead to robust and reproduceable emergence of spatial patterns and solve functional tasks in morphogenesis. This train of thoughts is pursued in the following steps: (i) First, the critical connectivity of Random Threshold Networks is calculated analytically by combination of a new combinatorial approach to ``damage spreading'' with an annealed approximation. A non-trivial dependence of damage propagation on the in-degree of network nodes is identified that may have important consequences for the evolution of network topologies. (ii) Then, a model of topological network evolution is studied in the context of evolving gene regulatory networks. This model leads to evolution of network topologies close to criticality, however, with a degree-distribution and activity pattern that deviate from random networks, in good agreement with statistical data obtained for real gene networks. (iii) Thereafter, morphogenesis by coupled regulatory networks is studied. Starting from experimental observations in Hydra, a gene network model capable of {\em de novo pattern formation} and regulation of pattern proportions is developed, providing a network-based, alternative scenario to gradient-based explanations of these morphogenetic processes. (iv) Robustness of this model with respect to two type of perturbations often found in biological organisms, stochastic update errors and cell flow, is studied. It is shown that noise-induced control contributes to pattern stabilization, even under cell flow. A first order phase transition is found at vanishing noise, a second order phase transition at increased cell flow. (v) Finally, a two-dimensional extension of the morphogenetic model is studied. Here, a competition between inititial symmetry breaking and neighborhood-dependent state changes of cells leads to sharp and localized boundaries of spatial activity domains, as required for robust self-organization of position information in a two-dimensional tissu

Transcriptional memory emerges from cooperative histone modifications

Background
Transcriptional regulation in cells makes use of diverse mechanisms to ensure that functional states can be maintained and adapted to variable environments; among them are chromatin-related mechanisms. While mathematical models of transcription factor networks controlling development are well established, models of transcriptional regulation by chromatin states are rather rare despite they appear to be a powerful regulatory mechanism.
Results
We here introduce a mathematical model of transcriptional regulation governed by histone modifications. This model describes binding of protein complexes to chromatin which are capable of reading and writing histone marks. Molecular interactions between these complexes and DNA or histones create a regulatory switch of transcriptional activity possessing a regulatory memory. The regulatory states of the switch depend on the activity of histone (de-) methylases, the structure of the DNA-binding regions of the complexes, and the number of histones contributing to binding. 
We apply our model to transcriptional regulation by trithorax- and polycomb- complex binding. By analyzing data on pluripotent and lineage-committed cells we verify basic model assumptions and provide evidence for a positive effect of the length of the modified regions on the stability of the induced regulatory states and thus on the transcriptional memory.
Conclusions
Our results provide new insights into epigenetic modes of transcriptional regulation. Moreover, they implicate well-founded hypotheses on cooperative histone modifications, proliferation induced epigenetic changes and higher order folding of chromatin which await experimental validation. Our approach represents a basic step towards multi-scale models of transcriptional control during development and lineage specification. 
&#xa

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 today's 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.Comment: 8 pages, 6 figures, IEEE/ACM Symposium on Nanoscale Architectures, NANOARCH 2008, Anaheim, CA, USA, Jun 12-13, 200