Statistical physics of dynamical networks and morphogenesis

Rohlf, Thimo

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 tissue



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Rohlf, Thimo: Statistical physics of dynamical networks and morphogenesis. 2004.


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