REPOSITORY > RESULTS

Abstract

Complex dynamical systems are often modeled as networks, with nodes representing dynamical
units which interact through the network’s links. Gene regulatory networks, responsible for the
production of proteins inside a cell, are an example of system that can be described as a network of
interacting genes. The behavior of a complex dynamical system is characterized by cooperativity
of its units at various scales, leading to emergent dynamics which is related to the system’s
function. Among the key problems concerning complex systems is the issue of stability of their
functioning, in relation to different internal and external interaction parameters.
In this Thesis we study two-dimensional chaotic maps coupled through non-directed networks
with different topologies. We use a non-symplectic coupling which involves a time delay in the
interaction among the maps. We test the stability of network topologies through investigation
of their collective motion, done by analyzing the departures from chaotic nature of the isolated
units. The study is done on two network scales: (a) full-size networks (a computer generated
scalefree tree and a tree with addition of cliques); (b) tree’s characteristic sub-graph 4-star, as a
tree’s typical dynamical motif which captures its topology in smallest possible number of nodes
and is suitable for time-delayed interaction. We study the dynamical relationship between these
two network structures, examining the emergence of cooperativity on a large scale (trees) as a
consequence of mesoscale dynamical patterns exhibited by the 4-star.
We find a variety of coherent dynamical effects on the networks, which include: regular motion
(emergent periodicity), weakly chaotic behavior (different from the uncoupled case), and selforganized
motion characterized by close to zero Lyapunov exponents and anomalous diffusion in
the phase space. Dynamical regions given as the intervals of coupling strength with distinctive
motion and stability patterns are identified, suggesting a mesoscale interpretation of collective
tree’s dynamics in terms of 4-star’s behavior. The system shows dynamical clustering in form of
the structured phase space organization of orbits for all coupling strengths. Furthermore, various
manifestations of the non-symplectic coupling are explored, including quasi-periodic orbits and
strange attractors with weakly positive Lyapunov exponents. In our extended 4-star system
whose dynamical units are driving each other, for certain coupling strengths we find the evidence
of strange nonchaotic attractors displaying quantitative features which are known to appear in
non-periodically driven maps.
We employ the same two-dimensional chaotic maps for studying the stability of a real directed
gene regulatory network of bacterium Escherichia Coli, the data on which are empirically known.
The main cooperative effects including stability, clustering and long-range correlations are still
present in the network’s emergent dynamics. However, with increase of coupling strength the
motion destabilizes on a sub-network of specific genes, although still maintaining some coherent
properties. For comparison, a two-dimensional Hill model of gene interaction is implemented
on the same Escherichia Coli network. We find the system to exhibit stable attractors and the
flexibility of response to external stimuli, along with the robustness to fluctuations of the environmental
inputs.