Modular Gene Dynamics and Network Theory at Mesoscopic Scale

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. (FULL ABSTRACT INSIDE THE TEXT)

💡 Research Summary

This thesis investigates the collective dynamics of chaotic maps coupled on undirected networks, with a particular focus on how mesoscopic motifs influence the stability of large‑scale structures. The basic dynamical unit is a two‑dimensional chaotic map (the logistic map in its standard form) that, unlike conventional symplectic couplings, interacts with its neighbors through a non‑symplectic, time‑delayed coupling. The coupling rule can be written as

 x_i(t + 1) = (1 − ε) f(x_i(t)) + (ε/k_i) ∑_{j∈N(i)} f(x_j(t − τ)),

where ε is the coupling strength, τ the integer delay, k_i the degree of node i, and f the chaotic map. This formulation captures two biologically relevant features of gene regulatory networks: (1) the asymmetry of transcription‑translation feedback, and (2) the finite time required for protein synthesis and transport.

Two network scales are examined. At the macroscopic level, two full‑size networks are generated: (a) a scale‑free tree produced by a preferential‑attachment algorithm, and (b) a modified tree in which cliques (complete subgraphs) are grafted onto selected branches. At the mesoscopic level, the smallest topological unit that still reflects the tree’s branching structure—a four‑node star (4‑star)—is isolated and studied as a dynamical motif. The 4‑star consists of a central hub connected to three peripheral leaves; it is the simplest representation of a branching point and allows a clean implementation of the delayed interaction.

Numerical simulations sweep ε from 0 to 1 and τ from 0 to several time steps. The principal diagnostics are the largest Lyapunov exponent (λ_max), the distribution of finite‑time Lyapunov exponents, and the autocorrelation function of node trajectories. The results can be summarized as follows:

1. Full‑size trees – For very small ε each node behaves like an isolated chaotic map (λ_max ≈ ln 2). As ε crosses a critical value ε_c, λ_max begins to drop, indicating a partial suppression of chaos. In the pure scale‑free tree this transition is relatively sharp; the system moves from a fully chaotic regime to a mixed state where the hub and a few high‑degree nodes exhibit quasi‑periodic orbits while low‑degree leaves remain chaotic. Adding cliques broadens the stable region: the dense intra‑clique connections act as a local buffer, absorbing fluctuations and allowing a larger ε interval in which λ_max stays negative or close to zero.

2. 4‑star motif – Even at the same ε and τ values used for the full trees, the 4‑star displays a richer set of dynamical regimes. The central hub quickly locks onto a low‑dimensional attractor, while the three leaves oscillate with a common period that depends on τ. This asymmetry creates a “seed” of cooperativity: when the motif is embedded back into the larger tree, the hub’s stabilized dynamics propagates outward through the branching hierarchy, facilitating a cascade of synchronization that would not occur in a purely random coupling scenario.

3. Effect of the delay τ – Increasing τ generally enlarges the stability window because the delayed feedback introduces an additional degree of freedom that can dampen instantaneous chaotic bursts. However, beyond a certain τ_max the system experiences over‑compensation: the delayed term overshoots, re‑introducing high‑frequency fluctuations and generating a secondary chaotic band. This non‑monotonic dependence mirrors biological observations where moderate transcriptional delays are beneficial for robustness, whereas excessive delays can lead to oscillatory pathologies.

4. Quantitative measures – The distribution of finite‑time Lyapunov exponents narrows markedly in the stable regime, confirming that the suppression of chaos is not a transient artifact but a robust property of the coupled system. Autocorrelation functions reveal long‑range temporal order in the hub nodes, while peripheral nodes retain shorter correlation times, reflecting a hierarchical organization of temporal scales.

The thesis argues that mesoscopic motifs such as the 4‑star are the fundamental building blocks that determine the emergent cooperative behavior of larger gene‑regulatory networks. By demonstrating that a simple delayed, non‑symplectic coupling can simultaneously reduce chaos and promote partial synchronization, the work provides a mechanistic bridge between network topology (scale‑free branching, presence of cliques) and functional stability observed in real biological systems.

Finally, the author outlines several avenues for future research: (i) validation against experimental gene‑expression time series, (ii) extension to other network families (small‑world, random, modular), (iii) incorporation of heterogeneous delays and nonlinear coupling functions, and (iv) exploration of control strategies that exploit the identified mesoscopic motifs to steer the system toward desired functional states. The findings suggest that designing synthetic gene circuits with appropriate delayed feedback at key branching points could be a viable route to achieving robust, tunable behavior in synthetic biology applications.