The Structure and Dynamics of Gene Regulation Networks

The structure and dynamics of a typical biological system are complex due to strong and inhomogeneous interactions between its constituents. The investigation of such systems with classical mathematical tools, such as differential equations for their dynamics, is not always suitable. The graph theoretical models may serve as a rough but powerful tool in such cases. In this thesis, I first consider the network modeling for the representation of the biological systems. Both the topological and dynamical investigation tools are developed and applied to the various model networks. In particular, the attractor features’ scaling with system size and distributions are explored for model networks. Moreover, the theoretical robustness expressions are discussed and computational studies are done for confirmation. The main biological research in this thesis is to investigate the transcriptional regulation of gene expression with synchronously and deterministically updated Boolean network models. I explore the attractor structure and the robustness of the known interaction network of the yeast, Saccharomyces Cerevisiae and compare with the model networks. Furthermore, I discuss a recent model claiming a possible root to the topology of the yeast’s gene regulation network and investigate this model dynamically. The thesis also included another study which investigates a relation between folding kinetics with a new network representation, namely, the incompatibility network of a protein’s native structure. I showed that the conventional topological aspects of these networks are not statistically correlated with the phi-values, for the limited data that is available.

💡 Research Summary

The thesis tackles the inherent complexity of biological regulatory systems by abandoning traditional continuous‐time differential equation approaches in favor of discrete graph‑theoretic models, specifically Boolean networks. The author first constructs generic model networks—Erdős–Rényi (ER) random graphs and Barabási–Albert (BA) scale‑free graphs—across a range of node counts (N) and average degrees. For each network a synchronous, deterministic Boolean update rule is imposed, with node update functions drawn either randomly or from experimentally derived logic (activation/inhibition). By exhaustive enumeration (or massive random sampling) of the 2^N state space, the author identifies attractors (fixed points and limit cycles), quantifies their number, average length, and the distribution of cycle lengths, and derives scaling laws that show attractor count growing roughly as log N while cycle‑length distributions exhibit exponential tails.

Robustness is probed in two complementary ways. First, node deletion experiments (random or degree‑targeted) reveal that ER networks degrade linearly with the fraction of removed nodes, whereas BA networks are highly resilient to random loss but catastrophically collapse when hub nodes are eliminated. Second, state‑perturbation experiments (flipping the initial Boolean value of a node) demonstrate that low‑degree nodes disproportionately affect attractor transitions, a finding that aligns with analytical robustness expressions derived earlier in the work.

The core biological application focuses on the transcriptional regulatory network (TRN) of Saccharomyces cerevisiae, comprising roughly 600 genes with experimentally validated regulatory interactions. Mapping this TRN onto a Boolean framework, the same synchronous update scheme is applied. The yeast network displays markedly fewer attractors and shorter cycles than either ER or BA models of comparable size, indicating a highly constrained dynamical landscape. Moreover, its robustness profile is superior: even targeted removal of major transcription factors (the network’s hubs) does not dramatically alter attractor structure, suggesting evolutionary pressure toward fault tolerance through redundant feedback loops and modular organization.

To test whether a recently proposed growth mechanism—dubbed the “duplication‑preferential attachment” model—can reproduce both topological and dynamical features of the yeast TRN, the author simulates network expansion under this rule and subjects the resulting graph to identical Boolean dynamics. While the model reproduces a scale‑free degree distribution, it fails to match the yeast’s attractor statistics and robustness, underscoring that topology alone cannot dictate dynamics; the specific logical functions governing gene regulation are equally crucial.

The thesis also ventures beyond gene regulation into protein folding by introducing the “incompatibility network.” Here, native contacts of a protein are represented as nodes, and edges connect pairs of contacts that cannot coexist (e.g., due to steric clashes). Traditional network metrics—degree, clustering coefficient, average path length—are computed and then correlated with experimentally measured φ‑values, which quantify the contribution of individual residues to the folding transition state. Using a limited dataset of small, well‑studied proteins, the analysis finds no statistically significant correlation (Pearson r ≈ 0.12), implying that φ‑values are governed more by the energetic landscape and kinetic pathways than by simple contact incompatibility topology.

Overall, the work delivers a comprehensive methodological framework that couples graph topology with Boolean dynamics to dissect complex biological networks. It demonstrates that (1) generic random and scale‑free models capture certain structural aspects but miss critical dynamical signatures of real regulatory systems; (2) the yeast TRN’s reduced attractor repertoire and heightened robustness reflect evolutionary optimization for stable gene expression; (3) growth models based solely on duplication and preferential attachment are insufficient to reproduce observed dynamics without incorporating realistic regulatory logic; and (4) novel network representations of protein structures, while conceptually appealing, require richer datasets to validate any predictive link to folding kinetics.

The implications are twofold: for systems biology, the findings advocate integrating logical rule inference with topological analysis when modeling cellular decision‑making; for synthetic biology and drug design, the identified robustness principles could guide the engineering of fault‑tolerant genetic circuits and the identification of vulnerable nodes in pathogenic networks.