Topological Phase Transitions and Phase-Locking Dynamics in Proteomic Networks: A Deterministic Model of Structural Robustness

Background This study investigates the deterministic constraints governing the structural integrity of proteomic networks. We propose a theoretical framework where protein stability is modeled through a hierarchy of universal fractal attractors, specifically focusing on the convergence of the Golden Ratio (\(\:\varphi\:\approx\:1.618\)) and the Feigenbaum constant (\(\:\delta\:\approx\:4.669\)). Methods We define two quantitative metrics: the Coherence Coefficient (\(\:{C}_{b}\)) and the Collective Phase-Coherence Coefficient (\(\:{C}_{coll}\)). These metrics measure the divergence between observed fractal dimensions (\(\:{D}_{obs}\)), derived from PDB atomic coordinates via mass-radius scaling laws, and theoretical geometric targets derived from the Feigenbaum-Fibonacci scale. The model was tested against a comprehensive dataset of 1,000 protein structures (curated from the RCSB PDB latest releases), categorized into four functional classes to assess the invariance of the topological parameters. Results The analysis identifies a critical regime of structural stability for \(\:0.89<{C}_{b}<0.95\), consistent with the dynamics of self-organized criticality (SOC) at the "edge of chaos." A discontinuous phase transition is observed: values below \(\:{C}_{b}<0.89\) correlate with increased entropic disorder (Topological Gap), while values exceeding \(\:{C}_{b}>0.985\) (e.g., in amyloid aggregates and prions) indicate a shift toward pathological rigidity. A Topological Robustness Factor, defined by the product \(\:\varphi\:\cdot\:\delta\:\approx\:7.55\), is proposed as a universal scaling constant for structural buffering across the dataset. Conclusions The results suggest that proteomic stability can be described as a state of geometric homeostasis maintained by non-linear oscillators. This framework provides a formal mathematical method to quantify structural robustness and predicts phase transitions in protein folding networks without relying on purely heuristic interpretations.