Quantum Graph-Based Differential Models with Fractional Calculus and Topological Data Analysis for Dynamic Characterization of Protein-Protein Interaction Networks

A. Understanding the intricate dynamics of Protein-Protein Interaction Networks (PPINs) is essential to decode complex biological processes and disease mechanisms.Existing graph-theoretic approaches often fall short in capturing the temporal and spatial intricacies of dynamic PPINs.To address these limitations, this study introduces a novel framework based on Quantum Graph-Based Differential Models (QGDM) integrated with Fractional Calculus and Topological Data Analysis (FC-TDA).The quantum graph formalism models PPINs with probabilistic edge dynamics, while fractional differential equations account for memory effects and long-range dependencies in protein interactions.TDA is used to extract persistent topological features and detect critical transitions in the network structure over time.The objective is to provide a high-fidelity and mathematically robust system for dynamically characterizing PPINs, enabling better insights into protein behavior under varying cellular conditions.Results from simulations on benchmark datasets such as yeast and human interactomes demonstrate superior accuracy in detecting functional modules and predicting interaction disruptions compared to existing graph and machine learning models.This integrated mathematical approach offers a powerful tool for systems biology with potential applications in drug target identification and precision medicine.