Advanced Mathematical Models, Biomathematics and Spatial Bioinformatics for the Discovery of Novel Therapeutic Targets

Rational therapeutic target discovery and systems biology face a decisive epistemological challenge: the fragmentation between the kinetic dynamics of intracellular pathways and the spatial-geometric complexity of macromolecular interactions. This work outlines an integrated analytical architecture that overcomes this dichotomy by unifying theoretical biomathematics, discrete differential geometry, and interpretable computational intelligence into a single multiscale predictive paradigm. The framework is structured across three synergistic methodological pillars: Continuous-Time Deterministic Dynamics: Using nonlinear systems of ordinary differential equations (ODEs), we formalize the temporal evolution of signal transduction cascades and metabolic fluxes, quantifying asymptotic stability, state bifurcations, and transient pathological trajectories. Geometric Deep Learning and Discrete Topological Curvature: Extending structural analysis beyond Euclidean space, we apply discrete curvature metrics (including Ricci curvatures on graphs and polygonal meshes) to molecular manifolds and protein-protein interaction networks. This approach enables the identification of cryptic allosteric sites, conformational deformations, and functional interfaces previously inaccessible to conventional docking models. Biologically Informed Neural Networks (BINNs): Transcending the opacity of black-box models, we embed physical principles, mass conservation laws, and biological topologies directly into the loss functions of neural network architectures. BINNs constrain computational learning within biophysically plausible solution spaces, enabling the accurate inference of parameters unmeasurable in vivo. The convergence of these domains yields a quantitative map of the molecular etiology underlying complex pathologies, distinguishing causal regulatory nodes from correlative epiphenomena. This architecture redefines target validation: it shifts therapeutic prediction from empirical screening to rational engineering, accelerating the in silico design of selective molecular perturbations with high efficacy and minimized systemic toxicity.