Algorithmic Topology of RNA Secondary Structures: Free-Energy Minimization and Pseudoknot Prediction in Genomic Sequences

We present a theoretical and computational study of RNA secondary structure algorithmic topology, free-energy minimization, and pseudoknot prediction in complex genomic sequences using graph theory and dynamic programming. Moving beyond traditional thermodynamic algorithms, we formalize RNA structures as planar and non-planar graphs over nucleotide sequences, develop advanced dynamic programming methods incorporating stack energies and transition probability matrices, and prove complexity bounds for optimal conformations with pseudoknots. Our analysis examines interval graph representations, arc diagrams, thermodynamic state partition functions, and derives an original analytical estimate for the probability of pseudoknot formation as a function of sequence length and GC-content. Furthermore, we outline hybrid algorithmic implementation protocols and validate benchmark accuracy against standard genomic datasets.