Mathematical Analysis of PDB-to-SMILES-to-PDB Transformations: Structural Optimization, Coordinate Invariance, and Molecular Interaction Fidelity

The bidirectional transformation of molecular structures between the Protein Data Bank (PDB) format and the Simplified Molecular Input Line Entry System (SMILES) is widely used to rectify structural distortions in molecular geometry. This process inherently minimizes energy, optimizes bond lengths, angles, and torsional constraints while introducing a global coordinate shift. In this paper, we provide a mathematical breakdown of PDB-to-SMILES and SMILES-to-PDB transformations and formally prove that while these transformations rectify the molecular distortions, the resulting coordinate shift does not alter molecular interaction properties. We will prove this using molecular mechanics force fields, rigid-body transformations, and quantum mechanical energy functionals, confirming the preservation of key interaction terms governing molecular behaviour.