  Recently, machine learning (ML) has established itself in various worldwide
benchmarking competitions in computational biology, including Critical
Assessment of Structure Prediction (CASP) and Drug Design Data Resource (D3R)
Grand Challenges. However, the intricate structural complexity and high ML
dimensionality of biomolecular datasets obstruct the efficient application of
ML algorithms in the field. In addition to data and algorithm, an efficient ML
machinery for biomolecular predictions must include structural representation
as an indispensable component. Mathematical representations that simplify the
biomolecular structural complexity and reduce ML dimensionality have emerged as
a prime winner in D3R Grand Challenges. This review is devoted to the recent
advances in developing low-dimensional and scalable mathematical
representations of biomolecules in our laboratory. We discuss three classes of
mathematical approaches, including algebraic topology, differential geometry,
and graph theory. We elucidate how the physical and biological challenges have
guided the evolution and development of these mathematical apparatuses for
massive and diverse biomolecular data. We focus the performance analysis on the
protein-ligand binding predictions in this review although these methods have
had tremendous success in many other applications, such as protein
classification, virtual screening, and the predictions of solubility, solvation
free energy, toxicity, partition coefficient, protein folding stability changes
upon mutation, etc.
