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Geometric, topological and graph theory modeling and analysis of biomoleculesare of essential importance in the conceptualization of molecular structure,function, dynamics, and transport. On the one hand, geometric modeling providesmolecular surface and structural representation, and offers the basis formolecular visualization, which is crucial for the understanding of molecularstructure and interactions. On the other hand, it bridges the gap betweenmolecular structural data and theoretical/mathematical models. Topologicalanalysis and modeling give rise to atomic critical points and connectivity, andshed light on the intrinsic topological invariants such as independentcomponents (atoms), rings (pockets) and cavities. Graph theory analyzesbiomolecular interactions and reveals biomolecular structure-functionrelationship. In this paper, we review certain geometric, topological and graphtheory apparatuses for biomolecular data modeling and analysis. Theseapparatuses are categorized into discrete and continuous ones. For discreteapproaches, graph theory, Gaussian network model, anisotropic network model,normal mode analysis, quasi-harmonic analysis, flexibility and rigidity index,molecular nonlinear dynamics, spectral graph theory, and persistent homologyare discussed. For continuous mathematical tools, we present discrete tocontinuum mapping, high dimensional persistent homology, biomolecular geometricmodeling, differential geometry theory of surfaces, curvature evaluation,variational derivation of minimal molecular surfaces, atoms in molecule theoryand quantum chemical topology. Four new approaches, including analyticalminimal molecular surface, Hessian matrix eigenvalue map, curvature map andvirtual particle model, are introduced for the first time to bridge the gaps inbiomolecular modeling and analysis.

A review of geometric, topological and graph theory apparatuses for the modeling and analysis of biomolecular data