Topological Data Analysis and its Application to Chemical Systems

Abstract

Topological data analysis techniques are applied to distinct problems in chemistry, to determine their efficacy and gain new understanding of chemical systems. The mapper algorithm is utilised to understand the underlying descriptor space of a solubility prediction data set. Insight from the resulting topological summaries was able to create more consistent solubility models. Persistent homology is then used to create a series of metric spaces for molecular shape. It is shown that these metric spaces correlate with other molecular descriptors, and also allow for the accounting of molecular flexibility.

This molecular flexibility is further explored with persistent homology. By constructing a point cloud of individual conformers, a technique to characterise the conformational spaces of various molecules is developed. Alanine dipeptide is shown to have a toroidal conformational space, and persistence is then used to locate extrema on its torsional free energy surface. Pentane is then studied, and shown to also have a toroidal conformational space, or a Mobius band when symmetry is taken into account. The conformational space of cyclooctane is shown to be non-manifold, and the separate manifold components separated. It is found that there are separate spherical and Klein bottle components, before the single point energy landscape of the sphere is also analysed and extrema located.

Finally, simulated water networks are analysed through persistent homology. The general use of persistence to analyse simulations is studied, and persistence is shown to be a well-behaved descriptor. A size-agnostic persistence descriptor is generated, and used with a support vector machine to understand the differences in simulated water networks. Atomistic and coarse-grained water potentials are compared, and similarities between potentials are related to topological features.

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Identifiers

ORCID for Jeremy G. Frey: orcid.org/0000-0003-0842-4302

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