  In this paper, we systematically review weighted persistent homology (WPH)
models and their applications in biomolecular data analysis. Essentially, the
weight value, which reflects physical, chemical and biological properties, can
be assigned to vertices (atom centers), edges (bonds), or higher order
simplexes (cluster of atoms), depending on the biomolecular structure,
function, and dynamics properties. Further, we propose the first localized
weighted persistent homology (LWPH). Inspired by the great success of element
specific persistent homology (ESPH), we do not treat biomolecules as an
inseparable system like all previous weighted models, instead we decompose them
into a series of local domains, which may be overlapped with each other. The
general persistent homology or weighted persistent homology analysis is then
applied on each of these local domains. In this way, functional properties,
that are embedded in local structures, can be revealed. Our model has been
applied to systematically studying DNA structures. It has been found that our
LWPH based features can be used to successfully discriminate the A-, B-, and
Z-types of DNA. More importantly, our LWPH based PCA model can identify two
configurational states of DNA structure in ion liquid environment, which can be
revealed only by the complicated helical coordinate system. The great
consistence with the helical-coordinate model demonstrates that our model
captures local structure variations so well that it is comparable with
geometric models. Moreover, geometric measurements are usually defined in very
local regions. For instance, the helical-coordinate system is limited to one or
two basepairs. However, our LWPH can quantitatively characterize structure
information in local regions or domains with arbitrary sizes and shapes, where
traditional geometrical measurements fail.
