  In this paper, we introduce multiscale persistent functions for biomolecular
structure characterization. The essential idea is to combine our multiscale
rigidity functions with persistent homology analysis, so as to construct a
series of multiscale persistent functions, particularly multiscale persistent
entropies, for structure characterization. To clarify the fundamental idea of
our method, the multiscale persistent entropy model is discussed in great
detail. Mathematically, unlike the previous persistent entropy or topological
entropy, a special resolution parameter is incorporated into our model. Various
scales can be achieved by tuning its value. Physically, our multiscale
persistent entropy can be used in conformation entropy evaluation. More
specifically, it is found that our method incorporates in it a natural
classification scheme. This is achieved through a density filtration of a
multiscale rigidity function built from bond and/or dihedral angle
distributions. To further validate our model, a systematical comparison with
the traditional entropy evaluation model is done. It is found that our model is
able to preserve the intrinsic topological features of biomolecular data much
better than traditional approaches, particularly for resolutions in the mediate
range. Moreover, our method can be successfully used in protein classification.
For a test database with around nine hundred proteins, a clear separation
between all-alpha and all-beta proteins can be achieved, using only the
dihedral and pseudo-bond angle information. Finally, a special protein
structure index (PSI) is proposed, for the first time, to describe the
"regularity" of protein structures. Essentially, PSI can be used to describe
the "regularity" information in any systems.
