  Biomolecular structure comparison not only reveals evolutionary
relationships, but also sheds light on biological functional properties.
However, traditional definitions of structure or sequence similarity always
involve superposition or alignment and are computationally inefficient. In this
paper, I propose a new method called persistent similarity, which is based on a
newly-invented method in algebraic topology, known as persistent homology.
Different from all previous topological methods, persistent homology is able to
embed a geometric measurement into topological invariants, thus provides a
bridge between geometry and topology. Further, with the proposed persistent
Betti function (PBF), topological information derived from the persistent
homology analysis can be uniquely represented by a series of continuous
one-dimensional (1D) functions. In this way, any complicated biomolecular
structure can be reduced to several simple 1D PBFs for comparison. Persistent
similarity is then defined as the quotient of sizes of intersect areas and
union areas between two correspondingly PBFs. If structures have no significant
topological properties, a pseudo-barcode is introduced to insure a better
comparison. Moreover, a multiscale biomolecular representation is introduced
through the multiscale rigidity function. It naturally induces a multiscale
persistent similarity. The multiscale persistent similarity enables an
objective-oriented comparison. State differently, it facilitates the comparison
of structures in any particular scale of interest. Finally, the proposed method
is validated by four different cases. It is found that the persistent
similarity can be used to describe the intrinsic similarities and differences
between the structures very well.
