  Proteins are the most important biomolecules for living organisms. The
understanding of protein structure, function, dynamics and transport is one of
most challenging tasks in biological science. In the present work, persistent
homology is, for the first time, introduced for extracting molecular
topological fingerprints (MTFs) based on the persistence of molecular
topological invariants. MTFs are utilized for protein characterization,
identification and classification. The method of slicing is proposed to track
the geometric origin of protein topological invariants. Both all-atom and
coarse-grained representations of MTFs are constructed. A new cutoff-like
filtration is proposed to shed light on the optimal cutoff distance in elastic
network models. Based on the correlation between protein compactness, rigidity
and connectivity, we propose an accumulated bar length generated from
persistent topological invariants for the quantitative modeling of protein
flexibility. To this end, a correlation matrix based filtration is developed.
This approach gives rise to an accurate prediction of the optimal
characteristic distance used in protein B-factor analysis. Finally, MTFs are
employed to characterize protein topological evolution during protein folding
and quantitatively predict the protein folding stability. An excellent
consistence between our persistent homology prediction and molecular dynamics
simulation is found. This work reveals the topology-function relationship of
proteins.
