  Characterization of transient intermediate or transition states is crucial
for the description of biomolecular folding pathways, which is however
difficult in both experiments and computer simulations. Such transient states
are typically of low population in simulation samples. Even for simple systems
such as RNA hairpins, recently there are mounting debates over the existence of
multiple intermediate states. In this paper, we develop a computational
approach to explore the relatively low populated transition or intermediate
states in biomolecular folding pathways, based on a topological data analysis
tool, Mapper, with simulation data from large-scale distributed computing. The
method is inspired by the classical Morse theory in mathematics which
characterizes the topology of high dimensional shapes via some functional level
sets. In this paper we exploit a conditional density filter which enables us to
focus on the structures on pathways, followed by clustering analysis on its
level sets, which helps separate low populated intermediates from high
populated uninteresting structures. A successful application of this method is
given on a motivating example, a RNA hairpin with GCAA tetraloop, where we are
able to provide structural evidence from computer simulations on the multiple
intermediate states and exhibit different pictures about unfolding and
refolding pathways. The method is effective in dealing with high degree of
heterogeneity in distribution, capturing structural features in multiple
pathways, and being less sensitive to the distance metric than nonlinear
dimensionality reduction or geometric embedding methods. It provides us a
systematic tool to explore the low density intermediate states in complex
biomolecular folding systems.
