Topological Methods for Exploring Low-density States in Biomolecular   Folding Pathways
  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. 
Yuan Yao, Jian Sun, Xuhui Huang, Gregory R. Bowman, Gurjeet Singh,   Michael Lesnick,