  Two types of osmolytes, i.e., trimethylamin N-oxide (TMAO) and urea,
demonstrate dramatically different properties in a protein folding process.
Even with the great progresses in revealing the potential underlying mechanism
of these two osmolyte systems, many problems still remain unsolved. In this
paper, we propose to use the persistent homology, a newly-invented topological
method, to systematically study the osmolytes molecular aggregation and their
hydrogen-bonding network from a global topological perspective. It has been
found that, for the first time, TMAO and urea show two extremely different
topological behaviors, i.e., extensive network and local cluster. In general,
TMAO forms highly consistent large loop or circle structures in high
concentrations. In contrast, urea is more tightly aggregated locally. Moreover,
the resulting hydrogen-bonding networks also demonstrate distinguishable
features. With the concentration increase, TMAO hydrogen-bonding networks vary
greatly in their total number of loop structures and large-sized loop
structures consistently increase. In contrast, urea hydrogen-bonding networks
remain relatively stable with slight reduce of the total loop number. Moreover,
the persistent entropy (PE) is, for the first time, used in characterization of
the topological information of the aggregation and hydrogen-bonding networks.
The average PE systematically increases with the concentration for both TMAO
and urea, and decreases in their hydrogen-bonding networks. But their PE
variances have totally different behaviors. Finally, topological features of
the hydrogen-bonding networks are found to be highly consistent with those from
the ion aggregation systems, indicating that our topological invariants can
characterize intrinsic features of the "structure making" and "structure
breaking" systems.
