  Improvements in experimental and computational technologies have led to
significant increases in data available for analysis. Topological data analysis
(TDA) is an emerging area of mathematical research that can identify structures
in these data sets. Here we develop a TDA method to detect physical structures
in a cell that persist over time. In most cells, protein filaments (actin)
interact with motor proteins (myosins) and organize into polymer networks and
higher-order structures. An example of these structures are ring channels that
maintain constant diameters over time and play key roles in processes such as
cell division, development, and wound healing. The interactions of actin with
myosin can be challenging to investigate experimentally in living systems,
given limitations in filament visualization \textit{in vivo}. We therefore use
complex agent-based models that simulate mechanical and chemical interactions
of polymer proteins in cells. To understand how filaments organize into
structures, we propose a TDA method that assesses effective ring generation in
data consisting of simulated actin filament positions through time. We analyze
the topological structure of point clouds sampled along these actin filaments
and propose an algorithm for connecting significant topological features in
time. We introduce visualization tools that allow the detection of dynamic ring
structure formation. This method provides a rigorous way to investigate how
specific interactions and parameters may impact the timing of filamentous
network organization.
