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

Topological data analysis approaches to uncovering the timing of ring structure onset in filamentous networks