Traveling Salesman Finds Random Walk: A Curve Reconstruction Algorithm for Supercoiled DNA

DNA supercoiling plays an important role in a variety of cellular processes, including transcription, replication, and DNA compaction. To fully understand these processes, we must uncover and characterize the dynamics of supercoiled DNA. However, supercoil dynamics are difficult to access because of the wide range of relevant length and timescales. In this work, we present an algorithm to reconstruct the arrangement of identical fluorescent particles distributed around a circular DNA molecule, given their three-dimensional trajectories through time. We find that this curve reconstruction problem is analogous to solving the traveling salesman problem. We demonstrate that our approach converges to the correct arrangement with a sufficiently long observation time. In addition, we show that the time required to accurately reconstruct the fluorophore arrangement is reduced by increasing the fluorophore density or reducing the level of supercoiling. This curve reconstruction algorithm, when paired with next-generation super-resolution imaging systems, could be used to access and thereby advance our understanding of supercoil dynamics.