Fig. 2: Automated tracing and topological determination for complex DNA molecules from AFM images.

Pre-processed AFM images are masked to identify molecules (of N = 1010 total DNA plasmid, knot and catenane molecules imaged) using (a) classical image processing and b deep learning methods (U-Net masking). c An enhanced skeletonisation method, using height-biasing, reduces binary masks to single pixel traces along the centre of the molecule, and d locates the central point or node of the crossings (black), minimising misalignment of emanating branches. e Emanating branches (purple, pink, blue and dark green) are paired based on their propagating vectors and a height trace spanning the crossing segment (light green and pink) is obtained. Arrows show the propagation direction of the segments. f The full-width half-maximum (FWHM) of each crossing is used to determine the crossing order of the segments, with the maximal FWHM designated as over-passing. The over-passing segment can be seen in green while the under-passing segment can be seen in pink. Dotted lines illustrate their FWHM measurements. g, h Branch pairing at each node enables separation of entangled molecules, shown here in magenta and cyan. i Under- and over-passing crossing classifications allow for topological classification which is output in Rolfsen knot notation format⁸², where 4²₁ correctly denotes the 4-node DNA catenane formed of two circular molecules that cross one another a minimum of four times. Scale bar: a–c: 50 nm, d, e: 10 nm, g–i: 50 nm. Height scale (c): −2 to 6 nm. N = 1 repeat for this image. Source data are provided as a Source Data file.