Abstract

This talk presents a very concrete application of knot theory and geometric computation to a problem in biology, together with a number of new and interesting open questions for mathematicians. The DNA of Salmonella Typhimurium is packed tightly into an ellipsoidal nucleoid, with the relative dimensions of about 250 meters of fishing line packed into a ball the size of a grapefruit. During replication, this cord splits down the middle, forming two entangled copies which must be mechanically separated from each other. This task is done by specialized enzymes called topoisomerases, which cut strands of DNA and pass them through each other. How many enzyme actions are required to unlink the DNA is determined by the knot type of the genome, and estimating this knotting is therefore an important problem.

We present results from OLIGOStorm; a new microscopy technique which can get partially labeled 3d point cloud data from the DNA in individual S. Typhimurium cells, with errors measured in tens of nanometers. This data is far too coarse to use conventional curve reconstruction techniques to resolve the underlying knot type of the DNA precisely, but we present computational evidence that analysis of even very coarse subsamples of curve ensembles can distinguish between underknotted and randomly knotted curves. Our results provide direct experimental evidence supporting the hypothesis that S. Typhimurium DNA is strongly underknotted, implying a robust biological mechanism controlling the packing of DNA into the nucleoid.

This talk covers joint work with Clayton Shonkwiler (Colorado State), Henrik Schumacher (Aachen), the Aiden and Nir labs (University of Texas Medical Branch), Ronaldo Oliveira (Rice and UFTM, Brazil) and Angel Rodriguez (UCR, Costa Rica).