Topology of macromolecules

The structure and entanglement of macromolecules, such as proteins and DNA, is essential for their function. Yet, methods to characterize and quantify structure complexity and entanglement at multiple length scales are lacking. Under some conditions, we can see these biopolymers as mathematical curves in 3-space. We create and employ novel methods from topology to assess entanglement in those systems and their effects to material properties and function. One of the challenges is the very definition of knotting and linking of mathematical curves, when they are not closed, as it is the case in most biopolymers. To this end, we create a new framework in mathematics that extends the study of knots and links to open curves in 3-space. We also develop the computational methods required for the application of these mathematical methods to structures obtained from experiments. When applied them to proteins, these  new metrics reveal aspects of their structure relevant to protein folding. Coarse grained models of macromolecules, such as random walks and polygons, enable to obtain scaling complexity results relevant to material properties. We will prove how the topological complexity, as it is captured by the second Vassiliev measure, varies with confinement and molecular weight. This reflects how entangled  biopolymers in the cell are and whether there are mechanisms that control entanglement.