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Abstract We study and characterize the topology of connectivity circuits observed in natively folded protein structures whose coordinates are deposited in the Protein Data Bank (PDB). Polypeptide chains of some proteins naturally fold into unique knotted configurations. Another kind of nontrivial topology of polypeptide chains is observed when, in addition to covalent bonds connecting consecutive amino acids in polypeptide chains, one also considers disulfide and ionic bonds between non‐consecutive amino acids. Bonds between non‐consecutive amino acids introduce bifurcation points into connectivity circuits defined by bonds between consecutive and nonconsecutive amino acids in analyzed proteins. Circuits with bifurcation points can form θ‐curves with various topologies. We catalog here the observed topologies of θ‐curves passing through bridges between consecutive and non‐consecutive amino acids in studied proteins. 

Random knot models are often used to measure the types of entanglements one would expect to observe in an unbiased system with some given physical property or set of properties. In nature, macromolecular chains often exist in extreme confinement. Current techniques for sampling random polygons in confinement are limited. In this paper, we gain insight into the knotting behavior of random polygons in extreme confinement by studying random polygons restricted to cylinders, where each edge connects the top and bottom disks of the cylinder. The knot spectrum generated by this model is compared to the knot spectrum of rooted equilateral random polygons in spherical confinement. Due to the rooting, such polygons require a radius of confinement R ⩾ 1. We present numerical evidence that the polygons generated by this simple cylindrical model generate knot probabilities that are equivalent to spherical confinement at a radius of R ≈ 0.62. We then show how knot complexity and the relative probability of different classes of knot types change as the length of the polygon increases in the cylindrical polygons. 

The goal of this chapter is to describe different techniques used to measure knotting in open curves. Note that there is no "agreed upon" definition for describing knotting in open curves. As a result, we describe the context motivating each definition and then describe some advantages and disadvantages of the different approaches.