A tile model of circuit topology for self-entangled biopolymers.

Building on the theory of circuit topology for intra-chain contacts in entangled proteins, we introduce tiles as a way to rigorously model local entanglements which are held in place by molecular forces. We develop operations that combine tiles so that entangled chains can be represented by algebraic expressions. Then we use our model to show that the only knot types that such entangled chains can have are [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and connected sums of these knots. This includes all proteins knots that have thus far been identified.

Topological descriptions of protein folding.

How knotted proteins fold has remained controversial since the identification of deeply knotted proteins nearly two decades ago. Both computational and experimental approaches have been used to investigate protein knot formation. Motivated by the computer simulations of Bölinger et al. [Bölinger D, et al. (2010) PLoS Comput Biol 6:e1000731] for the folding of the [Formula: see text]-knotted α-haloacid dehalogenase (DehI) protein, we introduce a topological description of knot folding that could describe pathways for the formation of all currently known protein knot types and predicts knot types that might be identified in the future. We analyze fingerprint data from crystal structures of protein knots as evidence that particular protein knots may fold according to specific pathways from our theory. Our results confirm Taylor's twisted hairpin theory of knot folding for the [Formula: see text]-knotted proteins and the [Formula: see text]-knotted ketol-acid reductoisomerases and present alternative folding mechanisms for the [Formula: see text]-knotted phytochromes and the [Formula: see text]- and [Formula: see text]-knotted proteins.

Predicting knot or catenane type of site-specific recombination products.

Site-specific recombination on supercoiled circular DNA yields a variety of knotted or catenated products. Here, we present a topological model of this process and characterize all possible products of the most common substrates: unknots, unlinks, and torus knots and catenanes. This model tightly prescribes the knot or catenane type of previously uncharacterized data. We also discuss how the model helps to distinguish products of distributive recombination and, in some cases, determine the order of processive recombination products.