Erica Flapan

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 $$3_1$$3 1 , $$4_1$$4 1 , $$5_1$$5 1 , $$5_2$$5 2 , $$6_1$$6 1 , $$6_2$$6 2 , $$6_3$$6 3 , $$7_7$$7 7 , $$8_{12}$$8 12 and connected sums of these knots. This includes all proteins knots that have thus far been identified.

Let $G$ and $H$ be disjoint embeddings of complete graphs $K_m$ and $K_n$ in $\mathbb{R}^3$ such that some cycle in $G$ links a cycle in $H$ with non-zero linking number. We say that $G$ and $H$ are weakly linked if the absolute value of the linking number of any cycle in $G$ with a cycle in $H$ is $0$ or $1$. Our main result is an algebraic characterisation of when a pair of disjointly embedded complete graphs is weakly linked. As a step towards this result, we show that if $G$ and $H$ are weakly linked, then each contains either a vertex common to all triangles linking the other or a triangle which shares an edge with all triangles linking the other. All families of weakly linked pairs of embedded complete graphs are then characterised by which of these two cases holds in each complete graph.

We show that the minimum number of sticks required to construct a non-paneled knotless embedding of $K_4$ is 9 and of $K_5$ is 12 or 13. We use our results about $K_4$ to show that the probability that a random linear embedding of $K_{3,3}$ in a cube is in the form of a M\"{o}bius ladder is $0.97380\pm 0.00003$, and offer this as a possible explanation for why $K_{3,3}$ subgraphs of metalloproteins occur primarily in this form.

We classify all groups which can occur as the topological symmetry group of some embedding of the Heawood graph in $S^3$ .

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 ${\mathrm{6}}_1$-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 $3_1$-knotted proteins and the $4_1$-knotted ketol-acid reductoisomerases and present alternative folding mechanisms for the $4_1$-knotted phytochromes and the $5_2$- and $6_1$-knotted proteins. Significance Knotting in proteins was once considered exceedingly rare. However, systematic analyses of solved protein structures over the last two decades have demonstrated the existence of many deeply knotted proteins. Conservation of knotting across some protein families strongly suggests that knotting can be important for protein structure and function, and hence, significant interest has arisen around how protein knots form. We build on results of previous computer simulations and prior theories of protein knot formation to obtain theoretical pathways for protein knotting that could apply to any knotted protein. By comparing our theoretical pathways with structural data on solved proteins, we determine which of our pathways may be feasible for each of the known protein knot types.