A tile model of entangled proteins

Abstract: It has been established that knots, slipknots, and other complex topologies exist in cellular proteins and have been conserved throughout evolution. Yet the global topology of a chain only provides a limited view of its topological entanglement. In particular, even if the chain is unknotted, there can be tangling of sites which may be distant on the amino acid sequence but are twisted together and held in place by molecular forces. We model this type of entanglement by starting with a collection of $1$-string and $2$-string tiles that are based on observed protein chains, which we then combine according to a well defined set of operations. Using the algebra of tiles and operations we symbolically represent all entanglements that can be built up in this way, and determine which knot types can occur. These include all knots that have been thus far identified in proteins.

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