Circuit topology

A simple example of a folded chain is a chain with two hard contacts. For a chain with two binary contacts, three arrangements are available: parallel (P), series (S), and crossed (X). For a chain with n contacts, the topology can be described by an n by n matrix in which each element illustrates the relation between a pair of contacts and may take one of the three states, P, S and X. Multivalent contacts can also be categorised in full or via decomposition into several binary contacts. Similarly, circuit topology allows for the classification of the pairwise arrangements of chain crossings and tangles, thus providing a complete 3D description of folded chains. Furthermore, one can apply circuit topology operations to soft and hard contacts to generate complex folds, using a bottom-up engineering approach.

Both knot theory and circuit topology aim to describe chain entanglement, making it important to understand their relationship. Knot theory considers any entangled chain as a connected sum of prime knots, which are themselves undecomposable. Circuit topology splits any entangled chains (including prime knots) into basic structural units called soft contacts, and lists simple rules on how soft contacts can be put together. An advantage of circuit topology is that it can be applied to open linear chains with intra-chain interactions, so-called hard contacts. This enabled topological analysis of proteins and genomes, which are often described as unknot in knot theory. Finally, circuit topology enables studying interactions between hard contacts and entanglements and can identify slip knots, while knot theory typically overlooks hard contacts and split knots. Thus, circuit topology serves as a complementary approach to knot theory.

Circuit topology has implications for folding kinetics and molecular evolution and has been applied to engineer polymers including molecular origami. Circuit topology along with contact order and size are determinants of the folding rate of linear polymers. It has also been applied to quantify the conformational organisation of intrinsically disordered proteins. In protein structure prediction, coarse-grained generative approaches recover global topological features of protein folds and enable rapid contact map prediction. Circuit topology can be applied to characterise the topology of multi-chain systems as well, including biomolecular condensates and aggregates. For example, it has been used to characterise coil–globule transitions and aggregation in polymers, revealing multichain topological motifs during collective structural transitions. In addition, circuit topology has been applied to classify conformational substates in amyloid polypeptides, providing insight into aggregation mechanisms and their modulation by small molecules. Finally, the topology approach can also be used for medical applications including disease analysis, and drug response predictions.