Topological Entanglement in Intrinsically Disordered Proteins: Sequence, Structural, and Functional Determinants

Intrinsically disordered proteins (IDPs) populate heterogeneous conformational ensembles that are difficult to characterize using conventional structural descriptors. As a result, it remains unclear which ensemble features meaningfully connect sequence composition to biological function. Here, we employ entanglement-based measures derived from knot theory to provide complementary insight into IDP organization. Using the human IDRome database, we analyze two continuous entanglement descriptors, the writhe and the second Vassiliev invariant $\left(V_2\right)$ , across more than 28,000 simulated disordered sequences. We show that these entanglement measures exhibit structured, low-dimensional variation across the database and display distinct relationships with sequence composition and ensemble geometry. Writhe primarily reflects compaction-dependent coiling tendencies that are largely recoverable from coarse sequence and structural features, whereas $V_2$ captures higher-order topological organization that is less predictable from simple metrics. Embedding the resulting distribution features reveals functionally enriched regions of entanglement space, and ortholog simulations demonstrate that these signatures are evolutionarily conserved. Together, these results establish entanglement as a biologically relevant dimension of IDP organization and provide a rigorous, complementary framework for linking its sequence, ensemble structure, and molecular function.