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We study here global and local entanglements of open protein chains byimplementing the concept of knotoids. Knotoids have been introduced in 2012 byVladimir Turaev as a generalization of knots in 3-dimensional space. Moreprecisely, knotoids are diagrams representing projections of open curves in 3Dspace, in contrast to knot diagrams which represent projections of closedcurves in 3D space. The intrinsic difference with classical knot theory is thatthe generalization provided by knotoids admits non-trivial topologicalentanglement of the open curves provided that their geometry is frozen as it isthe case for crystallized proteins. Consequently, our approach doesn't requirethe closure of chains into loops which implies that the geometry of analysedchains does not need to be changed by closure in order to characterize theirtopology. Our study revealed that the knotoid approach detects protein regionsthat were classified earlier as knotted and also new, topologically interestingregions that we classify as pre-knotted.

Studies of global and local entanglements of individual protein chains using the concept of knotoids