  Conformational transitions are ubiquitous in biomolecular systems, have
significant functional roles and are subject to evolutionary pressures. Here we
provide a first theoretical framework for topological transition, i.e.
conformational transitions that are associated with changes in molecular
topology. For folded linear biomolecules, arrangement of intramolecular
contacts is identified as a key topological property, termed as circuit
topology. Distance measures are proposed as reaction coordinates to represent
progress along a pathway from initial topology to final topology. Certain
topological classes are shown to be more accessible from a random topology. We
study dynamic stability and pathway degeneracy associated with a topological
reaction and found that off-pathways might seriously hamper evolution to
desired topologies. Finally we present an algorithm for estimating the number
of intermediate topologies visited during a topological reaction. The results
of this study are relevant to, among others, structural studies of RNA and
proteins, analysis of topologically associated domains in chromosomes, and
molecular evolution.
