In a natively folded protein of moderate or larger size, the protein backbone may weave through itself in complex ways, raising questions about what sequence of events might have to occur in order for the protein to reach its native configuration from the unfolded state. A mathematical framework is presented here for describing the notion of a topological folding barrier, which occurs when a protein chain must pass through a hole or opening, formed by other regions of the protein structure. Different folding pathways encounter different numbers of such barriers and therefore different degrees of frustration. A dynamic programming algorithm finds the optimal theoretical folding path and minimal degree of frustration for a protein based on its natively folded configuration. Calculations over a database of protein structures provide insights into questions such as whether the path of minimal frustration might tend to favor folding from one or from many sites of folding nucleation, or whether proteins favor folding around the N terminus, thereby providing support for the hypothesis that proteins fold co-translationally. The computational methods are applied to a multi-disulfide bonded protein, with computational findings that are consistent with the experimentally observed folding pathway. Attention is drawn to certain complex protein folds for which the computational method suggests there may be a preferred site of nucleation or where folding is likely to proceed through a relatively well-defined pathway or intermediate. The computational analyses lead to testable models for protein folding.

Among proteins of known three-dimensional structure, only a few possess complex topological features such as knotted or interlinked (catenated) protein backbones. Such unusual proteins offer potentially unique insights into folding pathways and stabilization mechanisms. They also present special challenges for both theorists and computational scientists interested in understanding and predicting protein-folding behavior. Here, we review complex topological features in proteins with a focus on recent progress on the identification and characterization of knotted and interlinked protein systems. Also, an approach is described for designing an expanded set of knotted proteins.