Exploring the correlation between the folding rates of proteins and the entanglement of their native states

The folding of a protein towards its native state is a rather complicated process. However, there is empirical evidence that the folding time correlates with the contact order, a simple measure of the spatial organization of the native state of the protein. Contact order is related to the average length of the main chain loops formed by amino acids that are in contact. Here we argue that folding kinetics can also be influenced by the entanglement that loops may undergo within the overall three-dimensional protein structure. In order to explore such a possibility, we introduce a novel descriptor, which we call 'maximum intrachain contact entanglement'. Specifically, we measure the maximum Gaussian entanglement between any looped portion of a protein and any other non-overlapping subchain of the same protein, which is easily computed by discretized line integrals on the coordinates of the C-alpha atoms. By analyzing experimental data sets of two-state and multi-state folders, we show that the new index is also a good predictor of the folding rate. Moreover, being only partially correlated with previous methods, it can be integrated with them to yield more accurate predictions.