Figure 3: Quantitative analyses of power law scaling in folk songs and in protein-folding dynamics.

(a) First line of the song ‘Gedenk Mit Hochgefuehl An Jene’ with accession code ‘elsass15’. Names of notes are underneath, along with the division into segments based on every occurrence on the note C. (b) Empirical frequencies of different segments showing clear evidence of power law scaling along with the theoretical slope predicted by a fifth-order random walk model. (c) Diagram of a first order random walk model built based on all 8,466 songs. Grey lines indicate possible transitions, while the lengths of outgoing arrows are proportional to transition probabilities. The notes and rest are also labelled with their overall frequencies in the data. (d) Predicted slope of the scaling relationship according to random walk models of different orders. An order-five model provides the best match to the empirical slope of the relationship, as obtained by linear regression. (e) Diagram of a 3,683-node random walk model of G-protein folding³¹. Nodes represent protein conformations and links are possible transitions, with stepping probabilities estimated by molecular dynamics simulations. Colours indicate different basins of the energy landscape (see next panel). (f) Nodes are ranked by their mean first passage time to the native state, and the cut-based free energy is calculated. These were manually separated into seven different energy basins (red, orange,..., grey) between which there is a sharp increase in free energy. (g) Although all seven power law distributions have slopes close to −1, they are not all the same. Intriguingly, the slopes appear strongly related to the activation free energies of each basin.