One metaphor that I have found to explain how proteins fold so quickly to a native shape is that of the blind golfer.
I have made a video to illustrate this metaphor. This shows how the shape of a slope can determine precisely where individual elements will end up.
The video represents many zero-dimensional points each following its own one-dimensional path down a three-dimensional slope to end up in a two-dimensional arrangement. Or almost: the one-dimensional paths are curves in three-dimensional space, the final resting surface is not flat, and one marble sits on top of three others.) The movement takes place through the fourth dimension of time.
Proteins form three-dimensional shapes, so the "slope" that their components slide down must be in an extra dimension.
Atoms have been observed to jump through a crystal lattice, without any indication that, in travelling from their start point to their end point, they pass through the three-dimensional space that we can observe. Is it conceivable that the component parts of a protein move through an extra dimension as the protein "folds"?
There are very simple protein chains whose native folded shape is well known. Is there any data available to suggest what path the different molecules in the protein follow during the folding process? Or do they appear to "jump"?
I imagine that it would be possible to create a mathematical model in n-dimensions to describe the simplest "slope" that the protein molecules could follow as the protein collapses into its native state.
What mathematical work is currently being done in this area?