In grad school I got interested in knots while doing structural biology and asked my advisor (extremely well known chemist) if it would be possible to make proteins that fold into knots and he pointed out (correctly) that entropy makes it very unlikely this would happen, as the process of folding into a knot typically decreases the degrees of freedom a protein can adopt. It was a reasonable point and I ignored the problem for some time.
It turned out later that a small number of proteins are found in a knotted conformation experimentally, https://en.wikipedia.org/wiki/Knotted_protein and alphafold even helped design a knotted protein. I can imagine there are also enzymatic modifications you can make to proteins that would generate knots without paying the entropic cost of threading the needle, so to speak.
Note I am talking about entropy of "folding" as a verb- the process by which a fold is adopted. Not the entropy of the final state (which is confusing called a "fold" by some practioners, although that habit is decreasing).
the argument is that the folding process would require the protein to go through an intermediate: forming a loop and then getting one end to go through the loop requires a very specific set of conformations, and reducing the number of conformations has an "entropic cost". Note also that the final post-folding form of a knotted protein isn't really a continuum of many different conformations, but probably one "locked" one which means there really is only a small set of conformations the protein can adopt once locked into the folded, knotted state.
In protein folding, there is not "only one way it can not be knotted"; you have to include all the degrees of freedom in the backbone available in the unfolded state (I am ignoring hydrophobicity and other important sources of entropy which help proteins fold rapidly to the "correct" final state). We already know this pretty well from more basic polymer physics,
but, entropy being subtle, I'm sure there are many different forces at play.
I think with proteins the ways it can be "not knotted" are basically infinite. Topologically-speaking, yeah, it's just a deformed ring. But the deformations (folds) are the huge thing that makes a protein do its job.
So there are tons of folding configurations, and entering a knotted state restricts a lot of the freedom-to-fold a protein would have
Topology gets its utility from collapsing lots of degrees of freedom into a small number of equivalence classes. That's great for reasoning about topological properties, where those degrees of freedom are not relevant, but it doesn't mean those degrees of freedom just go away.
Oh, cool- they turned a not-truly-a-knot (open ends) protein into a truly knotted one (connected ends) and then unfolded it using urea and it stayed knotted. That's what you'd expect but it's also pretty cool.
A category theory textbook opens the "motivation and use cases" section with the following poem:
There's a tiresome young man in Bayshore. / When his fiance cried 'I adore / the beautiful sea' / he replied 'I agree, / it's pretty, but what is it for?'
It turned out later that a small number of proteins are found in a knotted conformation experimentally, https://en.wikipedia.org/wiki/Knotted_protein and alphafold even helped design a knotted protein. I can imagine there are also enzymatic modifications you can make to proteins that would generate knots without paying the entropic cost of threading the needle, so to speak.