Why is that the challenge? You wrote in an earlier post that the challenge was to evolve, for example, a flagellum protein from a protein with another function. But a protein with another function would not constitute a “distant nonfunctional sequence”. it’s not clear that a protein with another function would even be distant.
You brought up the example of the filament proteins of the flagellum as a supposed example of one of these proteins that would have been too unlikely to evolve from some other function because you could find homologoues of the filament protein with as little as 20-30% amino acid sequence similarity.
I then pointed out that there are filament proteins that simultaneously function as adhesive proteins, and even more hilariously, there are filament proteins that are active enzymes. Filament proteins (flagellin), just like antibodies, have a hypervariable region which mutates a lot (for the reason that filament proteins are often the target of immune system antibodies in multicellular eukaryotes).
See:
Moens S, Vanderleyden J. Functions of bacterial flagella. Crit Rev Microbiol.
1996;22(2):67-100. Review. PubMed PMID: 8817078.
Eckhard U, Bandukwala H, Mansfield MJ, Marino G, Cheng J, Wallace I, Holyoak T, Charles TC, Austin J, Overall CM, Doxey AC. Discovery of a proteolytic flagellin family in diverse bacterial phyla that assembles enzymatically active flagella. Nat Commun. 2017 Sep 12;8(1):521. doi: 10.1038/s41467-017-00599-0
This is evidence that the “islands” constituting different functions in “sequence space” considerably overlap, and evidence against the claim that they are impossibly rare and isolated. This means three known functional “islands” overlap in the flagellum flagellin protein. It’s a filament protein, an adhesive protein, and an enzyme at the same time.
At some point, an entirely new fold had to appear even if one has to go back to the origin of life.
An entirely new fold is not the same as a new function. But proteins with novel folds typically evolve de novo from non coding DNA.
In the middle of sequence space, none of the processes you cite will help a search find an exceedingly rare functional island.
This idea that functional islands are exceedingly rare and isolated is a fantasy. Taking an existing enzyme, then mutating it until it stops working doesn’t say anything about how frequent functional proteins are in sequence space.
More importantly, you have still not acknowledged your complete misunderstanding of the Bershtein et al 2006 paper you keep referencing.