Your jigsaw puzzle analogy should work like this.
Put all the bits in a bag and shake them up.
Close your eyes and draw two bits randomly from the bag. See if they fit.
If they do, place them to one side. If not, put them back in the bag and draw two new bits from the bag.
Keep doing this until there are no more bits in the bag.
Through a random process combined with selection you have moved a significant step towards assembling the whole.
Now repeat the process with the groups of two pieces. Choosing two pairs at random and seeing if they fit. If they do put them to one side and if not put them back and chose two more.
You now have clumps of 4 .
Run the process on the clumps of 4 until you have clumps of 8. then 16, 32,64,128,256,512 and finally 1000 you assembled picture or proto-protein if you prefer.
Now imagine you have 1000 copies of the jigsaw in the bag and 1000 people choosing. I think most people would be surprised as to how quickly the first completed image would emerge.
(RAZD: what are the probabilities on this?)
then add a thousand new jigsaws each with a thousand copies and a thousand new choosers. It might take a bit longer to get going but it's near as damn it garaunteed that you will end up with all the jigsaws fully assembled