See this thread for a complete explanation of this one:
http://www.manhattangmat.com/forums/sc-og-11th-ed-122-t684.html?highlight=mcclintock
You're right that comparing clauses requires "as." But we're just comparing genes to pearls, not "genes can jump."
Don't forget that you have to have two clauses to use "as," not just one. If you look at the second part of the sentence "(like / as) pears moving mysteriously from one necklace to another" - that's a phrase, not a clause.
The "ing" form of a verb can function as a verb, noun, adjective, or adverb! The easiest way to deal with this is to remember that the "ing" form only functions as a verb when it is paired with a form of "to be" - such as, "she is going to the store." Since we don't have that here, "moving" does not function as a verb, and nothing else in that big after "jump" is a verb, so we have a phrase, not a clause. Therefore, we can't use "as."