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I'm not familiar with the "RNA World" hypothesis.. but basically, some people think that RNA--being simpler than DNA and more common in single celled organisms, and nonliving organic vectors (like viruses)... it must have come first then, due to the previous. Having come first, then, it should be present at higher levels in older organisms and proto-organisms.
As agreene says, the RNA world hypothesis has it that the earliest living organisms had only RNA. DNA evolved later as a specialized, more efficient, stabler information storage medium and proteins also evolved later as specialized, more efficient molecular machinery. RNA has both information storage and transmission capabilities (it is an info containing sequence) and mechanic properties like proteins, so theoretically life could function with RNA alone. I'd just like to add that this is a relatively recent theory: for the last sixty years, everyone has followed the "central dogma" of biology: DNA --> RNA --> Protein. But the discovery of many different types of RNA molecule with many different functions are the basis of this newer (and still controversial) theory.
The way is because from RNA you can actually create DNA via reverse transcription. RNA can encode a protein called a reverse transcriptase who can synthesize DNA off of the RNA strand, and then in the case of reverse transcription that DNA can be inserted into other DNA molecules using a protein called an integrase. RNA both stores information and can create products to act off of the information stored in it, thereby avoiding the chicken/egg question because it doesn't do either or, but both. And Blues, while you say this is controversial, reverse transcription is a very real thing; it's how HIV infects cells and that's how (although it's in the form of etrotransposons) our genome is filled with 50% junk.
Reverse transcription simplified for you: a tRNA will "prime" the DNA synthesis complimentary to the RNA strand that RNA strand will be degraded, leaving a single strand DNA that DNA will have a complimentary strand synthesized across from it to make a double-stranded DNA
Thank you for the simplified description of reverse transcription. When I said that the RNA world hypothesis is new and controversial, I meant that it was proposed relatively recently. If 1986 counts as relatively recent. And by "controversial" I meant that my former dean of science absolutely gutted an assistant prof who mentioned it in a lecture. Some people are not tolerant of ideas that were not in the intro book when they were in school. That is all.
i think that the reason it can be controversial is the fact that no living organism uses reversetranscriptase. There are a handful of retrovirus that use it--but nothing else. As it happens all of these virus tend to be "new" to the pathology lexicon and so some think this might be an example of a new niche--not an old one.