I'm a fan of privacy, so I wouldn't like people secretly taking skin samples and distributing facts about my DNA, but I wouldn't like the ability to license parts of my own DNA and expect the government to enforce IP laws to cover my DNA.

That's privacy, not copyright. Completely unrelated.

Not really, the incentive to copyright incentivizes the invasion of privacy. I can walk behind you with a trash bag and voila. Sure, they are different but related.

I imagine there would be trouble with the logistics of copyrighting molecular structures whose primary distinction is their propensity to copy themselves.

Software is commonly accepted as copyrightable, and no exception is made for quines.

1. Create computer virus 2. Spread computer virus 3. Sue victims for copyright infringement

See Monsanto.

Or drug companies, like here: http://en.wikipedia.org/wiki/BRCA1#Patents.2C_enforcement.2C....

Patent != Copyright

I'm not sure we want to go down this road. For one, it's not "your own" DNA. Barring the occasional mutation, every bit of it came from your parents, grandparents...

I'm not comfortable with the idea that my cells contain someone else's intellectual property.

What about twins?

Maybe your parents should have it?