That's not simplistic at all, and you are being completely naive. So what happens when you go to the trouble at "inventing" a new synthetized DNA, only to some years later find out another living being with the same genetic code? Because you invented this code by yourself, are you still allowed to state that you "own" it? This is patently ridiculous, since you run towards the exact same problem as if you only "discovered" it. Notice the brackets I place in those two words, there's a reason for it. Who gets to know whether if a company really synthetized a genetic code or just "found it"? And what is research and development, if not "finding out" what is possible in the gene landscape?
So you see, you are trying to make a distinction where none is possible. DNA copyright is something that is completely abhorrent, and the more people realise this sooner the better. Because companies are already copyrighting DNA sequences that are human. Imagine the consequences. Dystopia is just a corner away.
You're talking to a guy whose [first] university degree reads with the words "Molecular Genetics" featured prominently in the title. Saying I'm naive on this subject would be exceedingly foolish on your part.
Synthetic genes and non-synthetic genes look completely different in codon structure. At the present time, it would be near-impossible for someone to create a gene that coincidentally matched with a naturally-occurring gene; any such match would not be a coicidence, but intentional patterning after nature, and thus (I believe) should not be patentable.
At present, the knowledge to effectively "design" a gene doesn't exist for practical application; any synthetic genes are based off naturally-occurring ones. This is not to say that we are unlikely to be able to create designer genes in the future; hence, those who make the investment to do so should receive the rewards for doing so - a patent.
For now, existing patents on DNA sequences lean to discovery (and application of said discovery), which I tend to think is wrong. As an example, BRCA1 is/was under patent, making any test (not just the one sold by the patent holder) for that gene illegal without royalty payment. In Canada, that [American] patent has not been recognized and tests are routinely done for it without royalty payment.
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To address the more recent posts, any chemical is made up of elements - chemical formulations are subject to patent (and the same argument about natural-occurrence would apply). There is nothing fundamentally different about nucleic acids that should make patent law apply any differently - come up with something novel, patent should apply. Use/discover what already exists, absolutely not. And I'm also looking at this from a serious [molecular] biologists standpoint.