First thing I clear that living organisms are made up of several biochemically active components which together makes it ‘living', but technically we can't say them living individually because together they make a functional biological system which is living.
Here you are considering DNA as a living, its okay but you are referring it also as an organism which is not appropriate or technically correct.
Second thing Nucleus is comprised of DNA and RNA (95% and 5% respectively). So being an nucleus is not likely to be an living organism. If it is then every atom on earth is living organism.
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