The patent office (soon to be renamed "Intellectual Property office") website explains that for a patent to be granted
Your invention must:
* be new
* have an inventive step that is not obvious to someone with knowledge and experience in the subject
* be capable of being made or used in some kind of industry
* not be:
a scientific or mathematical discovery, theory or method
a literary, dramatic, musical or artistic work
a way of performing a mental act, playing a game or doing business
the presentation of information, or some computer programs
an animal or plant variety
a method of medical treatment or diagnosis
against public policy or morality.
Your DNA does not involve an "inventive step" and the exclusion of "presentation of information" would prevent a patent offering any protection against simply recording a sequence of nucleotide bases. "DNA patents" that have been granted by patent offices tend to involve the identification of genes within these sequences, in which cases it may be argued that there is an inventive step.