What exactly does someone who is a genetic engineer do?
Potentially lots of different things, depending on how you want to define it, I guess. Things like bioinformatics would be applicable (which, in this context, would involve working with technology to multiply/analyze DNA sequences and interpreting the results with a computer), also genomics and gene therapy (where chosen genes can be transported into an organisms DNA to allow for extra, desired functions).
These are broad topics, to give a better answer it would help to know: what is the reason/context for/of your interest?
Manipulating DNA in mice.
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