I was actually pretty impressed with how they presented PCR (other than the whole "we didn't use enough" point) and electrophoresis. That was the best depiction I have seen in a TV show. Of course. . .
How would they PCR alien DNA with six bases (A/T, G/C, and let's say B/Q) when the lab would only have A/T and G/C in stock? We know other nucleotides exist (and have edited then into bacterial genomes) but they aren't going to be a staple of a hospital lab.
Hell, how does the alien DNA replicate in Scully's cells when they will not have a natural supply of B/Q? Did the conspirqcy guys include DNA that codes for an anabolic pathway that makes B/Q when they gave her alien DNA?
Actually, once Scully gives everyone this alien DNA, how will it replicate?
I was under the impression that the sequence itself is defined as "alien" but it still follows the A/T G/C code. Otherwise they would have been able to sequence that part of her genome.
The alien sequence could just be artificial instead of literally being taken from one of the aliens. If they have a different genetic code, maybe the scientists managed to translate whatever is needed into a humanized sequence/protein.
The described scenario would actually not be so hard to achieve with our current tech and you would not need special alien proteins.
They mentioned that it's probably an analog to the CRISPR/Cas system (or maybe even CRISPR/Cas itself but they kept it for themselves for decades after finding out about it from the aliens).
The CRISPR/Cas DNA constructs can be integrated permanently into the genome of a cell. It can be used to cut out specific genes like genes for the immune system. Expression of the constructs can be triggered later by e.g. adding certain antibiotics.
To get it into human cells, you would package it into e.g a lentiviral expression system (basically empty viruses that can infect cells and integrate whatever DNA you want). That can be hidden in the smallpox shots and over time most cells would be altered (mainly you need to target the the immune stem cells and sperm/eggs or alternatively just keep infecting each generation).
One problem would be how to get a healthy immune system to ignore that virus long enough to reach the necessary infection level.
The cure seems to be another alien/artificial DNA construct administered only to chosen people (that are also infected by the stuff above) which expresses something that would block the effect of the immune-system-targeting alien DNA.
Though it would make more sense if the cure is the CRISPR/Cas and the "AIDS" construct is something that just blocks genes/proteins of the immune system (e.g. via silencing RNA) because otherwise they would not be able to stop it if the immune system genes were actually cut out from the genome. In that case they could produce a cure using again a retrovirus system (though obviously not within hours as it was depicted in the episodes).