Different isoforms of the Fas protein coming from the same pre-mRNA is best explained by alternative splicing. In this case it is the inclusion or exclusion of exon 6 from the same pre-mRNA that makes these 2 functionally diverse proteins that can respond to different environmental circumstances.
After RNA polymerase produces a transcript from a gene, the pre-processed transcript is called pre-mRNA. This pre-mRNA still contains introns and has not undergone post-transcriptional processing. The pre-mRNA becomes capped first on the 5' end with a 7-methylguanosine cap, typically before transcription is finished. A polyadenylated (poly-A) tail is then added to the 3' end of the mRNA by the polyadenylation complex following recognition of the poly-A sequence. Both the 5'-cap and poly-A tail serve to protect the pre-mRNA from degradation and increase stability of the transcript. The pre-mRNA is still not yet fully processed because it contains introns which need to be excised via splicing by spliceosomes. Different transcripts can be produced via alternative splicing in which varying exons are retained or removed to generate different protein isoforms.
Answer 1: Base excision repair is a mechanism of repairing damaged DNA.
Answer 2: Histone deacetylation is the removal of the acetyl group from histones causing the positive charge on the lysine residues in the histone to interact with the negatively charged phosphates groups on DNA resulting in a tightly bound histone-DNA complex that is less likely to be transcribed.
Answer 3: A DNA missense mutation is a mutation in which a point mutation occurs leading to the changing of an amino acid coded for. This could or couldn't change the function of the protein though does not do so via the mechanism described in the question.
Answer 4: Post-translational modifications is modification of a protein that is already made. This could result in a functionally different protein, though the mechanism requested is to explain the finding in the setting of transcription changes (an exon).