Someone asked this in my class and my instructor wasn't sure in her answer, doesn't anyone know what happens in protein synthesis if a mutation causes mRNA to not possess a stop codon? Would the protein eventually stop? Would it keep coding into the poly-A chain and insert a bunch of phenylalanine?
No, this will not happen. mRNAs are inspected in the nucleus before they are exported into the cytoplasm (at least in eukaryotes), where transcription and translation don't happen at the same place. This ensures that no mRNAs without stop codons or premature stop codons are exported. This phenomenon is called "mRNA surveillance". mRNAs that do not pass this quality check, are degraded. See the Wiki reference for some basic information and the references below for more.
Yes you are correct. These mRNAs, that lack stop codon will cause translation to continue into the poly-A tail (it will result in addition of lysines not phenylalanine). Since no stop codon is present, the ribosome remains attached to the mRNA. Under these circumstances, a pathway known as non-stop decay is activated.
An important protein in this pathway — Ski7 detects a stalled ribosome and initiates the decay process for both the peptide and the mRNA. The poly-lysine tail of the peptide has been shown to fasten the peptide decay process.
In prokaryotes too, the non-stop mutations cause the ribosome to run through the stop codon and to eventually stall. However, there are no poly-A tails and ribsosome recovery pathway is different from that of eukaryotes. An RNA called tmRNA (a hybrid of tRNA and mRNA) binds to the the A-site using its tRNA-like domain. Then the translation proceeds through the mRNA domain of tmRNA (this is called trans-translation) which causes an addition of a peptide tag (that acts like a degradation signal) to the stalled polypeptide and finally leads to ribosome release when the stop codon is reached.
Klauer and van Hoof. Degradation of mRNAs that lack a stop codon: A decade of nonstop progress. Wiley Interdiscip Rev RNA. 2012 Sep-Oct; 3(5): 649–660.
Jannsen and Hayes. The tmRNA ribosome rescue system. Adv Protein Chem Struct Biol. 2012; 86: 151–191.
In fact, it is rare that stop codon mutation causes translation of polyA sequence in mammalian cells because you will find another in-frame stop codon downstream. And some stop codon mutations still producing the protein stably(1).
However, as WYSIWYG mentions, in the absence of alternative in-frame stop codon, mRNAs would be degraded. It is also called none-stop decay(2). I am not sure that this phrase gets popularity though.