How many ways can an Apoptosis mechanism be made disfunctional or irreparably damaged? If a cell has damaged Apoptosis mechanisms and it divides will its daughter cells have such damage?
Most of us have had the apoptotic process in our B-lymphocytes disrupted when we had infectious mononucleosis, caused by the EBV virus. The EBV virus (pardon the virus-virus) encodes proteins, including one that mimics a host cell protein, Bcl-2, which plays an important role in apoptosis. The set of virus 'decoy' proteins forces the infected cell to survive and be a host to produce new virus, whereas the cell's normal response would be to apoptose. Wart viruses such as HPV may have similar mechanisms.
So in summary, excessive growth or transformation to an immortal cell typically requires activation of mechanisms that promote cell growth and the inibition of mechanisms that promote cell death. "Transforming viruses' often have mechanisms to affect both cell growth and cell death.
This answer isn't to the question 'how many ways can [the] apoptosis mechanism [..] be damaged ', but answers 'what's a common way that apoptosis pathways can be disrupted'.... large DNA transforming viruses.
Lot's of ways. Apoptosis is complex, but falls under two pathways ending up at caspase 3. Anywhere in the pathway may there be a problem but also in things that trigger the pathway. For example in cancer there is loss of tumour suppressors which ensure a damaged cell undergoes apoptosis or prevents replication and oncogenes which allow controlled replication. Damage to these genes allows a cell to divide in the absence of signals to divide and also forget to check that it's DNA isn't damaged before replicating. This is then passed on to the daughter cancer cells. As each check is removed the cell permits more and more mutations meaning more and more likelihood for the next mutation to occur. Cancer is caused literally by one cell having defects and every daughter cell also possess the defects. Those that have defects which accidentally kill the cell are selected out.