Are there any instances in real life of protein half-life regulating gene expression? For example, in a cell, Gene A produces a starting population of protein P, after which the expression of the gene is turned off. The starting population then degrades over time. When the numbers of protein P are reduced to roughly half, the transcription of Gene B, which had been repressed until then, is turned on (perhaps a byproduct of the protein degradation acts as a transcription factor for Gene B?)
My apologies if the question does not make sense. I am a sci-fi author, trying to design a synthetic virus for my novel - a bioweapon of sorts. The scenario is that the viral genome, upon making its way into the nucleus of a human cell, goes into a lysogenic phase, where the genes for its replication are turned off. The virus needs to initiate its replication only after a certain amount of time has elapsed, say X days. X is the half-life of protein P, which is produced by the viral genome while lysogenic. After X days, transcription of the rest of the viral genome is turned on, and the virus starts making copies of itself. Only this time, the copies go straight into lytic mode upon infecting a cell and start multiplying exponentially. My fictional virus has mechanisms to suppress interferon signalling and evade the immune system (for some time at least, until it gets out of control in the host body).
I am aware that some real-life viruses that infect bacteria can switch between the two phases, but the trigger for the switch is always an external event - like UV damage - and not something controlled by the virus itself. In my scenario, I want the virus to behave like a programmed timer, turning itself on after a fixed elapsed time.