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How to make an animal which can only breed for a certain number of generations?

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What would be the best way in which to genetically modify (or create through some other method) an animal which breeds normally for a certain number (50?) of generations before becoming sterile? Generations 1-49 would be normal, but generation 50 should be unable to breed.

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Polyploidy.

I don't think the telomere approach will work at all. If it did, we would have died out. Or rather, would never have existed. Short telomeres are a normal thing, happens to everybody (except the ones who die in an accident), yet nature "just makes it" so gametes are good to go anyway. Sperm from a 100 year old is, well, not precisely as good as sperm from a 17 year old in terms of numbers and mobility and such, but... whatever. It still works, and whatever comes out is a perfectly good individual with perfectly good telomers.

Now, polyploidy might be a better strategy. While it can be somewhat troublesome on higher animals, it's a well-established thing for plants, some frogs and amphibian stuff, a couple of marine vertebrae, and some worms.

If you have a population of, say, 2n males and 4n females, the offspring will be 3n. For a 3n individual, undergoing meiosis is, uh, troublesome. Because 3 doesn't divide by 2 so well.
So, that's that. No gametes, no offspring. Unless you do it like wild dandelion, which despite being unable to reproduce just tells you "Yeah, you know what, f... off!" and simply goes agamosperm, reproducing anyway. But for an animal, that's reasonably unlikely to happen.

The real challenge is that you want not 2 or 3, but 50 generations which are OK, so you would need to find a number which only results in an odd pairing after 50 generations. That's probably a tough one.
It's fiction though, so... got some leeway, I'd say. Might just say "poly" and not mention how many.

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Express mtDNA-specific cytidine deaminase in oocyte mitochondria.

Mitochondria are cellular organelles which, among other things, convert hydrocarbons like glucose into fuel that the cell can use, typically ATP. Mitochondria actually contain their own DNA, called mtDNA, which is separate from the cell's nuclear DNA. Because mitochondria originate from bacteria, they lack the sophisticated DNA protection and repair mechanisms provided to nuclear DNA. Because mitochondria are only passed on by the mother (the ones in sperm do not make it to the oocyte), mutations can build up over generations. This is especially likely given the fact that the internal environment of the mitochondrion is highly toxic to DNA due to high concentrations of free radicals. To prevent this, oocyte mitochondria are kept in a highly inactive state, limiting DNA damage.

So... what if you could speed up the mutation rate of mtDNA? There is a class of enzymes called cytidine deaminases which mutate nucleic acids, including DNA and RNA. It converts a nucleic base called cytidine to uridine, causing a mutation. This mutation is quickly corrected in nuclear DNA, but not in mtDNA. If you were to express a slow-acting cytidine deaminase in the mitochondria of maternal oocytes (egg cells), then over time, sufficient mutations would accumulate in the mitochondria of offspring that the embryos would eventually become non-viable. Because these mutations would be passed along in each generation, there would be a gradual reduction in viability.

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The encoding would need to be in the individuals phenotype and control the expression of individuals gametes on entering sexual maturity.

That way when two generational limited animals mate and their offspring carry their genomes, the "decrement to sterility" could be part of the genetic instructions that generate the animals eggs or sperm.

And with each generation, barring mutation, the number of generations that genome could sire would be one less than the previous generation.

There might be tricks to work around the "˜decrement to sterility' genetic encoding. Take the case were a 1st generation male impregnates a 4th generation female. The offsprings genotype would contain two different expressions of the generational limit. If the trait is dominant, then the child would be 2nd generation, but if it was recessive the child would be 5th generation.

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Tandem repeat mutation disease and genetic anticipation.

Some genetic diseases occur earlier and earlier with each generation. Most of these diseases are caused by mutations in genes with tandem repeats. The number of repeats gradually expand, and affected children manifest the disease younger than their parents did.

Expandable DNA repeats and human disease. Mirkin SM.

One of the central principles of classical (mendelian) genetics is that mutations are stably transmitted between generations. As long ago as 1918, however, a different type of inheritance was described for a human neurological disorder, myotonic dystrophy1. This type of inheritance was characterized by increased expressivity: that is, a decreased age of onset and increased severity in individuals of subsequent generations. A similar hereditary pattern was later observed for other neurological diseases: for example, Huntington's disease, spinal and bulbar muscular atrophy, and several ataxias. The penetrance "” that is, the probability that a given mutation results in disease "” can also increase in successive generations, as was first demonstrated for fragile X syndrome2. This unusual type of inheritance "” characterized by a progressive increase in the expressivity and, sometimes, the penetrance of a mutation as it passes through generations "” was called genetic anticipation.

The gradual expansion of the mutant gene is your timer counting down to 50 generations. With each generation is gets slightly longer. A (probably neurologic) disease which initially would not manifest within an individual's lifetime begins to show up in the very aged. With each generation, younger individuals develop the disease. At the 50th generation the disease occurs at the time of puberty, and these individuals do not live to bear children.

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Of the existing answers I like this one by Aos Sidhe best, but I would probably do it slightly differently.

Make your animal susceptible to a poison/virus which causes infertility. The poison should be airborne. Set 3 or 4 (for redundancy) meteorites moving towards you plant so they arrive in 50 generations, they carry the poison/virus, and will air burst as they enter in the atmosphere.

There is less risk of genetic drift devaluing the poison/virus as it does not have a counterpart on the planet. Winds should delivery the poison/virus world wide in a year or so, even if only a single meteorite arrives. Faster if the animal has developed intelligence and travel.

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