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If you search on the genetics of Down's Syndrome, you'll find plenty of results explaining the genetic mechanisms which can result in triploidy. However, I can't find any information on the inheritance combinations of triploidy - i.e. what are the possible genetic outcomes in the event that an individual with triploidy should reproduce.

It might seem unlikely, which is presumably why it's glossed over. However, I'm just curious about it from an academic standpoint.

So: how does one calculate the inheritance probabilities for a trait associated with triploidy, and what are the outcomes?

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Triploids are characteristically sterile, but people with Down's Syndrome aren't triploid(they're anueploid technically). There's only one chromosome duplicated, and it's the smallest one. Women with Down's can have kids just fine, but men tend to be sterile(according to WebMD at least).

See here for a full coverage of triploid genetics and why triploids are nearly always sterile. In short: gametes formed during meiosis pair off but there's always one copy of each chromosome that chooses a daughter cell at random. If one daughter cell gets none of the extra copies, you have a haploid gamete and a diploid zygote and everything goes according to plan. If one daughter cell gets all of the copies you have a diploid gamete and you'll have a triploid zygote and everything will be fine-ish, but you'll have another triploid generation.

Statistically this happens for 2(1 for females with asymmetric gametes, but let's fudge it a little) of the possible sorting patterns, of which there are 2^n where n is the number of chromosomes. For a triploid human male(bear with me) mating with a diploid female(the best possible combination mathematically) instead of the more typical 200 million viable sperm he would produce about 50 viable sperm(About 2 in 8 million sperm have a chance.). If you account for 23-triploidy that number goes up to about 100, but that's still worse than a millionth of the fertility.

Other combinations are vastly worse. A human female only has about 2 million eggs lifetime total, so even in the best of circumstances a triploid female has maybe a fifty fifty shot at having a single viable egg in her entire life. The calculation for the probability of a two triploid humans(or any n=23 organism) breeding successfully is left as an exercise for the reader but it's very low indeed.

If you were working in some other organism with maybe n=2 chromosomes the numbers would be more forgiving and it might be worth thinking about making Punnet squares. They'd work much like Punnet squares for diploid creatures, but the sides would be a little longer to accommodate the larger number of gamete genotypes, and you'd have to adjust the probability of some of the squares to account for aneuploidy-related fatalities and other weirdness. For instance, with two triploid parents the probability of getting a successful triploid zygote from any random pair of zygotes is something like 1/2^n instead of the 4/2^2n or whatever it is for diploid or quadruploid offspring. This is counterbalanced by aneuploid gametes perhaps dying(or even being slightly less fit).

If n=1 the math is more tractable, however(surprisingly, this exists. haplodiploidy complicates triploidy considerations so I'm pretending it doesn't exist). Examples like AAaxAa give strange results(only 1/12 are recessively phenotyped) but AaaxAa produces 1/4 recessive phenotypes and 3/4 dominant ones, just like AaxAa.

In sum, I see why triploids are just declared infertile and not dealt with in the fullest, most complex sense.

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