When does a gene frequency attain the golden ratio, 61.8%? Read on and see.
As we learned in high school, girls have two X chromosomes, denoted XX, while boys have an X and a Y, denoted XY. The father passes an X or a Y to his child at random, and the mother passes an X, because that's all she has. The child has an even chance of being a boy or a girl, which preserves the gender balance of the population.
Imagine a planet where the girl has two different sex chromosomes. I'll call them Z and W, not to be confused with X and Y. The boy is ZZ and the girl is ZW. The mother passes Z or W to the child, which determines whether that child is a boy or a girl. The ZW system seems just as viable as the XY system, so perhaps this is how gender is determined somewhere in the universe.
As it turns out, you don't have to go to another planet to find ZW, just look out your window. Birds use the ZW system, along with some reptiles and insects. Genetic evidence indicates that ZW evolved independently of XY. In fact the Z chromosome in some birds is similar to our chromosome 9, as though chromosome #9 was co-opted for the purpose of sex determination.
This has implications for parthenogenesis, wherein a female has a child without the benefit of a father. This is rare, but it happens in certain insects, amphibians, and reptiles. Since a sperm is not forthcoming, the haploid DNA in the egg is replicated to produce a full complement of DNA. The egg is now a clone of the mother, almost. If the mother carries one copy of a recessive gene, this gene has a 50% chance of being part of the egg. If present, it is copied, so that the recessive gene sits on both sides of the DNA ladder, and the child will have the recessive trait. This would not happen in a proper clone of the mother. With this in mind, let's return to the sex chromosomes. If the animal uses the XY system, then the mother has XX, and the child has XX, and is always female. Parthenogenic reproduction always produces a daughter. If there are no males nearby, the green frog can produce daughter frogs, which produce more daughter frogs, and so on down the line. Of course sexual reproduction is preferred for its genetic variability; parthenogenesis is a last resort.
Parthenogenesis has never been observed in mammals in the wild, and certainly not in humans. Jesus was not born of a virgin, and if he was, if that was a one in 10 billion event, then he would be a female. Jesus was a man, hence his Y chromosome came from Joseph - end of story.
The komodo dragon shows us another aspect of reproduction. She is capable of parthenogenesis, and she uses the ZW system. Her haploid DNA contains Z or W at random, and the resulting egg contains ZZ or WW. The former is a male, and the latter is not viable, and does not develop to term. Thus all the parthenogenic children of a komodo dragon are boys.
Other sex determination systems exist, such as XO, wherein the female is XX, and the male is X alone, denoted X0, with no corresponding sex chromosome from the father. Alligators determine the sex of their offspring by the temperature of the nest. The clutch is entirely female at 30°c or less, and entirely male at 34°c or greater. These and other systems are beyond the scope of this article.
Move to another planet, where three sexes come together to make a child. This is far more complicated than any bisexual system, (I have no idea how three strands of DNA would zip together ), and gene shuffling is not particularly superior in such a system, so trisexual reproduction almost certainly wouldn't evolve. Setting this aside, let's continue with the thought experiment.
Extend the XY system as follows. XXX is a girl, XXY is a boy, and XYY is an alpha, whatever that is. Since a girl always contributes an X, the child will be one of these three genotypes. The system is well defined, but it isn't well balanced. 2/9 of the children are girls, and 2/9 are alphas, and 5/9 are boys. That's a preponderance of boys in the population.
Bring in a third chromosome Z. XXX is a girl, XXY is a boy, and XXZ is an alpha. Since XYZ could emerge, the system is not well defined. I'm just going to wave my hand and say XYZ is not viable; it only happens 11% of the time anyways. Now the child is girl, boy, or alpha, but still the population is not in balance. Half the kids are girls, one fourth are boys, and one fourth are alphas.
Let A B and C be the sex chromosomes, with A dominating B dominating C. This is familiar territory; Y dominates X in mammals to create a boy. If A is present the child is an alpha. If A is not present, B implies a boy, and CCC is a girl. There are five genotypes: CCC, BCC, BBC, ACC, and ABC. The girl always contributes C, and there is at most one A, coming from alpha. Right off the bat we see that a third of the children are alphas. Let's try to balance the phenotypes (girl boy alpha), and then the genotypes.
Let s be the fraction of boys that are BCC, and let t be the boys that are BBC. Thus s + t = 1. In the same way, u is the fraction of alphas that are ACC, and v is ABC, whence u + v = 1. Write the following equation to insure 1/3 girls in the population.
4su/9 + 2sv/9 + 2tu/9 + tv/9 = 1/3
4su + 2sv + 2tu + tv = 3
2su + (2su + 2sv) + tu + (tu + tv) = 3
2su + 2s + tu + t = 3
su + 2s + u + t = 3
su + s + u = 2
Assuming this can be satisfied, alphas, girls, and boys are in balance, so let's look at genotype. If you know the child is alpha, the odds of ACC are 2s/3 + t/3, or (s+1)/3. In other words, (s+1)/3 = u, or 3u = s + 1. Substitute for s in the previous equation and get the following quadratic.
u2 + u - 1 = 0
Let q be the square root of 5, whence u = (q-1)/2. This is known as the golden ratio, approximately 0.618. Subtract, and v = 0.382.
Since s = 3u-1, s = (3q-5)/2. This is approximately 0.854. Subtract, and t = 0.146. If you check the probabilities of obtaining the various genotypes, the math works out. For example, the fraction of boys with BBC is (vs/9 + 2vt/9) over 1/3, which agrees with t. The sexes, and the genotypes, are all in balance. Three sexes is mathematically possible (even if the biology is a bit suspect), so open up your word processor and start writing that science fiction story.
Well I suppose that's true. If random drift causes the boys or the alphas in a population to have more C's than expected, then s or u increases, su + u +s increases, and there are more than a third girls in the next generation. That by itself isn't a problem. Here on earth, if a population has too many girls, the next generation is evenly divided between girls and boys. The system is self-correcting. But on this planet, the girl boy alpha distribution isn't the only thing that is perturbed. The genotypes of the boys and the alphas drift out of balance as well. The alphas for instance have more C's in the next generation, because the boys contribute more C's into the gene pool. This pushes u higher and higher, until it approaches 2/3, while s moves towards 1. There is, I suspect, another equilibrium, this time a stable equilibrium, with more C's, but at these frequencies there are to many girls. You may wish to compute this equilibrium, but you will need another variable g for the proportion of girls in the population. We can't assume g = 1/3 any more. Or s and u could drift downward, approaching an equilibrium with too few C's and too many boys. Enjoy.