I am trying to understand the meaning of the dominance coefficient. I'll be more specific to what I don't understand, in a moment.
Let $A_{11}$, $A_{12}$ and $A_{22}$ be genotypes with fitnesses $1$, $1-sh$ and $1-s$ respectively. I can understand the meaning of the selection coefficient $s$ as its sign clearly determines which allele is more advantageous.
Though, I cannot figure out the meaning of the dominance coefficient $h$. I understand the following:

  • if $h>1$ we have underdominance i.e. both homozygotes are more fit than the heterozygote
  • if $h<0$ we have overdominance i.e. the heterozygote is more fit than the homozygotes
  • if $0 < h <1$ we have that the heterozygote is only more fit than the deleterious homozygote

The last point is called incomplete dominance as opposed to complete dominance in the cases $h=0$ and $h=1$. Here, I'm having trouble to understand how this is related to dominance. I mean, I can clearly state that $h=0$ implies that the fitnesses of $A_{11}$ and $A_{12}$ are equal, but why does this imply that $A_1$ is dominant and $A_2$ recessive (the situation $h=1$ is symmetrical of course)?

For this terminology, I'm referring to Gillespie's Population Genetics: a concise guide. A table resuming what I'm talking about is also on Wikipedia here.

I'd like to point out that I'm not a biologist, but just a mathematician, so it may be something simple in how dominance works that I'm missing. Also, for the same reason, please, forgive me if I have worded something not appropriately. Thanks to those who will help.

EDIT: I'm not asking what the biological process behind dominance is. I'm just asking how $A_{11}$ and $A_{12}$ having the same fitnesses (which, for me, reads as having the same chance at surviving) relates to $A_1$ being dominant. As it was pointed out in the comments, this is the definition of dominance. Though, I thought that dominance of one allele means only which phenotype will prevail in the heterozygote case and I cannot find the relation between the two definitions.

  • $\begingroup$ The answer to I mean, I can clearly state that h=0h=0 implies that the fitnesses of A11A11 and A12A12 are equal, but why does this imply that A1A1 is dominant and A2A2 recessive is just, because it exactly how dominance is defined. $\endgroup$ – Remi.b May 1 '17 at 20:12
  • $\begingroup$ If your question is about the mechanism behind dominance relationships (which seems to be what your last paragraph is asking for), then you should have a look at Why are some genes dominant over others? What is the mechanism behind it? and eventually Evolution of dominance of alleles. $\endgroup$ – Remi.b May 1 '17 at 20:13
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    $\begingroup$ Possible duplicate of Why are some genes dominant over others? What is the mechanism behind it? $\endgroup$ – Remi.b May 1 '17 at 20:13
  • $\begingroup$ Thanks for your comment. I think I'm just missing the big picture then. I thought dominance was only related to which phenotype will show. If fitnesses of $A_{11}$ and $A_{12}$ are equal, doesn't it mean just that $A_{11}$ and $A_{22}$ have the same chance at surviving? Isn't this what "fitness" measures? $\endgroup$ – Harnak May 1 '17 at 20:24
  • $\begingroup$ Fitness is indeed some function of survival and reproductive success. The exact definition of this function may slightly vary depending on the model used.One will not that fitness is typically considered as a phenotypic trait. Note that when modelling fitnesses with $s$ and $h$, $s$ is typically positive and $h$ is typically between 0 and 1. For cases of overdominance, authors often prefer to use other 2 variables to express the fitnesses of all three genotypes. I am not sure this comment is of any help though... $\endgroup$ – Remi.b May 2 '17 at 3:28

More often than not selection acts at the phenotypic level, and dominance refers to the phenotype. However, when we talk about genotypic fitness ($W$), without any information about the phenotype that each genotype produce, we disregard phenotypic quantity and look at phenotypic quality (fitness). By saying that $W_{A_{11}}=W_{A_{12}}>W_{A_{22}}$, we assume that the phenotypic quality of $A_1$ dominates over the phenotypic quality of $A_2$ because $W_{A_{12}}=W_{A_{11}}$, and not $W_{A_{12}}=W_{A_{22}}$. But this is not always the case.

We cannot say that $A_1$ is dominant over $A_2$ because we don't have the information about the phenotype. Let's look at two scenarios. In scenario 1, we have a population in which the following genotypes produce the following phenotypes: $$A_1A_1 \rightarrow GREEN \\ A_1A_2 \rightarrow YELLOW \\ A_2A_2 \rightarrow RED \\$$ Let's assume that this species cannot distinguish $GREEN$ from $YELLOW$ and these are also the colors that have the highest reproductive success, which means that the following fitness values can apply: $$W_{A_1A_1} \rightarrow 1.0 \\ W_{A_1A_2} \rightarrow 1.0 \\ W_{A_2A_2} \rightarrow 0.9 \\$$ If we don't have the information about the phenotypes described above, one could mistakenly assume that $A_1$ is dominant over $A_2$ based on the fitness values only.

In scenario 2, assume the following phenotypes and fitness values:

$$A_1A_1 \rightarrow GREEN \rightarrow 1.0\\ A_1A_2 \rightarrow GREEN \rightarrow 1.0\\ A_2A_2 \rightarrow RED \rightarrow 0.9\\$$

In this case, assuming that $A_1$ is dominant over $A_2$ is correct, but one cannot be certain of that, unless we have the information about the phenotypes.

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