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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.

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  • $\begingroup$ 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
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Thank you for asking such a well-thought out question, I really enjoyed how hard this made me think about the concept. I think there might be an issue here that's quite common to biology, in my experience- poor definition of terminologies.

I think the core question can be summed up by your comment:

I thought dominance was only related to which phenotype will show. If fitnesses of $A_{11}$ and $A_{22}$ are equal, doesn't it mean just that $A_{11}$ and $A_{22}$ have the same chance at surviving?

In my thinking, the definition of dominant could mean two things:

  1. The expression of a particular characteristic
  2. The fitness associated with the expression of a characteristic

Of course, the second definition depends entirely on the first- if the phenotype of the heterozygote ($A_{12}$) is the same as the homozygote ($A_{11}$ or $A_{22}$), then their fitness must be the same- evolution acts on phenotypes, not genotypes.

Let me break this down using Gillespie's tables. For a given value of $s$ (between 0 and 1), we can imagine the following scenarios for the value of $h$:

$h$ Phenotype $A_1A_2$ fitness Dominance scenario
h=0 $A_1A_2$ phenotype = $A_1A_1$ phenotype $1-s\times 0 = 1 = A_1A_1 fitness$ $A_1$ is dominant allele
h=1 $A_1A_2$ phenotype = $A_2A_2$ phenotype $1-s\times 1 = 1 - s = A_2A_2 fitness$ $A_2$ is dominant allele

This is assuming the phenotype for $A_1A_2$ is the same as the phenotype of the homozygote, in each scenario. It may be possible that the phenotypes are different, but the fitness is the same- what we are actually measuring with $h$ and $s$ are fitness, not phenotypes. But if the phenotypes are the same, the fitnesses must also be the same- natural selection works on phenotypes, not genotypes.

Where the break between fitness and phenotype becomes complete is when $h$ is a value other than 0 or 1 (not complete dominance).

$h$ Dominance Type Scenario
0<h<1 Incomplete dominance The heterozygote ($A_1A_2$) more fit than $A_2A_2$ and less fit than $A_1A_1$
h<0 Overdominance The heterozygote ($A_1A_2$) more fit than homozygotes ($A_1A_1$ and $A_2A_2$)
h>1 Underdominance The heterozygote ($A_1A_2$) less fit than homozygotes ($A_1A_1$ and $A_2A_2$)

In each of these cases, the phenotype of the heterozygote is something different than either of the homozygotes (hence the term incomplete dominance). What the value $h$ is measuring here is not the difference in phenotypes, it's the effect it has on fitness- which might be different in different contexts.

Let's think about the classic example of overdominance, sickle cell disease. In terms of fitness, individuals with one copy of the sickle cell allele have the sickle cell trait. The sickle cell trait offers resistance to malaria, and therefore higher fitness in regions with endemic malaria. An individual with two copies of the sickle cell allele are less fit because they have sickle cell disease. An individual with no copies of the sickle cell allele is less fit because they don't have resistance to malaria infection.

But this measure of dominance, where we define dominance as the fitness associated with a the expression of a characteristic, is context-dependent- individuals with the sickle cell trait are prone to other diseases, and in the absence of endemic malaria it might be they have lower fitness than an individual with no copies of the sickle cell allele ($0<h<1$).

If we think about dominance as simply the expression of a characteristic, we would actually describe the sickle cell heterozygote as co-dominant- when you look at the blood of individuals with sickle cell trait (heterozygotes), they have a mix of normal-shaped red blood cells and sickle-shaped red blood cells.

So to conclude:

  1. Dominance sometimes refers to the expression of a trait in relation to an allele, and sometimes refers to the fitness associated with the expression of a trait in relation to an allele.
  2. Differences in fitness imply differences in phenotype (because natural selection acts on phenotypes, not genotypes)
  3. Differences in phenotype do not necessarily imply differences in fitness

I wrote also some notes on the following, but as they're not directly relevant to the question (but might be useful for Googlers later on) I will put them here below the cut:

Table of relative fitness measures with dominance coefficient $h$ and selection coefficient $s$

Genotype $A_1A_1$ $A_1A_2$ $A_2A2$
Relative fitness $1$ $1-hs$ $1-s$
  • The choice of alleles assigned to $A_1$ and $A_2$ are arbitrary- in reality these are As, Ts, Cs, and Gs. By convention the first allele is the dominant one, but if you're not sure which is dominant before you test it may be the second!
  • Similarly, by convention we assume the individual with $A_2A_2$ is not more fit than individual with $A_1A_1$ (in other words, $0<s<1$). But you might have it backwards!
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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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