Assumptions of the models for haploid and diploid selection

Question

For a bi-allelic locus, the model for haploid Natural Selection is:

$$\frac{dp}{dt} = \frac{pW_A}{pW_A + (1-p)W_B}$$

, where $p$ is the frequency of the allele $A$, which relative fitness is $W_A$. $W_B$ is the fitness of the allele $B$, which frequency is $1-p$. $\frac{dp}{dt}$ is the change in allele frequency through time.

The same model for diploid selection is:

$$\frac{dp}{dt} = \frac{p^2W_{AA} + p(1-p)W_{AB}}{p^2W_{AA} + 2p(1-p)W_{AB} + (1-p)^2W_{BB}}$$

, where $W_{AB}$ is the fitness of the diploid genotype carrying the allele $X$ on one chromosome and the allele $Y$ on the other chromosome. We define $W_{AB} = W_{BA}$ and therefore we use only the symbol $W_{AB}$ in the above calculations. Note: This equation assumes Hardy-Weinberg equilibrium.

Those equations can be found in any introductory courses in population genetics. For example, here is a source where you can find this same equation for diploid selection. They just express $q$ instead of $1-p$.

My questions are:

• Do these models assume constant population size? Why?
• Do these models assume threshold selection? Why?

Thank you!

Answer 1

Unless I am missing something the equations you have posted are incorrect. The second equation should be:

$$p' = \frac{p^2 W_{AA} + p(1-p)W_{AB}}{p^2W_{AA}+ 2p(1-p)W_{AB}+(1-p)W_{BB}} $$

$p'$ is not interpreted as $\frac{dp}{dt}.$

$p'$ is the frequency of allele A in the next generation. If zygotic frequencies via random mating are

$p^2(AA) + 2pq(Aa)+q^2(aa) = 1$

then all of AA individuals and 1/2 the Aa gametes bear the A allele. So we can find $p$ in the next generation as

$p^2+pq = p^2+p(1-p) = p^2+p-p^2 = p$ which shows that in a Hardy-Weinberg population the frequencies are stable.

The assumptions here are those underlying the HW model, which includes infinite population size (hence constant).