1
$\begingroup$

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!

$\endgroup$
  • $\begingroup$ See my updated question! $\endgroup$ – Sulawesi May 31 '14 at 9:52
1
$\begingroup$

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).

$\endgroup$
  • 1
    $\begingroup$ Yes indeed you'r right. I was curious about how changing in population size may change selection pressures and change allele frequencies, but the model I gave assumes infinite population size. And yes, dp/dt is not the same as p'. Thank you! $\endgroup$ – Sulawesi May 31 '14 at 11:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.