We are searching data for your request:

**Forums and discussions:**

**Manuals and reference books:**

**Data from registers:**

**Wait the end of the search in all databases.**

Upon completion, a link will appear to access the found materials.

Upon completion, a link will appear to access the found materials.

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!

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).

## Wright-Fisher Model

### The Wright–Fisher Model

The Wright–Fisher (WF) model describes a population with discrete, nonoverlapping generations. In each generation the entire population is replaced by the offspring from the previous generation. Parents are chosen via random sampling with replacement. In a haploid population of constant size *N*, the probability that an allele present in *i* individuals will be present in *j* individuals in the next generation is then given by the binomial sampling probability

The transition probabilities *P*_{ij} define a discrete-time Markov process on the state space of allele frequencies *x*(*t*)=*i*(*t*)/*N*. Expected allele frequencies remain constant across generations, whereas the variance per generation is Var[*x*]=*x*(1−*x*)/*N*. The probability that an allele eventually becomes fixed is simply its initial frequency. In particular, the fixation probability of a new mutation present in a single copy is 1/*N*. It is straightforward to extend the WF model to diploid systems by exchanging *N*→2*N*, as well as to nonconstant population sizes by taking larger or smaller samples in each generation.

Several alternatives to the WF model have been proposed for describing random genetic drift, notably the Moran and the Cannings models ( Ewens, 2004 ). The Cannings model is a generalization of the WF model that can incorporate arbitrary variance in allele frequency between generations, which can span a wide range in biological populations. The Moran model is a continuous-time model with overlapping generations. Additional levels of biological complexity that one may wish to incorporate include different sexes, age structure, demography, and population substructure. The study of drift in such models is often facilitated by the concept of an effective population size, which specifies the population size of an idealized WF model that approximates evolutionary patterns in the real scenario ( Charlesworth, 2009 ).

## Assumptions of the models for haploid and diploid selection - Biology

What are the equilibria of the following equations?

=1 and =0 are the only equilibria.

In the continuous model, dp/dt = (r A -r a ) p (1-p)=0 only when p or 1-p equal zero (or r A =r a ), as in the discrete model.

and and to be greater than zero.

If the denominator is negative, then the numerators must both be negative for and to be greater than zero.

This implies that there are only two cases in which the polymorphic equilibrium is valid:

- W Aa > W aa and W Aa > W AA

- W AA = 1, W Aa = 1/2, W aa = 1/4: