Population & evolution formulas

$p$ = frequency of allele $A$, $q$ = frequency of allele $a$ in the gene pool.

Valid whether or not the population is in Hardy-Weinberg equilibrium: each heterozygote carries one copy of each allele.

At equilibrium: $p^2$ = frequency of $AA$, $2pq$ = frequency of $Aa$, $q^2$ = frequency of $aa$.

Homozygote frequencies are the squares; each heterozygote frequency is twice the product of its two allele frequencies.

The fraction of males showing an X-linked recessive trait estimates $q$ directly. The fraction of affected females is $q^2$, which is why such traits are far more frequent in males.

Expected counts $E$ come from $p^{2}N$, $2pqN$, $q^{2}N$, using $p$ and $q$ estimated from the same data, which costs one degree of freedom. With two alleles, df = 1 (critical value 3.84 at the 5% level).

$F$ = inbreeding coefficient (the probability that the two alleles in an individual are identical by descent). Inbreeding raises homozygosity without changing allele frequencies. $F=1-\frac{H_{\text{obs}}}{2pq}$.

Fitnesses $AA=1$, $Aa=1$, $aa=1-s$. Selection is slow when $q$ is small, because almost all copies of $a$ are hidden in heterozygotes.

From $q^{\prime }=q/(1+q)$, each generation adds 1 to $1/q$. $n$ is the number of generations to go from $q_0$ to $q_n$.

Fitnesses $AA=1-s$, $Aa=1$, $aa=1-t$; $q$ = frequency of $a$. The equilibrium is stable. The allele whose homozygote pays the larger cost stays rarer.

$\mu$ = mutation rate per generation towards the deleterious allele; $s$ = selection coefficient against the affected genotype. These approximations hold when $\mu \ll s$.

$m$ = fraction of the recipient population's genes that come from migrants each generation; $q_m$ = allele frequency among migrants.

$N_e$ = effective population size; $t$ = generations. The smaller $N_e$, the faster variation is lost.

$N_m$, $N_f$ = numbers of breeding males and females. $N_e$ is always ≤ the census number of breeders, and is dominated by the rarer sex.

$S$ = selection differential (mean of selected parents − population mean); $R$ = response (mean of offspring − population mean). $h^2$ = narrow-sense heritability (additive variance / phenotypic variance); $H^2$ = broad-sense heritability (total genetic variance / phenotypic variance). Only $h^2$ predicts the response to selection.

In a diploid population of $N$ individuals, $2N\mu$ new neutral mutations arise per generation, and each is eventually fixed with probability $1/(2N)$. The neutral substitution rate $k$ therefore equals the neutral mutation rate $\mu$, independent of population size. This is the basis of the molecular clock.

$d_N$ = non-synonymous substitutions per non-synonymous site; $d_S$ = synonymous substitutions per synonymous site. $\omega <1$: purifying (negative) selection; $\omega \approx 1$: neutral evolution; $\omega >1$: positive (adaptive) selection.