The simplest formulas of natural selection

The simplest formulas of natural selection

Annotation

The dependences are found for haploid population:

• •    time to fixation of a allele as a function of population size and allele's relative fitness;

• •    allele frequency as a function of time;

• •    natural selection's speed as a function of time;

• •    natural selection's speed as a function of allele frequency.

We can affirm that the beneficial allele's frequency increases regularly. Therefore, time of change of beneficial allele's frequency is a function of some variables. We can write this function for a haploid population.

Let the allele X replaces the allele x in a haploid population. Let the population size is A . Then the ratio of the number of alleles X to the 1

number of alleles x is _A at the moment t ₁ , when allele X is one only.

A

The ratio of the number of alleles X to the number of alleles x is — — A

1

at the moment t ₂ , when allele x is one only. We need to find time to fixation of allele X T — t 2 — t 1 .

Let number of alleles x is a , and number of alleles X is b at a moment t . Let number of alleles x is c , and number of alleles X Is d at the moment t + 1. Then the progeny of one individual with the allele x is at unit time. The progeny of one individual with the allele X is ab at unit time.

cd

The ratio--i--is the relative fitness of allele x. Manifestly ab cd

--i--< 1. We must say that this relative fitness is used in population ab genetics usually. But we will use the relative fitness of allele X. The dc    dc relative fitness of allele X is W =--i— . Manifestly--i— > 1 and ba    ba d c_(c d V b"a "Ia"bv ■ dc    db

So, W =-- i--, hence W =-- i--. ba    ca

b is the ratio of the number of alleles X to the number of alleles x

a at a moment t .

d is the ratio of the number of alleles X to the number of alleles x

c at the moment 1 + 1.

Hence w shows how much the ratio of the number of alleles X to the number of alleles x increases at unit time. Hence the ratio of the wn number of alleles X to the number of alleles x is at the moment

A

• 1 1 + n , and the ratio of the number of alleles X to the number of alleles

TT ww x is --- at the moment L = 1 + T . Hence A =---, and

_A           21           _A

T =

2lnA ln w

The ratio of the number of alleles X to the number of alleles x Is

wt

_A at a moment t . Let the frequency of allele X is p at the moment t .

t wt    p           w

Hence --- =----- , and p =------ -.

A 1 - p      A + w

Let the frequency of allele X Is m at a moment 1 1 , and let the frequency of allele X is n at a moment 1 2 ( m <  n ) . Hence

ln n (1 - m)

w1           w2             .               m (1 - n)

------— — m, ------— — n , and At — t^ — t, =            . We see

A + wt1     A + wt2            21 In w that time for which frequency of allele X changes from m to n , isn't function of population size.

Manifestly p is natural selection's speed V :

r

V — p —

wt ln w   w2t ln w

1              1 —

A + w t   ( A + w t')'

P ^—

t

w

A + wt

hence V — p ln w — p 2 ln w.

Aspeed

We see that natural selection's speed has a maximum when the frequency is equal to ₂ , and natural selection's speed tends to zero when the frequency tends to zero or one. And it is known.