## RATES OF EVOLUTION

The rate of adaptive evolution of a population depends on several factors, including the rate of appearance of beneficial mutations and their probability of fixation.
The fate of new beneficial mutations within a population was first addressed by Fisher, Haldane, and Wright in the 1920's and 1930's.

In his 1927 paper, Haldane proved the classic result that the probability of fixation of a new beneficial mutation (P) is approximately 2s in a large population of constant size.

Natural populations do not, however, remain constant in size, but experience expansions, contractions, or fluctuations over time.

With changing population size, what is the probability of fixation of an allele and how does this affect the rate of evolution of a species?

## HALDANE'S RESULT

Where does 2s come from?
Haldane used a branching process to calculate P, noting that for an allele to be lost, all offspring copies of the allele must also be lost.

By assuming that the number of offspring per parent is, on average, one in a population of constant size, is 1+s for a new beneficial mutant, and is Poisson distributed,

When s is small,

## Exponential Growth

If a population is growing or shrinking exponentially at rate r, the average number of offspring per parent is not one, but (1+r) for wildtype individuals and (1+s)(1+r) for individuals carrying the rare beneficial allele.
In this case,

When r>0, beneficial mutations will fix at a faster rate.

When r<0, beneficial mutations will fix at a slower rate.

Why? In a growing population, every individual has more offspring on average, which protects the new mutation from loss while rare.

## Logistic Growth

In a population with density dependent growth, the probability of fixation of an allele depends on how far the population currently is from carrying capacity (K).
By solving the inhomogeneous branching process,

This reduces to 2(s+r) when the population is far from carrying capacity and to 2s at carrying capacity.

## General Results

The fixation probability of an allele depends on changes in the population size over the time period when the allele is rare and susceptible to loss.
We therefore define a * fixation effective population size* (N_{e}) such that:

## General Results

The fixation effective population size depends on s.
Importantly, N_{e} may be used in Kimura's more general diffusion equation for the probability of fixation:

This equation may be used to determine the probability of fixation for

- alleles that start in more than one copy
- deleterious alleles
- small population sizes

Example: In a diploid population of size 100 with exponential growth:

## CONCLUSIONS

The rate of adaptive evolution ultimately depends on whether beneficial mutations can be incorporated into a population.
The probability of incorporation depends strongly on the current population size relative to future population sizes.

Growing populations incorporate more beneficial mutations and fewer deleterious mutations than predicted based on current population size.

Shrinking populations incorporate fewer beneficial mutations and more deleterious mutations than predicted based on current population size.

Evolutionary forces are likely to reinforce, rather than counteract, externally caused changes in population size, such as those currently wrought by humans.