# Why is the strength of genetic drift inversely proportional to the population size?

I saw a concept on the Internet that says "the strength of genetic drift is inversely proportional to the population size". I don't know why they are inversely proportional? Can somebody explain? Thank you all!

• Both the answers are excellent but just to simplify - smaller the population larger is the probability of "chance events" having a profound impact. – biogirl Jan 21 '14 at 14:02

## 2 Answers

Plane Crash Analogy

4 people in a plane crash

In a small aeroplane, there are 2 people that wear a blue shirt and 2 people that wear a green shirt. The plane crashes, half of the people died. The 2 survivors are those wearing the green shirt… well, nothing so surprising!

400 people in a plane crash

In a very big aeroplane, there are 200 people that wear a blue shirt and 200 people that wear a green shirt. The plane crashes, half of the people died. The 200 survivors are those wearing green shirt… This is quite surprising!

Genetic Drift

The same logic applies to genetic drift. Genetic drift is caused by events that modify the reproductive success of individuals in a random way (independently of their genotype). We usually referred to this as random sampling. At each generation, individuals are randomly chosen to reproduce and some genotypes might just happen to be chosen more often than others at a given draw (=at a given generation). Genetic drift pushes the frequency of allele slightly away from what would be predicted. According to the Wright-Fisher model, the frequency of alleles (of a bi-allelic gene) in a haploid population (to make it easier) in the next time step is given by:

$$p' = \frac{p \cdot WA}{p \cdot WA + (1-p) \cdot Wa}$$

where $$p$$ is there frequency of an allele at time = $$t$$ and $$p'$$ is the frequency at time = $$t+1$$. $$WA$$ is the fitness of the genotype which frequency is $$p$$ and $$Wa$$ is the fitness of the genotype which frequency is $$1-p$$. If the population is infinite, the predictions of this equation are exactly correct.

Now if we say that meteorites fall and half of the individual get killed. The probability of getting killed by a meteorite obviously does not depend on genetic predisposition, it is a question of chance! If you look at a population of 1 million individuals, half of them having the genotype $$A$$, the other half having the genotype $$a$$. It is very unlikely that more than 60% of all individuals that get killed are of the same genotype. Therefore, the meteorites won't change much the frequency of the genotypes. If you look at a population of 4 individuals 2 are $$a$$ and 2 are $$A$$, 2 of them get killed by a meteorite. Well, you have a probability of one half that the two survivors are of the same genotype and that the genotypes frequency would have changed drastically.

Genetic drift refers to these changes in allele frequency which are due to random events (such as meteorites) and the strength of genetic drift indeed depends on the population size for probabilistic reasons. The greatest the population size, the lowest is the strength (or the relative importance) of genetic drift.

How to model genetic drift

There are three famous models of genetic drift that all lead to very similar expectations. I shall just name them here but I will not develop the underlying mathematics.

• Wright-Fisher model of genetic drift

• Not to confuse with the Wright-Fisher model of selection written above
• It models genetic drift as a random sample of the previous generation and hence uses a binomial distribution.
• Moran model

• It is based on a birth-death model (a type of Markov model).
• Kimura's diffusion equation model

• It is an extension of the above two models for a case of continuous time.
• One comment; I think the statement "...random fluctuation in fitness..." can be misleading/cause confusion, since drift is usually defined in terms of selectively neutral alleles. Your statement could be read as implying fluctuating selection due to randomly varying fitness optima between years ("...fluctuation in fitness"), which would rather increase the genetic variation than decrease it. I'm sure this is not what you ment though. Sure, random deaths could be labelled as random fitness, but I find this slightly misleading and it is better to leave fitness out of it. – fileunderwater Jan 21 '14 at 14:46
• Indeed this sentence might be a bit misleading. I did not want to restrict genetic drift to "sampling" effects as it rather highlight the Fisher's mathematical model rather than the conceptual reality of biological populations. And I did not want to restrict drift on neutral allele as it does also applies on alleles that are (weakly) selected. – Remi.b Jan 21 '14 at 15:21
• Yes sure, drift acts both with and without selection. And I agree that it's useful to think about the actual processes that cause the sampling. However, my main objection to using fitness in relation to drift is that the (expected) fitness of e.g. a genotype is unchanged under drift. That different individuals with the same genotype contribute differentially to the next generation is a property of the probabilistic process, but it does not modify the fitness of the genotype. – fileunderwater Jan 21 '14 at 15:45
• Yes, I totally understood the issue. I'd like to keep the fisher's thing. If you think you can make my answer more protected against missunderstandings and confusion between predisposed fitness and what is achieved at a given generation due to random events, you're welcome to edit. – Remi.b Jan 21 '14 at 15:51
• My comment was only regarding the phrasing at the beginning of the paragraph (which you've edited - now referring to "...the fitness of individuals") - I didn't mean to imply that you should remove the wright-fisher equation. Fitness is tricky, since it's used in so many ways by different people. – fileunderwater Jan 22 '14 at 12:20

Genetic drift refers to changes in allele frequencies that are due to random sampling effects, and not selection. If you sample alleles from a finite population (e.g. caused by the fact that only some individuals in the population reproduce each year), the resulting frequencies will deviate from the original frequencies due to random chance. If your sample is large (a large population) this deviation will be small, and if your sample is small the effect will be substantial.

This is exactly the same process as flipping a coin. If you flip a coin 10 times there is a large probability that the outcome will deviate a lot from 50% heads/tails (the outcome will follow the binomial distribution). This is analogous to the deviation in allele frequency due to genetic drift. If you flip the coin 10000 times the outcome will deviate little from the expected 50% distribution of heads/tail. Therefore the effect is described as inversely proportional.

There exist several models of genetic drift, but one way to look at it is as for how the variance in allele frequency changes over time, which can be described by:

$$V_t \approx p*q(1-e^{-t/2N_e})$$ where $$p$$ and $$q$$ are the allele frequencies and $$N_e$$ is the effective population size. Alternatively, the change in heterozygozity ($$H$$) over time due to drift is described by (Crow & Kimura, 1970):

$$H_t = H_{t0}\left( 1-\frac{1}{2N_e} \right)^t$$ In both these functions, the inverse relationship between genetic drift and population size is seen clearly (population size is found in the denominator in both equations).

The concept of genetic drift is closely related to the founder effect, but in that case, the sampling is only done once, when a small subpopulation of individuals establishes a new population.