Explain genetic drift and the bottleneck - founder effects

WHY does drift exist at all?

Think of a jar with 50 red and 50 blue marbles (allele frequencies $p=q=0.5$ ). To make the next generation you scoop only 4 marbles and refill by copying them. You might scoop 3 red, 1 blue — now $p=0.75$ . Repeat, and soon you hit all-red or all-blue. That drift to fixation is unavoidable in finite populations.

WHAT is genetic drift, precisely?

HOW strong is drift? Derive it from scratch

We model the next generation's allele count as picking $2N$ gene copies (for $N$ diploid individuals) from a parent pool with allele- $A$ frequency $p$ .

Step 1 — set up the sampling. Number of $A$ copies in offspring $= X \sim \text{Binomial}(2N, p)$ . Why this step? Each of the $2N$ gene copies is an independent "success/failure" draw of allele $A$ with probability $p$ — that is the textbook definition of a binomial.

Step 2 — mean of the new frequency. $E[p'] = \frac{E[X]}{2N} = \frac{2Np}{2N} = p$ Why this step? $E[X]=2Np$ for a binomial. So drift has no direction — on average frequency is unchanged. Drift is a variance effect, not a mean effect.

Step 3 — variance of the new frequency. Binomial variance $\text{Var}(X)=2Np(1-p)=2Npq$ . Dividing a random variable by a constant $2N$ divides its variance by $(2N)^2$ : $\boxed{\;\text{Var}(p') = \frac{2Npq}{(2N)^2} = \frac{pq}{2N}\;}$

Step 4 — probability of fixation. Because $E[p']=p$ every generation (a martingale), the long-run probability an allele eventually fixes equals its current frequency: $P(\text{fixation of }A) = p$ Why this step? The expected final frequency must equal the starting frequency $p$ ; but the only possible final values are $0$ and $1$ , so $1\cdot P_{\text{fix}} + 0\cdot(1-P_{\text{fix}}) = p \Rightarrow P_{\text{fix}}=p$ .

Step 5 — loss of heterozygosity. Heterozygosity $H$ (genetic variation) decays each generation: $H_t = H_0\left(1-\frac{1}{2N}\right)^t$ Why this step? Each generation two random gene copies have chance $\tfrac{1}{2N}$ of being identical-by-descent (same parent copy), so a fraction $\tfrac{1}{2N}$ of variation is lost per generation. Small $N$ ⇒ fast loss.

Figure — Explain genetic drift and the bottleneck - founder effects

The two special cases

Worked examples

Common mistakes (steel-manned)

Active recall

Recall Test yourself (hide and answer)

Why does $E[p']=p$ under pure drift?
What is $\text{Var}(p')$ and why does small $N$ increase it?
Difference between bottleneck and founder effect?
Why can't recovering population size restore lost diversity?
Fixation probability of a neutral allele at frequency $p$ ?

Recall Feynman: explain to a 12-year-old

Imagine a bag with 50 red and 50 blue candies — that's your whole village of genes. To make the next village, you only grab a tiny handful and copy it. If you happen to grab mostly red, the new village is mostly red — not because red is better, just luck! If you grab a super tiny handful, you might get all red and lose blue forever. A bottleneck is when a disaster kills almost everyone, leaving a tiny random handful. A founder effect is when a few candies wander off to start a new bag somewhere else. Small handfuls = big luck swings.

Flashcards

Genetic drift is defined as :::random change in allele frequencies due to sampling error in finite populations, independent of selection.

The variance of allele frequency after one generation of drift

\text{Var}(p')=\dfrac{pq}{2N}

Why does drift have no directional effect on the mean allele frequency?

Because

E[p']=p

; sampling is unbiased, so on average frequency is unchanged — only variance grows.

How does drift strength depend on population size?

Inversely — smaller

N

gives larger variance and faster fixation/loss.

Probability that a neutral allele at frequency

p

eventually fixes

exactly

p

(martingale argument).

Fixation probability of a single new neutral mutation in

N

diploids

\dfrac{1}{2N}

Bottleneck effect

sudden drastic reduction in population size, leaving survivors with a random subset of alleles and reduced variation.

Founder effect

a small group founds a new population whose gene pool reflects only the founders' (chance-biased) alleles.

Why can't a bottlenecked population recover diversity by breeding?

Lost alleles are permanently gone; new variation only arises slowly via mutation/migration.

Heterozygosity decay per generation under drift

H_t=H_0\left(1-\frac{1}{2N}\right)^t

Classic real example of a bottleneck

cheetahs — extreme genetic uniformity.

Classic example of the founder effect

Ellis–van Creveld syndrome frequency in Amish populations.

Connections

Hardy-Weinberg Equilibrium — drift is a violation of the "infinite population" assumption.
Natural Selection — directional force; contrast with directionless drift.
Gene Flow (Migration) — counteracts drift and founder effects by adding alleles.
Speciation — drift + isolation (founder events) can seed new species.
Effective Population Size (Ne) — the $N$ that actually governs drift.
Inbreeding & Heterozygosity — both erode variation in small populations.
Mutation — the only long-term source restoring lost variation.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, genetic drift ka matlab hai — allele frequency ka chance se badalna, selection se nahi. Har generation me offspring parents ke gametes ka ek random sample hote hain. Agar population chhoti hai (N small), to sample me proportion original se kaafi alag aa sakta hai — bilkul jaise sirf 4 marble uthao to red-blue ka ratio wild ho jaata hai. Isiliye formula banta hai $\text{Var}(p') = pq/2N$ : N jitna chhota, variance utna bada, aur allele utni jaldi fix (frequency 1) ya lost (frequency 0) ho jaata hai.

Important baat — drift ki koi direction nahi hoti, kyunki $E[p']=p$ . Yaani average me frequency same rehti hai, sirf uska "wobble" badhta hai. Isliye ek accha allele bhi bad luck se gayab ho sakta hai, aur ek slightly harmful allele fix ho sakta hai. Yeh natural selection se bilkul ulta concept hai jahan fitness decide karta hai.

Bottleneck effect tab hota hai jab koi disaster (bimari, shikaar, aapda) population ko achanak bahut chhota kar de — survivors sirf ek random subset alleles carry karte hain, variation gir jaati hai. Cheetah iska best example hai. Founder effect tab, jab thode se individuals nayi jagah jaakar nayi population start karein (jaise island colonize karna, ya Amish community) — naya gene pool sirf founders ke alleles dikhata hai. Dono actually ek hi cheez hain: tiny N ki wajah se strong drift.

Yaad rakhna: population dobara badi ho jaaye tab bhi kho chuke alleles wapas nahi aate — variation sirf mutation se dheere-dheere aati hai. Numbers badhna aur diversity badhna do alag baatein hain!

Test yourself — Population Genetics & Speciation

Connections