Explain genetic drift and the bottleneck - founder effects

5.5.3 · Biology › Population Genetics & Speciation

Drift exist kyun karta hai?

Socho ek jar mein 50 red aur 50 blue marbles hain (allele frequencies $p = q = 0.5$ ). Agla generation banane ke liye tum sirf 4 marbles scoop karte ho aur unhe copy karke refill karte ho. Tum 3 red, 1 blue scoop kar sakte ho — ab $p = 0.75$ . Repeat karo, aur jald hi all-red ya all-blue ho jata hai. Finite populations mein fixation ka yeh drift anavoidable hai.

Genetic drift precisely KIYA hai?

Drift kitni strong hai? Scratch se derive karo

Hum agali generation ka allele count model karte hain $2 N$ gene copies pick karke ( $N$ diploid individuals ke liye) ek parent pool se jismein allele- $A$ ki frequency $p$ hai.

Step 1 — sampling set up karo. Offspring mein $A$ copies ki sankhya $= X \sim Binomial (2 N, p)$ . Yeh step kyun? $2 N$ gene copies mein se har ek allele $A$ ka independent "success/failure" draw hai probability $p$ ke saath — yahi binomial ki textbook definition hai.

Step 2 — nayi frequency ka mean. $E [p^{'}] = \frac{E [ X ]}{2 N} = \frac{2 N p}{2 N} = p$ Yeh step kyun? Binomial ke liye $E [X] = 2 N p$ . Toh drift ki koi direction nahi — average pe frequency unchanged rehti hai. Drift ek variance effect hai, mean effect nahi.

Step 3 — nayi frequency ka variance. Binomial variance $Var (X) = 2 N p (1 - p) = 2 N pq$ . Ek random variable ko constant $2 N$ se divide karne par uska variance $(2 N)^{2}$ se divide ho jata hai: $Var (p^{'}) = \frac{2 N pq}{( 2 N ) ^{2}} = \frac{pq}{2 N}$

Step 4 — fixation ki probability. Kyunki $E [p^{'}] = p$ har generation (ek martingale), kisi allele ke eventually fix hone ki long-run probability uski current frequency ke barabar hoti hai: $P (fixation of A) = p$ Yeh step kyun? Expected final frequency starting frequency $p$ ke barabar honi chahiye; lekin sirf possible final values $0$ aur $1$ hain, isliye $1 \cdot P_{fix} + 0 \cdot (1 - P_{fix}) = p \Rightarrow P_{fix} = p$ .

Step 5 — heterozygosity ka loss. Heterozygosity $H$ (genetic variation) har generation decay karta hai: $H_{t} = H_{0} (1 - \frac{1}{2 N})^{t}$ Yeh step kyun? Har generation mein do random gene copies mein $\frac{1}{2 N}$ chance hota hai identical-by-descent hone ka (same parent copy), isliye variation ka $\frac{1}{2 N}$ fraction har generation lost ho jata hai. Chhota $N$ ⇒ fast loss.

Figure — Explain genetic drift and the bottleneck - founder effects

Do special cases

Worked examples

Common mistakes (steel-manned)

Active recall

Recall Khud ko test karo (hide karke answer do)

Pure drift ke under $E [p^{'}] = p$ kyun hota hai?
$Var (p^{'})$ kya hai aur chhota $N$ ise kyun badhata hai?
Bottleneck aur founder effect mein kya fark hai?
Population size recover karna lost diversity restore kyun nahi kar sakta?
Frequency $p$ par ek neutral allele ki fixation probability kya hai?

Recall Feynman: ek 12-saal ke bacche ko explain karo

Ek bag imagine karo 50 red aur 50 blue candies ke saath — yeh tumhara poora genes ka village hai. Agla village banane ke liye, tum sirf ek bahut chhoti handful lete ho aur copy karte ho. Agar zyadata red milte hain, naya village mostly red hoga — isliye nahi ki red better hai, sirf luck! Agar bahut super tiny handful lo, toh sab red mil sakta hai aur blue hamesha ke liye khho sakta hai. Bottleneck tab hota hai jab koi disaster almost sabko maar deta hai, ek tiny random handful chhodke. Founder effect tab hota hai jab kuch candies wander off ho jaayen aur kahin aur ek naya bag start karen. Chhoti handfuls = bade luck swings.

Flashcards

Genetic drift ki definition kya hai

random change in allele frequencies due to sampling error in finite populations, independent of selection.

Ek generation ke drift ke baad allele frequency ka variance

Var (p^{'}) = \frac{pq}{2 N}

Drift mean allele frequency par koi directional effect kyun nahi daalta?

Kyunki

E [p^{'}] = p

; sampling unbiased hai, isliye average par frequency unchanged rehti hai — sirf variance badhti hai.

Drift ki strength population size par kaise depend karti hai?

Inversely — chhota

N

bada variance aur faster fixation/loss deta hai.

Frequency

p

par ek neutral allele eventually fix hone ki probability

exactly

p

(martingale argument).

N

diploids mein ek single naye neutral mutation ki fixation probability

\frac{1}{2 N}

Bottleneck effect kya hai

population size mein achanak drastic reduction, survivors ko alleles ka random subset aur reduced variation ke saath chhodkar.

Founder effect kya hai

ek chhota group ek nayi population found karta hai jiska gene pool sirf founders ke (chance-biased) alleles reflect karta hai.

Ek bottlenecked population breeding se diversity kyun recover nahi kar sakti?

Lost alleles permanently chali gayi hain; nayi variation sirf mutation/migration se slowly aati hai.

Drift ke under har generation heterozygosity decay

H_{t} = H_{0} (1 - \frac{1}{2 N})^{t}

Bottleneck ka classic real example

cheetahs — extreme genetic uniformity.

Classic founder effect ka example

Ellis–van Creveld syndrome frequency in Amish populations.

Connections

Hardy-Weinberg Equilibrium — drift "infinite population" assumption ki ek violation hai.
Natural Selection — directional force; directionless drift se contrast karo.
Gene Flow (Migration) — drift aur founder effects ko counteract karta hai alleles add karke.
Speciation — drift + isolation (founder events) nayi species seed kar sakte hain.
Effective Population Size (Ne) — woh $N$ jo actually drift govern karta hai.
Inbreeding & Heterozygosity — dono small populations mein variation erode karte hain.
Mutation — lost variation restore karne ka single long-term source.