Population Genetics & Speciation

Question 1

(a) Derivation & proof (6)

• Allele frequencies in gamete pool: $A = p$, $a = q$. (1)
• Random mating = random union of gametes → genotype probabilities from expansion of $(p+q)^2$: $AA = p^2$, $Aa = 2pq$, $aa = q^2$. (2)
• Proof of stationarity: allele frequency of $A$ in offspring $p' = p^2 + \tfrac{1}{2}(2pq) = p^2 + pq = p(p+q) = p.$ (2)
• Since $p'=p$, genotype frequencies next generation are again $p^2:2pq:q^2$ → equilibrium reached in ONE generation and stays. Assumptions: infinite population, random mating, no selection, no mutation, no migration, non-overlapping generations/diploid. (1)

(b) Calculation + chi-square (8)

• $q^2 = 36/400 = 0.09 \Rightarrow q = 0.3$. (1)
• $p = 1 - 0.3 = 0.7$. (1)
• Expected carriers $= 2pq \times 400 = 2(0.7)(0.3)(400) = 0.42\times400 = 168$. (1)
• Expected counts: $AA = p^2\cdot400 = 0.49\times400 = 196$; $Aa = 168$; $aa = 36$. (1)
• $\chi^2 = \frac{(202-196)^2}{196} + \frac{(162-168)^2}{168} + \frac{(36-36)^2}{36}$ $= \frac{36}{196} + \frac{36}{168} + 0 = 0.1837 + 0.2143 = 0.398$. (2)
• df = (#genotypes − 1 − #alleles estimated) = 3 − 1 − 1 = 1. (1)
• $0.398 < 3.84 \Rightarrow$ fail to reject $H_0$; population consistent with H–W equilibrium. (1)

(c) Selection recurrence (8)

• Mean fitness $\bar w = p^2(1) + 2pq(1) + q^2(1-s) = 1 - sq^2$. (2)
• Frequency of $a$ next generation: $q' = \frac{pq(1) + q^2(1-s)}{1 - sq^2} = \frac{q(1-sq)}{1-sq^2}.$ (2)
• $\Delta q = q' - q = \dfrac{q(1-sq) - q(1-sq^2)}{1-sq^2} = \dfrac{-s q^2(1-q)}{1-sq^2}.$ (2)
• For small $q$, $\Delta q \approx -sq^2 \to 0$ as $q\to0$: removal slows dramatically because deleterious recessive alleles are "hidden" in heterozygotes and not exposed to selection. Hence recessive alleles are never fully eliminated by selection alone. (2)

Question 2

(a) Definitions (6)

• Genetic drift = random change in allele frequencies between generations due to sampling error, strongest in small populations. (2)
• Bottleneck = drastic reduction in population size (e.g. by disaster) followed by recovery, reducing genetic diversity — example: northern elephant seal / cheetah. (2)
• Founder effect = new population started by a small number of colonising individuals carrying a non-representative allele sample — example: Amish (polydactyly) / Galápagos colonisers. (2)

(b) Half-life of heterozygosity (6)

• $\tfrac12 = (1 - 1/(2\cdot50))^t = (0.99)^t$. (2)
• $\ln(0.5) = t\ln(0.99)$. (1)
• $t = \dfrac{\ln 0.5}{\ln 0.99} = \dfrac{-0.6931}{-0.01005} \approx 68.97$. (2)
• ≈ 69 generations. (1)

(c) Simulation pseudocode + fixation (8)

import numpy as np
def wright_fisher(p0, N, T):
    p = p0
    traj = [p]
    for _ in range(T):
        # number of A alleles drawn ~ Binomial(2N, p)
        k = np.random.binomial(2*N, p)
        p = k / (2*N)
        traj.append(p)
    return traj

• Correct binomial sampling of $2N$ alleles each generation. (3)
• Estimating fixation probability: run $R$ replicate simulations to a large $T$ (or until $p=0$ or $p=1$); fixation probability $\approx$ (number of runs ending at $p=1$)/$R$. (3)
• Theoretical expectation: for a neutral allele the probability of fixation equals its initial frequency, $P_{fix} = p_0$. (2)

Question 3

(a) Species concept (3)

• A species = a group of actually or potentially interbreeding natural populations that are reproductively isolated from other such groups (produce fertile offspring). (2)
• Limitation: cannot apply to asexual/clonal organisms, fossils, or ring species / cases with ongoing hybridisation. (1)

(b) Speciation & isolation (7)

• Allopatric: population split by a geographic barrier, gene flow stopped, divergence in isolation. (2)
• Sympatric: speciation without geographic separation (same area) — e.g. via polyploidy, disruptive selection, host shift. (2)
• Classification: (i) temporal = prezygotic; (ii) hybrid sterility = postzygotic; (iii) gametic incompatibility = prezygotic; (iv) hybrid inviability = postzygotic. (3, ½ each rounded)

(c) Cladistics + tectonics of change (8)

• Using Z (all 0) as outgroup, character polarity: C1,C2 derived in W,X,Y; C3 derived in X,Y; C4,C5 derived in Y only. (2)
• Nested synapomorphies:
  • C1+C2 → clade (W, X, Y) [ingroup].
  • C3 → clade (X, Y).
  • C4+C5 → autapomorphies of Y.
• Most parsimonious cladogram: (Z,(W,(X,Y))) — W branches first within ingroup, then X and Y are sister taxa. (3)
• Gradualism = evolutionary change accumulates slowly and steadily → tree would show roughly proportional, even branch lengths. Punctuated equilibrium = long stasis punctuated by rapid change at speciation events → most morphological change concentrated at nodes with short intense bursts and long unchanging branches. (3)

[
  {"claim":"q=0.3, p=0.7, carriers=168 under HWE for aa=36/400","code":"q=sqrt(Rational(36,400)); p=1-q; carriers=2*p*q*400; result=(q==Rational(3,10) and p==Rational(7,10) and carriers==168)"},
  {"claim":"chi-square statistic approx 0.398","code":"chi=Rational((202-196)**2,196)+Rational((162-168)**2,168)+Rational(0,36); result=abs(float(chi)-0.398)<0.01"},
  {"claim":"heterozygosity half-life at N=50 is about 69 generations","code":"import sympy; t=log(Rational(1,2))/log(Rational(99,100)); result=abs(float(t)-68.97)<0.1"},
  {"claim":"delta q simplifies to -s*q**2*(1-q)/(1-s*q**2)","code":"s,q=symbols('s q'); p=1-q; dq=simplify((q*(1-s*q))/(1-s*q**2)-q); target=simplify(-s*q**2*(1-q)/(1-s*q**2)); result=simplify(dq-target)==0"}
]