Level 5 — MasteryProkaryotic vs Eukaryotic Cells

Prokaryotic vs Eukaryotic Cells

⏱ 60 minutes50 marksprintable — key stays hidden on paper

Mastery Examination (Level 5)

Time limit: 60 minutes Total marks: 50 Instructions: Answer ALL questions. Show full working for calculations. Use $...$ notation where appropriate. Diagrams may be used to support answers.

Question 1 — Surface-Area-to-Volume Constraint & Why Cells Stay Small (20 marks)

A biologist models a cell as a perfect sphere of radius $r$ (in µm). Nutrient uptake occurs across the surface at a rate proportional to surface area, while metabolic demand is proportional to volume.

(a) Derive an expression for the surface-area-to-volume ratio (SA:V) of a sphere as a function of $r$ , and state how it changes as $r$ increases. (3)

(b) A spherical bacterium has $r = 1\ \mu\text{m}$ and a spherical eukaryotic cell has $r = 10\ \mu\text{m}$ . Calculate the SA:V ratio for each (units µm $^{-1}$ ), and state the factor by which the bacterium's ratio exceeds the eukaryote's. (4)

(c) Suppose nutrient supply rate $S = k_S \cdot (\text{surface area})$ with $k_S = 2\ \text{units·µm}^{-2}\text{·s}^{-1}$ , and metabolic demand $D = k_D \cdot (\text{volume})$ with $k_D = 0.5\ \text{units·µm}^{-3}\text{·s}^{-1}$ . Determine the maximum radius $r_{max}$ at which supply still meets demand ( $S \ge D$ ). (5)

(d) Write a short pseudocode (or Python) function is_viable(r) that returns True when $S \ge D$ for a sphere of radius r using the constants above. (3)

(e) Explain, using your results from (a)–(c), TWO structural adaptations real cells use to overcome the SA:V constraint without simply staying spherical and small. (5)

Question 2 — Endosymbiotic Theory: Build the Argument (18 marks)

(a) State the endosymbiotic theory in one or two sentences. (2)

(b) Present FOUR independent lines of evidence supporting the theory, and for EACH explain why it points to a prokaryotic origin of mitochondria/chloroplasts. (8)

(c) A mitochondrion contains a circular DNA molecule of $1.6 \times 10^4$ base pairs. If a typical prokaryotic gene averages 1000 bp and the mitochondrion encodes 13 proteins, estimate what fraction of the genome (by length) is protein-coding, and comment on what this implies about gene transfer to the nucleus during evolution. (4)

(d) Construct a logical counter-argument a skeptic might raise against endosymbiotic theory, then refute it with evidence. (4)

Question 3 — Comparative Cell Architecture (12 marks)

(a) Complete a comparison identifying which of the following are present in: prokaryotes, plant cells, animal cells — for each of: nucleoid, true nucleus, cell wall, chloroplast, plasmid, centriole, large central vacuole, 70S ribosome. Present as a table. (6)

(b) A bacterial cell wall differs chemically from a plant cell wall. Name the main structural polymer in each, and explain how this difference is exploited by (i) an antibiotic and (ii) the human immune system. (4)

(c) Explain the functional significance of a bacterial plasmid in the context of antibiotic resistance spreading through a population. (2)

Answer keyMark scheme & solutions

Question 1

(a) For a sphere: surface area $A = 4\pi r^2$ , volume $V = \tfrac{4}{3}\pi r^3$ . (1 mark) $\text{SA:V} = \frac{4\pi r^2}{\tfrac{4}{3}\pi r^3} = \frac{3}{r}$ (1 mark) As $r$ increases, SA:V decreases (inversely proportional to $r$ ). (1 mark)

(b)

Bacterium: $3/1 = 3\ \mu\text{m}^{-1}$ (1 mark)
Eukaryote: $3/10 = 0.3\ \mu\text{m}^{-1}$ (1 mark)
Correct units stated (1 mark)
Factor: $3 / 0.3 = 10$ times greater (1 mark)

(c) Supply $S = k_S \cdot 4\pi r^2 = 2\cdot 4\pi r^2 = 8\pi r^2$ . (1 mark) Demand $D = k_D \cdot \tfrac{4}{3}\pi r^3 = 0.5\cdot\tfrac{4}{3}\pi r^3 = \tfrac{2}{3}\pi r^3$ . (1 mark) Set $S \ge D$ : $8\pi r^2 \ge \tfrac{2}{3}\pi r^3$ . (1 mark) Divide by $\pi r^2$ (r>0): $8 \ge \tfrac{2}{3} r \Rightarrow r \le 12$ . (1 mark) $r_{max} = 12\ \mu\text{m}$ (1 mark)

(d) (3 marks: correct area/volume formulas 1, correct comparison 1, returns bool 1)

import math
def is_viable(r):
    kS, kD = 2, 0.5
    S = kS * 4 * math.pi * r**2
    D = kD * (4/3) * math.pi * r**3
    return S >= D

(e) Any two, well explained (2.5 each):

Flattening / elongation (e.g. flattened epithelial cells, elongated neurons): non-spherical shapes increase SA relative to V, restoring exchange capacity. (why: for fixed volume, sphere has minimum SA; deviating raises SA:V)
Membrane folding / microvilli / cristae: internal or surface folds add exchange area without increasing volume.
Compartmentalisation (organelles): shortens diffusion distances by localising reactions.
Vacuoles: metabolically inert volume that reduces active cytoplasm needing supply.

Question 2

(a) Mitochondria and chloroplasts originated as free-living prokaryotes that were engulfed by (or entered) a host cell and, rather than being digested, entered a permanent mutually beneficial endosymbiotic relationship. (2 marks)

(b) Four of (2 marks each = evidence 1 + reasoning 1):

Own circular DNA resembling bacterial genome → prokaryotes have circular DNA; nucleus does not.
70S ribosomes (bacterial type) inside them, unlike the cell's 80S → indicates independent prokaryotic translation machinery.
Double membrane — inner from original prokaryote, outer from host engulfment vesicle → consistent with phagocytosis origin.
Binary fission — they divide independently of the cell cycle like bacteria → retained bacterial reproduction.
Sensitivity to antibiotics that target bacteria (e.g. chloramphenicol) → machinery is bacterial.

(c) Coding length $= 13 \times 1000 = 13000$ bp. (1 mark) Fraction $= 13000 / 16000 = 0.8125 \approx 81\%$ . (2 marks) Comment: very high coding density (little "junk") is characteristic of streamlined prokaryotic genomes; the original endosymbiont had far more genes, so most were transferred to the host nucleus over evolutionary time, retaining only genes needed locally. (1 mark)

(d) (Counter-argument 2 + refutation 2) Skeptic: "If organelles were once free-living bacteria, they should still be culturable/self-sufficient." Refutation: gene transfer to the nucleus made them obligately dependent — they no longer carry all genes needed for autonomy, which is expected under long co-evolution, not evidence against it.

Question 3

(a) (0.75 per correct row set; full marks 6)

Feature	Prokaryote	Plant	Animal
Nucleoid	✓	✗	✗
True nucleus	✗	✓	✓
Cell wall	✓	✓	✗
Chloroplast	✗	✓	✗
Plasmid	✓	✗	✗
Centriole	✗	✗(usually)	✓
Large central vacuole	✗	✓	✗
70S ribosome	✓	✗*	✗*

(80S in cytoplasm; 70S only inside organelles — accept either with reasoning.)

(b) Bacterial wall = peptidoglycan (murein); plant wall = cellulose. (1 mark each) (i) Antibiotics e.g. penicillin inhibit peptidoglycan cross-linking, weakening the bacterial wall → lysis; harmless to human cells lacking peptidoglycan. (1 mark) (ii) Immune system recognises peptidoglycan/bacterial wall components as foreign antigens (PAMPs), triggering immune response. (1 mark)

(c) Plasmids carry resistance genes and can be transferred between bacteria by conjugation (horizontal gene transfer), spreading resistance rapidly through a population even without cell division. (2 marks)

[
  {"claim":"SA:V of sphere r=1 is 3 and r=10 is 0.3, ratio 10", "code":"r=symbols('r'); sav=(4*pi*r**2)/(Rational(4,3)*pi*r**3); b=sav.subs(r,1); e=sav.subs(r,10); result=(b==3 and e==Rational(3,10) and simplify(b/e)==10)"},
  {"claim":"r_max where 8*pi*r^2 >= (2/3)*pi*r^3 is 12", "code":"r=symbols('r',positive=True); sol=solve(Eq(8*pi*r**2,Rational(2,3)*pi*r**3),r); result=(12 in sol)"},
  {"claim":"coding fraction 13000/16000 approx 0.8125", "code":"frac=Rational(13*1000,16000); result=(frac==Rational(13,16) and abs(float(frac)-0.8125)<1e-9)"}
]