Answer key: Bacterial Growth & Doubling Time Activity
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Goal: Model bacterial growth by binary fission with N = N₀ × 2ⁿ, calculate generation time from two counts, and explain why real cultures stop growing exponentially.
Completed results table
| Culture (1,000 cells to start) | Generations n | Final cells N | Time to one million cells |
|---|---|---|---|
| E. coli · 20 min | 12 | 4.1 × 10^6 | 3.32 h |
| Vibrio natriegens · 10 min | 24 | 1.68 × 10^10 | 1.66 h |
| M. tuberculosis · 20 h | 8.4 | 337,794 | 8.3 days |
Answers
E. coli divides every 20 minutes. How many generations fit into 4 hours, and how many cells do 1,000 become? Show the calculation.
n = t ÷ g = 240 ÷ 20 = 12 generations, so N = 1,000 × 2¹² = 1,000 × 4,096 = 4,096,000 cells, about 4.1 × 10⁶.
Vibrio natriegens divides twice as fast, every 10 minutes. Does it make twice as many cells in 4 hours? Explain.
No, far more. It completes 24 generations instead of 12, so it grows by 2²⁴ instead of 2¹²: 2¹² = 4,096 times as many cells, about 1.68 × 10¹⁰.
M. tuberculosis divides only about every 20 hours. How long do 1,000 cells take to reach a million? How does this help explain why tuberculosis develops slowly?
It needs log₂(1,000,000 ÷ 1,000) = 9.97 generations, which takes 9.97 × 20 h ≈ 199 hours, about 8.3 days. Because the bacteria multiply so slowly, an infection takes weeks to months to build up, and lab cultures take weeks to grow.
A culture grew from 100 to 6,400 cells in 2 hours. Find the generation time.
6,400 ÷ 100 = 64 = 2⁶, so 6 generations happened in 120 minutes: g = 120 ÷ 6 = 20 minutes.
Why does a real culture in a flask stop growing exponentially? Name the phase that follows the log phase.
Nutrients run out and waste products build up, so the division rate falls until new cells only replace those that die: the stationary phase. Eventually the culture enters the death phase.
Challenge: on the log-scale graph, exponential growth is a straight line. Why?
log₁₀ N = log₁₀ N₀ + n × log₁₀ 2, and n grows in step with time, so log N rises by the same amount (0.301) every generation, which makes a straight line.