GATE 2022 BT Q.65: solution and answer

GATE 2022 · BTQ.65 · 2 marks · NAT

A bacterial strain is grown in nutrient medium at 37 $^{\circ}$ C under aerobic conditions. The medium is inoculated with $1 0^{2}$ cells from a seed culture. If the number of cells in the culture is $1 0^{5}$ after 10 hours of growth, the doubling time of the strain (rounded off to nearest integer) is ________ h.

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Answer and solution

ANSWER

The doubling time is about 1.003 h, which rounds to 1 h.

Official answer: 1Source: Official answer key, GATE 2022 BT (IIT Kharagpur)

Assume exponential growth over the whole 10 h, with no lag phase. Each doubling multiplies the cell count by 2, so $N = N_{0} 2^{n}$ , where $N_{0}$ is the inoculum, $N$ is the final count and $n$ is the number of generations.
Take the ratio of final to initial cells: $\frac{N}{N _{0}} = \frac{1 0 ^{5}}{1 0 ^{2}} = 1 0^{3}$ .
Number of generations: $n = lo g_{2} (1 0^{3}) = \frac{3 ln 10}{ln 2} = \frac{3 \times 2.30259}{0.69315} \approx 9.966$ .
Doubling time is the elapsed time divided by the number of generations: $t_{d} = \frac{t}{n} = \frac{10 h}{9.966} \approx 1.003 h$
Check with the specific growth rate $μ$ (in $h^{- 1}$ ), defined by $ln (N / N_{0}) = μ t$ : $μ = \frac{ln 1000}{10 h} = 0.6908 h^{- 1}$ , so $t_{d} = \frac{ln 2}{μ} = \frac{0.6931}{0.6908 h ^{- 1}} \approx 1.003 h$ .
Rounded to the nearest integer, the doubling time is $1 h$ .

KEY IDEA

In exponential growth the cell count doubles once per doubling time, so the number of generations is $n = lo g_{2} (N / N_{0})$ and $t_{d} = t / n$ . The same result comes from $t_{d} = ln 2/ μ$ , with $μ = ln (N / N_{0}) / t$ . A 1000-fold rise is almost exactly 10 doublings, because $2^{10} = 1024$ .

WATCH OUT FOR

Dividing by the wrong quantity. The common slips are using $lo g_{10}$ in place of $lo g_{2}$ (which gives 3 "generations" and $t_{d} \approx 3.3$ h), using the cell increase $1 0^{5} - 1 0^{2}$ in place of the ratio, or reporting $μ$ ( $0.69 h^{- 1}$ ) as the doubling time. The temperature and aerobic conditions are context only and do not enter the calculation.

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GATE 2022 BT Q.65: solution and answer

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