Q.41 The generation time of E. coli is 20 minutes. If there are 106 E. coli present in an
exponentially growing synchronous culture, then the average time (in minutes) required to
obtain a final population of 4×106 E. coli is ________.

E. coli Population Growth Calculation: 10^6 to 4×10^6 in 20-Minute Generation Time

The average time required for an exponentially growing synchronous culture of E. coli to increase from 106106 to 4×1064×106 cells, with a 20-minute generation time, is 40 minutes. This result follows the standard bacterial exponential growth formula, where population doubles every generation.​

Growth Formula

Bacterial populations in exponential phase follow Nt=N0×2nNt=N0×2n, where N0=106N0=106 is the initial population, Nt=4×106Nt=4×106 is the final population, and $n$ is the number of generations. Solving for $n$ gives n=log⁡2(Nt/N0)=log⁡2(4)=2n=log2(Nt/N0)=log2(4)=2 generations. Total time equals $n\times$ generation time = $2\times 20=40$ minutes.​

Synchronous Culture Role

In a synchronous culture, all cells divide simultaneously, producing a stepwise population increase every generation time, unlike asynchronous cultures with smooth exponential curves. The question specifies synchronous growth, but the calculation uses the same exponential formula since the average time aligns with generation multiples for exact doublings. E. coli‘s 20-minute generation time holds under optimal conditions (37°C, rich medium).​

Step-by-Step Solution

• Initial population (N0N0): 106106 cells.

• Final population (NtNt): 4×1064×106 cells (4-fold increase).

• Growth factor: Nt/N0=4=22Nt/N0=4=22, so 2 generations needed.​

• Time per generation: 20 minutes.

• Total time: $2\times 20=40$ minutes.​

Common Errors Explained

No options are provided, but typical mistakes include:

• Assuming 4 generations (confusing 4-fold with 2424): 80 minutes (wrong).

• Using continuous growth t=ln⁡(Nt/N0)kt=kln(Nt/N0) where $k=\ln (2)/20\approx 0.0347/\text{min}$, yielding ~39.6 minutes (near but not exact; discrete generations fit fill-in question).​

• Ignoring synchronous aspect or overlapping replication (relevant for very fast growth but not here).​