Bacterial growth & doubling time

Bacteria divide by binary fission, so N = N₀ × 2ⁿ, where n = t / g is the number of generations in time t and g is the generation (doubling) time. Starting with 1,000 E. coli cells at g = 20 minutes, 4 hours (240 minutes) is 12 generations, giving 1,000 × 4,096 = 4.1 × 10⁶ cells. The graph uses a log scale, so this exponential growth appears as a straight line.

Rearrange the growth equation: the number of generations is n = log₂(N / N₀), and the generation time is g = t / n. If 100 cells become 6,400 in 120 minutes, N / N₀ = 64 = 2⁶, so there were 6 generations and g = 120 / 6 = 20 minutes. The same numbers give a growth rate constant k = 3 generations per hour.

In rich broth at 37 °C, E. coli divides about every 20 minutes, so one cell could become 2²⁴ ≈ 17 million in 8 hours of log-phase growth. In minimal medium or at lower temperature the generation time stretches to an hour or more, and in the gut it is estimated at many hours. Vibrio natriegens is faster (about 10 minutes), while Mycobacterium tuberculosis takes 15 to 20 hours.

Doubling (generation) time is g = t / n, where n = log₂(N / N₀) is the number of generations in time t. With base-10 logs, n = 3.3 × log₁₀(N / N₀). If a culture grows from 10³ to 10⁶ cells in 200 minutes, n = 3.3 × 3 ≈ 10 generations (exactly 9.97), so g ≈ 20 minutes. The growth rate constant is k = n / t = 1 / g generations per unit time.

Exponential growth only describes the log phase of a culture. If E. coli kept doubling every 20 minutes for two days, 144 generations would produce about 2 × 10⁴³ cells, thousands of times the mass of the Earth, which is clearly impossible. In a real flask nutrients run out and waste accumulates, so the curve levels off into the stationary phase and then falls in the death phase; slow growers such as Mycobacterium tuberculosis (g ≈ 20 hours) never get close.