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.

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.