Exponential Growth

Bacteria reproduce by binary fission: one cell divides into two. The time taken for a population to double is the generation time (doubling time). Under ideal conditions it can be as short as 20 minutes for E. coli.If every cell divides once per generation, the population doubles each time: 1, 2, 4, 8, 16 and so on. This is exponential growth. After n generations:N = N₀ × 2ⁿwhere N₀ is the starting number of cells and N is the number after n generations. The number of generations is the time elapsed divided by the generation time.Worked example: 100 bacteria with a generation time of 20 minutes grow for 2 hours, so n = 120 ÷ 20 = 6.N = 100 × 2⁶ = 100 × 64 = 6400 cellsWorking backwards, the number of generations can be found from two counts using logarithms, where log₁₀ 2 = 0.301:n = (log₁₀ N − log₁₀ N₀) ÷ 0.301The mean growth rate constant, k, is the number of generations per unit time:k = (log₁₀ N − log₁₀ N₀) ÷ (0.301 × t)For example, if a population rises from 10³ to 10⁶ cells in 5 hours, n = 3 ÷ 0.301 = 9.97, so k is about 2.0 generations per hour.Exponential growth cannot continue for long in a closed culture, because nutrients run out and waste builds up. This is why it lasts only for the log phase.