Exponential growth of a bacterial population
When bacteria have plenty of nutrients, space and time, each cell divides on a regular schedule, so the population grows in proportion to its own size. This is exponential growth, and it is the reason a few cells in a culture can become millions within hours. The calculator predicts the population after a given time from three quantities: the starting population, a growth rate per hour, and the elapsed hours.
It is useful for planning culture inoculation, estimating how long to incubate before a plate reaches a countable density, and for understanding how quickly contamination or infection can expand in principle. Keep in mind that the model describes only the exponential phase. Real cultures pass through a lag phase first and later slow into a stationary phase when nutrients run short or waste accumulates, so the number returned is an upper-end projection for unrestricted growth.
The exponential growth model
The calculator evaluates the continuous exponential growth equation:
- N = N0 × ert
- N0 is the initial population (cells, or CFU per mL).
- r is the growth rate per hour as a continuous rate (the natural-log rate, not a percentage). It relates to the doubling time by td = ln 2 / r, about 0.693 / r.
- t is the elapsed time in hours, and e ≈ 2.71828.
- The result is printed in scientific notation with three significant digits, for example 1.48e+5.
Worked example
Start with N0 = 1,000 cells and a growth rate r = 0.5 per hour for t = 10 hours. The exponent is r × t = 0.5 × 10 = 5, and e5 ≈ 148.413.
The population is N = 1,000 × 148.413 ≈ 148,413 cells, which the calculator displays as 1.48e+5. The doubling time for this rate is ln 2 / 0.5 ≈ 1.39 hours, so in 10 hours the culture doubles about 10 / 1.386 ≈ 7.21 times, and 27.21 ≈ 148 confirms the roughly 148-fold increase.
Common mistakes and how to interpret the result
- Entering the rate as a percentage or as a doubling time. The input is a continuous rate per hour. If you know the doubling time td in hours, enter 0.693 divided by td.
- Extending the projection too far. Exponential growth quickly produces impossible numbers. After the nutrients are consumed, the real population levels off, so long times give unrealistic results.
- Mixing time units. The rate is per hour, so the time must be in hours. Ten minutes is 0.1667 hours, not 10.
- Ignoring the lag phase. A fresh inoculum may not divide for a while. Subtract the lag time from the total time to compare against a real culture.