What this calculator does
Bacteria in exponential phase double at a constant interval, so the population after time t is the starting count multiplied by 2 raised to the power of t divided by the doubling time. Everything else follows from that one expression.
The numbers get away from intuition very quickly. E. coli under good conditions doubles roughly every 20 minutes, so a hundred cells become 800 in an hour, 6,400 in two hours, and over 400,000 in four. This is why a small contamination left at room temperature becomes a large one overnight.
The formula
The number of generations is elapsed time divided by doubling time. The population is the starting count multiplied by two raised to that power. The growth rate constant k is the natural logarithm of 2 divided by the doubling time, which is the same growth expressed in continuous rather than doubling terms. The model describes exponential phase only.
| Term | Meaning |
|---|---|
| Doubling time (td) | The interval in which the population doubles during exponential growth. |
| Generation | One doubling. Generations elapsed is time divided by doubling time. |
| Growth rate constant (k) | ln2 divided by doubling time, the continuous-growth equivalent of the doubling rate. |
| Exponential phase | The period of constant doubling, after lag phase and before nutrients run short. |
The inputs explained
| Field | What to enter |
|---|---|
| Initial population (cells) | Starting cell count or density. |
| Doubling time (min) | Doubling time in minutes. About 20 for E. coli in rich medium and much longer for slow-growing organisms. |
| Elapsed time (min) | Elapsed time in minutes. |
When to use it
Planning a culture
Working out how long a culture needs to reach a target density saves either checking repeatedly or overshooting into stationary phase.
Understanding a food safety interval
The growth from a small initial contamination over a few hours at room temperature is the reasoning behind most time and temperature rules.
Comparing growth rates
Doubling times are the standard way of comparing how fast organisms grow, and the growth constant k is what appears in the differential form of the same model.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How fast does a bacterial population grow?
A starting population of 100 cells with a 20 minute doubling time, at increasing elapsed times.
| Elapsed time | Population | Generations elapsed | Growth multiple |
|---|---|---|---|
| 20 min | 200 cells | 1.00 | 2.00× |
| 40 min | 400 cells | 2.00 | 4.00× |
| 60 min | 800 cells | 3.00 | 8.00× |
| 120 min | 6,400 cells | 6.00 | 64.00× |
| 240 min | 409,600 cells | 12.00 | 4,096.00× |
Questions
What is a typical bacterial doubling time?
About 20 minutes for E. coli in rich medium at 37°C, which is among the fastest. Many environmental bacteria take hours, and Mycobacterium tuberculosis takes around 24 hours, which is why TB cultures take weeks to read.
Does this model hold indefinitely?
No, only during exponential phase. Real cultures show a lag phase first while cells adapt, then exponential growth, then stationary phase as nutrients run out and waste accumulates, then decline. Projecting exponential growth far forward gives absurd results very quickly.
What is the difference between doubling time and growth rate constant?
They describe the same growth in different terms. Doubling time is how long a doubling takes; k is the continuous rate, equal to ln2 divided by the doubling time. A 20 minute doubling corresponds to a k of about 0.0347 per minute.
How do I find doubling time from experimental data?
Plot the natural log of cell density against time during exponential phase. The slope of the straight portion is k, and doubling time is ln2 divided by it. Using the whole curve rather than just the exponential portion will underestimate the growth rate.
For microbial reduction rather than growth, see the log reduction calculator. For enzyme reaction rates, see the Michaelis-Menten calculator.