The doubling time formula
During the exponential phase of growth, a population of cells or bacteria doubles at a fixed interval, the doubling time. From two counts N0 and N taken a time t apart:
Doubling time = t × ln 2 ÷ ln(N ÷ N0)
The other useful quantities are:
- Number of doublings (population doublings): n = log2(N ÷ N0).
- Specific growth rate: μ = ln(N ÷ N0) ÷ t, per hour.
- Projected count after an extra time: N × e^(μ × time).
Example
A culture grows from 100,000 to 800,000 cells in 72 hours. N ÷ N0 = 8 = 2³, so there were 3 doublings, and the doubling time is 72 ÷ 3 = 24 hours. If growth stays exponential, there will be about 1.6 million cells 24 hours later.
Typical doubling times
| Organism or cell type | Doubling time |
|---|---|
| E. coli in rich medium at 37 °C | about 20 minutes |
| Baker’s yeast | about 90 minutes |
| Many cancer cell lines, such as HeLa | about 20 to 30 hours |
| Human fibroblasts | about 24 to 48 hours |
| Mycobacterium tuberculosis | about 15 to 20 hours |
Growth phases
A batch culture goes through four phases: a lag phase while cells adapt, the exponential or log phase, a stationary phase when nutrients run out or space fills up, and a decline phase. The formula only applies to the exponential phase. Counts taken during the lag or stationary phase underestimate the true growth rate.
Tracking cell lines
Cell biologists record the population doubling level (PDL) at each passage by adding the number of doublings. Normal human cells stop dividing after roughly 40 to 60 doublings, the Hayflick limit, while immortalized and cancer lines can divide indefinitely.
Tips for accurate counts
- Count viable cells, for example with trypan blue exclusion, rather than total cells.
- Seed at a density that keeps the culture well below confluence during the measurement.
- Use several time points and fit a line to ln(N) against time: the slope is μ and the doubling time is ln 2 ÷ μ.