Generation time, and the four phases it belongs to
Generation time is the interval a bacterial population needs to double, and it is meaningful only during the exponential phase of a batch culture. Bacteria reproduce by binary fission: one cell becomes two, two become four, four become eight. After n such rounds a population of N0 cells has become N0 × 2n, and that single relationship generates every equation on this page.
A batch culture passes through four phases. In lag phase the inoculum adapts — synthesising enzymes, repairing damage — and the count barely moves. In exponential (log) phase every cell divides at the maximum rate the medium and temperature allow, and the log of the count rises linearly with time. In stationary phase a nutrient runs out or a waste product accumulates, division and death come into balance, and the count plateaus. In death phase the viable count falls, often exponentially.
Only the exponential phase has a generation time. Measuring across a lag will make the culture look slow, and measuring into stationary phase will make it look slower still, because the formula has no way to know that part of your interval was not exponential. That is the single most important practical point on this page: the arithmetic cannot detect a badly chosen interval, and neither can the number it produces.
Generation time is the same quantity that cell culture calls population doubling time and that bioprocess engineering expresses as specific growth rate μ. The conversions are exact — μ = k × ln 2 and g = 1 ÷ k — so the cell doubling time calculator and this page always agree on the same data. Only the convention differs: microbiology works in base-10 logs and minutes, cell culture in base-2 and hours.
Why the formula divides by 0.301
Start from Nt = N0 × 2n and take base-10 logs of both sides: log₁₀Nt = log₁₀N0 + n log₁₀2. Rearranging gives n = (log₁₀Nt − log₁₀N0) ÷ log₁₀2, and log₁₀2 = 0.30103. The constant is not empirical and has nothing to do with bacteria — it is simply how many base-10 logs one doubling is worth.
The reason microbiology uses base-10 logs at all is that counts span orders of magnitude and plate-count data is reported and plotted in logs anyway. Divide a log difference by 0.301 and you have the number of doublings; a one-log increase is 3.32 doublings, and a six-log increase is 19.93.
From n, the other two quantities are definitions rather than derivations. Generation time g = t ÷ n is how long each doubling took on average. The mean growth rate constant k = n ÷ t is doublings per unit time, and it is the exact reciprocal of the generation time: a 30-minute generation time is 2 generations per hour. Working in k rather than g is convenient because rates add and averages behave sensibly, whereas averaging generation times directly gives the wrong answer.
One conversion catches people out. The specific growth rate μ used in the Monod equation and in chemostat theory is defined on natural logs, not base 2: μ = ln(Nt/N0) ÷ t = k × ln 2. A culture with k = 2 generations per hour has μ = 1.386 h⁻¹, not 2. Mixing the two is a 44% error and it is common in student write-ups.
Worked example: 10³ to 10⁹ CFU/mL in 10 hours
You inoculate broth, wait out the lag, and take plate counts at the start and end of a ten-hour log-phase window: 1.0 × 10³ CFU/mL rising to 1.0 × 10⁹ CFU/mL.
- Take the logs. log₁₀(10⁹) = 9.000 and log₁₀(10³) = 3.000.
- Difference. 9.000 − 3.000 = 6.000 logs, a millionfold increase.
- Generations. n = 6.000 ÷ 0.30103 = 19.93 generations.
- Generation time. g = 10 h ÷ 19.93 = 0.5017 h = 30.1 minutes.
- Growth rate constant. k = 19.93 ÷ 10 = 1.99 generations per hour, which is 1 ÷ 0.5017 as it must be.
- Specific growth rate. μ = 1.99 × 0.6931 = 1.381 h⁻¹.
Sanity-check the answer without a calculator. Twenty doublings is 2²⁰ = 1 048 576, near enough a millionfold, and twenty doublings in ten hours is one every half hour. The exact answer and the rough one agree, which is the check worth doing every time.
Use it forward. To go from 1.0 × 10⁹ to 1.0 × 10¹⁰ CFU/mL at the same rate takes one more log, which is 3.32 generations, which is 3.32 ÷ 1.99 = 1.67 hours — though in practice a culture at 10⁹ is usually entering stationary phase, and that projection is exactly the kind that the exponential model gets wrong.
How to read g, and what makes it change
Generation time is not a fixed property of a species. It is a property of that organism in that medium at that temperature, and the same strain can differ several-fold between rich and minimal medium. Compare a measured g against your own value for the same organism under the same conditions before comparing it with anything published.
Four variables move it most. Temperature: rate rises with temperature up to the organism's optimum and falls steeply above it, so a culture at 30 °C for a mesophile with a 37 °C optimum grows measurably slower. Medium richness: a cell that must synthesise its own amino acids divides less often than one supplied with them. Aeration: for aerobes, oxygen transfer is frequently the binding constraint in a static flask. pH and osmolarity: both narrow the range over which growth is near-maximal.
In food microbiology the same number drives risk directly. A pathogen with a 20-minute generation time at 37 °C but a 6-hour generation time at 10 °C is the entire reason refrigeration works — cold does not usually kill the organism, it lengthens g until the population cannot reach an infectious dose within the food's shelf life. That framing also explains why a modest temperature abuse matters so much: shortening g from 6 hours to 1 hour turns a 3-fold increase overnight into a 10⁵-fold increase.
Two diagnostics are worth applying to your own result. If g is much longer than you expect, suspect that the interval included lag or stationary phase before suspecting the organism. If g is implausibly short, suspect a unit mismatch between the two counts — this calculator flags anything under five minutes for that reason.
Generation time, growth rate constant and 8-hour expansion
| Generation time g | k (generations per hour) | Generations in 8 h | Fold increase in 8 h |
|---|---|---|---|
| 20 min | 3.00 | 24.0 | 16 800 000 |
| 30 min | 2.00 | 16.0 | 65 500 |
| 40 min | 1.50 | 12.0 | 4 100 |
| 60 min | 1.00 | 8.0 | 256 |
| 90 min | 0.67 | 5.33 | 40 |
| 2 h | 0.50 | 4.0 | 16 |
| 3 h | 0.33 | 2.67 | 6.3 |
| 4 h | 0.25 | 2.0 | 4 |
The fold-increase column is why generation time dominates food safety arithmetic: halving g from 40 to 20 minutes multiplies the 8-hour outcome by more than four thousand.
What goes wrong with a generation time measurement
- Starting the clock at inoculation. Lag phase is not growth. Take the first count once the culture is demonstrably in log phase, and record the clock time of that count rather than of the inoculation.
- Ending inside stationary phase. Plateau flattens the average and lengthens the apparent g. If the final count is near the density this organism normally plateaus at in this medium, shorten the interval.
- Comparing a viable count with a total count. A plate count measures culturable cells; a chamber count or an OD reading measures particles including dead ones. Use the same method at both ends or the ratio is meaningless.
- Reading OD as if it were linear at high density. Optical density saturates and understates growth once the culture is dense. Calibrate OD against plate counts before using it for a rate.
- Reporting μ as if it were k. They differ by ln 2. A culture at 2 generations per hour has μ = 1.386 h⁻¹.
- Relying on two counts alone for a published figure. Two plate counts each carry their own error. For a rate you intend to report, take five or more counts and take k from the slope of log₁₀N against time.
- Assuming one culture speaks for the strain. Generation time depends on medium, temperature, aeration and pH. Record all four alongside the number.
Measuring the counts that go into the calculation
The rate is only as good as the two counts. Three methods dominate, and they measure different things.
Viable plate counts are the reference method: dilute, plate, incubate, count colonies inside the countable window, and convert with the CFU per mL calculator. They count only cells able to reproduce, which is usually the population you care about, at the cost of an overnight wait and the pipetting error of the dilution series.
Optical density at 600 nm is fast, non-destructive and continuous, and it is what fermentation monitoring actually uses. It measures light scattering by all cells, alive or dead, and it departs from linearity at high density, so calibrate it against plate counts for your organism and instrument before converting — the OD600 to cell density calculator handles that conversion.
Direct microscopic counts in a Petroff-Hausser chamber give an immediate total count and reveal morphology and clumping that a number would hide; the geometry is the same as in the hemocytometer cell count calculator, with a shallower chamber suited to bacteria.
Whichever you use, the best practice for a rate is to sample repeatedly rather than twice, plot log₁₀N against time, and take k as the slope of the straight portion. That approach shows you where log phase actually began and ended, which is information no two-point calculation can contain — and the straight-line plot is also the clearest evidence that the growth really was exponential.
Three notations, one quantity
Textbooks differ over what to call the growth rate constant. Some define k as generations per hour, as this calculator does, so k = 1 ÷ g. Others use k for the specific growth rate on natural logs, which is μ here. A few use μmax for the value attained in unrestricted growth. Check which convention a source uses before comparing numbers: the two differ by a factor of ln 2 = 0.693, which is large enough to matter and small enough to go unnoticed.
Key terms
- Binary fission
- Division of one bacterial cell into two, the process that makes the population double and puts the 2ⁿ in the growth equation.
- Lag phase
- The adaptation period after inoculation during which the count barely changes. Generation time is undefined here.
- Exponential (log) phase
- The interval in which log₁₀N rises linearly with time. The only phase in which g and k are meaningful.
- Mean growth rate constant (k)
- Generations completed per unit time, the reciprocal of generation time. A culture with g = 30 min has k = 2 per hour.
- Specific growth rate (μ)
- The natural-log growth rate, μ = k × ln 2, used in the Monod equation and in chemostat theory.
- Stationary phase
- The plateau at which division and death balance because a nutrient is exhausted or a product has accumulated.