Binary fission and N = N₀ × 2^(t / g)
Bacteria reproduce by splitting one cell into two. After g minutes the population doubles, after 2g it quadruples, and so on. The formula reads "double the starting count once per generation."
Four growth phases
- Lag phase: cells adapt to new media, no division.
- Log phase: pure exponential division.
- Stationary phase: division balanced by death.
- Death phase: viable count falls exponentially.
Lag phase length varies with how different the new environment is from the old one. Cells transferred between two flasks of identical warm broth show almost no lag, while cells revived from a frozen stock can take 2 to 4 hours to resume dividing, since they must first repair cold-damaged membranes and rebuild depleted enzyme pools before entering log phase.
This calculator models the log phase. Once stationary phase begins, switch to the logistic growth calculator with a carrying capacity K.
Generation time across species
Generation time varies enormously across bacteria. E. coli grown in rich media at 37°C divides roughly every 20 minutes; Bacillus subtilis takes about 25 to 30 minutes; Clostridium perfringens, among the fastest known bacteria, can divide in under 10 minutes. Slow growers sit at the other extreme: Mycobacterium tuberculosis needs close to 24 hours per generation. As a result, tuberculosis cultures take weeks to show visible colonies while an E. coli plate shows growth overnight. The same doubling math used in the doubling time calculator applies here, just measured in generations rather than calendar periods. Temperature matters as much as species identity: E. coli's 20 minute generation time at 37°C stretches to several hours near typical refrigerator temperature, since most bacterial enzymes work far more slowly in the cold. Because of this, refrigeration slows food spoilage instead of stopping it outright.
Measuring bacterial growth in the lab
Microbiologists rarely count colonies one by one during log phase. Instead they track optical density (OD600), the amount of light a culture scatters at 600 nanometers, which rises in step with cell count. A typical growth curve experiment samples OD600 every 15 to 30 minutes, plots the log of OD against time, and reads the slope of the straight-line log phase segment directly as the growth rate. Converting that slope into a generation time uses g = ln(2) / slope, the same relationship this calculator's chart is built on. Plate counts, where a diluted sample is spread on agar and colonies are counted the next day, give an absolute cell count to calibrate the OD readings against, since OD alone cannot distinguish live cells from dead ones.
Food safety and the danger zone
Food safety guidance leans on this same math. The rule of keeping perishable food out of the 40°F to 140°F range for more than two hours exists because pathogens like Salmonella have generation times of roughly 20 to 30 minutes at room temperature. Left out for four hours, a starting contamination of just 100 cells with a 25 minute generation time reaches 100 × 2^9.6, about 77,600 cells, comfortably past the dose needed to cause illness for some pathogens, so commercial kitchens track holding time as carefully as holding temperature.
PCR amplification
Polymerase chain reaction doubles DNA copies each cycle. The same equation applies with g = 1 cycle: 30 cycles yield 2³⁰ ≈ 1.07 × 10⁹ copies from a single starting strand. Real-time PCR (qPCR) turns this doubling into a quantification tool: the cycle at which fluorescence crosses a threshold, called the Ct value, is inversely related to the starting copy number, since a sample crossing threshold 10 cycles earlier started with roughly 2^10 ≈ 1,024 times more template.
FAQ
What is the bacterial growth formula?
The bacterial growth formula is N(t) = N₀ × 2^(t / g), where N₀ is the starting cell count, t is elapsed time and g is the generation time. Each generation doubles the population through binary fission. E. coli divides roughly every 20 minutes, while Mycobacterium tuberculosis takes about 24 hours per generation, so the same equation produces very different curves depending on the species.
What is generation time?
Generation time is the interval between one cell division and the next, the bacterial equivalent of doubling time. It depends on species, temperature, nutrient concentration and pH, and it controls how steep the exponential curve rises. A culture with a 10 minute generation time reaches any given cell count six times faster than one with a 60 minute generation time, all other conditions being equal.
What are the four phases of bacterial growth?
The four phases are lag, log, stationary and death. In lag phase, cells adapt to new media and barely divide. Log phase is pure exponential division following N = N₀ × 2^(t/g). In stationary phase, division slows to match the death rate as nutrients run low. In death phase, the viable count falls off. Only the log phase behaves as true exponential growth.
How fast can bacteria grow?
Very fast under ideal lab conditions. Starting from 1,000 E. coli cells with a 20 minute generation time, six hours equals 18 generations, so the population reaches 1,000 × 2^18, about 262 million cells, a roughly 262,000 fold increase in a single afternoon. Real cultures slow down sooner because nutrients deplete and waste accumulates, pushing the culture into stationary phase before that ceiling is reached.
How does PCR relate to exponential growth?
PCR doubles a DNA template every thermal cycle, matching the bacterial growth equation with a generation time of one cycle. Thirty cycles give 2^30, about 1.07 billion copies, starting from a single template molecule. So a few dozen PCR cycles can turn an undetectable trace of DNA into enough material for gel detection or sequencing, the same math that describes a growing colony.
When does bacterial growth stop being exponential?
Bacterial growth stops being exponential once the culture enters stationary phase, usually when nutrients run low or waste products like lactic acid or ethanol build up to inhibitory levels. At that point the logistic growth model, which includes a fixed carrying capacity, fits the population curve far better than the pure exponential N = N₀ × 2^(t/g) equation used during log phase.