Continuous model. N0 — initial population (OD₆₀₀ or cells/mL) at time t0; μ — specific growth rate (h⁻¹ or min⁻¹); t — elapsed time in the same unit.
Doubling time
Td=μln2=ln(Nt/N0)(tf−t0)ln2
Time required for the population to double once. Derived by setting N(t0+Td)=2N0 and solving for Td.
Specific growth rate
μ=tf−t0ln(Nt/N0)
The slope of lnN vs t during the exponential phase. Proportional to the net balance between cell division and death rates:μ=μmax−kd, where kd is the specific death rate.
Number of generations
n=log2(N0Nt)=ln2ln(Nt/N0)
Count of complete binary fission cycles between t0 and tf. Base-2 logarithm reflects the doubling nature of binary fission. Non-integer values are valid: they represent the average across an asynchronous population.
1. The four phases of bacterial growth
A batch culture follows a stereotyped trajectory with four distinct phases. This calculator is valid only during the exponential phase.
Lag phaseCells adapt to the new medium — synthesising enzymes, repairing DNA, adjusting metabolism. No net increase in viable count. Duration: minutes to several hours depending on inoculum history and medium shift.
Log phaseExponential growth. All cells divide at the maximum rate allowed by nutrients and temperature. μ is constant and equal to μmax. This is the window where T_d is defined and this calculator applies.
StationaryNutrient depletion or waste accumulation limits growth. Cell division equals death rate — net μ≈0. Population is constant but metabolically stressed.
Death phaseCell death exceeds division. Population declines exponentially. Some species form endospores or produce secondary metabolites at this stage.
2. Measuring growth: OD₆₀₀ and cell counts
OD₆₀₀ (optical density at 600 nm) is the standard proxy for bacterial biomass. It measures light scattering, not absorbance, so it reflects cell number and size rather than pigmentation.
▸Linear range: OD₆₀₀ 0.1–0.6 for most spectrophotometers with a 1 cm path length. Beyond 0.6 the Beer–Lambert law breaks down and readings underestimate true cell density. Dilute before measuring.
▸Typical inoculation: start at OD₆₀₀ 0.05–0.1, harvest at 0.4–0.6 to stay within log phase and linear detector range.
▸Conversion: 1 OD₆₀₀ unit ≈ 8×10⁸ cells/mL for E. coli, but this factor is species- and instrument-dependent. Always calibrate with direct cell counts (haemocytometer or flow cytometry) for your strain.
3. Reference doubling times
Typical T_d values under optimal conditions. Use these to sanity-check your results.
Organism / cell type
T_d (optimal)
Conditions
E. coli
~20 min
LB broth, 37 °C, aerated
S. cerevisiae
90–120 min
YPD, 30 °C
B. subtilis
~25 min
LB, 37 °C
HeLa (human)
24–30 h
DMEM + 10 % FBS, 37 °C
CHO (hamster ovary)
18–24 h
DMEM/F12, 37 °C
M. tuberculosis
~18 h
7H9 medium, 37 °C
4. Beyond the exponential model: Monod kinetics
In batch culture, μ is not constant forever — it depends on substrate concentration [S] via the Monod equation (analogous to Michaelis–Menten kinetics):
μ=μmaxKs+[S][S]
where Ks is the half-saturation constant (substrate concentration at μ=μmax/2). When [S]≫Ks (nutrient excess), μ≈μmax— the exponential phase assumption holds. As [S] falls, μ decreases and the culture transitions to stationary phase. This calculator assumes the nutrient-excess regime.
5. Assumptions and limitations
①Exponential phase only. The formula breaks down in lag, stationary, or death phases. Confirm your culture is in log phase before using this calculator.
②Homogeneous, well-mixed culture. The model assumes all cells experience identical conditions. Gradients in oxygen, pH, or nutrients (common in large bioreactors or biofilms) produce heterogeneous μ values.
③OD₆₀₀ linearity. Values above 0.6 underestimate true density. The calculator warns when N_t exceeds this threshold in OD mode.
④N_t / N_0 ratio. Ratios above 100 (more than ~6.6 generations) in a single measurement window are biologically implausible without intermediate readings — the culture has almost certainly left log phase.
⑤Synchrony. The model describes an asynchronous population. Synchronised cultures (e.g. after cell sorting) show stepwise growth, not the smooth exponential curve plotted here.