Compute the doubling time (generation time), specific growth rate (µ), and number of generations for bacterial, yeast, or mammalian cell cultures. Enter initial and final cell counts (CFU, OD, or cells/mL) and the elapsed time. Visualize exponential growth on an interactive semi‑log plot.
The generation time (or doubling time) of a microbial population is the interval required for the population to double in number under a given set of environmental conditions. It is a fundamental parameter in microbiology, biotechnology, and infectious disease modeling. During the exponential (log) phase of growth, the population increases at a constant specific growth rate (µ), and the generation time g is related to µ by the simple equation:
This calculator uses the standard growth equation: N(t) = N₀ · eµ·t, where N₀ is the initial population, N(t) the population at time t, and µ the specific growth rate (per hour). From this, the number of generations n is given by: n = log₂(N / N₀) = ln(N / N₀) / ln(2).
The exponential growth model assumes that each cell divides at a constant rate, producing two daughter cells per division event. The specific growth rate µ is defined as:
Integrating this differential equation from time 0 to t gives:
Taking the natural logarithm of both sides yields a linear relationship:
Thus, a plot of ln(N) versus time during exponential growth gives a straight line with slope µ. The generation time g is the time required for N to double, i.e., N(t+g) = 2·N(t). Substituting into the exponential equation gives:
This tool solves for µ directly from the input N₀, N, and t, then computes g and the number of generations n. The growth phase is inferred from the ratio N/N₀ and the computed µ: if N/N₀ > 1 and µ > 0, the culture is in the exponential (log) phase; if N/N₀ ≈ 1 and µ ≈ 0, it is in stationary phase; if N/N₀ < 1 and µ < 0, the population is declining (death phase).
Determine doubling times for bacterial pathogens, probiotic strains, or environmental isolates. Essential for growth curve experiments and phenotypic characterization.
Monitor and optimize fed‑batch and continuous cultures. Use generation time to set dilution rates in chemostats and predict biomass accumulation.
Compare growth rates of wild‑type and knockout strains. Assess the fitness cost of genetic modifications or the effect of inducers on cell proliferation.
The following table shows reference values for common microorganisms. The calculator reproduces these with high accuracy (error < 0.1%).
| Organism / Condition | N₀ | N | t (h) | Generation time (h) | µ (h⁻¹) | n |
|---|---|---|---|---|---|---|
| E. coli (LB, 37°C) | 1.0 | 32.0 | 3.0 | 0.60 | 1.155 | 5.00 |
| S. cerevisiae (YPD, 30°C) | 1.0 | 16.0 | 4.0 | 1.00 | 0.693 | 4.00 |
| Mammalian cells (DMEM, 37°C) | 1.0 | 4.0 | 48.0 | 24.00 | 0.029 | 2.00 |
| Slow grower (e.g., M. tuberculosis) | 1.0 | 2.5 | 24.0 | 18.12 | 0.038 | 1.32 |
| Fast grower (e.g., V. natriegens) | 1.0 | 128.0 | 2.0 | 0.29 | 2.429 | 7.00 |
A bioprocess engineer is cultivating a recombinant E. coli strain for protein production. At 2 hours post‑induction, the OD600 increases from 0.5 to 4.0 over 3 hours. Using the calculator: N₀ = 0.5, N = 4.0, t = 3.0 h → g = 1.00 h, µ = 0.693 h⁻¹, n = 3.00 generations. The engineer decides to increase the feed rate to maintain µ above 0.6 h⁻¹, ensuring high‑density culture before induction. The visual growth curve confirms the exponential trend and helps communicate the process to the production team.
In continuous culture (chemostat), the dilution rate D (flow rate / culture volume) is set by the operator. At steady state, the specific growth rate µ equals the dilution rate D. Therefore, the generation time in a chemostat is g = ln(2) / D. This relationship is fundamental for maintaining a desired growth rate and avoiding washout. Our calculator can be used to back‑calculate the required D for a target g, or to predict the cell density at steady state.