Give this calculator a starting cell count and a growth rate, and it predicts the population after any time, draws the growth curve and reports the generation time. Turn it around and it solves the other unknowns: how long a culture needs to reach a target density, what growth rate two counts imply, or how many doublings per hour a given rate means. Everything runs on the exponential growth equation used in microbiology.
In the log phase every cell divides on the same schedule, so the population multiplies rather than adds. The count after time t is N(t) = N₀ × e^(r·t), where N₀ is the starting count and r the growth rate constant. Two other numbers fall straight out of r: the generation time g = ln 2 / r, and the number of generations log₂(Nₜ/N₀). Because the curve is exponential, a change that looks small at the start becomes enormous later; the chart on the page shows that bend, and hovering over it reads the count at any moment.
The Example button loads these numbers so you can see each mode before typing your own.
The curve runs from N₀ to the final count, with the time axis picked to fit the span: minutes for short runs, hours or days for longer ones. The green dot marks the start and the orange dot the target or the final population. A steeper curve means a larger r; the same culture at half the rate takes twice as long to reach any count, which is easy to see by editing r and watching the curve flatten.
Exponential growth describes only the log phase. A fresh culture spends time adapting first, the lag phase, and later slows and plateaus as nutrients run out or waste builds up, the stationary phase. Estimates far beyond the log phase overshoot reality, so use the calculator to plan sampling times, to convert between counts and doublings, or to compare growth rates between conditions, not to predict a culture days ahead. For allele frequencies in populations rather than cell counts, see the Hardy-Weinberg calculator.
During exponential growth the population follows N(t) = N₀ × e^(r·t): N₀ is the starting count, r the growth rate constant per hour and t the elapsed time. Written with doublings the same relationship is N(t) = N₀ × 2^(t/g), where g is the generation time.
Generation time is the time the population takes to double: g = ln 2 / r. If you counted cells at two times, first get r = ln(Nₜ/N₀) / t, then g from it. For 1,000 cells growing to 8,000 in 3 hours, r = 0.693 per hour and g = 1 hour.
They describe the same speed in two ways. The growth rate r is the exponent in the growth equation, per hour; the generation time g is how long one doubling takes. They are linked by g = ln 2 / r, so a faster rate means a shorter generation time.
The number of doublings is log₂(Nₜ/N₀). Going from 1,000 to 32,000 cells is 5 generations because 2⁵ = 32. The calculator prints this for every mode that has a start and an end count.
No. It models the exponential (log) phase only, where nutrients are not limiting. Real cultures start with a lag and eventually plateau, so use it for the growth window between those phases, or to plan when a culture will reach a target density.