Bacterial growth is calculated using an exponential growth equation: N(t) = N(0) × e^(rt), where N(0) is the starting number of bacteria, r is the growth rate, and t is time. If you know any three of those variables, you can solve for the fourth. In practice, most calculations come down to finding either the growth rate or the doubling time from experimental data, then using those values to predict population size at any point.
The Core Growth Equation
Bacteria reproduce by splitting in two, so under ideal conditions a population doubles at a regular interval. This makes their growth exponential, and the math follows a standard exponential formula:
N(t) = N(0) × 2^n
- N(t) = number of bacteria at time t
- N(0) = number of bacteria you started with
- n = number of times the population has doubled (generations)
If you start with 1,000 bacteria and they double 5 times, you have 1,000 × 2^5 = 32,000 bacteria. That version is intuitive, but it requires you to know the exact number of doublings. The more flexible version uses a continuous growth rate (r) instead: N(t) = N(0) × e^(rt). Here, r is the per-bacterium replication rate per unit time, and t is the elapsed time in whatever units r uses (usually hours). This form lets you plug in any time point, not just whole generation intervals.
How to Find the Growth Rate
The growth rate constant (μ, often used interchangeably with r) is what you’ll calculate most often in a lab or homework problem. The formula rearranges the exponential equation into something you can solve directly from two data points:
μ = (ln N(t) − ln N(0)) / (t − t₀)
Take the natural log of your cell count at two different time points during exponential growth, subtract them, and divide by the time between measurements. If you prefer base-10 logarithms (common in textbooks), the equivalent formula is:
μ = (log₁₀ N(t) − log₁₀ N(0)) × 2.303 / (t − t₀)
The 2.303 converts between base-10 and natural logarithms. For example, say you measure 5 × 10^5 cells/ml at hour 2 and 4 × 10^7 cells/ml at hour 6. Using natural logs: μ = (ln(4 × 10^7) − ln(5 × 10^5)) / (6 − 2) = (17.50 − 13.12) / 4 = 1.10 per hour.
Converting Growth Rate to Doubling Time
Once you have the growth rate, the doubling time (also called generation time) is a single step:
Doubling time = ln(2) / μ = 0.693 / μ
Using the example above, 0.693 / 1.10 = 0.63 hours, or about 38 minutes per generation. For reference, E. coli growing in rich broth at 37°C under optimal aeration doubles roughly every 20 minutes, reaching densities above one billion colony-forming units per milliliter overnight. Most other common lab species are slower, with doubling times ranging from 30 minutes to several hours depending on species and conditions.
The Four Phases of the Growth Curve
These equations only work during the exponential (log) phase. A bacterial culture actually passes through four distinct phases, and choosing the wrong time window for your calculation will give you a meaningless growth rate.
In the lag phase, bacteria are adjusting to their new environment. They’re synthesizing enzymes and taking up nutrients, but not yet dividing. Cell numbers stay flat. This phase lasts anywhere from minutes to hours depending on how different the new conditions are from where the cells came from.
The log (exponential) phase is when cells are dividing at their maximum rate and the population doubles at a constant interval. This is the only phase where the exponential growth equation applies cleanly. On a graph of log(cell count) versus time, this phase appears as a straight line.
In the stationary phase, nutrients run low, waste products accumulate, and the rate of cell division roughly equals the rate of cell death. Total numbers plateau. Finally, in the death phase, cells die faster than they divide and the population declines.
How to Identify Exponential Growth on a Plot
The most reliable way to pick out the log phase is to plot your data on a semi-logarithmic scale: time on the x-axis (linear), and the logarithm of cell numbers or biomass on the y-axis. During exponential growth, this transformation turns the curve into a straight line. The slope of that straight line equals μ, the specific growth rate.
If you plot raw cell counts on a regular (linear) scale instead, the exponential phase looks like a steep upward curve and it’s difficult to tell exactly where it begins and ends. The semi-log plot makes those boundaries obvious because the line bends at the transitions into lag and stationary phases. Any data points you use for growth rate calculations should come from the straight-line region only.
Measuring Cell Numbers in the Lab
Plate Counts and CFU/ml
The gold standard for counting viable bacteria is the plate count. You serially dilute your sample, spread a small volume onto agar plates, incubate overnight, and count the colonies that appear. Each colony grew from a single living cell, so results are reported as colony-forming units per milliliter (CFU/ml).
To calculate the original concentration, multiply the number of colonies on your countable plate by the inverse of the final dilution factor. The final dilution factor accounts for every dilution step: any initial sample dilution, the serial dilution series, and the volume actually plated. For example, if you count 200 colonies on a plate where the total dilution factor was 1/4,000, the original sample contained 200 × 4,000 = 800,000 CFU/ml (8 × 10^5).
Plates with between 30 and 300 colonies are generally considered countable. Fewer than 30 introduces too much statistical error; more than 300 makes colonies hard to distinguish.
Optical Density (OD600)
For faster, real-time measurements, most labs use a spectrophotometer set to 600 nm wavelength. Bacteria in liquid culture scatter light, and denser cultures scatter more. The reading, called OD600, gives you a relative measure of cell density without waiting overnight for colonies to grow.
As a rough benchmark, an E. coli culture at OD600 of 0.1 contains approximately 10^8 cells/ml, though this varies significantly. Cells grown in different media can give very different conversions: the same strain in Luria-Bertani broth at OD600 0.1 may have only about 2 × 10^7 cells/ml, a fivefold difference. OD600 is also sensitive to cell size, cell shape, and the specific spectrophotometer you’re using. You need to calibrate the relationship between OD600 and actual cell count for your particular strain and instrument, typically by plating serial dilutions at several OD values and building a standard curve.
Putting It All Together: A Worked Example
Suppose you inoculate a flask with E. coli and take OD600 readings every 30 minutes. After calibrating your spectrophotometer, you know that for your strain, OD600 of 0.1 equals 1 × 10^8 cells/ml. At hour 1, you read OD600 = 0.05 (5 × 10^7 cells/ml). At hour 3, you read OD600 = 0.8 (8 × 10^8 cells/ml). You’ve confirmed from your semi-log plot that both time points fall on the straight-line portion of the curve.
Growth rate: μ = (ln(8 × 10^8) − ln(5 × 10^7)) / (3 − 1) = (20.50 − 17.73) / 2 = 1.39 per hour.
Doubling time: 0.693 / 1.39 = 0.50 hours, or 30 minutes.
To predict the population at hour 5 (assuming exponential growth continues): N(5) = 5 × 10^7 × e^(1.39 × 5) = 5 × 10^7 × e^(6.95) = 5 × 10^7 × 1,043 ≈ 5.2 × 10^10 cells/ml. In reality, the culture would likely hit stationary phase well before reaching that density, which is why these predictions only hold while nutrients remain abundant.
Why Temperature Matters for Growth Calculations
Growth rate is highly sensitive to temperature. Each bacterial species has an optimal temperature where it divides fastest, and deviations in either direction slow growth substantially. Some researchers have tried to model this relationship using the Arrhenius equation, borrowed from chemistry, which describes how reaction rates change with temperature. However, bacterial growth involves many simultaneous enzyme-driven reactions, and a single rate equation doesn’t capture that complexity well. Empirical models, where you simply measure growth rates at several temperatures and fit a curve, tend to predict real-world behavior more accurately. The practical takeaway: always record and control your incubation temperature, because even a few degrees of drift can change your calculated growth rate enough to throw off downstream predictions.