# Generation Time Calculator

Exponential growths model many phenomena, from biology to finance. With this bacterial **generation time calculator** you will discover how to calculate bacterial growth over time, its main features, and parameters. Here you'll learn:

- The rules of
**bacterial population**growth; - How to calculate the
**bacterial growth rate**; and - What is
**generation time**of bacteria populations.

If you want to find out more about **bacterial population growth**, why it is important, and about an interesting bacterial experiment, then keep on reading!

## What is exponential growth?

Exponential growth models are used when a quantity, a function, or, in our case, the size of a bacteria population increases over time by a **constant percent increase per time unit**, with the size of the increment depending on the value of the function at the last step.

Exponential growth models often describe functions with "lazy" beginnings followed by explosive increases; exponentials are, in fact, the fastest growing functions in mathematics.

We got a taste of exponential growth during the coronavirus pandemic: a few cases one day, a little bit more the day after, and then things went out of control: without precautions, the initial phases of an epidemic follows the exponential - then luckily it slows down.

## How do we calculate the generation time of bacteria?

The equation that controls the exponential growth is:

where:

- $N(t)$ -
*Population*at time $t$ - $N(t_0)$ -
*Initial number*of bacteria, at the starting time, $t_0$; - $r$ -
*Growth rate*, that is the**increment**per time unit; and - $t-t_0$ -
*Elapsed time*.

Often, the time $t_0$ is set to $0$, which simplifies the equation to:

This is how to calculate the bacterial **growth rate**, $r$, we rearrange the formula:

## What is generation time?

A commonly used quantity in the study of populations is the **generation time**, $t_d$, that is, the required time for the population to double in size through *binary fission*:

The doubling time is:

We have a tool that teaches you how to calculate generation time in a cell culture.

## What if we look at things in reverse?

The exponential model for bacterial population growth can be used to model a **reduction in the number of individuals**, similarly to the log reduction model.

Researchers ; not all of the individuals would survive in the face of a growing viral infection. In mathematics, this translates to a negative growth rate, $r$, associated with an **exponential decay**.

The doubling time in this "reversed model" correspond to the *half life*; you can try our half life calculator, too!

## Testing our generation time calculator

On the 24th of February 1988, in a laboratory at Michigan State University, the longest evolutionary experiment in history began. Twelve identical populations of *E. Coli* bacteria were left to evolve independently. In 2021 the experiment reached over 70 thousand generations, witnessing mutations on every possible nucleotide of the bacteria's genetic code.

Daily, 1% of each population is transferred and primed to grow for another day: the curbing of 99% of the individuals daily is necessary because of exponential growth: let's try our bacterial growth calculator with this experiment.

Let's start with just 12 bacteria, one for each population. The growth rate of *E. coli* in the experiment is $\thicksim0.36$, which in turn corresponds to a doubling time of $\thicksim3.61$ hours. Let's also say that the bacterial population is allowed to grow without limitations.

Now we input all of the values in the generation time calculator, assuming a day has passed:

It may not look impressive, it's the population of a small village, after all. But the day after this number would increase to 100,000, a modestly sized city. And at the end of the third day, we would have 10 million bacteria, as big as Tokyo. After a week (168 hours), the number of bacteria would be bigger than the number of the stars in the Milky Way (we use this number in the Drake equation calculator):

The higher the growth rate, the shorter the generation time of bacteria. Remember to keep an eye on your colonies every now and then!

## FAQ

### What is exponential growth?

Exponential growth is a phenomenon where a quantity grows following an increment controlled by the exponent, and not a multiplicative coefficient. This implies slow initial increases, followed by an explosive growth.

### What is bacteria growth?

Bacterial growth is the process with which a population of microorganism increases. The initial phase of the growth follows an exponential law, however, due to the limitedness of resources, this soon plateaus.

### How fast do bacteria grow?

The speed with which a bacterial population grows is controlled by its generation time, that is the time required for a doubling in size of the population. Escherichia coli, a commonly studied bacteria has a doubling time of about 20 minutes.

### How do I calculate the doubling time of a population?

The doubling time of a population depends on its original size, on the population at a given time $t$, and on the value of $t$ itself, following the rule:

$t_d = t * \frac{\ln(2)}{\ln(\frac{N(t)}{N(0))}}$