Binary Fission & Bacterial Growth Calculations

How one bacterium becomes two

Bacteria are prokaryotic cells. Their main genetic material is a single loop of DNA rather than chromosomes enclosed inside a nucleus. They reproduce by binary fission, a form of simple cell division:

1. The bacterial DNA is copied.
2. The cell grows, and the two DNA copies move towards opposite ends.
3. The cytoplasm divides and new cell wall material forms between the two parts.
4. Two daughter cells are produced, each receiving a copy of the genetic material.

Binary fission is not mitosis: mitosis is the division of a eukaryotic nucleus, whereas bacteria have no nucleus. Both processes pass copied genetic material to new cells, but their mechanisms are not the same.

Why the population doubles

One bacterium becomes two in one division cycle. If every resulting cell divides again, two become four; after another cycle, four become eight. Starting with one bacterium, the sequence is therefore

$1,\ 2,\ 4,\ 8,\ 16,\ldots$

Each complete cycle multiplies the population by 2, so this is exponential growth, not growth by adding a fixed number each time. After $n$ divisions, one starting bacterium produces a population of

$N=2^n$

More generally, if the initial population is $N_0$,

$N=N_0\times 2^n$

where $N$ is the population after the stated time.

This mathematical pattern describes growth under suitable conditions. Bacteria need enough nutrients and a suitable temperature, and some species can divide as often as once every 20 minutes under optimum conditions. In a real culture, such rapid doubling cannot continue indefinitely because nutrients eventually become scarce and waste products accumulate.

Finding the number of divisions

The mean division time is the average time taken for a bacterium in the population to divide. Find the number of division cycles using

$n=\frac{\text{total time}}{\text{mean division time}}$

The two times must first be converted to the same unit. For example, suppose a population begins with 4 bacteria, grows for 3 hours and has a mean division time of 30 minutes. Three hours is 180 minutes, so there are $180\div30=6$ divisions. The final population is

$N=4\times2^6=4\times64=256\text{ bacteria}$

Notice the two separate stages: dividing the times finds the number of divisions, while raising 2 to that power finds the multiplication factor.

Answers in standard form — Higher Tier

Repeated doubling can produce very large populations. Higher Tier students must be able to express the answer in standard form:

$a\times10^m \qquad 1\leq a<10$

For instance, if one bacterium divides every 20 minutes for 6 hours, first convert 6 hours to 360 minutes. The number of divisions is $360\div20=18$, giving

$N=2^{18}=262\,144=2.62144\times10^5\text{ bacteria}$

The coefficient, $2.62144$, lies between 1 and 10, so the expression is in correct standard form. Only round when the question requests a particular degree of accuracy.

Binary fission

• Bacterial DNA is copied.
• The cell grows and the DNA copies move to opposite ends.
• The cytoplasm and cell wall divide.
• One bacterium forms two daughter cells.
• Binary fission is simple prokaryotic cell division, not mitosis.

Population calculation

Make the time units match, then calculate

$n=\frac{\text{total time}}{\text{mean division time}}$