### Unlimited growth: Exponential growth

### Binary division

A formula for the number of bacteria after *n* generations

The discrete model of binary division makes it relatively easy to perform calculations about bacterial colonies. The time required for cell division is called the **generation time.** It is a measure of the growth rate of the population: the shorter the generation time, the faster the population growth. The generation time is in fact equal to the time required to achieve a doubling of the number of bacteria and is therefore also called the **doubling time** \(t_d\). For many bacteria \(t_d\) is between 10 and 30 minutes.

Suppose that at time \(t=0\) the number of bacteria present equals \(A_0\). Then the number of bacteria

after one generation, \(t=t_d\), equals \(A_0\times 2\);

after two generations \(t=2t_d\), equals \(A_1\times 2=A_0\times 2\times 2=A_0\times 2^2\);

after three generations, \(t=3t_d\), equals \(A_2\times 2=A_0\times 2\times 2\times 2=A_0\times 2^3\);

...

After \(n\) generations \(t=n\times t_d\) , equals \(A_0\times 2^n\).

A general formula for the growth of bacteria Despite the fact that cell division takes place step by step, microbial growth can be regarded as a process of continuous growth, because of the large numbers of cells and because cell division of those large numbers generally does not take place simultaneously. Therefore, we can also calculate the bacterial population at times that are not integer multiplications of the generation time. Assuming a population of \(A_0\) cells at time \(t=0\), the population size \(A(t)\) at any time \(t\) can be calculated with the following formula: \[A(t)=A_0\times 2^{\frac{t}{t_d}}\] We can also write the formula as \[A(t)=A_0\times \bigl(2^{\frac{1}{t_d}}\bigr)^t\]

So the general model for bacterial growth is an exponential function with base \(2^{\frac{1}{t_d}}\). Bacterial growth is an example of a model for unlimited growth that is known as **exponential growth.**