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The Biology Project > Biomath > Applications > Exponential Population Growth

Exponential Population Growth

Bacterial growth

In this section we will return to the questions posed in the first section on exponential and logarithmic functions. Recall that we are studying a population of bacteria undergoing binary fission. In particular, the population doubles every three hours.

We would like to know the following:

  1. How many bacteria are present after 51 hours if a culture is inoculated with 1 bacterium?
  2. With how many bacteria should a culture be inoculated if there are to be 81,920 bacteria present on hour 42?
  3. How long would it take for an initial population of 6 to reach a size of 12,288 bacteria?

Animation of bacterial culture

Why are we using exponential functions to answer these questions?

The population of bacteria in our example doubles every 3 hours. What exactly does that mean? Imagine you inoculate a fresh culture with N bacteria at 12:00 pm. At 3 pm, you will have 2N bacteria, at 6 pm you will have 4N bacteria, at 9 pm you will have 8N bacteria, and so on. If these cell divisions occur at EXACTLY each of these time points the cells are said to be growing synchronously. If this were the case, the growth process would be geometric. A geometric growth model predicts that the population increases at discrete time points (in this example hours 3, 6, and 9). In other words, there is not a continuous increase in the population.


Graph depicting geometric growth


However, this is not what actually happens. Returning to our example above, imagine you take a small sample of the culture every hour and count the number of bacteria cells present. If bacterial growth were geometric, you would expect to have N bacteria between 12 pm and 3pm, 2N bacteria between 3 pm and 6 pm, etc. However, if you perform this experiment in the laboratory, even under the best experimental conditions, this will not be the case. If you go a step further and make a graph with the number of bacteria on the y-axis and time on the x-axis, you will get a plot that looks much more like exponential growth than geometric growth.

Graph depicting exponential growth


Why does bacterial growth look like exponential growth in practice?

The answer is because bacterial growth is not completely synchronized. Some cells divide in fewer than 3 hours; while others will take a little longer to divide. Even if you start a culture with a single cell, synchronicity will be maintained only through a few cell divisions. A single cell will divide at a discrete point in time, and the resulting 2 cells will divide at ABOUT the same time, and the resulting 4 will again divide at ABOUT the same time. As the population grows, the individual nature of cells will result in a smoothing of the division process. This smoothing yields an exponential growth curve, and allows us to use exponential functions to make calculations that predict bacterial growth. So, while exponential growth might not be the perfect model of bacterial growth by binary fission, it is the appropriate model to use given experimental reality.

Now try solving the 3 problems posed at the beginning of this section

Problem 1-Calculate the number of bacteria in a culture at a given time

Problem 2-Caluclate the number of bacteria you will need to start a new culture

Problem 3-Calculate the time that a culture needs to grow to reach a given size


Next Application: Carbon Dating


The Biology Project > Biomath > Applications > Exponential Population Growth

Credits and Citation

The Biology Project
Department of Biochemistry and Molecular Biophysics
The University of Arizona
December 2005
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