Doublings: The exponential function with base 2
UBC Calculus Online Course Notes

Exponential Growth: The Andromeda Strain

"The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. That is not particularly disturbing until you think about it, but the fact is that bacteria multiply geometrically: one becomes two, two become four, four become eight, and so on. In this way it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth."
Michael Crichton (1969) The Andromeda Strain, Dell, N.Y. p247

The frightening statement in Crichton's book may seem like science fiction, and if we take the pragmatic approach of a biologist, it sounds implausible and unrealistic. However, with the assumptions of "ideal circumstances", the mathematics behind this statement is surprisingly robust. We will actually determine its accuracy below, but our main focus will be not just to see if Crichton exaggerates, but also to understand the nature of so-called geometric growth. Better known to us in more recent terminology as exponential growth, this process hardly fails to surprise us with its potential for rapid increase.


Goals:

  • To investigate a simple example of exponential growth in both discrete and continuous growth situations.
  • To understand the idea of doubling time.
  • To develop intuition about exponents with various bases.
  • To motivate the need for an inverse operation (the logarithm) and investigate its relationship with an exponential function.


For your consideration:

  • Bacterial growth involves a process of doubling, and then doubling again and again. The time taken for the population to double is called, naturally enough, the doubling time. We will use the letter $ \tau $ to represent the doubling time. What is the doubling time for E. coli according to Crichton?

  • Expressing all quantities in the same units is important. Since we are interested in growth over a period of hours, express the doubling time in terms of hours.

  • How many doublings take place in 1 hour? In one day?

  • How big is the population of cells after n doublings?


To appreciate the alarming growth, we might tabulate the first few values of the exponential expression $ 2^n $ for n=0,1,2...

Positive Powers of 2

Power $ 2^1 $ $ 2^2 $ $ 2^3 $ $ 2^4 $ $ 2^5 $ $ 2^6 $ $ 2^7 $ $ 2^8 $ $ 2^9 $ $ 2^{10} $
Value 2 4 8 16 32 64 128 256 512 1024


We could graph the first few on a plot of $ y=2^n $ versus $ n $ (not all the values will fit conveniently on the same scale since the values grow so rapidly.) The plot would look like this:

 
The diagram should look like this: If you do not have Java, 
you may not see it. 

		|   * 
		|            2^n 
		|   
		|   * 
		| 
		| * 
		* 
		| 
		---------- n 


Chosing a convenient time unit for the problem

Without actually thinking about it, you may have already "converted" the time units of minutes, hours, or days to time expressed in terms of multiples of $ \tau $ , i.e. multiple of 20 minutes. Of course, this makes it much simpler to "count the number of doublings" and thus calculate how big the population would be after some time. This step, trivial as it might seem, is actually an example of a fundamental scientific method: the idea is that many natural processes have some kind of "intrinsic" timescale associated with them. If we look at these processes in their "own set of units", they look simplest.

To give a more convincing argument for why this might be useful, consider the fact that many kinds of species and populations could, in "ideal circumstances" undergo explosive (geometric) growth. Even though they may reproduce in more complex ways (not just by splitting into two copies), the population might also be characterized by a doubling time - for example, the population of rabbits in Australia doubled every few years after they were first introduced.

If we know the doubling time associated with the growth of some particular species, and we convert our calendar units (days, weeks, years, etc) to multiples of the doubling time, then the process of growth (on that scale) would look identical to the one we already studied for bacteria in this example. This means that by looking at just one case, and making the correct reinterpretation to various species, we have said all there is to say about the phenomenon of uncontrolled growth. This shows the power of mathematical reasoning. But it can also lead to a pitfall if we are not prepared to carefully evaluate the implications of our results.


For your consideration:

  • Assume that Crichton's ideal circumstances hold and determine whether his statement is correct. Consider that the mass of an E. coli bacterium is roughly $ 10^{-12} gm $ and that the mass of the earth is $ 5.9763 \times 10^{24} kg $ . Find the mass of a colony of E. coli after 1 day of geometric growth. (You can use the calculator below. Here are instructions for its use.)

  • How long would it take to reach a mass equivalent to planet earth?

  • What is meant by "ideal circumstances" in the quotation from Crichton?

  • Are there any situations in which ideal circumstances actually occur? If so, give an example. If not, explain why not.

  • If the circumstances are "realistic" rather than "ideal", what might be different about the process being described? Would the bacteria keep reproducing?

  • Growth of bacteria like E. coli has a doubling time of roughly twenty minutes. (We noted that this time period also defines a natural choice of the units of time for the process.) Suppose that human beings, under "ideal conditions," could double their population size every 50 years and that there are 6 billion humans on earth in the year 2000. Determine what the human population would be in the year 2500 under the uncontrolled growth scenario.

    In the year 2500, how many people would have to inhabit each square kilometer of the planet? (Take the radius r of the earth to be 6400 km for the purpose of computing its surface area which is $  S = 4\pi r^2  $ .) Notice that this includes the earth's oceans and uninhabitable regions.

  • Do you think this could happen? Is the assumption that humans double their population every 50 years reasonable? Why or why not? You might want to investigate what the experts say.

  • Estimate what the doubling times would be for other kinds of populations such as elephants, rabbits, and flies.


Shortcuts and recycling your envelopes!

Many students think that mathematics is all about "getting the right answer" and being accurate. This is vital in some cases (when we are designing a bridge or administering powerful medication) but superfluous in others. Applied mathematicians are famous for their "back of the envelope calculations" (sometimes scribbled on napkins at the diner, or on the back of ticket stubs at the opera). These are supposed to be rough estimates that get the answer in right ball park, without going overboard or getting overly bored with detailed calculations.

You might want to notice from our powers of 2 table that

$ 2^{10} = 1024 \approx 1000 = 10^3 $

This is a useful little conversion from powers of 2 to powers of 10. It is not exact, but it helps establish an "order of magnitude" approximation. Now that you know this handy fact, go back to the problem about human crowding on planet earth and see if the back of an envelope will do.