| UBC Calculus Online Course Notes |

## Exponential Growth: The Andromeda Strain
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
- 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 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 for n=0,1,2...
Positive Powers of 2
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 , 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
and that the mass of the earth is
.
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 .) 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.
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
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. |