Download Microbiology: In search of the exact doubling time

MathBench- Australia
In search of ... the exact doubling time
Dec 2015
page 1
Microbiology:
In search of ... the exact doubling time
URL: http://mathbench.org.au/microbiology/matt-gets-messy-in-search-of-exact-doubling-time/
Learning Outcomes
When you have completed this module, you should be able to:


Use the growth equation to calculate generation time from a set of data
Know how to use logarithms (logs) to solve equations with exponents
So far, we've been guesstimating the doubling or generation time from a graph of the cell
numbers over time. But there is a problem...
Guesstimating a parameter from a graph is not really accurate scientific procedure. It
would be hard to imagine publishing a report in an academic journal that started out "we
looked at the graph, squinted a little, and decided that the doubling time was 23 minutes...".
It is possible to look-up the doubling time in a table of data, but only if the table
happens to include an exactly doubled population. If the table does NOT contain a
doubled population entry, then you're in trouble. In other words, we can figure out the exact
doubling time for the “time series with convenient numbers below, but not for the one with
any general sort of number:
Time
10:00 am
10:20 am
convenient
general
10 million 10 million
20 million 15 million
Clearly, the convenient example doubled in 20 minutes. And at first glance, it appears that the
more general example got "halfway" to doubling, so doubling time should be 40 minutes.
Let's see if that's true -- if the population keeps growing at the same rate, will it double in 40
minutes?
time
10:00
10:20
10:40
general
10 million
... multiply by 1.5 to get
15 million
... multiply by 1.5 to get
22.5 million
MathBench- Australia
In search of ... the exact doubling time
Dec 2015
page 2
Oops, we got to 22.5, not 20 million. Granted, this is a small difference, but it's still not the
right answer. As the old joke goes, a million here, a million there, pretty soon you're talking
about a real epidemic. What we need is a foolproof way to determine exactly what the
doubling time is.
Finding doubling time with general numbers
Recall that we started out with an equation:
Nt = N0 2n
Remember the exponential part of this equation is that we raise 2 to the power of the number
of generations, n. Another way of saying this is that, for every generation, we multiply the
original population by 2.
Let's say I told you that N0 was 10 million and Nt was 80 million one hour later. Luckily
for you, these are nice neat numbers. You would probably think to yourself, "well, the
population doubled three times, from 10 to 20 million, from 20 to 40 million, and from 40 to
80 million. Since three doublings took 1 hour, each doubling took 20 minutes."
Another somewhat fancier way of getting the same answer would be to substitute the
information that you know (Nt and N0) into the equation above, and solve for the information
that you don't know (number of generations, n). Here's how that would look with some
conveniently chosen special numbers:
What is the doubling time? (10 million -> 80 million in 1 hour)
Hint
Explanation
Substitute in the known info
80 million = 10 million x
2n
Divide both sides of the equation by 10 million 8 = 2n
Use trial and error!
n must be 3
3 doublings in 60 minutes means...
60min/3doublings =
20min/doubling,
so the doubling time (g)
is 20 min
Let's try the same basic procedure with more general numbers. Let's say the population
increased from 10 to 70 million in one hour.
MathBench- Australia
In search of ... the exact doubling time
Dec 2015
page 3
What is the doubling time? (10 million -> 70 million in 1 hour)
Hint
Explanation
Substitute in the known info
70 million =
10 million x
2n
Divide both sides by the initial population (10
7 = 2n
million)
Use trial and error...?
n must be ...
hmmm... a
little less
than 3...
We have not solved our problem yet, and what's in the way is that 2 is raised to some power
in order to get 7 -- but we don't have a way to calculate what that power is. In other words,
the exponent is the trouble. We need a way to get the "exponential" out of "exponential
growth" so we can deal with the maths. But how?
Logarithms = Exponent-busters
In a word: when you have a problem with exponents, try logs. The great thing about logs is
that they help make sense of very large and very small numbers. They also help in solving
equations with exponents.
If you're already comfortable with log expressions, just keep going. Otherwise, click on the
link here for a quick reminder and an attempt to explain that exponent-busting magic.
Now let’s try out our knowledge of logs
So let's try using these rules where we ran into trouble before. Recall that we got as far as
finding the log in our equation -- now we can bust right past it...
MathBench- Australia
In search of ... the exact doubling time
Dec 2015
page 4
What is the doubling time? (10 million -> 70 million in 1 hour)
Hint
Explanation
Substitute in the known info
70 million = 10
million x 2n
Divide both sides by 10 million
7 = 2n
Bust that exponent
log(7) = n x
log(2)
This looks complicated, but really all you need is 0.84 = n x 0.30
a calculator
Now some rearranging...
n = 0.84 / 0.30 =
2.8
So there were 2.8 generations in 60 minutes, so.... 60 / 2.8 = 21.4
minutes each!!
Bottom line: Logs plus a calculator are a biologist's best friend....
Let's bust some exponents
So, give that a try on your own.
Let's say the oysters were infected by a single Vibrio at 3 pm, and by the next day at
3pm, when they were harvested, there were 10,000 bacteria per oyster. Assuming
exponential growth, how fast was the population doubling?
 Start with the basic equation for exponential growth: Nt = N0 x 2n, where n means the number
of generations.
 Remember that you already know both population sizes, so plug them in
 You should get to an equation that has only 1 variable, but that variable is an exponent. Time
to bust an exponent!
 Once you've got an equation with only one variable and a few logs, use your calculator to get
the value of the logs.
 Once you know n (how many generations or doublings there were), you should be able to use
common sense to figure out how long each generation was (g).
Answers: No. of generations (n) = log (10000)/log (2) = 13.3
Generation time (g) = 24 x 60 / 13.3 = 108 minutes
Extended problem #1: Exponential growth ends (with a whimper)
Here's a second set of data on bacterial growth rates.
Your instructions are 1) to determine the exponential growth rate of the bacteria, and 2) to
determine when growth stops being exponential.
MathBench- Australia
In search of ... the exact doubling time
Time (min)
No. of cells
0
20
40
60
80
100
120
4.3 * 106
9.7 * 106
22 * 106
48 * 106
97 * 106
116 * 106
118 * 106
67 * 106
140
Dec 2015
page 5
Just look at the data first... what can you see with your bare eyes, so to speak?
About doubling time: The population doubles a little more than once in the first 20 minutes,
so doubling time is probably between 15 and 20 minutes.
About the general shape of the population trajectory: In general the population raises, and
then levels out (around 100 minutes), and then falls (around 120 minutes).
About when growth begins to slow down: Each 20 minutes, the population more than
doubles, until 80 minutes, when it just barely doubles. So maybe at 80 minutes it is no longer
exponential?
The online version of this module contains an
interactive applet, which allows you plot bacterial
growth and calculate doubling time. To find this
applet go to:
http://mathbench.org.au/microbiology/mattsholiday-nightmare/6-looking-at-data/
Extended Problem #2: How long before the oysters are unsafe??
Based on lab tests, each oyster has 400,000 Vibrio at 9am. Assuming it was infected by a
single cell at 5pm, then
1. What is the doubling rate of the Vibrio cells?
2. How long till the infective dose of 1 million is reached?
MathBench- Australia
In search of ... the exact doubling time
Dec 2015
page 6
Doubling rate?



1*219 = about 524,000, so just under 19 doublings
5pm to 9am = 16 hours = 960 min
960 minutes / 19 doublings = about 51 minutes/doubling
Answer: about 51 minutes
If the Vibrio keep multiplying at this rate, how long before they reach the infective
dose?



400K --> 800K --> 1.6 million
Less than 2 doublings
2*51 = 102 minutes
Answer: less than 2 hours
MathBench- Australia
In search of ... the exact doubling time
Dec 2015
page 7
Summary
In order to find the exact doubling time, you first need to understand how to use logs to get
rid of exponents. You need to know this, because the number of doublings (n) is the
exponent in the exponential growth equation:
Nt = N0 2n
So, to determine the exact doubling time of a population, you need a two-step procedure:
1. Substitute the two population sizes into the equation, then divide both sides of the
equation by N0 and finally take the log of both sides. This gives you the number of
doublings.
2. Divide the amount of time passed by the number of doublings to find the doubling
time.
This same procedure works on any kind of exponential growth, as we saw in examples
about world population and about the growth of technology.
Learning Outcomes
Now that you have completed this module, you should be able to:

Use the growth equation to calculate generation time from a set of data
 Know how to use logarithms (logs) to solve equations with exponents