Table of Contents

• Pre-quiz
• Part I: Exponential Growth and Log transforms
• 1: Building the basic growth equation
• 2: What does exponential growth look like?
• 3: Why does the semi-log plot unbend the plot of exponential growth?
• 4: Do ALL curves get 'unbent' by taking the log?
• Part II: Exponential Growth and Log transforms
• 5: But how long is a generation?
• 6: Finding generation time on a graph
• 7: Is this growth exponential?
• Part III: The Numbers get messy
• 8: In search of ... the exact doubling time
• 9: Finding doubling time with messy numbers
• 10: Doubling time practice for fun and profit
• 11: In Search of ... a more convenient equation
• 12: Down and dirty with e-to-the-t
• 13: Comparing apples and oranges
• Part IV: The Numbers get messy
• 14: Fun with Rhizobia
• 15: Extended problem #1: E. coli ate my lab report
• 16: Extended problem #2: Exponential growth ends (with a whimper)
• 17: More practice with exponential growth

In Search of ... a more convenient equation

So now you know how to look at the data on a culture and find the doubling time. What about starting with the doubling time and predicting how the culture should grow? For example, if you know that the doubling time is 22 minutes and the culture grows for 66 minutes, it's pretty obvious that it doubles 3 times. For example, if you started with an N₀ of 2 million cells/ml, you would end with (approximately) 16 million cells/ml.

But can you predict the population size after 55 minutes? Well, yes, you could write:

N_t = N₀ * 2^g, so

N_t = 2 million * 2^{(2.5 generations)} [since 55 minutes = 22*2.5, or 2.5 doubling times]

If we then want to compare that to something that doubles in 25 minutes, we would write

N_t = 2 million * 2^{(2.2 generations)} [since 55 = 25*2.2]

What would be really nice would be to have population size as a function of minutes, rather than as a function of the number of generations that have gone by. Then we could simply write

N_t = (some function)⁵⁵

for each population. But how to do this?

And while we're at it, mathematicians generally prefer to state growth equations with a base of "e", rather than a base of 2. Why? What's so special about e?

First of all, e is just a number -- actually it's a bit less than 3, a decimal that goes on forever without repeating. But it's a pretty special number -- you know it has to be, seeing as it has a name. You probably know another named number, pi, which is very useful for calculating quantities related to circles, such as their circumference or area. Like e, pi is simply a number (in this case a bit more than 3), but it's so useful that mathematicians gave it a name.

Pi is useful for circles, whereas e is useful for exponential equations. The reason is that when you graph the function

y = e^x,

you get a curve in which the y value and the slope are always equal. Or, you could say, the population size (y) and the number added per unit time are the same. For example, if the population is 1000, then it is also adding 1000 individuals per unit of time (at least for a brief moment -- as soon as some individuals get added, the rate of adding also goes up).

Obviously this situation, where the y value and the slope are the same, is very unusual -- you need a very special base in the exponential equation to make it happen. That base is "e". Here's one way to determine the exact value of "e" graphically:

 

"e" is a great invention if you're a mathematician, because it makes the curve growth easier to analyze. For everyone else, we use "e" because mathematicians use "e".

Of course, just like very few circles have a diameter of exactly 1, very few populations follow the e^x curve exactly. Luckily, with a few modifications, we can make this equation fit our needs.