# Bacterial growth: even more exponential than you thought.

I have long wrestled with exponents.

Not physically. And not really for myself.

I wrestle on behalf of my students.

We tell them that exponentiation is just repeated multiplication. Then, as if we cannot hear our own nonsensical self-contradiction, we tell them that 2^{0.5} and 2^{-3} and 2^{0} all make perfect sense. It’s a nasty bait-and-switch.

In August, giving an invited address at MathFest in Indianapolis, I mentioned one way to these strange exponents a bit more legible to learners.

Picture a bacterial blob, doubling in size every hour.

Now, how big is it after 0 hours (that is, at the starting time)? Its original size, of course. So if–and this is a big “if–but if we want the exponential notation to refer to* any moment*, not just whole hours, then we are left to conclude that 2^{0} = 1.

Now, what about at -3 hours, i.e., three hours before we started the clock? It was 1/8 its original size. Thus, invoking that same “if,” we must say 2^{-3} = 1/2^{3}.

And so on.

Anyway, at that talk, I met the lovely John Chase. He emailed me later:

The fact that you use “blobs” rather than discrete bacteria is important, since the aggregate growth for a bacterial colony depends on the individual splitting time for bacteria in a non-obvious way.

“Non-obvious” is gentle language for a shocking fact. The harsh truth, in a paper that John wrote with collaborator Matthew Wright: average *doubling *time is faster than average *dividing *time.

How so? Well, the paper begins with “the classic problem”:

Suppose a bacterium has an average division time of 1 hour. Write a model that gives the population size after t hours if the initial population is 1 bacterium.

The traditional answer: Population = 2^{t}.

But this, John and Matt compellingly argue, is wrong.

Think about it. Do you mean to say that the division time is always and precisely 1 hour? That’s no good. Then, the bacterial population becomes a step function, remaining constant for a whole hour, and then doubling in the final instant.

Obviously that’s not how a bacterial colony works. After 24 hours, it instantaneously leaps from 8 million to 16 million?

No. You must be picturing some kind of randomness in the doubling times. Some kind of distribution of times for a given bacterium to split, with 1 hour as an average. You’re relying on the randomness to smooth the ugly discrete jumps into something more plausible.

But watch out. If the *average* splitting time is 1 hour, then for a wide variety of possible distributions–including basically all the ones you’d naturally imagine–the population’s average doubling time will be less than 1 hour.

The growth is faster than it “should” be. You double faster than you divide.

Here’s an off-the-cuff question for further research: What if the splitting time does not have an *arithmetic* mean of 1 hour, but rather a *geometric* mean (natural in cases of multiplicative growth), or even a *harmonic* mean (natural in cases where we average rates)?

Does one of those means more nicely correspond to our intuition that an average splitting time should tell us the average doubling time?

Or is our intuition, as in all things exponential, simply garbage?

The details are in John and Matt’s paper, published last year in Mathematics Magazine.