I understand how e is used to model interest with money. But what I don't understand is how e can also be used to model bacterial growth. I understand how with money even the smallest accumulation of interest begins to earn interest if compounded continuously. I don't understand how the same thing can be said about bacteria where the cells have to mature to a certain point before they can begin dividing. The split must be complete before another split begins. I guess I don't understand how e can model both continuous systems and biological systems that are more discrete.
[Math] Exponential growth and bacteria
algebra-precalculus
Best Answer
If we were looking at very tiny sums of money the discreteness might seem a little weird also. If we deposited one penny in a bank (some imaginary bank with no minimum balance charges) with a 6.932% annual interest rate compounded continuously we could see what we'd have in 10 years using the continuous interest formula: $$ A=P e^{rt} $$ Plugging in $P=\$0.01$, $r=0.06932/yr$, and $t=10 \ yr$ we should see that around the 10 year mark we'd have two pennies in the bank. The continuous interest formula assumes that $A$ and $P$ are real numbers, not necessarily discrete integers, so if you plug in $t=5\ yr$ you'll find a balance of $\$0.0141425$ (or 1.41 pennies). The bank isn't going to go handing out fractions of pennies, but the equation we're working with treats it as if there is a fraction of a penny there that is itself gaining compound interest.
Usually, when we're working with more realistic dollar amounts this discreteness-of-pennies problem doesn't really have any ill effects, and the same is true for bacteria populations. Most of the time, we're not thinking about individual E. coli cells, but rather millions and millions of them. The basic assumption is that the life stages of the individual bacteria in the entire colony are randomly distributed so that at any one time a certain number are currently dividing while all the rest are maturing. Multiplying the penny example by a million, at 5 years we wouldn't consider ourselves as having 1 million 1.41th pennies, we'd just say that 414,251 of our pennies had matured and divided already.