# Exponential growth and bacteria

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.

• bacteria and money they reproduce the same way; every bacteria and every contribute the same way. – abel Feb 12 '15 at 19:53

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.

• My interpretation was that e is an educated, clever way of approximating populations that grow far too fast to try to model any other way. By modeling a more discrete growth with continuous growth, we are able to approximate the maximum possible population at any time. Is my thinking correct on this? – Snerd Feb 13 '15 at 12:30
• That's pretty close. The exponential function ($e^x$) is a useful mathematical function because how fast it grows (the steepness of the slope) is directly tied to how big it currently is. That is why it works so nicely in compound interest computations, because with compound interest we're getting more growth from the money we've since added to the total. – AmateurDotCounter Feb 15 '15 at 16:16
• With money we set the interest rate, and then bankers can use the formula to compute how much they owe you at any point in the future. With bacteria growth it is different, we watch a bacteria population for a while and see how big it is at a few different time intervals and try to fit the growth to a function $y=a e^{bx}$ the same way you might try a best-fit line to a $y=mx+b$ function. So what is happening is we're really pretending it is continuous from the start when we determine the best-fit parameter $b$. – AmateurDotCounter Feb 15 '15 at 16:23
• If there are enough bacteria (millions of them) and we've watched the colony for long enough to see that we best-fit $y=a e^{bx}$ really accurately, then we can be pretty sure that we can use that formula to predict the exact number of bacteria (not necessarily a minimum or a maximum) for any future time. – AmateurDotCounter Feb 15 '15 at 16:28
• If we get really discrete, like in your main question, we can still try to work things out. Day 1: 1 bacteria, Day 2: 2 bacteria, Day 3: 4 bacteria, 8, 16, 32, ... Day N: $2^N$ bacteria. The new predictive equation would look like $y=2^N$, but we're only allowed to put an integer number of days into the equation for N (since midway through day 3 we should still only have exactly 4 bacteria). We could play around with logarithms and see that $y=e^{ln(y)}=e^{ln(2^N)}=e^{ln(2)N}$ is another way to write out our formula, but we still have that problem of only putting integer numbers in for N. – AmateurDotCounter Feb 15 '15 at 16:59