# Did a Swedish statistician reveal the secret of a 100% germ free toilet bowl?

I guess we have all seen the ads for bleach that claim they kill 99% of germs dead. Maybe you have wondered about the other 1%? Killing the last 1% may be harder than you think. Naively you might think that bleach just kills bacteria, and that’s the end of it. But scientists working on bacteria like E. Coli have found that the individual bacteria are surprising diverse, in the following sense. Consider a population of millions of  E. Coli, and assume that they are all clones, i.e., are genetically identical. If they are all descendents of a single bacterium then this would be true. You might assume that these genetically identical clones would all behave identically, e.g., would all require the same dose of something nasty like bleach, to kill them.

If so, you’d be wrong. The clones are surprisingly heterogeneous, some kick the bucket with just a whiff of something toxic, while others tough out much larger doses. Crudely speaking, what may be happening is that bacteria have evolved to to be heterogeneous as a survival strategy. In a population, some reproduce rapidly but are sensitive to toxins. This fraction of the population are best suited to a friendly environment. Others reproduce much more slowly but are much tougher. These allow the bacterial population to survive even hostile environments, e.g., someone squirting bleach around to keep their toilet fresh. I guess the message here is that when you are cleaning your bathroom, you are fighting evolution, and the best you can hope for is a draw, you can’t win.

But where does the Swedish statistician come in? If you want to kill 100% of the germs, then you need a dose of bleach large enough to kill the toughest bacteria in the population. So how do you work out what this dose is? Well for a particular strain of bacteria, then you’d probably have to determine the number via experiments, but the pioneering Swedish engineer and statistician Wallodi Weibull* may have the answer to how this 100%-kill dose varies with the size of the population of bacteria. If there is upper limit, dMAX, beyond which none survive his formula would say that as the population size  decreases you can drop the dose as d = dMAX-c/Nxwith c an unknown constant, and x an unknown exponent**.

Anyway, sadly I don’t think this prediction is useful. You might as well (in the absence of knowledge of the value of dMAX) squirt the bleach around, and then if you want give it another go for luck. But it is a genuinely novel (as far as I know) prediction. You read it here first. And it is does prove that to look for applications of physics you don’t need to look up to the stars.

* Weibull was Swedish (see his Wikipedia page), although his name always strikes me as being not-at-all Swedish.

** This is derived from the distribution that bears Weibull’s name, and basically just relies on assuming that that the distribution of doses to which bacteria succumb is a simple smooth function of dose, near the upper limit of dose that the bacteria can tolerate. The derivation is an application of a branch of statistics called extreme-value statistics.