Friday, 5 July 2013

Boxi toxi

When I was up in The University yesterday one of the MSc students, on whose shoulders I have just been hefted to Adjuncthood, drew my attention to "Interesting Fact of the Week" which he'd written out on the fridge door in the lab.

"Four cups of botox is enough to kill every person on the planet."

To my shame (sorry Jamie) I had a bit of a rant at this: asking, among other things how big a cup was he talking about.  Because the size of the cup is irrelevant if you care, as I claim to, about being correct to the right order of magnitude rather than to the 15th decipoint.  The 4x difference between a 75ml egg-cup or a 300ml coffee cup doesn't impinge on the truth of the statement as much as other factors that need to be measured or estimated.  The other thing I mentioned was that Polonium is pretty toxic too (my toxin is bigger than your toxin? what are we like?) as I found out from my lectures about Radon.  Perhaps the first thing to concede is that, in the World According to Wikipedia, the poison from Clostridium botulinum (botox to his many frozen-faced friends) is the most toxic substance known to science. Botox requires only 2ng/kg (say 150ng per person) for a median lethal dose if injected or 10ng/kg if inhaled as an aerosol.

Polonium z drugiej strony has a mld of 1μg per person - about 10x less toxic that botox. And HCN/cyanide/Prussic Acid/the fatal smell of bitter almonds that Hercule Poirot is always twirling his moustaches at?  That requires 250mg (a quarter gram) to knock one person off or about 1750 tons to pull down the curtains on the human race.  Just as well delusional Jim Jones had less than 1000 followers to take with him when he HCN+kool-aided his sect to death in 1978. The logistics for a significantly larger number become considerable.

We're all agreed that the population of the world has just topped 7 billion. A nanogram (ng) is a billionth of a gram. The only figure we need to guesstimate is the median weight for humans to convert 2ng/kg into the dose that will convert a person to a body.  Wolframalpha gives a median weight of 85kg for males and 71kg for females, but admits that these are based on a couple of thousand US citizens. We'll have to adjust that downwards to account for the fact that most people live on $2 a day and so can't afford to get up to the weight of an average American - or an average European for that matter. Lets say 70kg for males and 60kg for females or 65kg on average. Wolframalpha also reveals that 27% of the world's people are children, so we'll call them half-pints.

(65kg * 73% + 65kg/2 * 27%) * 2ng * 7 billion people = 800g or 3.6 (8fl.oz. American cooking) cups.  In other words, what it says on the fridge:

"Four cups of botox is enough to kill every person on the planet."

I particularly relish calculations where a lot of zeroes (a billion, a billionth) cancel out and give us an answer in the is it bigger than a bread box? range.There is a handy measure for the amount of any chemical called a mole, aka gram molecular weight.  Carbon has a molecular weight of 12, so chemical theory states that 12g of carbon contains 6 x 10^23 ( 6.02×1023 for those who prefer to look scientific) molecules.  Calcium is more massive so you need 20g of that to have the same "Avogadro's number" of molecules.  Botulinum toxin is a cocktail of at least 7 proteins with a much larger molecular weight: you need 150,000,000g or 150 tons to find 6 x 10^23 botox molecules.  The mdl for botox is 150ng (and you can see where this is going) which contains
(6 x 10^23 / 150 x 10^15) = 10 million molecules. Which sounds like a lot until you realise that there are 100 billion nerve cells in your brain - bo is toxic because it dings neurons.


  1. In the USA, where I live, a cup is a precisely defined measurement though - it is 8fl oz, or half an American pint. Perhaps that's what the factoid was referring to?

  2. Quite so, I'm sure you're right, as all (to a close approximation) factoids originate in America. We use our nested US cup measures every day in the kitchen - usually filled with flour or raisins rather than cyanide, though.