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## 'Weekly Challenge 48: Quorum-sensing' printed from http://nrich.maths.org/

Rudolph's nose glows because it is home to a species of bacteria,

*Vibrio rudolphi*, that luminesces when it reaches a certain population density. It detects the size of its population by quorum sensing: each bacterial cell releases a signal molecule, X, at a rate of $1$ per minute
and if the concentration of X is greater than or equal to $10^{11}$ cells/ml, the bacteria will glow. X decays with a half life of ten minutes but the bacteria divide every 30 minutes.

Sadly, Rudolph catches a nasty cold, which, by the time he is better, has killed all of the bacterial cells in his nose except for one. Santa is worried: there are only 24 hours left until Christmas. Will Rudolph's nose be glowing again in time?

If you need any data that is not included, try to estimate it: Santa wants an answer now, so that he can make alternative plans if need be.

Did you know ... ?

The mathematics of rates and half-lives are of great importance in mathematical biology where growth factors are often in competition with decay factors.