Abstract

We look at two related problems where noise and small numbers provide limitations on the behaviour of cells. Firstly “The Poisson box”: For a simple birth death process (constant birth rate, exponential deaths) it is well known that the variance equals the mean. We conjecture that for two coupled birth death processes it is not possible for both processes to simultaneously beat this bound. That is, if X is controlling Y, and vice versa, then in order for the variance in Y to be reduced below the Poisson limit then the variance in X must be above it. For cell biology, this suggests that large fluctuations in the population of one molecular species might be a natural consequence of it being implicated in regulating a second. The conjecture is known to hold in some circumstances – a general proof remains elusive though. Secondly, Optimal clocks: How do you make accurate clocks from small numbers of independent random events (such as the production, degradation or modification of a molecule). If the number of events/molecules is fixed then the answer is well known – you line up the events, one after the other, all with the same rate. If the number of events is itself random then the optimal topology can be much more complex. However, for many distributions the optimal answer is well approximated by a simple mechanism, one which we have implemented as part of a synthetic oscillator in E-coli.