While this statement is obvious to anyone who has ever looked at Anabaena, it is important to make plain what "random" is, so that we would recognize it if we saw it in a mutant. Click here to see the actual distribution of interheterocyst distances and how it compares with a random distribution, that is, the distribution if cells chose whether to differentiate or not, without respect to the choice of other cells. It is the shape of the curve, not the peak spacing, that tells us whether differentiation is random.

This may seem counterintuitive. Most people think that if spacing is both rare and random, that the peak spacing should be somewhere way out there -- they're surprised to learn that there is no peak, just an exponential decline. Let's see why.

Random spacing means independent choice: every cell has the same chance of becoming a heterocyst. Call that chance a. A cell has a probability of a that it will differentiate and a probability of (1-a) that it won't. If heterocysts appear at random, then the probability of any given interheterocyst distance is shown below:

Probability of random interheterocyst spacing of given length. Green cells represent vegetative cells (V), and yellow cells represent heterocysts (H). Heterocysts arise with a probability of a and so the probability that a cell is a vegetative cell is (1-a).

Since (1-a) must be less than 1, each longer interheterocyst distance must be less probable than a shorter distance. The frequency of interheterocyst spacing, a*(1-a)ⁿ, falls exponentially as a gets larger. Since this is certainly not what happens with actual filaments, the spacing is certainly not random.