Box 4.

Why Care About the Rest of the Poisson Distribution?

Emphasis here has been on calculating the fraction of bacteria expected to be unadsorbed by bacteria, that is, with r = 0 (Fig. 3). The calculation of p(1), or p(2), etc., however, can also be relevant to experiments with phages. For example, what if there is a desire to keep multiple phage adsorptions to a single bacterium minimal? This can be relevant when performing one-step growth experiments,43,70 where ideally few multiple-phage adsorptions will be occurring. Using Excel®, the relevant function is POISSON.DIST(r,n,False).57 Thus, for MOIactual = 1, the fraction of bacteria adsorbed by only a single phage is just 0.37, whereas the fraction adsorbed by more than one phage is expected to be 0.26, that is, about one-quarter of the bacteria. This, though, is actually about 42% of the phage-adsorbed bacteria that are multiply phage adsorbed! The fraction of bacteria that are expected to be multiply adsorbed, however, drops to less than 1% (0.005) with an MOIactual = 0.1, but still this is 5% of the bacteria that are phage adsorbed. It is only on dropping to an even lower MOIactual, such as of 0.01, that the fraction of phage-infected bacteria that are multiply adsorbed drops to less than 1%, that is, 0.005 in that case. Of course, crucial to such calculations are that one considers only those phages that have succeeded in adsorbing, hence Benzer et al.'s admonishment quoted earlier (the second quotation, i.e., from their p. 114, i.e., “is never 100%”).40

Note that the fraction of bacteria that are phage adsorbed is equal to 1 − POISSON.DIST(0,n,False). The fraction of bacteria that have been adsorbed by more than one phage is equal instead to 1 − POISSON.DIST(0,n,False) − POISSON.DIST(1,n,False), where n in both cases again is equal to MOIactual. To calculate the fraction of phage-adsorbed bacteria that are expected to be multiply adsorbed, I simply divided the second expression by the first expression.