Poisson Distributions of Phage Adsorption to Bacteria, from Actual Multiplicity of Infection (MOIactual)
by Stephen T. Abedon Ph.D. (abedon.1@osu.edu)
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poisson.phage.org · Abedon’s Books · DOI: 10.5281/zenodo.21036111
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When free phages are mixed with susceptible bacteria, individual adsorption events are essentially independent and random. Because a single bacterium can be adsorbed by more than one phage, the number of phages that land on any given bacterium is not fixed but is distributed — and the default description of that distribution is the Poisson distribution. It resembles the familiar normal distribution, except that its values are discrete (integers only: 0, 1, 2, 3…) and never negative. Rather than a single average, it tells you what fraction of bacteria are expected to have been adsorbed by exactly zero phages, by exactly one, by exactly two, and so on. More than fifty years ago, Mark Adams urged that the student of phage should be familiar with the Poisson distribution, and that advice still holds: it is the bridge between how many phages you add and what actually happens to the bacteria.
The Poisson distribution is driven by one parameter, the average number of phages adsorbed per bacterium — the actual multiplicity of infection, MOIactual. The most practically important quantity that falls out of it is the zero term: the expected fraction of bacteria adsorbed by no phages at all, which equals simply e−MOI. If phages cannot or do not replicate, that is also the fraction of bacteria expected to survive. Its complement, 1 − e−MOI, is the fraction expected to be adsorbed — and, for obligately lytic phages, the fraction expected to be killed. At MOIactual = 1, this means 37% of bacteria are expected to remain unadsorbed even though, on average, there is one phage per cell; at MOIactual = 5, fewer than 1% are expected to remain unadsorbed; at MOIactual = 10, survival drops to roughly e−10 ≈ 5 × 10−5, a better than 10,000-fold reduction.
Because adsorption is distributed rather than uniform, two things happen that simple averaging hides. First, at low MOI there are expected to be more multiply-adsorbed bacteria than a naive "one phage each" picture suggests. Second, and more important for antibacterial use, at high MOI there is more bacterial survival than expected: to drive the unadsorbed fraction toward zero, most surviving-target bacteria must by necessity be adsorbed many times over. You cannot simply supply "enough" phages and assume total killing — the Poisson distribution predicts a surviving tail unless MOIactual is pushed high enough that e−MOI times the starting population falls below one cell.
This calculator reports two views of the same distribution. The first is the fraction across all adsorbable bacteria. The second considers only those bacteria expected to be adsorbed at least once, obtained by dividing each n ≥ 1 frequency by (1 − e−MOI). The second view is often the more experimentally realistic one, since many experiments observe only infected cells. It also makes a useful point vivid: even at modest MOI, a meaningful share of adsorbed bacteria are expected to have been hit by more than one phage — relevant to phenomena such as lysis from without, multiplicity reactivation, and superinfection.
A persistent source of confusion is that "multiplicity of infection" often gets used to mean multiplicity of addition — this is phages added per bacterium (MOIinput) — rather than phages that have actually adsorbed (MOIactual). Only the latter drives the Poisson distribution. Adsorption takes time, and over realistic intervals not every added phage adsorbs, so a ratio of phages added is not the same as a ratio of phages adsorbed. When MOIactual is paired correctly with the Poisson zero term, it underpins related quantities such as phage killing titers and the inundative phage densities needed to reduce a bacterial population by a target amount over a defined interval. In short, the Poisson distribution is what turns a phage titer into a prediction about bacterial survival.