Box 2.

Let's Analyze Some Data

We can better appreciate the benefits of using Poisson distributions when we apply this framework to real data. Let us consider, for example, the dose-response data provided by Chang et al.30 They delivered phage PEV31 to mouse lungs that had been challenged with 2 × 104 CFU of Pseudomonas aeruginosa 2 h earlier. At the point of phage addition, the total number of CFUs found in the mouse lungs was, on average, a bit less than 105 CFU, which we will call exactly 105 CFU for the sake of being conservative in our MOIinput estimations. Mice were dosed at this point, that is, 2 h after bacterial challenge, with 7.5 × 104, 5 × 106, and 5 × 108 PFUs, which seems to have resulted in somewhat fewer phages in the first group, for example, by about one fewer log as measured in sacrificed mice (which I speculate is due to relatively fixed losses in numbers of phages that are somewhat independent of the amount of phages applied), but about the same number of phages in the second and third groups. That discrepancy makes the next step a little difficult, but let us say nevertheless—for the sake of ease of consideration—that for MOIinput we have values of 0.5, 50, and 5000, respectively. For example, 5 × 108/105 = 5 × 103 for the latter.

The next step is to calculate the number of bacteria that are expected to still be alive assuming that MOIactual ≈ MOIinput. From e−n we thus, respectively, have estimated log10 bacterial survival rates of −0.21 (or 0.6 in non-log units), −22 (or 10−22, i.e., effectively zero), and basically negative infinity (i.e., also effectively zero). Thus, for the three doses, we expect bacterial survival rates in absolute terms, per lung, of not quite 105 CFU (though close to that number), zero, and zero. Actual numbers of bacteria after 24 h, however, were 2.3 × 108 CFU (for the no-phage control), about 1.1 × 107 CFU for the lowest phage dosage, 6.5 × 106 CFU for the middle phage dosage, and 3.1 × 106 CFU for the highest phage dose. All of these represent increases in the original pulmonary load, that is, by at least one log in each case, and thereby not quite the “significant reduction in pulmonary bacterial load” described in this study's abstract. Thus, rather than decreases in CFUs as estimated quantitatively based on assumptions of Poisson distributions, instead increases in CFUs were observed, as relative to the initial bacterial numbers,28 though slightly smaller increases in CFUs with the higher phage doses.

We can infer from these observations that bacteria were able to replicate in situ despite phage treatments. As an immediate following step, one therefore should be asking whether this ability of bacteria to continue to replicate was due to bacterial evolution of phage resistance. In fact, such evolution was found to be the case, though fractions of bacteria that were phage resistant after 24 h of phage treatment were not 100% but instead 30%, 74%, and 91%, respectively, for the increasing phage doses. What does this suggest? First, clearly the treatment phages in all cases failed to reach and/or otherwise kill all of the targeted bacteria, and those still phage-sensitive bacteria thereby appear to have been able to replicate within what I will guess were phage-free microenvironments (see Appendix A1 for my arguments against phenotypic resistance as an alternative explanation). Those microenvironments likely were found either within the lumens of the treated mouse lungs or instead external to the lumen but still in association with the mouse lungs. Second, the higher phage doses probably reduced the number of phage-free microenvironments, or perhaps reduced the ability of bacteria to reach these phage-free microenvironments, resulting in greater fractions of the bacteria that divided, after the start of phage treatment, being phage resistant when phage doses were higher. The second inference, by the way, could be indicative of just how difficult it may be for phages applied into a lung to reach all of the bacteria infecting that lung.

As a conclusion, clearly in this case the impact of phages on bacterial numbers, at least after 24 or so hours of treatment, was substantially less than would be predicted assuming Poisson distributions of adsorptions by all treatment phages. The differences, furthermore, I suspect were due to inefficiencies in phage penetration to targeted bacteria after dosing into the murine lung. This, I should mention, is not an argument against the validity of Poisson distributions in describing phage adsorption to bacteria, but instead that for Poisson distributions to be valid, then all bacteria found in a phage-targeted population must be equally likely to become phage adsorbed. When Poisson distributions fail to describe experimental results, in other words, that can be telling us something. See Appendix A2, along with Appendix A1, for additional discussion of this study.

CFU, colony-forming units; PFUs, plaque-forming units.