Box 1.

Calculating Half-Lives with Respect to Phage Adsorption

The not-phage-adsorbed half-life of a bacterium can be approximated as 1/kP, whereas the not-yet-adsorbed half-life of a phage can be approximated as 1/kN. Here, k is the phage adsorption rate constant, P is phage titer, and N is bacterial concentration. I tend to set k to 2.5 × 10−9 mL−1 min−1.43,51 Note, though, that actual half-lives in fact are 0.69-fold smaller than this, where 0.69 = −ln(0.5) = ln(2). That is, ln(2)/kP and ln(2)/kN, respectively, are actual half-life calculations whereas the simpler expressions presented at the start of this paragraph technically describe instead mean free times. For our purposes, though, there is little meaningful difference between 1 and 0.69 and therefore between mean free times and phage or bacterial half-lives, both as defined in terms of rates of phage adsorption. Feel free, therefore, to use this mean free time simplification when predicting half-lives on your own.

For concentrations (per mL) of 109, 108, 107, 106, 105, 104, or 103, we have mean free times of 24 s (109), 4 min (108), 40 min (107), 7 h (106), 3 days (105), 4 weeks (104), and 9 months (103), respectively (or 17 s, 3 min, 28 min, 5 h, 2 days, 3 weeks, and 6 months, also respectively, for half-lives). Thus, if you have a more or less constant 107 phages per millimeter with phage numbers substantially exceeding bacterial numbers, then half of the co-located, phage-susceptible bacteria will be lost to phage adsorption every half or so hour, again given k equal to 2.5 × 10−9 mL−1 min−1. Similarly, if you have a constant 107 bacteria per milliliter, then half of the co-located phages to which those bacteria are susceptible will be lost to adsorption also every half or so hour.

In short, neither individual phages will adsorb nor individual bacteria will be adsorbed very quickly given concentrations of phages (determining bacterial survival) or of bacteria (determining free phage persistence) in the range of 106/mL or less. As a result, in terms of calculating MOIactual, it really does not matter how many bacteria are present, unless you have somewhat more than 106 bacteria per milliliter adsorbing phages, and even then phage numbers might be boosted via either in situ phage replication or multiple phage dosing, resulting at least potentially in phage titers remaining more or less consistent despite their ongoing absorption to bacteria. It also can be foolish to count on phage titers that persist at densities of around 106/mL or fewer to eradicate targeted bacteria (e.g., Fig. 2). Even phage titers persisting in a range of 107/mL we can predict will not result in especially rapid bacterial eradication, with this latter point considered further after the heading of, “Inundative phage densities.”

In addition, if very few bacteria are present, then even a large MOIinput of, say, 1000, might not result in all that rapid loss of bacteria to phage adsorption. For example, starting with 103 bacteria per milliliter will mean that the associated 106 phages/mL (103 bacteria × 1000 phages bacteria−1 = 106 phages) will take about a quarter of a day to reduce bacterial numbers only by about one half. Further, successful virion production by phage infection of only 103 bacteria per milliliter is unlikely to have much of an impact on phage titers across treated environments, though such in situ virion production still might have an impact over very short, for example, sub-millimeter distances, as such short distances between bacteria might be found within cellular arrangements, microcolonies, and/or biofilms.29 In any case, readers cannot have much appreciation of either phage or bacterial half-lives if dosing is indicated in publications as just MOIs and particularly as just MOIinput in combination with ambiguous reporting of target-bacterium concentrations.

How rapidly individual bacteria may be adsorbed by phages is, thus, a function especially of what phage titers have been achieved in situ. The corollary is that without that measure, as too often can be either obscured or outright lacking in phage therapy publications, there literally can be no understanding of how long it should take for targeted bacteria to be adsorbed by treatment phages.

MOI, multiplicity of infection.