Evidence for a Phage Proliferation Threshold?

Both experiments (5) and theory (3, 4) have suggested that for a population of phage to increase in numbers requires the host cell population to surpass a critical density termed the “replication threshold” or the “proliferation threshold.” However, recently in the Journal of Virology, Kasman et al. (1) argued that no such threshold exists. Why this discrepancy? For a population of phage to increase in numbers, not only must phage from the initial dose replicate but also progeny phage must survive long enough to sustain further replication. This in turn depends on the density of remaining uninfected cells and on the rate of loss of free phage. The proliferation threshold is that cell density above which the probability of a progeny phage replicating is greater than the probability of that phage being lost (4). From this we identify three ways to reconcile the apparent inconsistencies between Kasman et al. (1) and Wiggins and Alexander (5).

First, the rate of phage loss in vitro is many times lower than in natural systems such as in sewage or in vivo. Consequently, the proliferation threshold is expected a priori to be much lower in in vitro experiments such as those of Kasman et al. (1) than in any in vivo system, maybe even too small to measure. Wiggins and Alexander (5) assessed different rates of phage loss, but they were not reported by Kasman et al. If relevant parameter estimates were available, then the proliferation threshold could be predicted using a formula derived from kinetic theory (4). Second, where Kasman et al. use an actual multiplicity of infection of 10, the bacterial infection rate is so high that there are effectively no uninfected cells left for progeny phage to infect: inundation by the initial phage renders any subsequent phage replication or density threshold irrelevant. Kinetic theory predicts that the proliferation threshold is manifested only if the initial phage dose is much smaller than the actual multiplicity of infection of 10 (technically, the phage dose must be less than the “inundation threshold” but more than the “failure threshold” [4]). Third, in natural systems of interest the host cell density typically increases with time. Thus, if the initial cell density is low, it takes a certain time before the proliferation threshold is crossed and thereby made observable. Wiggins and Alexander made this transparent using explicit time series, whereas Kasman et al. took measurements at a fixed time point, which would hinder detection of any time-dependent threshold.

Therefore, that Kasman et al. (1) saw no proliferation threshold probably does not mean that no threshold exists but rather is an expected result of their specific scenario. Kasman et al. mention the profound implications for bacteriophage therapy, but therapeutically what really matters is the possible presence of thresholds in vivo, where phage loss is high. Merril et al. (2) highlighted the importance of the rate of phage loss for the efficacy of bacteriophage therapy in a mouse model, and the kinetic theory (3, 4) clarifies why this makes most in vitro measurements of in vivo processes and outcomes so misleading.