Overcoming the Phage Replication Threshold: a Mathematical Model with Implications for Phage Therapy

Abstract

Prior observations of phage-host systems in vitro have led to the conclusion that susceptible host cell populations must reach a critical density before phage replication can occur. Such a replication threshold density would have broad implications for the therapeutic use of phage. In this report, we demonstrate experimentally that no such replication threshold exists and explain the previous data used to support the existence of the threshold in terms of a classical model of the kinetics of colloidal particle interactions in solution. This result leads us to conclude that the frequently used measure of multiplicity of infection (MOI), computed as the ratio of the number of phage to the number of cells, is generally inappropriate for situations in which cell concentrations are less than 10 7 /ml. In its place, we propose an alternative measure, MOI actual , that takes into account the cell concentration and adsorption time. Properties of this function are elucidated that explain the demonstrated usefulness of MOI at high cell densities, as well as some unexpected consequences at low concentrations. In addition, the concept of MOI actual allows us to write simple formulas for computing practical quantities, such as the number of phage sufficient to infect 99.99% of host cells at arbitrary concentrations.It has long been observed that when bacteriophage are mixed with susceptible host bacteria, the number of phage in the culture supernatant does not increase until after an eclipse period of, generally, 30 to 40 min at 37°C has passed. This period of time is explained as the time the phage requires to inject its genome into the host, express its genes, and assemble progeny phage and release them into the environment. Additionally, when host cell densities are very low, it has been observed that there is a longer delay before phage numbers increase over the numbers of input phage. This period has been explained as the time needed for the host cells to reach a "replication threshold" (16) or "proliferation threshold" (7,8) density. This density has been reported to be approximately 10 4 cells per ml for the multiple phage-host combinations tested (16) and has been said to have broad implications for the propagation of phages in natural environments and in terms of their use as antimicrobial therapies (7,8). The mechanism of this delay in phage replication has not been widely investigated or discussed.One explanation for the apparent threshold density would be a requirement on the part of the phage for the host cell to be in a particular metabolic state and that this state is only reached when the cell density is 10 4 CFU/ml or more. Small molecules called autoinducers or quorum factors are known to be secreted into the environment by bacteria and, by their accumulation as the number of cells increases, to allow the bacteria to monitor their local population density (3). These soluble signaling molecules alter the expression of dozens of genes and thereby regulate the metabolic state when the sensing bacteria are expo...

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