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.

FIG. 1.

Effect of conditioned medium on transduction efficiency. Actively growing E. coli (ER2738) cells in LB containing 20 μg of tetracycline/ml, in order to maintain the F′ plasmid, were briefly chilled on ice before being diluted 10,000-fold in either fresh LB containing tetracycline or filter-sterilized conditioned medium isolated from logarithmic growth or saturated cultures of the same cells. Transducing M13K07 phage carrying plasmid pBlue-GFPuv were added to the corresponding cell suspensions. Duplicate transduction mixtures were set up in parallel that contained the same number of cells and phage but in 1/10 of the volume. All mixtures were incubated at 37°C for 30 min, and then aliquots were plated on LB agar containing IPTG in triplicate. After overnight incubation, all colonies were counted and the number of transduced colonies was detected by expression of green fluorescent protein. The mean percentage of colonies transduced under each condition is shown.

FIG. 2.

A fixed number of phage with serial dilutions of host cells. (A) Approximately 400 PFU of P1 transducing phage were incubated with serial dilutions of P1C600 host cells at the cell densities indicated for 30 min and then plated on kanamycin to select for cells that had been transduced with the reporter phagemid. (B) Approximately 400 PFU of transducing phage M13K07 were incubated with serial dilutions of ER2738 host cells at the cell densities indicated for 30 min and then plated on carbenicillin and X-Gal to select for cells that had been transduced with reporter phagemid pBlue-GFPuv. The expected number of transductants for each cell density was calculated as either N(1 − e^{−MOI_actual}) (○) or N(1 − e^−MOI_input) (□). ▴, observed numbers of transductants.

FIG. 3.

A fixed number of cells with serial dilutions of phage. Nine dilutions of transducing phage M13K07 lysate carrying phagemid pBlue-GFPuv were used to infect nine aliquots of 200 CFU of ER2738 host cells at a cell density of 1,000 CFU/cm³. After 30 min of incubation at 37°C, each reaction mixture was plated on nonselective LB agar containing IPTG and X-Gal. The percentages of blue colonies and green fluorescent protein-positive colonies were determined by direct counting and are plotted as the observed values (closed symbols). Expected values (open symbols) were calculated by finding the MOI_actual for each set of reaction conditions in the experiment and multiplying the total number of colonies observed in each sample by its respective value for 1 − e^{−MOI_actual}. The results of two independent dilution series are shown. (tu, transducing units.)

FIG. 4.

Phage replication at low host cell densities. Actively growing NovaBlue E. coli bacteria carrying the pBlue-GFPuv phagemid () or ER2738 E. coli bacteria carrying the pBluescript phagemid () were serially diluted in fresh medium and then infected with 10¹⁰ PFU of helper phage M13K07. Aliquots were removed at 30 and 60 min postinfection, the host cells were removed by centrifugation and filtered with a 0.2-μm-pore-size filter, and in the resulting supernatants, the titers of the progeny phage that transduced ampicillin resistance and β-galactosidase expression were determined. No progeny phage were detectable at 30 min. Titers at 60 min are plotted as a function of the initial cell concentration in each culture. The slopes of the lines were plotted by linear regression and were 1.103 and 0.904, respectively. (tu, transducing units.)

FIG. 5.

The P_min (M, N) function as related to cell density. The P_min function (equation 4) for phages M13 (•) and P1 (▪) is plotted as a function of host cell density for an adsorption time of 30 min, a volume of 1 cm³, and an MOI_actual of 10. Note that for all cell concentrations less than , P_min is essentially the same.