Phage Cocktail Development for Bacteriophage Therapy: Toward Improving Spectrum of Activity Breadth and Depth

Abstract

Phage therapy is the use of bacterial viruses as antibacterial agents. A primary consideration for commercial development of phages for phage therapy is the number of different bacterial strains that are successfully targeted, as this defines the breadth of a phage cocktail's spectrum of activity. Alternatively, phage cocktails may be used to reduce the potential for bacteria to evolve phage resistance. This, as we consider here, is in part a function of a cocktail's 'depth' of activity. Improved cocktail depth is achieved through inclusion of at least two phages able to infect a single bacterial strain, especially two phages against which bacterial mutation to cross resistance is relatively rare. Here, we consider the breadth of activity of phage cocktails while taking both depth of activity and bacterial mutation to cross resistance into account. This is done by building on familiar algorithms normally used for determination solely of phage cocktail breadth of activity. We show in particular how phage cocktails for phage therapy may be rationally designed toward enhancing the number of bacteria impacted while also reducing the potential for a subset of those bacteria to evolve phage resistance, all as based on previously determined phage properties.

Keywords: JavaScript; antimicrobial resistance; bacteriophage therapy; combating resistance; combination therapy; host range; online apps; quantitative biology.

Figure 1

Variations in phage cocktail breadth and depth of activity. Phage cocktails can be designed to impact multiple species (right), multiple strains of a single species (middle), or just a single bacterial strain (left). Our emphasis here primarily is toward combining the properties of the middle category with those of the leftward category. “Sur-mesure” means that a cocktail is custom-made or personalized; “Prêt-à-porter” is translated literally as ‘ready-to-wear’, but meaning here that it is not custom made for a specific patient. Spectrum of activity breadth can be considered in absolute terms, e.g., impacting one species versus impacting three species (see the arrow at the bottom of the figure). Spectrum of activity breadth can also be considered in relative terms, e.g., impacting 75% rather than 100% of the strains making up a single bacterial species. This latter concept we describe below as “Breadth₁”, read as “Breadth subscript 1”. Spectrum of activity depth refers to how many phages target a single, specific bacterial strain, especially given an inability of that bacterial strain to mutate to cross resistance to the different phages. To the left, the cocktail targets a single bacterial strain, designated as “Strain 1” in the figure, with a depth of 2. This means that Strain 1 is being targeted by two different phages, especially two phages that are found in two different cross-resistance group. As this is the only strain shown to the left, and therefore is 100% of bacterial strains shown there, in the figure this is indicated as “Depth = 2 for 100%”. In the middle, with the cocktail targeting multiple strains of a single bacterial species, relative breadth of activity is 75% ( = percent strains with Depth ≥ 1), as impacting “Strain 1” (Depth = 2), “Strain 2” (Depth = 1), and “Strain 4” (Depth = 1), but not “Strain 3” (Depth = 0). Thus, for one-quarter of the strains the depth is equal to 0 (“Depth = 0 for 25%”), for one-half depth instead is equal to 1 (Depth “= 1 for 50%”), and for another one-quarter depth is equal to 2 (Depth “= 2 for 25%”). To the right, the cocktail is impacting three different bacterial species, A, B, and C, using three different phages. Relative breadth of activity there is 100% (percentage of shown strains targeted with Depth ≥ 1), but depth of activity is only 1 for each of those strains (“Depth = 1 for 100%”).

Figure 2

Determination of phage cocktail breadth of activity in light of depth. Panel I shows a set of phage host-range data (in columns), with 0s indicating those bacteria (in rows) that are found outside of a phage’s host range and 1s indicating those bacteria that are found within a phage’s host range. Panel II shows the breadth of activity of combinations of 6, 5, 4, 3, or 2 phages—as assembled from phage isolates indicated in Panel I—that have been identified as cocktails having the broadest spectra (“Percent bacteria hit”). This is in terms of each bacterium being impacted by at least one phage and which thereby represent what we describe as ‘Breadth₁’. These Breadth₁ determinations are calculated using Equation (1). Panel III is equivalent to Panel II except bacteria must be ‘hit’ by at least two phages to be counted, rather than by at least one phage (the latter is calculated instead for Panel II). These ‘Breadth₂’ determinations are calculated using Equation (2).

Figure 3

Combining phages found in both different and the same cross-resistance groups into a single phage cocktail. Phages a and b, c and d, and e and f are found within three different cross-resistance groups, red, gold, and blue, respectively. The presented cocktail, dubbed “a b c d e f”, consists of six phages and three cross-resistance groups, with constituent phages thus found overall in combinations of the same and different cross-resistance groups.

Figure 4

Calculating Breadth₂ without all phages being found in different cross-resistance groups. Phages a through f are found in three different cross-resistance groups, red, gold, and blue, i.e., as presented equivalently in Figure 3. Thick lines between phages serve as guides as to what phage combinations (sub-cocktails) would be involved in Equation 2-based calculations. As examples, note the lines between phages a, c, and f (black-solid) or between phages b, d, and f (purple-dashed). The respective Depth_n (a c f) and Depth_n (b d f) values are either a 0, 1, 2, or 3. This is 0 if no phages from a given combination hit an individual tested bacterial strain (n), 1 if only one hits, 2 if two hit, and 3 if three hit. Considering all six of these phages, then Breadth₂ (a b c d e f) = { $${{\displaystyle \sum }}_1^n$$ True₂ [Depth_n (a c e), Depth_n (a c f), Depth_n (a d e), Depth_n (a d f), Depth_n (b c e), Depth_n (b c f), Depth_n (b d e), Depth_n (b d f)] }/n. This is every possible combination of phages acting on every bacterial strain tested, where each phage making up an individual phage combination has been sourced from a different cross-resistance group. The function True₂ returns a value of 1 if any Depth_n value for a tested bacterial strain is a 2 or higher. Alternatively, True₂ returns a value of 0 for a given bacterial strain if that strain is not hit by at least two phages sourced from different cross-resistance groups. Breadth₂ (a b c d e f) thus will have a value of between 0 and 1, ranging from no bacteria hit by at least two different phages sourced from different cross-resistance groups to all bacterial strains tested hit by at least two different phages sourced from different cross-resistance groups. For example, Breadth₂ (a b c d e f) = 0.5 would mean that half of the bacteria tested were individually hit by at least two different phages sourced from two different cross-resistance groups, while the other half were not. This specific example is not solvable as presented, however, as no actual phage host-range data is provided. See instead the main text, Figure 5, and then Figure 6 and Table S1.

Figure 5

Determination of Breadth₂ for phages found in both the same and different cross-resistance groups. Different bacterial strains, ‘1’ through ‘10′, are indicated in orange toward the bottom of the figure. Lines indicate that a bacterial strain is found within the host range of either phages x, y*, or z, with phage y* as indicated found in a different cross-resistance group from phages x and z. Depths of activity for each bacterial strain are indicated as the numbers 0, 1, or 2. These values cannot be greater than 2 in this example because there are only two phage cross-resistance groups, phages x and z versus phage y*. For the sub-cocktail consisting of phages x and y* (top row) these are Depth_n (x y*) values. For the sub-cocktail consisting of phages y* and z (middle row) these are Depth_n (y* z) values. Finally, for the combination of phages x, y*, and z (the full cocktail; bottom row) these are the activity depths for each individual bacterial strain tested. Shown to the right are percentages of bacterial strains hit for the different phage combinations. This is by at least one phage (bigger percentage = Breadth₁, e.g., 60% for the top row) or instead by at least two phages from different cross-resistance groups (smaller percentage = Breadth₂, e.g., 20% also for the top row). Carrying out calculations for each bacterial strain tested, the bottom digit in a column (“Activity depth”) is a 1 if at least one previous digit in the same column is a 1 and also no previous digits are larger than a 1 (this criterion is met in the figure for bacterial strains 1, 3, 8, and 9). Alternatively, the bottom digit is a 2 if at least one previous digit is a 2 (or in the case of different examples from this one, potentially instead a number greater than a 2). A Depth of 2 thus is the case for bacterial strains 2, 6, and 10. To calculate Breadth₁ (x y* z), simply count the fraction of 1s or higher found across the bottom row, i.e., as per the True₁ function, which is a process that is equivalent to solving Equation (1) as well as Equation (4). This comes out to 7 out of 10 or 70%. To calculate Breadth₂ (x y*) or Breadth₂ (y* z), both of which are based on Equation (2), count only the fraction of 2s as well as any higher values if they were present. Thus, e.g., Breadth₂ (x y*) = 2/10 or 20%. To calculate Breadth₂ (x y* z), as based on Equation (3), do the same, i.e., count the fraction of 2s or higher as per the True₂ function, but for the bottom row only. Thus, Breadth₂ (x y* z) = 3/10 or 30%. In other words, in this example a total of three bacterial strains of ten are hit by two phages from different cross-resistance groups. This is the breadth of activity of this cocktail in light of an activity depth for three specific bacteria that is sufficient to reduce their potential to mutate to phage resistance, of a total of ten bacterial strains tested.

Figure 6

Example of calculation of activity depth taking into account multiple cross-resistance groups with diverse phage membership. Shown to the left are the first rows of Table S1, Supplementary Materials. The number “1” found in the lower, left-hand corner refers to bacterial strain number 1, as indicated within the lower arrow. Demonstrated also within the arrows (top) is how an activity breadth of 3 is arrived at for bacterial strain number 1. Note that the “Total depth” for each cross-resistance group cannot exceed a value of 1 and thus is equal to either 0 or 1.