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

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## Abstract

**:**

## 1. Introduction

## 2. Cocktails Can Emphasize Different Spectra of Activity

#### 2.1. Emphasizing Breadth vs. Depth of Activity

#### 2.2. Multiple Species Targets

#### 2.3. Single Species Target: Emphasizing Cocktail Breadth of Activity

#### 2.4. Single Strain Target: Emphasizing Depth of Activity

#### 2.5. Sur-Mesure vs. Prêt-à-Porter Targeting

## 3. Cross Resistance

#### 3.1. Minimizing Cross Resistance toward Improving Depth of Activity

^{−7}mutations per bacterium per cell division if 10

^{7}bacteria are targeted. Demerec and Fano [36] measured rates of mutation by Escherichia coli B to phages T1, T3, T4, T5, T6, and T7, which ranged on average for a given phage type from roughly 10

^{−7}to 10

^{−8}mutations per bacterium per cell division. Rates of mutation by E. coli BW25113 to phage U136B appear to be somewhat higher, with a frequency of 10

^{−6}. Rates of mutation by Pseudomonas aeruginosa P14 to resistance to phage DMS3vir were found to be higher still, at nearly 10

^{−4}, though this may be due in part to the large mutational target of the multiple genes necessary for synthesis of the pilus this phage uses as its adsorption receptor [58]. See Silva et al. [52] for a review of phage receptor molecules.

^{−4}, 10

^{−6}, or 10

^{−8}mutations per bacterium per cell division, we would expect a bacterial population of 10

^{9}individual bacteria to contain roughly one hundred thousand (10

^{5}), one thousand (10

^{3}), or ten (10

^{1}) bacterial cells that are phage resistant, respectively. By contrast, rates of E. coli B mutation for resistance to phage T2 are much lower, owing to the independent recognition by this phage of two different primary receptor molecules displayed by that host strain [59]. Other dual receptor-using phages exist as well [46,60,61,62,63]. Though presumably phages targeting two different receptor molecules for their primary bacterial-attachment process can be useful for phage therapy purposes, for individual phage-bacterium combinations dual receptor recognition does not appear to be common.

^{−6}mutations per bacterium per cell division, then acquiring two such mutations independently should occur at a rate per cell division of 10

^{−6}× 10

^{−6}= 10

^{−12}. The reciprocal of this latter number ideally will exceed the number of bacteria present within an infection. In actuality, however, if rare single mutations conferring cross resistance can still occur [36], then the rate of mutation to both of these example phages may be greater than 10

^{−12}. Nevertheless, the lower the rates of bacterial mutation to resistance to two different phage types, then the more useful a combination of those phages should be toward reducing the potential for bacterial evolution of phage resistance during phage treatments, in this example lower than 10

^{−6}if not necessarily as low as 10

^{−12}.

#### 3.2. Dual Optimization

## 4. Developing Dual Spectrum-of-Activity Cocktails

#### 4.1. Defining A Phage’s Host Range

#### 4.2. Determining Cocktail Breadth of Activity

_{1}”, with the subscript “1” indicating that a cocktail’s depth of activity against each bacterium tested must be greater than or equal to 1 to be counted. In other words, this is the fraction of bacteria hit by at least one phage found in a cocktail. This we define as

_{1}= ([total] − [bacteria not phage impacted])/[total],

_{1}’ approach is how cocktail breadth of activity is generally as well as intuitively determined, though here stated with slightly more formality. Specifically, this is a calculation for the fraction of bacteria tested that are affected by at least one phage making up a given phage cocktail. In Figure 2 we provide a demonstration host-range data set (Panel I) along with breadth of activity (Breadth

_{1}) for various cocktails consisting of different phage combinations (Panel II).

#### Multiple Usages of Numbering, a Potential Source of Confusion

_{n}, Table 2); (e) as a description of a depth of activity of a phage cocktail for a specific bacterial strain (e.g., “Depth = 1”, Figure 1); (f) as a subscript (Breadth

_{1}and True

_{1}) referring to the minimum depth of activity counted also for a cocktail (Table 2); and (g) as the names of the first Equation, the first figure, and the first table. In addition, in terms of binary relationships, a ‘1’ could refer not only to a bacterium being found within a phage’s host range but also whether a given bacterial strain is hit by some minimum number of phages, i.e., as returned by True

_{1}and True

_{2}, etc. (Table 2). Please, therefore, take care when interpreting numbers and numbering usage.

#### 4.3. Taking into Account Cocktail Depth of Activity

_{1}). Toward greater cocktail depth of activity, however, each targeted bacterial strain instead must be hit by more than one phage (Depth ≥ 2, defining instead a Breadth

_{2}). For personalized phage treatment, i.e., sur-mesure, such a determination can be somewhat trivial. This is because one need identify only, e.g., two phages that target a single, specific bacterial pathogen.

_{2}”:

_{2}= ([total] − [bacteria impacted by less than 2 phages])/[total],

_{2}referring to depths of activity of ≥2).

_{2}) can be replaced with “less than 3” (defining a Breadth

_{3}), etc. The fraction of bacteria in our hypothetical panel impacted by at least two phages, for different phage combinations making up cocktails, is presented in Panel III of Figure 2 vs. as impacted by at least 1 phage (Panel II). In order to reduce the effort required to do breadth and depth calculations using Equation (2) as well as Equation (1), especially when working with large host-range data sets, we have developed an online, JavaScript-based, “Phage cocktail optimizer” calculator [31].

#### 4.4. Additional Observations

_{2}vs. Breadth

_{1}. This combination of greater breadth given lesser depth should represent a trend, and this is because by increasing depth we are imposing an additional constraint on a cocktail’s functionality. Indeed, as a general rule the breadth of activity of a cocktail possessing a greater depth of activity should never be greater than the breadth of activity of a cocktail possessing a lesser depth of activity. For example, breadth at a Depth such as of ≥1 will always include, e.g., those bacteria for which Depth instead is ≥2, but not vice versa. Based on observation of randomly generated host-range data sets (not presented), these suppositions appear to be born out. In short, and not unexpectedly, it should be harder to hit the same number of bacteria with two different phage types than with just a single phage type.

## 5. Taking Cross Resistance into Account

_{2}; Equation (2)). Limiting cocktails only to phages found in different cross-resistance groups, however, may be too constraining for cocktail development toward enhancement solely of breadth of activity (Breadth

_{1}). This section therefore elaborates on Equation 2-type calculations to assess the breadth of activity of individual cocktails that consist of mixtures of phages which are found in various combinations of both different and the same cross-resistance groups (Figure 3).

#### 5.1. Breadth_{2} Calculations for All Phage Combinations

_{n}values. Here, the subscript ‘n’ is the name of a single, tested bacterial strain, e.g., ‘14′ from Figure 2, panel I. The total number of Depth

_{n}calculations in turn would be equal to the number of phage combinations multiplied by the number of bacterial strains tested. For Figure 3, there would be a total of eight combinations of phages, i.e., ‘sub-cocktails’, consisting of a c e, a d e, a c f, a d f, b c e, b c f, b d e, and b d f. From Figure 2, panel I, there would a total of 45 strains. Thus, the total number of individual Depth

_{n}calculations in the case of Figure 3 could be 45 × 8 = 360, e.g., Depth

_{11}(a c e), Depth

_{17}(b d f), etc.

_{14}for those three different phages together as being equal to 2, as only two of those three phages hit the tested bacterium. Thus, in this example, Depth

_{14}(a c f) = 2. The next step involves determining whether at least one of the possible combinations of phages sourced from different cross-resistance groups results in a Depth

_{n}of at least 2 also for each bacterial strain tested, e.g., such as is the case for Depth

_{14}(a c f). We describe this latter step as representing a ‘True

_{2}’ function, which returns a binary value of 1 if at least one Depth

_{n}value for a given tested bacterial strain is equal to at least 2 (where the latter is the meaning of the ‘2’ subscript in True

_{2}). The number of True

_{2}positive results (that is, True

_{2}= 1) across all bacterial strains tested, divided by the total number of bacterial strains tested (n), defines this more elaborately determined Breadth

_{2}value, which in this case we call Breadth

_{2}(a b c d e f). We illustrate the first steps in how to do this in Figure 4, again using the hypothetical six-phage cocktail introduced in Figure 3.

#### 5.2. Example of Breadth_{2} Calculations Based on Limited Host-Range Data

_{n}(x y*) is set to 2 if both phages x and y* hit bacterial strain n; Depth

_{n}(y* z) is also set to 2 if both phages y* and z hit bacterial strain n; and True

_{2}[Depth

_{n}(x y*), Depth

_{n}(y* z)] returns a value of 1 for a given bacterial strain, n, if either Depth

_{n}(x y*) or Depth

_{n}(y* z)—and thereby activity depth for the combination of phages x, y*, and z for that bacterial strain—is equal to at least 2 (which, as noted, is the meaning of the ‘2′ subscript in both True

_{2}and Breadth

_{2}). This maximum calculated depth of activity for an individual bacterial strain—as equal in this example to Depth

_{n}(x y*) or Depth

_{n}(y* z), or both—we describe, as noted, as a cocktail’s ‘activity depth’ for that strain.

_{2}(x y* z) is calculated, whether by Equation (2) or using Equation (3), across all bacterial strains tested. A given cocktail thus has a specific, determinable activity depth for each individually-tested bacterial strain, which is a function of the number of different cross-resistance groups the phages hitting a given bacterium are a part of, while Breadth

_{2}is the fraction of bacteria tested for which a cocktail has an activity depth of at least 2. In Figure 5, we provide an example of how activity depth is calculated for the cocktail consisting of phages x, y*, and z (resulting in the bottom row) and how Equation (3), that is, Breadth

_{2}(x y* z), may be solved (resulting in the far-right value, also in the bottom row).

_{1}(x y* z) is also calculated, as 7/10 or 70% (second to last column, bottom row). This is solved as,

_{1}[Depth

_{n}(x y), Depth

_{n}(y z)] returns a 1 if either Depth

_{n}(x y*) or Depth

_{n}(y z) is equal to 1 or greater. Therefore, {${{\displaystyle \sum}}_{1}^{n}$ True

_{1}[Depth

_{n}(x y*), Depth

_{n}(y* z)]} is equal to {[total] − [bacteria not phage impacted]} as found in Equation (1). Breadth

_{2}(x y* z), by contrast, is calculated in Figure 5 and found to be equal to only 3/10, or 30% (last column, bottom row). This disparity between Breadth

_{1}(x y* z) (= 70%) and Breadth

_{2}(x y* z) (= 30%) is qualitatively similar to how calculated Breadth

_{2}values (panel III) are somewhat smaller than calculated Breadth

_{1}values (panel II), both as presented in Figure 2.

#### 5.3. Example of Breadth_{2} Calculations Based on More Extensive Cross-Resistance Data

_{2}calculations [32]. To further illustrate both such calculations and their results, we have taken the host-range data presented in Panel I of Figure 2, arbitrarily divided it into four cross-resistance groups, and then applied this app to the data. The resulting analysis is presented in Table S1 (Supplementary Materials) and also illustrated in part in Figure 6.

_{2}for that cocktail visually by using Table S1, which in part is summarized in Figure 6. This is because the analysis consists simply of determining whether at least one phage that is found within each cross-resistance group hits the targeted bacterial strain. One then counts the number of cross-resistance groups that contain at least one such phage and the resulting sum is equal to the activity depth for a given bacterial strain. To obtain Breadth

_{2}, one then divides [number of bacterial strains with activity depth ≥ 2] by [all bacterial strains tested]. This, as a reminder, is the fraction of bacterial strains hit by least two phages sourced from different cross-resistance groups, and therefore for which the potential for evolution of phage resistance is reduced relative to the total number of bacterial strains targeted. The calculation is also essentially a restatement of Equation (2) while taking into account cross resistance, and gives a result that is equivalent to that provided by Equation (3). Similar calculations can be made from this data set (Table S1) to obtain a Breadth

_{3}or Breadth

_{4}. Further discussion of these calculations can be found in Section S1 (Supplementary Materials).

_{1}will tend to be larger than Breadth

_{2}and that Breadth

_{2}will tend to be larger than Breadth

_{3}, etc. The general approach to calculating phage cocktail breadth of activity in light of depth of activity, taking cross resistance into account, will however remain the same when analyzing real host-range and cross-resistance data.

#### 5.4. Complications Involving Cross Resistance

## 6. Discussion

## 7. Conclusions

## Supplementary Materials

_{2}calculations based on more extensive cross-resistance data; Table S1: Depth of activity against various bacterial strains, taking cross resistance into account; S2: www.phage-therapy.org/calculators/cocktail_optimizer.html; S3: www.phage-therapy.org/ scripts/cocktail_optimizer.js; S4: www.phage-therapy.org/calculators/xresistance_avoider.html; S5: www.phage-therapy.org/scripts/xresistance_avoider.js.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**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 “Breadth

_{1}”, 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 ‘Breadth

_{1}’. These Breadth

_{1}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 ‘Breadth

_{2}’ 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 Breadth

_{2}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 Breadth

_{2}(a b c d e f) = { ${{\displaystyle \sum}}_{1}^{n}$ True

_{2}[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 True

_{2}returns a value of 1 if any Depth

_{n}value for a tested bacterial strain is a 2 or higher. Alternatively, True

_{2}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. Breadth

_{2}(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, Breadth

_{2}(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 Breadth

_{2}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 = Breadth

_{1}, e.g., 60% for the top row) or instead by at least two phages from different cross-resistance groups (smaller percentage = Breadth

_{2}, 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 Breadth

_{1}(x y* z), simply count the fraction of 1s or higher found across the bottom row, i.e., as per the True

_{1}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 Breadth

_{2}(x y*) or Breadth

_{2}(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., Breadth

_{2}(x y*) = 2/10 or 20%. To calculate Breadth

_{2}(x y* z), as based on Equation (3), do the same, i.e., count the fraction of 2s or higher as per the True

_{2}function, but for the bottom row only. Thus, Breadth

_{2}(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.

Cocktail Properties: | Single Strain | Single Species | Multiple Species |
---|---|---|---|

Targeting single patient? | Yes | No ^{1} | No ^{1} |

Breadth of activity goal? | Low | Medium | High |

Depth of activity goal? | High (ideally) | Not necessarily high | Not necessarily high ^{2} |

Prêt-à-porter ^{3} as goal? | Not necessarily | Yes | Yes (potentially ^{1}) |

Sur-mesure ^{4} as goal? | Yes | No ^{1} | No ^{1} |

Personalized medicine? | Yes | No ^{1} | No ^{1} |

Minimizing Resistance? | Readily achievable | Partially achievable ^{5} | Difficult with breadth |

Cross resistance an issue? | Yes (if depth is goal) | Yes (if depth is goal) | No (for between species) |

^{1}: Unless for treatment of mixed infections consisting of multiple strains of the same species or multiple species;

^{2}: “High” meaning > 1, i.e., two or more phages together impacting a substantial fraction of bacteria targeted;

^{3}: Idiomatically meaning ‘off the shelf’, i.e., the typical goal for phage cocktails for commercial use;

^{4}: Meaning ‘custom made’, e.g., a formulation for personalized medicine cocktail use;

^{5}: Such as a large fraction of bacterial strains being targeted by more than one phage making up a cocktail.

Notation | Usage | Units | Meaning |
---|---|---|---|

“Hit” | Term | NA | Referring to a bacterium being found within a phage’s host range |

Sub-cocktail | Term | NA | Referring to a subset of the phage types making up a cocktail, especially where none of that subset of phage types are found in the same cross-resistance groups; sub-cocktails are used in Depth_{n} calculations toward activity depth determinations |

0 | Number | NA | Refers to a binary output meaning ‘false’ as well as a description of depth of activity, in the latter case meaning that a bacterium is not hit by a given phage or is hit by no phages found in a cocktail |

1 | Number | NA | Refers to binary outputs meaning ‘true’, as well as a description of depth of activity, in the latter case meaning that a bacterium is hit either by only a single phage found in a cocktail or by at least one phage; we use ‘1’ also as a stand-in for the name of a bacterial strain |

Letters, lower case | Abbreviations | NA | Phage names, e.g., a, b, c, … j, k, l as designating columns in Figure 2, panel I |

Numbers | Abbreviations | NA | Bacterial strain names, e.g., 1, 2, 3, … n as designating rows in Figure 2, panel I |

Breadth_{1} | Variable | Fraction of bacteria tested | Breadth of activity of a phage cocktail without taking depth of activity into account; fraction of bacteria hit by at least one phage in a phage cocktail; see Equations (1) and (4) |

Breadth_{2} | Variable | Fraction of bacteria tested | Breadth of activity of a phage cocktail taking depth of activity into account; fraction of bacteria hit by at least two phages found in different cross-resistance groups and thus more limited in their potential to evolve phage resistance; see Equations (2) and (3) |

Breadth_{3} | Variable | Fraction of bacteria tested | Same as Breadth_{2} except fraction of bacteria hit by three or more phages; Breadth_{4} would be by four or more phages; the general case we indicate as Breadth_{x} |

n | Variable | Bacteria | A given tested bacterial strain or total number of tested bacterial strains |

Activity depth | Variable | Number of phages of a cocktail | Number of phage cross-resistance groups that hit a given, tested bacterial strain, e.g., equal to 2 if hit by phages from two different groups as based on Depth_{n} calculations made for all sub-cocktails; see Figures 5 and 6 and Table S1 |

Depth_{n} | Variable | Number of phages in a sub-cocktail | Depth of activity impacting a given, tested bacterial strain, n, that is associated with a specific sub-cocktail of phage types, e.g., phages a, c, and f, and thus Depth_{n} (a c f); can be used in Breadth_{x} determinations; see Equations (3) and (4) and Figures 4 and 5 |

True_{1} | Function | Binary (0 or 1) | Returns a value of 1 for a given tested bacterial strain if at least one phage in a cocktail hits that bacterial strain; can be used in Breadth_{1} determinations; see Equation (4) and Figure 5 |

True_{2} | Function | Binary (0 or 1) | Returns a value of 1 for a given tested bacterial strain if at least two phages in a cocktail from different cross-resistance groups hit that bacterial strain; can be used in Breadth_{2} determinations, i.e., see Equation (3) and Figures 4 and 5; True_{3} is for at least three phages, etc. |

x y z | Notation | NA | Description of the phages making up a cocktail, in this case phages x, y, and z; no indication of phage cross-resistance grouping is explicitly provided by this notation; commas between letters are implied but are not shown in order to reduce clutter |

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Abedon, S.T.; Danis-Wlodarczyk, K.M.; Wozniak, D.J.
Phage Cocktail Development for Bacteriophage Therapy: Toward Improving Spectrum of Activity Breadth and Depth. *Pharmaceuticals* **2021**, *14*, 1019.
https://doi.org/10.3390/ph14101019

**AMA Style**

Abedon ST, Danis-Wlodarczyk KM, Wozniak DJ.
Phage Cocktail Development for Bacteriophage Therapy: Toward Improving Spectrum of Activity Breadth and Depth. *Pharmaceuticals*. 2021; 14(10):1019.
https://doi.org/10.3390/ph14101019

**Chicago/Turabian Style**

Abedon, Stephen T., Katarzyna M. Danis-Wlodarczyk, and Daniel J. Wozniak.
2021. "Phage Cocktail Development for Bacteriophage Therapy: Toward Improving Spectrum of Activity Breadth and Depth" *Pharmaceuticals* 14, no. 10: 1019.
https://doi.org/10.3390/ph14101019