Using Poisson Distributions to Titer Phages and Predict Phage Therapy Outcomes
by Stephen T. Abedon Ph.D. (abedon.1@osu.edu)
phage.org | phage-therapy.org | biologyaspoetry.org | abedon.phage.org | google scholar
Version 2026.04.07
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The killing titer calculation assumes complete adsorption. Enter the phage adsorption rate constant (k) and adsorption duration (t) to estimate the fraction of phages that actually adsorbed. The adsorption rate constant for your phage can be determined at adsorption.phage.org.
Enter a known starting phage titer (P0) and bacterial concentration (N0) to predict expected bacterial survival after complete adsorption. Optionally enter your observed outcome to receive a diagnostic interpretation of what happened in your experiment — whether killing was consistent with passive treatment, fell short of prediction, or exceeded it.
Key assumption: calculations assume all bacteria are equally accessible to phage adsorption (no biofilm shielding, no spatial structure). For adsorption-completeness checking, use the optional k/t inputs below or the Phage Adsorptions Calculator.
Enter a known starting phage titer and bacterial concentration to predict expected bacterial survival — then optionally compare with your observed result to interpret what happened in your experiment.
The prediction above assumes all phages adsorb. Enter k and t to estimate whether adsorption was likely complete, and to compute the bacterial half-life and phage half-life at your conditions. Find your phage's adsorption rate constant at adsorption.phage.org.
In phage therapy pharmacology, the killing titer is a direct measure of phage antibacterial efficacy — specifically, the phage concentration required to achieve a given degree of bacterial killing. Three properties of phages together make killing titers a particularly meaningful pharmacodynamic quantity (Abedon & Thomas-Abedon, 2010): (1) single-hit killing kinetics — a single phage adsorption event is sufficient to kill a bacterium; (2) permanent adsorption — once a phage adsorbs to a bacterium, it does not dissociate, meaning every adsorption event is a committed killing event; and (3) multiple adsorption — more than one phage can adsorb to a single bacterium, though only one adsorption is necessary for killing. Together these properties mean that the fraction of bacteria surviving phage treatment is determined entirely by the Poisson probability of escaping all adsorption events — which is what this calculator computes.
A useful therapeutic benchmark is the "multiplicity of 10" rule (Abedon & Thomas-Abedon, 2010): an adsorbed MOI of approximately 10 phages per bacterium reduces a population of 106 bacteria to roughly 45 survivors (e−10 × 106 ≈ 45 × 10−6 × 106 ≈ 45), which approaches practical eradication especially when combined with immune clearance. Achieving this requires knowing the actual adsorbed phage concentration — which is exactly what a killing titer measures. As a worked example: if one begins with 106 bacteria and ends with 101 surviving CFUs, the actual MOI attained must have been −ln(101/106) ≈ 11.5 phages per bacterium. If 108 phages were added to achieve this, then that would mean that only about 11.5% of those phages are actually adsorbed — the rest contributed nothing to killing.
Thus, and crucially, only adsorbed phages contribute to killing. The killing titer therefore equals the adsorbed-phage titer, but which equals the added-phage titer only if all of the added phages adsorb. When adsorption is incomplete: killing titer = adsorbed titer < added titer. At its most concise: the killing titer is simply the number of adsorbed phages per mL, determined indirectly in terms of the number of bacteria that are killed (Abedon & Thomas-Abedon, 2010; Abedon & Katsaounis, 2018).
The relationship works in both directions. Given an observed Before and After CFU count, the killing titer is calculated as shown above. But given a known starting titer (P0) and a starting bacterial density (N0, in CFU/mL), the expected fraction of bacteria surviving can equally be predicted: survival fraction = e−P0/N0. When that starting titer equals the killing titer (P0 = PK), the prediction is exact by definition. For example, a starting titer of 7 × 107/mL applied to 108 bacteria/mL predicts a surviving fraction of e−0.7 ≈ 0.5, i.e., 50% of bacteria killed — and conversely, observing 50% killing from 108 bacteria/mL gives a killing titer of −ln(0.5) × 108 = 0.7 × 108 = 7 × 107/mL (Abedon, 2023).
Killing titer calculations can be used to predict the maximum possible impact of specific phage titers on bacterial populations, and also to assess the effectiveness of phage treatments given achievement of those titers in situ (Abedon & Thomas-Abedon, 2010; Abedon, 2023). The key diagnostic comparisons are:
If the calculated killing titer is less than the actual starting phage titer: phages are probably not efficiently reaching or bactericidally infecting target bacteria. The ratio of killing titer to starting titer gives an estimate of what fraction of phages actually adsorbed and killed. This is the most common situation when adsorption is incomplete, bacterial density is low, or the assay duration was too short.
If the calculated killing titer exceeds the actual starting phage titer: either (1) the phage suspension contains more killing-competent virions than standard plaque-based titer methods account for — which, as discussed above, is the expected situation when killing titer is compared to plaque titer rather than to a total virion count — or (2) phages replicated in situ during the assay, increasing the effective phage concentration beyond the original inoculum (and do not neglect the potential for multiplicity reactivation for UV-treated phages; see references below). The latter is a diagnostic indicator that some degree of active phage treatment occurred, that is, as involving also in situ phage replication, (Abedon & Thomas-Abedon, 2010; Abedon, 2023).
Killing titers are useful for any phage that can kill bacteria upon adsorption even if it is unable to complete a replication cycle under assay conditions. Two particularly important applications are: (1) UV-inactivated phages, which retain the ability to adsorb and kill (via DNA injection and host disruption) but are unable to replicate because their genome has been damaged; and (2) phages engineered to kill bacteria, such as those carrying phage-delivered lethal cargo genes that kill upon infection but prevent or eliminate productive replication (Bull & Regoes, 2006). Killing titers are also useful for obligately lytic phages being assayed under conditions that prevent replication — for example when a suitable indicator host for plaque formation is unavailable.
Killing titers can also serve as a cross-check on plaque-based titer determinations. If killing titer and plaque titer agree closely, this confirms adsorption went to completion during the killing titer assay. If the plaque-based titer significantly exceeds the killing titer, adsorption was likely incomplete — only a fraction of phages adsorbed during the assay period. Conversely, if the killing titer exceeds the plaque titer, this can indicate either that phages replicated during the killing titer assay (inflating the apparent phage concentration), or — importantly — that the phage has a high bactericidal ability but a low efficiency of plating (EOP): it kills bacteria effectively upon adsorption but forms plaques poorly or not at all on the indicator host used, such as the UV-inactivated or non-replicative killer phage particles discussed above. In this case the killing titer is the more meaningful measure of phage activity.
Killing titer calculations are generally not as reliable for temperate phages. At the multiplicities of infection typically used in a killing titer assay, a substantial fraction of infections by a temperate phage may result in lysogeny rather than killing — the bacterium survives as a lysogen and continues to form colonies. This means that the surviving CFU count after phage treatment reflects not only bacteria that escaped adsorption, but also bacteria that were adsorbed and lysogenized rather than killed. The Poisson-based back-calculation cannot distinguish between these two outcomes, and the resulting killing titer will underestimate the true phage concentration by an amount that depends on the lysogenization frequency — which itself varies with MOI and host physiological state. For temperate phages, plaque assays using conditions that favor lytic development (or instead using molecular quantification methods) are generally more appropriate.
The central assumption of the killing titer calculation is that all — or very nearly all — of the phages in the sample adsorb to bacteria before the surviving CFU count is taken. When this holds, the number of phages that adsorbed equals the number of phages present, and the Poisson-based back-calculation gives the correct phage concentration.
In practice, adsorption is a second-order process governed by the phage adsorption rate constant (k) and the bacterial CFU concentration (N). The half-life of free phages in the presence of bacteria is approximately ln(2) / (k × N). At 108 CFU/mL with a typical k ≈ 2 × 10−9 mL/min, the half-life is roughly 3–4 minutes, and > 95% adsorption is achieved in about 15–20 minutes. At lower bacterial CFU concentrations the required duration of adsorption increases dramatically — by orders of magnitude — so the assumption of complete adsorption should not be made casually when bacterial numbers are low.
CFU counts do not necessarily correspond one-to-one with individual bacteria. Bacterial clumps — groups of cells that remain physically associated — plate as a single colony-forming unit even though they contain many cells. Each CFU, whether a single cell or a clump, is the relevant unit of Poisson-distributed phage adsorption events for the purposes of this calculation: a single successful adsorption event anywhere on a clump can be sufficient to eliminate it as a CFU, because phages can replicate following adsorption to one cell and spread to kill the rest of the clump. A clump therefore behaves as a single killable target, just as a solitary bacterium does, and the Poisson calculation over CFUs is internally consistent as long as both Before and After measurements are in CFU units. This logic applies specifically to phages that both adsorb and replicate; for non-replicating phages (UV-inactivated, engineered killers), the same argument can fail to hold (that is, with UV-inactivated phages one must also take into account the process of multiplicity reactivation, which with multiplicities of infection exceeding 1 can convert multiple adsorbing non-replicative phages into a single replicative phage).
A complication arises when clumps are not uniform in size. A larger clump presents more adsorption targets per CFU — more cell surface, more phage receptors — and is therefore more likely to adsorb at least one phage than a smaller clump or a single cell. This means the effective adsorption rate per CFU is not uniform across the population: large clumps as CFUs are killed more efficiently than small ones or single cells. As a result, the survival fraction is no longer a clean single-component Poisson decay, but a mixture of Poisson processes with different rates. In practice, at moderate MOI and with reasonably well-dispersed cultures, this effect is likely small. However, in highly clumped cultures — for example, certain staphylococci or streptococci — clump-size heterogeneity may introduce a meaningful deviation from the Poisson model, and the calculated killing titer should be interpreted with corresponding caution (Abedon & Thomas-Abedon, 2010).
Note also that the killing titer calculated in this way is expressed in units of phages per CFU — which may represent phages per cell in a well-dispersed culture, but phages per clump in a clumped one. If the goal is to determine the absolute virion concentration in phages per unit volume independent of clump size, and if the average number of cells per CFU is both known and uniform, a correction can in principle be applied.
A natural question is: does it really matter whether 90%, 95%, or 100% of phages adsorb? The interactive tool below lets you explore this quantitatively. Enter the true phage concentration and a bacterial CFU concentration, then click the button to generate a table showing how the calculated killing titer changes across a range of adsorption efficiencies.
Killing titer occupies a specific position in the broader hierarchy of methods for quantifying phage particles (Abedon & Katsaounis, 2018). From most inclusive to most restrictive, the ordering is:
Killing titer therefore measures a subset of all phage particles — specifically, those that can adsorb and kill bacteria — but a superset of phages capable of productive infection. A phage that adsorbs and kills but cannot complete a productive infection cycle (e.g., a UV-inactivated phage with intact tail fibers and DNA injection machinery) will be counted by a killing titer assay but will not titer but not by plaque assay. This is precisely why killing titers are the method of choice for UV-inactivated and non-replicative engineered phages.
An important implication is that killing titer exceeding plaque titer is the expected and normal situation, not an anomaly requiring explanation (Abedon & Katsaounis, 2018). A phage stock that gives a killing titer of 109/mL but a plaque titer of only 108/mL simply contains a population in which 10× more particles can kill than can form plaques — which is biologically unremarkable and may reflect, for example, a subpopulation of phages with DNA damage that preserves adsorption and killing ability but prevents replication. Only if the killing titer is lower than the plaque titer is something unexpected occurring — incomplete adsorption during the assay is the most likely explanation, and the ratio of killing titer to plaquing titer in this case can give an estimate of adsorption efficiency.
A persistent source of error in phage work is the conflation of multiplicity of addition (the ratio of phages added to bacteria present) with multiplicity of infection (the ratio of phages that actually adsorbed to bacteria), an issue that has been appreciated by the phage community since at least the 1940s. Only adsorbed phages contribute to killing, and only MOI — not multiplicity of addition — correctly predicts bacterial survival via the Poisson formula (Abedon & Katsaounis, 2018). At lower bacterial densities (e.g., < 106/mL) and shorter incubation times, multiplicity of addition can wildly exceed actual MOI, leading to large overestimates of expected killing.
The killing titer calculation in this tool bypasses this problem by working backwards from observed bacterial survival — it infers MOI from the data rather than assuming it from the ratio of phages added. This is one of the key strengths of the killing titer approach.
At low MOI, most bacteria are either uninfected or singly infected — but the fraction of infected bacteria that are multiply infected can be substantially higher than the fraction of all bacteria that are multiply infected. This distinction matters for interpreting killing titer assays (Abedon & Katsaounis, 2018).
For example, at MOI = 0.01: only 0.0005% of all bacteria are multiply adsorbed, but approximately 0.5% of infected bacteria are multiply adsorbed — a 1000-fold difference. At MOI = 0.1: only 0.005% of all bacteria are multiply adsorbed, but about 4.9% of infected bacteria are multiply adsorbed. At MOI = 1: 26% of all bacteria but 42% of infected bacteria carry more than one phage. These distinctions matter because multiply adsorbed bacteria still count as a single killing event, meaning adsorptions beyond the first are "wasted" from a killing efficiency standpoint. The Poisson model correctly accounts for this — but it underscores why high MOI does not linearly increase killing efficiency.
Adsorption goes to completion most reliably when: (1) bacterial CFU concentrations are high — generally ≥ 108/mL — so that phage-bacterium adsorptions are frequent; and (2) the duration of adsorption is sufficient for multiple phage half-lives to elapse. The phage adsorption rate constant (k) governs how quickly this occurs: the half-life of free phages is approximately ln(2) / (k × N), where N is the bacterial CFU concentration. For adsorption rate constant values and calculations, see adsorption.phage.org. The Bacterial Half-Life Calculator can help estimate the required duration of adsorption for a given bacterial concentration and adsorption rate constant.
The MOIinput (phages per CFU) can also be used to check whether adsorption is likely complete: use the MOI Calculator to compare MOIinput to MOIactual. When adsorption goes to completion these two values should be approximately equal. Note that this killing titer calculator assumes they are equal regardless of the input values.
The Poisson model underlying the killing titer calculation assumes that all bacteria are equally accessible to phage adsorption — that is, that phages and bacteria are freely diffusing in a well-mixed environment and every bacterium has an equal probability of being adsorbed. This assumption breaks down for bacteria in biofilms (Abedon, 2011). In a biofilm, bacteria are embedded in extracellular polymeric substances, organized into microcolonies, and physically shielded from phage penetration — bacteria deep within the biofilm matrix may be largely inaccessible until overlying bacteria are lysed. The result is that the effective adsorption probability is not uniform across the population, the Poisson model does not apply cleanly, and the killing titer will systematically underestimate the degree of phage-bacterium mismatch.
For biofilm-associated bacteria, killing titer calculations should be interpreted with caution and ideally validated by comparing observed killing with predicted killing at the measured phage concentration and estimated adsorption fraction. Differences between observed and predicted killing nonetheless are informative about the degree to which spatial structure is reducing phage efficacy.
Key practical conclusion: Killing titer error is higher when the fraction of bacteria killed is smaller — i.e., when MOI (multiplicity of infection) is low. At very low MOI the Before and After colony counts are nearly identical, and the small difference between them is dominated by counting noise, making the killing titer estimate unreliable. At very high MOI the After count approaches zero and its relative counting error rises steeply. The optimal MOI is approximately 1.79 (see chart). Note that in killing titer assays it is colonies — not plaques — that are counted: the Before and After plates record surviving CFUs, not plaques.
This can be quantified rigorously: applying error propagation (the delta method) to the Poisson counting error in both the Before and After colony counts gives a theoretical minimum CV (coefficient of variation) for the killing titer as a function of MOI and colony count. See the Methodology tab for the full derivation. The key result is:
The interactive chart below plots CV(KT) against MOI for a chosen After plate count, showing directly how precision varies across the assay range and where the optimum lies.
A useful validation strategy — particularly applicable to phages that are genuinely unable to replicate under assay conditions, such as UV-inactivated or engineered non-replicative phages — is to run the assay at two different adsorption durations (for example, 10 minutes and 20 minutes). If the calculated killing titer is the same at both durations, adsorption was already complete at the shorter time and the result is reliable. If the killing titer increases between the two durations, adsorption was still incomplete at the shorter time, and the longer duration gives a better estimate. This approach does not work for phages that can replicate, because an extended adsorption period would also allow phage progeny to be produced and released — inflating the apparent phage concentration rather than simply reflecting more complete adsorption of the original inoculum.
For quantitative questions about the rate of bacterial killing, see the
Decimal Reduction Time Calculator at
phage-therapy.org.
For minimum phage titers required for a target reduction, see the
Inundative Phage Density Calculator at
The killing titer is derived from the Poisson distribution of phage adsorption events. Because phages adsorb randomly and independently across the bacterial population, the probability that any given bacterium (or CFU) escapes adsorption by all phages follows a Poisson process. The surviving CFU concentration after phage treatment therefore encodes information about how many phages were present — allowing the phage concentration to be back-calculated from the ratio of surviving to initial CFUs.
A useful way to understand the killing titer calculation is as a methodological inversion (Abedon, 2023): whereas the standard Poisson bacteriology calculation predicts bacterial survival given a known starting phage concentration, the killing titer runs the same equation in reverse — it infers the phage concentration from observed bacterial survival. The killing titer is therefore a phage titer determination based on bacterial survival, rather than a prediction of bacterial survival determined by knowledge of initial phage titers. In this sense the killing titer provides an estimate of what phage concentration would have been necessary to achieve the observed degree of bacterial killing — which can then be compared with the phage concentration that was actually present at the start of the experiment.
The Before CFU concentration (NB, where B stands for "Before") is the CFU concentration in the bacterial suspension at the moment phage addition begins — not the CFU concentration measured prior to dilution in the pre-addition bacterial stock. This distinction matters whenever the phage preparation is added as a non-negligible volume relative to the bacterial suspension. That is, the act of adding phages dilutes the bacteria, reducing the CFU concentration before adsorption starts. The correct value therefore is:
The Dilution Helper in the Calculator tab computes this automatically. If phage volumes are very small relative to bacterial volumes (e.g., adding 10 µL phage to 990 µL bacteria), the dilution factor is negligible (0.99×) and can reasonably be ignored. For larger phage volumes the correction becomes important.
The After CFU concentration (A, sometimes written NA) is the surviving CFU concentration measured after an adequate duration of adsorption under conditions that allow phage-bacterium adsorptions to proceed. This concentration represents only the bacteria (or bacterial CFUs) not adsorbed by any phage, and therefore still capable of forming colonies. Bacteria that have been adsorbed but not yet visibly lysed are still counted as killed for this purpose — they no longer form colonies — which is why CFU counts rather than total cell counts are the appropriate measurement.
"…the student of phage should be familiar with the Poisson distribution."
— Mark H. Adams, Bacteriophages (1959), p. 30
If phages adsorb randomly and independently across CFUs, the fraction of CFUs that escape adsorption by all phages follows the Poisson zero term. The applicability of the Poisson distribution to phage adsorption was established by Dulbecco (1949), who demonstrated that the distribution of phage infections across bacteria is indeed Poissonal under standard conditions:
The killing titer PK — the number of adsorbing, killing phages per unit volume — is thus equal to the number of phages per unit volume that must have been present to reduce the CFU concentration from NB to A, assuming all those phages adsorbed. The result is expressed in the same units as NB (e.g., phages/mL if CFUs were in CFU/mL).
One important practical caveat: even under ideal conditions — high bacterial CFU concentrations and adequate duration of adsorption — a small residual fraction of phages will typically fail to adsorb (Abedon, 2023). Complete adsorption in the strict sense is rarely achieved in practice, which means the killing titer is almost always a slight underestimate of the true phage concentration. For most applications this residual is small enough to be acceptable, but it should be borne in mind when precise absolute quantification is required.
If NA = NB, no killing occurred and the killing titer is zero — either no phages adsorbed, the phage concentration was below the detection threshold of the method, or the phages used do not kill bacteria upon adsorption under assay conditions. If NA > NB, the surviving CFU count exceeds the starting count and the killing titer is undefined — the calculation requires that CFUs decrease monotonically relative to the starting value.
When only a fraction f of phages adsorb (0 < f ≤ 1), the effective MOI driving bacterial killing is f × MOIinput. The killing titer PK will therefore recover only a fraction f of the true phage concentration. The Background tab provides an interactive table illustrating this underestimation across a range of adsorption efficiencies.
If adsorption efficiency is known independently, the true phage concentration can be recovered as:
This calculator does not perform this correction, as it assumes f = 1.
Because CFU plate counts follow a Poisson distribution, it is possible to derive a theoretical lower bound on the variance of the killing titer estimate — the minimum variance achievable given only Poisson counting noise, with no pipetting error, dilution error, or biological variability. This bound is useful as a benchmark: it tells you how precisely the killing titer could be determined in principle, and therefore how much of the observed variance in real experiments is attributable to controllable experimental error versus irreducible statistical noise.
Let the observed plate count on the After plate be NA and on the Before plate be NB, with associated plating/dilution factors fA and fB such that A = nA · fA and B = nB · fB. Since nA ~ Poisson(λA) and nB ~ Poisson(λB), their variances equal their means: Var(NA) = NA, Var(NB) = NB. Since A = NA · fA, we have Var(A) = fA² · Var(NA) = fA² · NA = A² / NA, and similarly Var(B) = B² / NB.
The delta method is a standard statistical technique for approximating how error in measured quantities propagates into a calculated result. Here ∂ denotes a partial derivative — informally, the sensitivity of the killing titer to a small change in one plate count while the other is held fixed. If you have not encountered calculus, the key takeaway is simply the final CV² formula, which can be applied directly without understanding the derivation. The killing titer PK = −ln(A/NB) · NB. Taking partial derivatives:
When the experimenter independently adjusts dilutions to achieve target plate counts on both plates — the optimal strategy — nA and nB are independent. In the practically important case where the After plate is the binding constraint (because at high MOI the After count would become very small if both plates used the same dilution), we set nA = nmin (the minimum reliable plate count, typically ~10–20 colonies) and allow NB to be independently optimized. If both plates are then plated at their own optimal dilutions for ~100 colonies each (nA = nB = n):
However, when the After plate count is constrained — because a very high MOI forces NA toward zero unless an inconveniently large plating volume is used — the effective constraint is nA = nmin and nB = nmin · eMOI (the Before count grows exponentially with MOI at fixed dilution). Substituting:
The practical implication is that the precision curve is broad near its minimum: MOI values between roughly 1 and 4 all give CV within about 20% of the optimum. MOI = 0.1 is catastrophically bad — about 10-fold worse CV than optimal. The MOI, Assay Precision section of the Background tab provides an interactive chart of this curve for any chosen plate count — allowing you to visualize how error varies with MOI and to find the optimal MOI for your experimental design.
The CV derivation above assumes unbiased plate count estimates. In practice, two common sources of bias work in opposite directions. Too Numerous To Count (TNTC) plates — where plaques overlap and cannot be individually resolved — introduce a downward bias if TNTC results are simply discarded: eliminating only high values from a data set systematically depresses the mean. Conversely, Too Few To Count (TFTC) results introduce an upward bias if they are discarded: eliminating low values inflates the mean. Both problems are best addressed by using a trimmed mean (removing the top and bottom fraction of values, e.g., 25%) or the median rather than the arithmetic mean — retaining all data points but reducing the influence of extreme values (Abedon & Katsaounis, 2018). The median in particular is robust to a single outlier in either direction, provided n ≥ 3, and is the recommended approach when plate counts span a wide range. In Microsoft Excel: AVERAGE, MEDIAN, and TRIMMEAN (note: TRIMMEAN uses the combined trim fraction, so a 25% trim from each end is specified as 0.5).
This calculator assumes 100% adsorption efficiency and does not correct for incomplete adsorption, phage replication during the assay, or errors in the volume measurements used to determine the starting CFU concentration. It also does not account for the possibility that the phage suspension itself contains bactericidal agents other than phage virions. The user is responsible for ensuring that experimental conditions meet the assumptions of the calculation and for verifying results by comparison with an independent titer method where possible.
Originally posted 2017.03.08 · Stephen T. Abedon · original page
Scientific experiments ideally will be hypothesis-driven, will carry some expectation, generate an observation, and then afford some comparison between the two. If observations and expectations do not coincide, then we reject or reformulate the hypothesis, or instead question the experiment itself. In many cases it is helpful for both expectations and observations to be quantitative, with agreement seen as convergence on a single number. Given how basic these concepts are, it can be surprising when a simple method exists for calculating experimental expectations quantitatively, but which nevertheless does not appear to be generally used. Here I discuss the calculation for killing titers in phage therapy experiments, and consider what information may be gleaned from killing titers in terms of phage therapy pharmacology — in particular, how killing titer determinations can be used to easily distinguish active treatments from passive ones, or either from otherwise inadequate treatments.
Phage therapy, or more generally phage-mediated biological control of bacteria, is the use of bacterial viruses to reduce numbers of pathogenic or otherwise problematic bacteria. The technique is based on the idea that the adsorption of a lytic phage to a bacterium will result in the death of that bacterium. These phages display single-hit killing kinetics (Bull and Regoes, 2006) such that bactericidal interactions are discrete — consisting of the interaction of a single phage virion with a target bacterium — rather than requiring the cumulative action of many antibacterial units.
Given more or less equivalent properties for target bacteria within a population, and equivalent properties for applied phages, the consequence of phages interacting as discrete entities with bacteria is that those adsorptions can be modeled statistically as a Poisson distribution (Dulbecco, 1949). With a Poisson distribution we can readily predict the fraction of bacteria that will have adsorbed no phages, the fraction adsorbed by exactly one phage, the fraction adsorbed by two phages, and so on. Here we are concerned solely with the zero-adsorption category, predicted to equal e−M, where M is the multiplicity of adsorption — the ratio of adsorbed phages to adsorbable bacteria (MOIactual). Passive treatment is phage therapy that is dependent entirely on traditionally dosed phages, with an assumption of no subsequent in situ phage replication.
A killing titer employs the Poisson distribution to estimate the number of bactericidal units within a phage stock. The utility of killing titers is their detection specifically of bactericidal particles rather than plaque-forming units. Thus, the number of virions present within a suspension can be calculated even if those virions are not capable of forming plaques by standard measures.
The assay requires knowledge of bacterial concentrations and demands that all phages successfully adsorb — or, more precisely, that only adsorbed virions are counted towards the killing titer. Ideally without allowing bacteria to substantially replicate over the course of adsorption, one then calculates a final bacterial viable count. The ratio of surviving bacteria to the original bacterial concentration equals e−M — the fraction of bacteria not bactericidally adsorbed. Multiplicity M therefore equals the negative natural log of that ratio.
Thus, if half of the bacteria are killed, then M = −ln(0.5) ≈ 0.69. If the original bacterial concentration was 108/mL, then the killing titer is M × 108 ≈ 6.9 × 107 killing phages/mL. This is precisely what the KT Calculator computes.
Based on these premises, it is possible to add a known quantity of phages to a known quantity of bacteria, measure surviving bacteria, and calculate both an expected killing titer (from the applied phage dose) and an observed killing titer (from the survival data). The PT Calculator performs exactly this comparison. Three outcomes are possible:
This framework provides a simple, quantitative measure of phage performance that allows direct comparison between prediction and observation in any phage experiment. It is important to ensure that all phage adsorptions occur within the experimental environment rather than during subsequent bacterial enumeration (Brown-Jaque et al., 2016).
A more refined expected multiplicity can be calculated without assuming complete adsorption, by incorporating the phage adsorption rate constant k and the duration of phage exposure t. The governing equation (Stent, 1963) is:
This derivation assumes bacterial density is low enough that phage depletion by adsorption is negligible. For higher bacterial densities, more complex formulations are required (Abedon, 2011). The PT Calculator implements this calculation, and adsorption.phage.org provides tools to determine k experimentally.
Using Stent's (1963) reference adsorption rate constant of 2.5 × 10−9 mL/min: applying 108 phages/mL for 10 min gives an expected multiplicity of adsorption of 2.5 × 10−9 × 108 × 10 = 2.5, predicting about 8% bacterial survival. For a food biocontrol scenario with 104 bacteria, 107 phages/mL, k = 10−9 mL/min, and 100 minutes:
If following incubation with phages more bacteria are observed than predicted, phages are failing to reach bacteria or bacteria are replicating. If fewer, either active treatment is occurring or bacteria are being physically removed — a potential confound in experimental design.
To reduce 104 bacteria to fewer than 1 requires a 105-fold reduction, hence M = −ln(10−5) ≈ 11.5. With k = 10−9 mL/min and t = 100 min:
Given real-world uncertainties, higher doses (109/mL or more) may be preferable in practice. The point is that this calculation makes it possible to rationally design experiments — and then compare observations with expectations.
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