Determine Phage Adsorption Rate Constants from Free-Phage Decline Data
by Stephen T. Abedon Ph.D. (abedon.1@osu.edu)
phage.org | phage-therapy.org | biologyaspoetry.org | abedon.phage.org | google scholar
Jump to: 📊 Rate Constant Calculator | 📂 Examples | 🔄 Unit Converter & Visualizer | ⚗️ k and Killing | 📖 Background & Methods | 🧮 More Calculators
adsorption.phage.org · Abedon’s Books · DOI: 10.5281/zenodo.21132369
How can I improve this page? contact: adsorption@phage.org
Click to upload or drag & drop a spreadsheet here
Accepts .xlsx, .xls, .csv, .tsv, .txt — needs at minimum a time column and a free-phage titer column
| # | Time | Free Phage Titer (PFU/mL) | Exclude from fit? | Ignore row? | Delete row |
|---|
The datasets below are digitized or obtained from published adsorption experiments. Click Load into Calculator on any card to transfer the data directly to the Rate Constant Calculator tab, where it will be graphed and analysed automatically. No judgement is made here about data quality or curve shape — all interpretation is left to the calculator and to you.
A rapid means of adsorption rate constant assessment can be achieved by measuring
rates of bacterial killing at high phage concentration. The assay assumes an
efficiency of plating (EOP) near 1 — that is, each phage-infected bacterium
gives rise to a plaque, so CFU decline reflects adsorption events directly.
At P₀ = 5×10⁷ PFU/mL and B₀ = 10⁷ CFU/mL over 5 minutes:
A measurable decline in bacterial numbers at these conditions is
indicative of an adsorption rate constant in the range of roughly 10⁻⁹ mL min⁻¹
or greater. No detectable decline suggests k ≪ 10⁻⁹ mL min⁻¹ (see EOP note below).
Plating note: Starting at B₀ = 10⁷ CFU/mL, a 10⁵-fold dilution
gives ~100 CFU/plate. At P₀ = 5×10⁷ PFU/mL the same dilution
leaves ~500 phages/plate, which should have no post-plating impact on colony counts.
EOP note: If EOP < 1, you cannot be sure of the killing phage
titer and therefore cannot reliably assess adsorption rates from bacterial killing.
Low EOP with high killing capacity will result in an overestimation of adsorption rates,
since the perceived phage titer (based on plaque counts) is lower than the actual
killing titer. At the extreme, a killing-positive but replication-negative phage
has an EOP approaching zero but killing ability upon adsorption approaching 100%,
resulting in a measured k that approaches infinity.
Enter k and initial bacterial concentration below to see predicted survival curves at various phage concentrations. Use these to choose P₀ for your experiment and to estimate how much dilution is needed before plating.
To ensure reasonable accuracy, diluting must take place immediately at specific time points, such as at 5 min. The zero-min time point can be taken from a phage-free culture, making sure to take into account any dilution that would result from subsequent phage addition. To minimize the impact of diluting errors, determine the zero-min time point using at least three dilution series. For other time points, perform an initial 100-fold dilution to stop adsorptions and then perform multiple subsequent parallel dilutions (many dilution series rather than many platings from a single dilution series).
Dilute into buffer so as to temporarily halt bacterial replication. Consider using median rather than mean determinations for titer estimations (Abedon and Katsaounis, 2021 — 10.1007/978-3-319-41986-2_17).
It is also possible to separate free phages from bacteria via centrifugation (filtration is not recommended since it is the bacteria that remain on the filter) or to preferentially inactivate free phages by various means. Note, though, that phage-infected bacteria will still release new phages on plates, though at ~100 bacteria per plate, that impact should be negligible.
Once you have measured B(t) at several time points, enter them below to calculate k. Do not include t=0 — B₀ is the reference value entered above.
| Time | B(t) (CFU/mL) | ln(B/B₀) | k from point (mL min⁻¹) |
|---|
The rate of phage adsorption to bacteria is governed by mass-action kinetics: the instantaneous rate at which free phages are lost from suspension is proportional to the product of phage concentration (P), bacterial concentration (N), and the adsorption rate constant (k). This gives the differential equation dP/dt = −kNP. When N is held approximately constant — as in a well-designed short adsorption assay — this integrates to the exponential decay expression used throughout this calculator.
The units of k are mL min⁻¹ (equivalent to cm³ min⁻¹). This reflects a "clearance" perspective: k describes the volume effectively swept clear of free phages by a single bacterium per unit time. Multiplying by N gives the first-order rate constant for free-phage loss (units: min⁻¹), and the reciprocal 1/(kN) is the mean free time — the average time a phage spends searching before it adsorbs.
Note that the rate at which an individual phage finds bacteria is determined by k × N, while the rate at which an individual bacterium acquires phages is determined by k × P. These two perspectives on the same constant are relevant to different practical questions — the former to free-phage clearance in adsorption assays, the latter to phage therapy dosing.
From the physics of diffusion-driven particle collisions, k can be decomposed as:
where S is a measure of bacterial target size (proportional to cell radius R, such that S = 4πR), C is the virion diffusion constant (larger virions diffuse more slowly; higher medium viscosity reduces C), and f is the efficiency of adsorption given collision — the probability that a phage–bacterium encounter actually results in irreversible attachment. The value of f reflects the density and affinity of phage receptor molecules on the bacterial surface.
In practice, k therefore tends to be larger for phages infecting bigger bacteria, for smaller (faster-diffusing) virions, and for phages with high receptor affinity. Measured values span roughly 10⁻⁷ to 10⁻¹¹ mL min⁻¹ across different phage–host pairs.
Because phage loss is exponential, plotting phage titers against time on a linear y-axis produces a sharply falling curve that quickly flattens near zero. On such a linear-linear plot it is nearly impossible to assess whether the decline is truly exponential, to determine the slope accurately, or to detect a change in adsorption rate. Plotting the same data with a logarithmic y-axis (semi-log or log-linear plot) converts the exponential decay into a straight line. The slope of that line is −kN, from which k follows directly after dividing by N. Non-linearities — whether from bacterial growth, virion release, phage aggregation, or a biphasic adsorption process — are far more visible on the semi-log scale. Despite this, linear-linear graphing remains common in the literature and is one of the most frequently cited methodological errors in adsorption studies.
Not all phage populations adsorb at a single constant rate. A biphasic adsorption curve arises when a fraction of phages adsorbs rapidly while the remainder adsorbs more slowly — or not at all. On a semi-log plot this appears as an initial steep linear decline followed by a shallower (or flat) second phase. On a linear-linear plot the two phases may be nearly invisible, making semi-log presentation critical for detecting this phenomenon.
Possible causes include phage population heterogeneity (e.g., a fraction that has lost tail fibers), a subpopulation of resistant or non-susceptible bacteria, reversible phage aggregation, or saturation of bacterial receptor sites at high multiplicities. The Load Biphasic Example button in Step 2 loads a simulated dataset illustrating this pattern, based on the example values used by Abedon (2023) (k dropping from 2.5 × 10⁻⁹ to 2.5 × 10⁻¹⁰ mL min⁻¹ at a breakpoint). When analyzing biphasic data, restrict your regression to the initial linear phase and exclude later points manually using the checkboxes.
R² (the coefficient of determination) equals the square of the Pearson correlation coefficient r: R² = r². The Pearson r ranges from −1 to +1 and measures the strength and direction of the linear relationship between ln(P) and time t; R² then measures the proportion of variance in ln(P) explained by that linear relationship, ranging from 0 to 1. For a declining adsorption curve, r will be negative, so it is conventional to report R² rather than r. An R² of 0.98 corresponds to r = −0.990; an R² of 0.99 corresponds to r = −0.995. Values below about 0.98 suggest the data depart meaningfully from a straight line on the semi-log plot.
The central experimental requirement for an adsorption assay is the ability to measure free-phage titers independently of phages that have adsorbed to bacteria. Three approaches are widely used, each with specific limitations:
Regardless of method, assay duration should generally not exceed 10 minutes. Longer assays allow bacterial growth (which increases N and accelerates adsorption over time, causing downward curvature on the semi-log plot) and risk virion release from lysing cells (which artificially inflates free-phage counts, causing upward curvature).
Much of the information in this calculator can be found in the following references. Please cite this tool as: Abedon, S.T. (2026). Adsorption Rate Calculator. adsorption.phage.org. DOI: 10.5281/zenodo.21132369.