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
Version 2026.04.07
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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 |
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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). Phage Adsorption Rate Calculator. adsorption.phage.org.
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