🔬 Lysogenization Frequency Calculator

The Model

Single-cell MOI is Poisson-distributed

The single-cell MOI, n, is the number of phages that actually infect one individual cell. Across a population it is Poisson-distributed with mean λ: P(n) = λⁿ·e^−λ ÷ n!. The reported experimental MOI is a nominal value M = (phages added) ÷ (cells present). The Poisson mean that governs infection is not M but λ = a·M, where a is an adsorption-efficiency / measurement factor (0 < a ≤ 1) that corrects for incomplete adsorption and for inaccuracy in measuring phage and cell concentrations. Using M in place of λ overstates the true mean multiplicity. The Poisson distribution itself contains no biology; all of the biology lives in the lysogenization probabilities below.

Reference curves (the Poisson bounds)

The fraction of cells with at least one phage is P(n≥1) = 1 − e^−λ (“every infection lysogenizes”). The fraction with two or more phages is P(n≥2) = 1 − e^−λ − λe^−λ (“every multiple adsorption lysogenizes”). The exactly-one term is P(1) = λe^−λ. On log-log axes at low MOI the single-infection term rises with slope 1 (∝ λ) while the multiple-infection term rises with slope 2 (∝ λ²). This slope difference is the entire diagnostic: any single-phage contribution rides on the λ¹ term and therefore diverges upward from the multiple-adsorption curve as MOI drops. As a rule of thumb, among infected cells the fraction that are multiply infected is approximately λ/2 at low MOI — so at λ = 0.01, singles outnumber multiples roughly 200:1 by population.

Single- versus multiple-infection lysogenization

Let Q_n be the probability that a cell infected by exactly n phages lysogenizes. The observed per-total-cell frequency is f(λ) = Σ_n≥1 P(n)·Q_n. Because the empirical Q_n saturates at n = 2 (see below), this reduces to a two-parameter form: f(λ) = P(1)·q₁ + P(n≥2)·q₂, where q₁ is the fraction of singly infected cells that lysogenize and q₂ is the fraction of multiply infected cells that lysogenize (the n = 2 value, assumed to hold for all n ≥ 2). The multiples-only reference is f_mult(λ) = P(n≥2)·q₂; the ratio of the full model to this reference grows as MOI falls, which is the single-phage signal made quantitative. This tool’s reference values (q₁ = 0.01, q₂ = 0.50) are round approximations of the Geng et al. fit.

Relationship to the Geng et al. Model

Geng et al. (2024) fit the general form f(λ) = Σ_n P(n)·Q_n, allowing a non-zero Q_n at every n rather than assuming a hard two-phage threshold. Their fitted Q_n was found to saturate at n = 2: Q₁ ≈ 0.005–0.009 (single-phage lysogenization across the two runs) and Q₂ = Q₃ = Q₄ ≈ 0.41–0.47. The reduced two-parameter model used here is therefore their result, not a competing one: fixing q₂ as a single saturated value and keeping a separate q₁ for single infections reproduces their fit in the regime where cells with n ≥ 3 are negligibly rare. The round numbers (q₁ = 1%, q₂ = 50%) are the approximations. This tool lets you set q₁ and q₂ directly (assume-and-check) and/or fit them to the data, as Geng did. The fit here minimizes the sum of squared residuals in log space (values span several orders of magnitude), with a soft constraint q₁ ≤ q₂.

Example Dataset

The built-in example is Geng et al. (2024), Fig. 4c, from the paper’s publicly available Source Data file (sheet Fig4c). Two independent runs; host log-phase E. coli MG1655; phage lambda (λ_ts, cI857 bor::kan^R), with lysogens scored by kanamycin resistance via OD growth dynamics. The tabulated MOIs are already adsorption-corrected values (λ = aM), so plotting the Poisson references directly at these λ is apples-to-apples. The lowest one or two points of each run sit near the assay resolution limit and are noisy; the single-phage claim rests on the trend, not on any one point.

References

What the Curves Mean

Data on the multiples-only (dashed) curve everywhere is consistent with no single-phage lysogenization — all lysogeny arising from n ≥ 2. This is roughly the classic Kourilsky (1973) picture: log-phase lysogeny at or below the multiple-adsorption floor.

Data diverging upward from multiples-only toward P(n≥1) at low MOI indicates a non-zero single-phage term (q₁ > 0). This is the Geng (2024) result, consistent with the single-phage inference of Kobiler et al. (2002).

Data above the P(n≥2) “every multiple” bound cannot be explained by multiples at any efficiency — it is direct evidence of single-phage lysogenization, because you cannot exceed 100% of multiples unless singles also lysogenize.

Magnitude. For log-phase lambda, single-phage lysogenization tops out around ~1% or less (Geng q₁ ≈ 0.5–0.9%). In stationary phase the single-phage frequency is far higher (Kobiler ≈ 41%); single-phage lysogenization is a strong function of host growth state, decreasing roughly exponentially with growth rate. This tool’s example is the log-phase regime.

Caveats the Tool Surfaces

Nominal versus effective MOI (the a factor)

Reported MOI is M = phages added ÷ cells; the Poisson mean that matters is λ = aM with a < 1. If a curve is plotted versus nominal M while the references assume a = 1, the Poisson floor is overstated and the single-phage signal is understated. Correcting (a < 1) lowers λ, lowers the floor, and moves the low-MOI data further above it — the adsorption correction helps the single-phage reading. Geng fit a and found it agreed with the theoretically predicted adsorption efficiency. Use the MOI-type control on the Calculator tab to convert nominal M to λ.

The missing MOI = 0 control (the key gap)

To claim single-phage lysogenization from the lowest-MOI points, one must exclude a phage-independent background of the resistance marker — spontaneous marker mutations, pre-existing resistant cells, incomplete selection, carryover. The only clean discriminator is an MOI = 0 well carried through the same selection, which measures that background directly. That control is not reported in Geng. The optional background parameter b is the honest sensitivity test: a nonzero b can absorb the low-MOI lift and drive the fitted q₁ toward zero. Ask whether the single-phage signal is robust to a plausible background. The tool flags data points that sit within a few-fold of b or of the resolution floor: those points cannot, by themselves, establish single-phage lysogenization.

Assay type: OD-inference versus plating

Geng did not plate colonies; lysogens were inferred from the kinetics of a kanamycin-selective culture, extrapolating the selective growth curve back to the dilution time. There is therefore no small-integer Poisson counting noise, but the back-extrapolation across a huge dynamic range at tiny lysogen fractions is an inference-level uncertainty, fractionally largest where the signal is smallest (low MOI) and plausibly biased upward. The robust inference comes from the global fit constrained by well-measured mid-range points, not the noisy low-MOI tail.

Artifacts that are mostly designed out

Premature lysis before selection is avoided in Geng by a 250× dilution at ~15 min post-infection, before the first burst (~50–60 min latent period at 30 °C). Secondary reinfection is negligible because cell density at infection is low (~10⁶–10⁷/mL) and the 250× dilution drops the second-order adsorption rate ~10⁴-fold. So the low-MOI deviation is likely inference rather than biology, which narrows the interpretation to genuine single-phage signal or a background/back-extrapolation floor — discriminated only by the MOI = 0 control.

Framing to Avoid Over-Claiming

Three studies — Kourilsky (1973), Kobiler et al. (2002), and Geng et al. (2024) — are consistent with a low, non-zero single-phage log-phase lysogenization frequency. Kourilsky could only bound it (data at the floor), while Kobiler and Geng resolve it, but in each case near the limit of the assay and without an MOI = 0 background control. State the single-phage number as a consistent upper-range estimate (~1% for log-phase lambda), not a firmly established value. For the underlying Poisson zero-class and survival math, see the companion Poisson Frequencies Calculator.

🧮 Phage Biology and Phage Therapy Calculators