🔬 Lysogenization Frequency Calculator

A Note on Terminology

The Model

How many phages infect a cell is Poisson-distributed

At a given MOI_actual, cells do not all receive the same number of phages. The population therefore splits into categories defined by how many phages infect a cell: the cells infected by no phage, the cells infected by exactly one phage, the cells infected by exactly two, and so on. We label each category by n (the number of infecting phages for cells in that category), and the fraction of the whole population falling in a category n follows the Poisson distribution, P(n) = MOIⁿ·e^−MOI ÷ n! (with MOI = MOI_actual). Grouped more coarsely, the cells are uninfected (n = 0), singly infected (n = 1), or multiply infected (n > 1). This split is pure counting statistics — it contains no biology. All of the biology enters later, as the predicted maximum frequency within a category at which a cell could go on to lysogenize.

Reference curves (the Poisson bounds)

Two of the plotted curves are those Poisson class fractions themselves, with no lysogenization factor applied — i.e., what the lysogenization frequency would be if every cell in the class lysogenized. The fraction of cells with at least one phage is P(n≥1) = 1 − e^−MOI (“every infected cell lysogenizes”). The fraction with two or more phages is P(n>1) = 1 − e^−MOI − MOI·e^−MOI (“every multiply infected cell lysogenizes”). The exactly one fraction is P(1) = MOI·e^−MOI. On log-log axes at low MOI the single-infection fraction rises in proportion to MOI, while the multiple-infection fraction rises in proportion to MOI² (MOI squared). Because MOI squared shrinks faster than MOI as MOI falls, the multiple-infection fraction drops away much faster than the single-infection fraction at low MOI. This difference is the entire diagnostic: any single-phage contribution rides on the fraction that scales with MOI (not MOI squared) and therefore diverges upward from the multiple-infection curve as MOI drops. If among infected cells the fraction that are multiply infected is about MOI/2 at low MOI, then at MOI = 0.01 singly infected cells outnumber multiply infected ones by roughly 200:1.

Single- versus multiple-infection lysogenization

Now attach the biology. Let Q_n be the probability that a cell infected by exactly n phages lysogenizes. The predicted lysogenization frequency (per total cell) is the sum, over the infected classes, of each class fraction times its lysogenization probability. Because the empirical Q_n saturates at n = 2 (see below), this reduces to a two-parameter form: lysogenization frequency = P(1)·q₁ + P(n>1)·q_>1, where q₁ is the fraction of singly infected cells that lysogenize and q_>1 is the fraction of multiply infected cells that lysogenize (the n = 2 value, assumed to hold for all n > 1). The first term is the single-phage contribution and the second is the multiple-phage contribution; the “All” curve in the tool is their sum (P(1)·q₁ + P(n>1)·q_>1), and the “multiples only” curve is just the second term (P(n>1)·q_>1). The ratio of All to multiples-only is expected to grow as MOI falls only if q₁ > 0 (that is, only if there is a non-zero single-phage contribution, P(1)·q₁), constituting the quantitative single-phage signal. This tool’s reference values (q₁ = 1%, q_>1 = 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 (lysogenization frequency = 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 q_>1 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%, q_>1 = 50%) are the approximations, to allow exploration of the impact of changing these per-n probabilities. This tool lets you set q₁ and q_>1 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 ≤ q_>1.

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 (MOI_actual = a × MOI_input), so plotting the Poisson references directly at these MOIs is apples-to-apples. The lowest one or two points of each run may sit near the assay resolution limit, perhaps explaining why they are noisy; the single-phage claim rests on the trend, not on any one point.

Several curves from Kourilsky (1973) are also built in, digitized from his figures (approximate). Because he reported greater than 95% adsorption, his “average phage input per cell” is within a few percent of MOI_actual, so his points sit on essentially the same axis as Geng’s. His replication-competent, starved λQ⁻ data (Fig. 1) reaches a Geng-like ceiling (~50%) with a ~2-phage threshold, making it a direct contrast to Geng’s resolved single-phage tail; his unstarved λQ⁻ shows the same threshold at a far lower ceiling (~2%); and his replication-defective λP⁻ requires ~3–4 phages, a steeper low-MOI rise that the two-parameter model here cannot represent (it needs a per-n treatment). Note that in some experiments Kourilsky himself observed curves matching the P(n≥1) distribution and concluded a single phage can sometimes lysogenize a cell, anticipating the resolved single-phage result.

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>1) “every multiple” bound cannot be explained by multiple infection however effectively multiples lysogenize — 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)

Nominal MOI is phages added ÷ cells; the effective MOI that matters for the Poisson math is a × nominal MOI, with a < 1. If a curve is plotted versus nominal MOI while the references assume a = 1, the Poisson floor is overstated and the single-phage signal is understated. Correcting (a < 1) lowers the effective MOI, 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 effectiveness. Use the MOI-kind control on the Calculator tab to convert nominal MOI to effective MOI.

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 the better-measured points at intermediate MOI, 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.

A Thought Experiment

To catch single-phage lysogenization you must sample very low MOI, which — as seen in the Geng et al. experiment — can result in reduced data precision. The proposed experiment represents an approach toward sidestepping this issue. Mix the permissive cells you want to score (targets) with a large excess of non-permissive but adsorption-competent cells (decoys). Decoys adsorb phage but make no new virions and are not scored, so they act as a phage sink: they lower the average number of phages that reach each target — the effective target MOI — while phage titers and target numbers stay high enough to measure and to yield scorable lysogens. Push the decoy excess high enough and essentially every infected target has received just one phage, so the lysogens scored among targets are overwhelmingly single-phage lysogens, giving a direct read on q₁. The same dilution does lower the per-cell frequency of lysogens, but that is recoverable: you can simply assay more target cells — plating a larger volume, or concentrating the targets before plating — so the practical limit is how many targets you can process, not a hard floor. This tab explores that trade-off numerically.

1. Adsorption setup

2. Lysogenization & detection

Defaults are the reference values; you can paste in a fit from the Calculator tab.

3. Plot

4. Readouts

What This Does

Some studies report just a single (MOI, lysogenization frequency) point — for example Kobiler et al. (2002), who found roughly 0.55% lysogenization at a stated MOI of about 0.01. One point cannot fit both q₁ and q_>1, but it can still answer the question that matters: does this point require single-phage lysogenization? This tab compares the observed value against the Poisson multiple-infection ceiling at that MOI (the most that multiple infection could give, if every multiply infected cell lysogenized), against the multiples-only expectation for an assumed q_>1, and then tests how robust the conclusion is to the two controls that matter for any such claim (as with the Geng et al. data): the accuracy of the MOI, and a possible phage-independent background.

1. The data point

2. Assumptions & controls

3. Plot

4. Readouts

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