Understand the relationship
The reasoning behind the result
Nominal multiplicity is a ratio of two recorded quantities
mnominal = U / C; U = assayed titre × recorded volume
U is a recorded or assay-derived quantity of functional infectious units, and C is the recorded target-cell quantity. With titre in IU/mL, a recorded volume in µL is divided by 1,000 before multiplication. Dividing concentrations alone would be wrong if the corresponding volumes differ.
Physical particles, genome copies and infectious units answer different measurement questions. Likewise, an infectious-unit assay depends on its system and conditions. This workbench does not assume a universal particle-to-IU conversion or infer effective infection from a nominal ratio.
A mean of one permits zero and multiple allocations
P(K=k) = exp(−m) × m^k / k!
K is the number of units allocated to a cell in an ideal independent, uniform Poisson model. The probability of zero is exp(−m), exactly one is m × exp(−m), and two or more is the remainder. Those mutually exclusive shares add to one.
Multiplying a probability by the entered cell quantity gives an expected count, which can be fractional. It is not a guaranteed integer outcome. The table lists zero through five and retains the complete six-or-more tail, rather than truncating the distribution silently.
The occupied population has a different conditional mean
P(K≥1) = 1 − exp(−m); E[K | K≥1] = m / (1 − exp(−m))
The overall mean includes cells with zero allocations. Restricting the denominator to occupied cells raises the mean among that subset. At m = 1, the ideal occupied share is about 0.6321 and the conditional mean is about 1.582 units per occupied cell.
At m = 0 no cells are occupied in the model, so the conditional mean is undefined. Very small positive means are evaluated with stable exponential-difference and tail calculations so a rare multiple allocation is not erased by subtraction rounding.
An observed occupied fraction supports only a conditional inverse
mfit = −ln(1 − fobserved)
An observed fraction of 75/100 gives mfit = −ln(0.25) = ln(4), about 1.3863. This is a point estimate under the ideal model, not the recorded nominal input and not a reconstruction of an original stock titre. The finite observed sample also has sampling uncertainty.
When every observed cell is occupied, the inverse has no finite estimate. That sample does not establish an underlying probability of exactly one. Heterogeneity, aggregation, interference and later events can invalidate the ideal allocation model even when the inverse is numerically finite.
Follow the numbers
One on average leaves many unoccupied
- 100,000 recorded IU divided by 100,000 recorded cells gives nominal m = 1 IU/cell.
- Under ideal independent allocation, P(0) = exp(−1) ≈ 0.367879 and P(1) = exp(−1) × 1 ≈ 0.367879.
- The two-or-more share is 1 − P(0) − P(1) ≈ 0.264241. The occupied share is 1 − P(0) ≈ 0.632121.
The nominal mean is one, while the ideal model allocates none to about 36.8% of cells and multiple units to about 26.4%. Actual occupancy must be measured.
Quick guide
How to use this calculator
- Choose recorded units, assayed titre and volume, a purely mathematical mean, or observed occupied cells.
- Keep functional infectious units distinct from physical particles and from the number of occupied cells.
- Read the nominal ratio separately from the conditional ideal-model distribution.
- Inspect zero, exactly-one and multiple shares, and keep a saturated observed sample distinct from a finite inverse estimate.
Calculation method
Calculation and interpretation
Understand why a mean of one does not mean every cell receives exactly one infectious unit.
Nominal m = recorded IU / recorded cells; P(K=k) = exp(−m) m^k/k!; P(K≥1) = 1 − exp(−m); ideal inverse m = −ln(1 − observed occupied fraction)
Worked example
One on average leaves many unoccupied
The nominal mean is one, while the ideal model allocates none to about 36.8% of cells and multiple units to about 26.4%. Actual occupancy must be measured.
Nominal m = recorded IU / recorded cells; P(K=k) = exp(−m) m^k/k!; P(K≥1) = 1 − exp(−m); ideal inverse m = −ln(1 − observed occupied fraction)
Supported inputs
Precision and limits
No prescribed infection conditions
No recommended multiplicity, titre adjustment, infection protocol, organism-specific parameter or propagation decision is supplied.
Model versus observation
Ideal random allocation is not guaranteed biological infection. The observed-fraction inverse assumes the same ideal model and does not infer stock titre or functional efficiency.
Continue calculating
Related calculators
