# How to analyze  Poisson Law Computation

Poisson distribution, named after the French mathematician Siméon Denis Poisson, is the probability of occurrence of a given number of events in a given (fixed) time period if the events occur at a constant rate (known) and are independent of the occurrence of the previous event. It is based on a discrete probability distribution, where the set of outcomes are discrete or finite, such as the toss of a coin or roll of dice.

In the context of a digital PCR experiment, the discrete outcomes are the presence or the absence of the target gene. The thousands of individual partitions produced for a digital PCR reaction are expected to follow a Poisson distribution considering the partitions are monodispersed and they contain the equivalent volume of the sample mix.

If these parameters are not met and the partitions exhibit polydispersity, the volume of sample mix in the partitions will vary largely and the larger partitions may contain more targets than the smaller ones, lowering the precision of the digital PCR reaction.

In this item, we walk you through the mathematical computation of the Poisson Law for a digital PCR experiment.

For a digital PCR experiment, a well containing the partitioned sample of interest, and a target gene to quantify, we first need to define the following variables:

• $$N$$: total number of analyzable partitions in the well
• $$p$$: number of positive partitions for the target gene
• $$v$$: volume of the partition (in µL), assumed to be constant
• $$d$$: dilution factor used to dilute the sample from the stock to the well

(e.g. $$d=10$$ means the sample has been diluted 10 times)

and then these additional ones:

• $$V = N \ v$$ : total partition volume injected in the well

• $$C_{0}$$  : concentration of target genes in the well (in copies/µL)

• $$C = d \ C_{0}$$ : concentration of target genes in the stock (in copies/µL)

• $$\lambda = C_{0} \ v$$ : average number of target genes per partition in the well

The distribution of the target genes encapsulated in the partitions of the well follows a Poisson distribution of parameter $$\lambda$$ :

Proba ( partition encapsulates $$\text{k}$$ target genes ) $$= \dfrac{\lambda^k}{k!} e^{-\lambda}$$

A partition is said:

• Positive partition” if it has encapsulated at least 1 target gene (in which case we will observe a fluorescent partition at the end point of the amplification process, so most of the uncertainty lies in this “at least one” condition)

• “Negative partition” if has encapsulated 0 target gene (in which case we will observe a non-fluorescent partition at the end point of the amplification process)

The distribution of positive partitions in the well follows a binomial distribution of probability $$1 – e^{-\lambda}$$:

• Probability (well contains $$\text{p}$$ positive partitions $$= {\rm C}_{N}^{p} (1 – e^{-\lambda})^p (e^{-\lambda} )^{N-p}$$
• Probability (partition is negative) $$= e^{-\lambda}$$
• Probability (partition is positive) $$= 1 – e^{-\lambda}$$

If $$N$$ is large enough:

• Proba (partition is positive) $$= \dfrac{p}{N}$$

So the formula for the estimated stock concentration is:

$$C = – \dfrac{d}{v} \ ln{\left(1 – \dfrac{p}{N} \right)}$$

If you need to automatically compute estimated concentrations of target genes, together with their confidence interval and relative uncertainty, an online tool is available: Try it!

For more information regarding the uncertainty curves, as well as the limits of detection and quantification, please see the item: Dynamic Ranges of Detection & Quantification.