Contents

1. XanthoMoClo-plasmid-copy-number.py contains analysis code to quantify plasmid copy numbers presented in Soltysiak, Ory et. al.
2. src/PoissonJoint.py contains the core logic of estimating the number of targets from a (noisy) serial dilution.

Limiting Serial Dilutions

The objective of this tool is to estimate with high confidence the number of targets (viable cells/genomes/detectable targets) from a limiting serial dilution of this starting material. A serial dilution is said to be limiting if we fully dilute out the starting sample, that is, we expect to see empty wells at some point in the dilution.

Serial dilutions are modeled as a Poisson process. We use the fact that for an expected number of events x, the probability of failure to observe an event is given by

Poisson(0 events | x) = e^(-x)

In a serial diution starting from x cells, the Probability of seeing observing any target, i.e. >0 events can thus be written as

Poisson(>0 events | x) = 1- e^(-x)

If we are performing an f fold serial dilution, the process can be represented as product_i=1^(i=k) (Poisson(growth observed well | x/f^k ) for k serial dilutions.

We construct the Likelihood function as follows

$$L = \prod_{r=1}^{r=R} \prod_{d=1}^{d=D} (1 - e^{-x/F^d})^{v_{rd}} (e^{-x/F^d})^{1-v_{rd}}$$

We find the number of cells x that maximizes the Log Likelihood.

Further, using the Cramer-Rao bound, we estimate the variance at the MLE to be

$$\text{Variance} = \frac{-1}{\frac{\partial^2 \ln L}{\partial x^2}}$$

where

$$\frac{\partial^2 \ln L}{\partial x^2} = \sum_{r=1}^{r=R} \sum_{d=1}^{d=D}- \frac{F^{- 2 d} v_{rd} e^{- F^{- d} x}}{1 - e^{- F^{- d} x}} - \frac{F^{- 2 d} v_{rd} e^{- 2 F^{- d} x}}{\left(1 - e^{- F^{- d} x}\right)^{2}}$$

Finally, we compute the 95% CI as MLE $$\pm 1.96 \sqrt \sigma$$

Usage

See ./test.py