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In microbial research, a common way to check growth rates of bacteria is by performing a dilution of the bacterial population and then plating the resulting dilution on a petri dish. After some time, the cells on the petri dish grow into visible colonies which you can then count to arrive at an estimate of the original population size ('colony forming unit' or CFU counts). For this method, it is important to use the right dilution so that your plates contain somewhere between 30-300 cells upon plating, which ensures that your counts are not too low (so that they are too affected by the stochasticity of your dilution process), nor too high (in which case the colonies are too close together, making it difficult to count them).

My question is quite fundamental: what is an appropriate statistical method to compare CFU counts? Say I have inoculated bacterial populations in test tubes and grown them under two experimental conditions (say, 25 and 30 degrees Celcius). How do I estimate (and test) the impact of temperature on bacterial growth? I have counts, so a Poisson GLM may be appropriate. However, what I measured (CFUs) is only a proxy for ACTUAL population size, for which I have no accurate measure. Also, even though I have counts, it is not quite clear to me whether they are generated by a Poisson process, where events happens with a certain constant probability per unit time. Cell divisions do happen with some probabilty per unit time, but every division generates more cells that can again divide - it intuitively seems to me that this exponential relationship complicates things.

Any insights on whether these considerations render a Poisson GLM problematic, and if so, what a better approach might be?

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what I measured (CFUs) is only a proxy for ACTUAL population size...

That's true of almost all technical measurements. A chemical concentration might be estimated by evaluating the optical absorbance at a particular wavelength. Expression of messenger RNA (mRNA) from a gene might be estimated by reverse-transcription into complementary DNA (cDNA) followed by a real-time quantitative polymerase chain reaction that is then monitored by the fluorescence of a probe.

In that respect, CFUs might be one of the assays closest in kind to what is most directly of interest. So long as the plating doesn't kill any bacteria, it provides a count that can be back-calculated to estimate the number of bacteria in the culture from which they were plated.

it is not quite clear to me whether they are generated by a Poisson process, where events happens with a certain constant probability per unit time...

The Poisson distribution is not limited to events occurring over time. It can be applied to any event that is rare over some extensive property: for example, few animals per unit area, or few cells per milliliter of volume.

The latter (few cells per milliliter, spread out to be a few cells per unit surface area) are how you think about using a Poisson distribution to describe your CFU data. You have taken a sample of the full bacterial culture and diluted it until it has only a small number of bacteria in the volume that you apply to the Petri dish. You then quickly spread out that volume over the entire surface of the dish so that no 2 bacteria are close to each other. Compared to the hour or so typical of a bacterial cell cycle, the few seconds needed for the dilution and spreading mean that any continuing replication of cells does not substantially affect the results.

The exponential growth in numbers over time comes into play after you have spread out the individual bacteria. You can't see a micrometer-scale single bacterium with the naked eye. After overnight culture on an adequately rich medium, the exponential growth means that each individual bacterium has produced a visible colony to count, separate from other colonies if you have the right dilution.

A Poisson model is thus an appropriate choice for such data. To get estimates for the original bacterial cultures from which you sampled, you should use an appropriate log-offset to represent the effective volume of original culture that you spread over the dish. This page and its links explain offsets in the context of time as the extensive variable, but the principle applies similarly to volumes or other extensive variables.

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