The post-doc is polishing up has latest manuscript, and on reading it over yesterday I realized that we have data to clarify a very old issue. Here's a link to an old post on this topic: http://rrresearch.fieldofscience.com/2011/05/fraction-competent-problem.html
In the old days, the best way to estimate the distribution of competence among the cells of a 'competent' culture was to measure the proportion of cells that became transformed by selectable markers that were on two separate DNA fragments (usually markers carried by a single donor strain, but far enough apart on the chromosome that they were never taken up on the same fragment.
Such assays typically find that cells that have been transformed by one marker are enriched for cells transformed by the other marker - the fraction of double transformants is higher than expected from the fractions of either single transformants - and this is used to estimate the fraction of the cells in the culture that are not competent.
Now, from the post-doc's work, we have data telling us what fractions of the cells have acquired one selected marker have also acquired one, two, three or more unselected fragments of donor DNA, and we want to use this data to unpack the relationship above.
1. all parts of the donor chromosome are equally likely to be taken up and recombined into the recipient chromosome,
2. all the cells in the competent culture are equally likely to take up DNA
3. taking up and recombining one DNA fragment does not alter the cell's probability of taking up and recombining another one,
4. we ignore that only a single strand is replaced at a recombination site, and that the donor and recipient strands at this site separate the first time the cell divides
5. we ignore the possibility of mismatch repair at these heteroduplex sites.
the frequency of cells transformed by two unlinked markers should be the product of the frequencies of cells transformed by each marker (under assumption 1 above, equal to the square of the single-marker transformation frequency).
For now I'm mainly interested in what happens if assumptions 2 and 3 are not valid. (The post-doc is more interested in 4 and 5, and the plan (hope) is that we'll pool our thinking and explain the whole business.)
Most traditional analysis has focused on relaxing assumption 2. If some cells in the culture are equally competent and the rest not competent at all (and all the other assumptions hold), then the fraction of the cells that are competent is the ratio of the product of the two observed single transformation frequencies to the observed double-transformation frequency. If the ratio is 1.0, then we'd conclude that all the cells are competent.
What if we instead relax assumption 3? What if taking up one DNA fragment uses up cellular resources and so reduces the cells' ability to take up more fragments? At the extreme, no cell would take up more than one fragment. The frequency of cotransformation would then be zero, and our ratio would be infinity. If the resource-depletion was not absolute, then the ratio would be smaller, but it would still be larger than 1.0. So if we observed ratio greater than 1.0, we'd conclude that some cells can only take up one DNA fragment.
But deviations from assumptions 2 and 3 push the ratio in opposite directions. The typical observation of a ratio less than 1.0 means that assumption 2 must be invalid but not that assumption 3 is valid. In principle, how far below 1.0 the ratio is sets a limit on how much of an effect deviations from assumption 3 can be having, but we haven't worked out the math to calculate this. For example, if the observed ratio were 0.5, could this be because a quarter of the cells were competent but only half of those were able to take up more than one fragment?
Conversely, seeing a ratio greater than 1.0 would mean that assumption 3 was invalid but not that assumption 2 was valid, and we could in principle calculate the range of deviations consistent with a particular observed ratio.
We could also view relaxing assumptions 2 and 3 a different way, thinking about variation in the levels of expression of competence genes leading to differences in how much machinery/resources cells have available. What if some cells are more competent than others? Some cells might not have enough resources to take up even a single fragment, some might have only enough resources to take up a single fragment, and some might have enough resources to take up two or more fragments. What would our ratio look like then?
So. For each of two selected markers, the post-doc's data gives us the actual distributions of cells that also took up no, one, two or three additional fragments: 0.35 had none, 0.40 had one, 0.21 had two, and 0.04 had three. What can we do with these numbers?
The single-segment transformation frequency for the two selected markers was each about 0.03. I forget the double-transformant frequency, but I remember that it predicted that only about 0.1 of the cells were competent (relaxing assumption 2), so I think it must have been about 0.01.
How far could assumption 3 be relaxed and still give the observed numbers? Could 0.2 of the cells have been competent if only 0.65 of these were able to take up more than one fragment?
Consistent with this, only two of 20 unselected clones had recombined donor segments.
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