There's a simple test, based on measuring the frequency of double transformants when cells are given DNA containing two unlinked selectable markers. This test always indicates that some of the cells are taking up multiple fragments of DNA and the others aren't taking up any DNA at all. The differences in competence of cells grown under different conditions, or carrying different mutations, appear to result from differences in the proportion of cells that are competent, not from differences in how much DNA the competent cells take up.
This is an odd result, and I've always been puzzled by it. I've also always mistrusted it because it's quite indirect, though I haven't been able to see how it could be wrong. The post-doc and I spent the day working on our Genome BC proposal, and towards the end we were both grappling with the problem of how to maximize the fraction of the cells we sequence that would be competent. This was difficult because we don't really know what to expect. But then I realized that we could use deep sequencing to settle the question once and for all.
Our plans are ambitious, but we're limiting the sequencing budget to an amount equivalent to the value of the fellowship the post-doc hopes to get from NIH (he'll hear soon) because this is our proposed 'matching funds'. But now, in addition to planning to do one lane of sequencing (= about 300-fold coverage) of two colonies that grew from cells we know were competent because they acquired a selectable marker from the donor DNA, we're going to sequence three random (unselected) colonies. If some cells are fully competent and the rest not competent at all, we predict that at least one of these three will have not acquired any donor DNA at all, and those that do have donor sequences will have replaced about 2% of their genomes. If the cells are all at least a bit competent, then the three unselected colonies will all have some donor DNA, but perhaps quite different amounts.
The library construction and sequencing will cost about $2000 per lane, and the 300X coverage is major overkill, but I think it's worth $6000 to put this question to rest, or to at least get enough preliminary information that we can ask NIH for more.