OK, I've done the third of the experiments I planned here, and the results cleanly show that nothing interesting is going on (at least nothing interesting that warrants investigation).
I wanted to know if the peculiarities I had long-ago noticed in DNA of competent cells were reproducible. So yesterday I grew some wildtype cells in rich medium. Some of them I collected during exponential growth ('log phase', cell density about 2 x 10^9 cells/ml), some when the culture was approaching its final density ('stationary phase', about 10^10 cells/ml), and some I transferred to the competence-inducing starvation medium while they were in log phase. I prepared chromosomal DNA from all three treatments, and then incubated it either in the standard DNA buffer TE (10 mM Tris pH 8, 1 mM EDTA) or in the mung bean nuclease buffer that had previously given anomalous results for competent-cell DNA. DNA in the mung bean nuclease buffer sat at room temperature for about an hour, with and without being heated to 65 °C for 10 minutes. Then I ran all the DNAs in a gel to check their condition.
You can see that there's no significant differences between the effects of the treatments on the different DNAs. The competent cell DNA is in slightly shorter fragments (the biggest marker band is 29 kb), but the difference between it and the other DNAs isn't changed by the MBN buffer. Heating the DNAs in the buffer causes the usual streaking of the bands (caused by having a lot of high-molecular-weight DNA running at the same position) to be a bit blurry, but again that applies equally to all three DNAs.
So I can put this old result out of my mind, and get to work on my planned competence and phage recombination assays.
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