Yes it is well understood. If you think about the substrates for the ligation reaction, they are ends of the DNA molecule. Let's say they were digested by some restriction enzyme like HindIII. Every DNA end that was created through HindIII digestion is functionally equivalent to any other end, so they all serve as substrates for ligation and every possible combination of ends can be joined (if they were all created using the same enzyme).

Pretend you have a ligation with 2 molecules of DNA, the vector and the insert. Each molecule was prepared by digestion and therefore is linear with two ends. Call the vector ends A and B, and the insert ends C and D. The ligation reaction could be A+B, in which case you get the vector only back. It could be C+D, in which case you get a non-replicating circular insert that is lost. it could be A+C giving a linear vector plus insert. Subsequently the ends of that molecule (B+D) could undergo ligation either with itself or with anything else, but if it does so by itself you get a vector plus insert molecule. And of course there are also all the other possibilities here as well (A+D, B+C, B+D followed by a the full range of second events).

So getting insert plus vector ligation product is both an intermolecular ligation followed by an intramolecular ligation.

The term isn't "false positive", it's an "empty vector" construct.

An "empty vector" is where the vector has ligated back together without the insert. You'd never actually observe a circular insert since you have no way to screen for it during transformation. Depending on your plasmid of choice and selection medium, you can see a lot of these.

And that is why you need to choose a method to reduct the chance that you'll waste time & $$ screening for empty vectors. These include phosphatase treat either your vector or your insert, use TOPO-TA cloning, blue/white screens, lethal genes, double-digested inserts, etc.

It's very important to ensure your restriction digests go to completion. This means you should use about 10 units of each enzyme per microgram of DNA to cut and run it for about 2-3 hours. Run the necessary controls to monitor reaction complete.

Also do gel purification of your vector and insert DNA prior to ligation. It's important to make sure as much background DNA as possible is removed. You may get a lot of loss during this step but that's ok, you really don't need very much DNA to do a successful ligation/transformation. Purity of vector/insert DNA is waaaaaay more important than yield in this case.

I would call this ligation and transformation "background," which I quantify by the number of colonies arising from a control 'vector alone' ligation that consists of the same mass of the same prep of linearized vector used in the vector + insert ligation. I do this every time since it is really not much more work and just looking at the plates from the two ligations/transformations can speak volumes about the likelihood that the colonies on the vector+insert plate have what I want. At the very least, it immediately informs me whether I am picking 4 colonies to screen, 10 or more colonies to screen, or whether I would rather re-tool and do the ligation over.

I strongly agree with fellow Californian Katie A S Burnette in that thinking through the cloning method and acting on the sources of background ahead of time can save a lot of time and money in the end. Vector phosphatase treatment for me is always a given when the insert and vector ends are the same.

Other sources of this 'vector only' background that get overlooked sometimes are 1) incomplete vector digestion and 2) vector contamination with the denatured supercoiled form. For the first background source, even a digestion efficiency of 99.999% can leave quite a large number of uncut, supercoiled vectors that wouldn't be affected by phosphate treatment and would transform very readily into the competent cells and produce background colonies. Usually, these molecules aren't a nuisance unless the insert+vector efficiency is low since there is a much higher mass of these in the transformation. If the insert+vector efficiency is inherently low, which can be for legitimate reasons, then gel purification of the linear plasmid form away from the supercoiled form can be very helpful if performed methodically such as with the prep gel running slowly.

Lastly, some plasmid preps either done incorrectly or just by nature of the plasmid itself can lead to the starting vector prep containing the denatured supercoiled form of the plasmid. The online Qiagen information online shows a good example of this, but it is often from the lysis and denaturation step of the prep going on too long and the result is a faint band running below the blob of supercoiled plasmid on a gel. Again, it could be a minority of the prep, but this form is resistant to restriction endonuclease digestion but transforms into the competent cells and causes background colonies just fine. Vector gel purification is helpful too in these circumstances. As I said above, I have just also used weird plasmids/plasmid-insert combinations that seem to be prone to yielding higher than normal denatured, supercoiled plasmids in preps done correctly or even with shortened lysis times.

For both of these sources, it can also be helpful to include a second control in the insert+vector ligation/transformation, and that is having a vector alone + no ligase control so that the mass of linearized vector prep and the buffer constituents/ligase protein mass that go into the competent cells matches that of the insert+vector ligation. If colony numbers from this control are similar to that in the insert+vector +ligase reaction, then some form of uncut vector is in the vector preparation and further steps should be taken. I'll use this control if using known problematic plasmids or enzymes or if I haven't been steadily cloning for a while.

Hope this helps you or any others that come across this thread.