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I have read the supercoiled plasmid DNA migrates fastest when subjected to agarose gel electrophoresis and this makes sense to me because it is small and experiences the least friction because of its supercoiled structure.

Linear DNA migrates faster than the circular form. I didn't understand why exactly. Shouldn't the circular one be considered to be smaller than the linear, and thus quicker?

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First let me supply an illustration of the situation described in the question, together with a reference.

Plasmid migration on agarose gel electrophoresis

Although you can see this sort of thing, just by searching for “plasmid migration on agarose gel”, this is one of my own from the last millennium (plasmid pBR322), appearing in a text from the last millennium, Adams et al. The Biochemistry of the Nucleic Acids (11th ed).

The poster’s attempt to rationalize the results, though expressing the quandary, is marred by the loose use of the word small, and the absence of any statement as to why ‘small’ pieces of DNA should migrate more rapidly.

The different forms of the plasmid have exactly the same molecular mass, so what is meant by small? My own initial way of looking at this is to consider the agarose gel as a mesh or sieve through which the molecules must pass, so that the larger the volume they occupy in space, the less likely they are to pass through the ‘holes’ in the mesh. The volume that they occupy in space might be regarded as that generated by rotating them from their midpoint in three dimensions. On this basis the supercoils, in which the DNA is collapsed into a short fat pseudo-linear shape should occupy the least volume in space, and the long thin linear the greatest. The diameter of the circle should place it somewhere between the other two.

So, we reach the same conclusion as the poster: the faster migration of the ‘linears’ than the nicked circles does not fit this simple model. In the text accompanying the figure I wrote “…a result that is a little unexpected” without attempting to explain it.

Am I able to explain this somewhat unexpected result now? An internet search produces either hand-waving or evasion (I stand to be corrected on this), so that I can only imagine that the result has something to do with the linear molecules having a greater chance of their ends being directed to the holes in the ‘mesh’ than the nicked circles, because of the open nature of the latter. One can perhaps imagine an hour-glass shaped portion of the sphere for the rotation of the linears that allows the ends to be directed at the holes.

More waving of hands? Perhaps. But the main point is that the physical explanation for the migration of molecules of different shapes during electrophoresis is more complex than a mere biologist might have hoped.

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