without causing the rest of the molecule to disintegrate?
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It depends on what species the DNA from. If you are talking about eukaryotic DNA in a test tube. Then single cut will give you two pieces of one linear DNA molecule and so on. DNA is a macromolecule, the entire genome can be in billions of bases in size. The only way to precisely remove something from a DNA molecule is by using restriction enzymes that form ends that can be stitched together again by a ligase enzyme. Or using a recombination event that substitutes one piece for another that has a partial similar sequence. Or the jumping genes or transposons that can cut themselves out inaccurately from a DNA molecule and sometimes take more than they are from the molecule with them. Disintegrate is not a suitable word because the DNA will not disintegrate only in presence of digestive nucleases that would chop it up entirely or in presence of strong chemicals that will disrupt the DNA bonding and backbone.