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A longer DNA molecule would have more phosphate groups, so it should have a greater negative charge, right? It was taught in my class that only terminal ends of DNA are charged and all the phosphates in the middle are not charged.

My teacher said that this is the reason that Electrophoresis separates the fragments. All DNA have same charge and so same force on all of them. Then difference in masses would give different acceleration and thus separation of fragments according to size. But isn’t Gel Electrophoresis more like a sieve where longer molecules move slowly and shorter pass through easily?

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Each phosphate group has a single negative charge. Not just the terminals. The reason gel electrophoresis works so well with DNA is that charge is linearly proportional to size. The longer the fragment, the slower it moves. A double stranded DNA molecule with 100 nucleotides has a charge of -200. If it were twice as long, it would have twice as many, just as if it were 3.4 times longer, it would have 3.4 times as many negative charges. This way, charge is essentially ignored, so DNA migrates through the gel at a rate inversely proportional to its size.

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