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That's an example of symmetric sequence
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and it happens that most restriction enzymes also recognize those spaces.
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If you cut and you open up a segment of DNA
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then you've left these sticky ends, for example
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and these sticky ends are capable of recognizing each other by the process of hybridization.
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These will naturally want to reform and they'll want to reform to re-establish this base pairing.
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They could be pasted back together,
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and the pasting process
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takes advantage of this natural process of complimentary hybridization.
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This gives you a biological mechanism for cutting,
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using restriction enzymes,
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and then you denature so that it falls apart,