What makes DNA molecules so stable?

The main bonding in DNA which renders the double helix structure so stable is that of hydrogen bonds. Between the complementary base pairs, hydrogen bonds connect the two strands of the helix. There are 3 H bonds between Guanine and Cytosine and 2 between Adenine and Thymine. As well as this there are hydrogen bonds between the bases and surrounding water molecules, and this combined with the even stronger phosphodiester bonds in the sugar phosphate backbone make DNA very stable. 

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