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Our school textbook states that base pairs confer stability to the B-DNA double helix in two ways:

1) Hydrogen bonding between purines and pyrimidines of opposite strands.

2) "Stacking of one base-pair plane over the other"

It's the second point that I'm not able to understand. I did try looking it up on the internet, but I'm not able to find anything relevant in that regard.

So could someone please try explaining that second point to me?

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When you carefully look at the structure of DNA helix, you will notice that consecutive bases are partially layered i.e. one base is "on-top" of the other base. Stacking is a chemical phenomenon in which the cloud of the pi-electrons in an aromatic molecule interact with that of the other. This interaction is non-covalent. Since the DNA nucleobases are aromatic molecules, they also exhibit stacking which stabilizes the structure of the DNA. A study by Yakovchuk et al. (2006) suggests that base stacking has higher contribution to DNA stability than base pairing. Sponer at al. (1997) found that there is no great difference between the stacking energies of B- and Z- forms of DNA helix.

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In short, because of the pi bonds in aromatic rings, they're attracted to each other when stacked (or in several other conformations). Explaining is maybe better done by the Wikipedia page, but this is what you're looking for:

https://en.wikipedia.org/wiki/Stacking_%28chemistry%29

For a more in-depth look you could take a look at this article for example:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1360284/

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