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So, i was given this cut piece of dna and i was required to choose the restriction enzyme used to cut it. I had to pick either XhoI(CTCGAG) or TaqI(TCGA), where the sequence between parentheses is their palindrome sequence. Which one is it and why? I've picked XhoI since i know that the enzyme cuts in between the palindrome

GAGCT        C
C        TCGAG

DNA after the cut.

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In order to keep this on topic, I will try and give a more general answer.

The most widely used restriction enzymes, and the ones typically taught first, all cut within their recognition sequences, however, this is not necessarily so as most Type I and Type III enzymes cut at a distance from their site, sometimes this is a fixed distance, sometimes it is not. It is only Type II enzymes that cut symmetrically within their sites, and even then, not all of them. However, cutting within the site is usually the most useful kind of cut for molecular genetics and thus these are the most common enzymes to encounter in the lab (see here for more).

For a particular enzyme, the best way to find the cut site is simply to google it or search directly on the NEB site (where you will likely find the top Google hit anyway), doing this for XhoI would get you a page containing information like this:

XhoI cut-site screenshot

Where the little arrows in the given recognition sequence indicate where each strand will be cut, whereas as for enzymes that cut outside the recognition site you will get something like this (for AlwI) where the $(N)_4$ part tells you that the enzyme cuts four bases away on the 5'-3' strand, and the $(N)_5$ part tells you it cuts one base further away on the opposite strand.

enter image description here

In order to identify which of a number of enzymes cut a certain sequence you need only find the appropriate information for the relevant enzymes so you can compare these cut diagrams for the relevant enzymes.

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