Nowadays there are many different techniques for looking at gene and protein structure and functions. You can make protein crystal structures, you can see what substrates the protein binds too, you can do various chemical assays to open the protein up and see what it looks like inside. The most classic way however is the scientific equivalent of hitting it until it stops working, and then seeing what you've damaged. This technique, in a slightly more sophisticated wording, seems to be the cornerstone of much of biochemistry, and probably developmental biology as well.
In most cases the 'hitting' is a lot less random than I've probably made it sound. Say you have a stretch of DNA that binds to a protein, and you want to know which parts of the DNA actually physically bind to the protein. The best way to do this is to get the sequence of interest and change the base pairs (that make up the DNA sequence) very specifically, to see which changes stop the protein binding.
Starting with the hypothetical DNA sequence AATATAT. In order to find which bases bind to your protein, you need to make a few very specific point mutations. The way most of the labs around me do this is with a kit from Stratagene called QuikChange (R). Your DNA sequence is likely to be inside a plasmid (small circular piece of DNA) so you design a small primer with the change you want to make, i.e AAGATAT. Adding this to the plasmid, along with some polymerase to expand the DNA, and some nucleotides to expand it with and you get a perfect copy of the plasmid, perfect except for the small difference of the T-G mutation.
QuikChange provides plasmids for you to put your DNA sequence in, and these plasmids have been methylated; some of the DNA base-pairs will have methyl groups attached. To get rid of this original plasmid (after all, you only want your mutated copy, not the un-mutated original) you use a restriction enzyme (DpnI) that literally chops up methylated DNA, leaving nothing behind but your mutated sequence.
Then you add your protein, and see how it binds. If the binding is still just as strong, then that clearly wasn't an important residue. If the binding is weaker, or if less of the protein binds, then that might have been one of the important ones.
(You can of course just use PCR with mutated primers to create single mutants. But you do run a risk of introducing other accidental mutations through the PCR process. And when it doesn't work it's incredibly irritating. I did some research over the summer which proved conclusively that it is possible for PCR mutagenesis to not work for a continuous period of over two months)
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