Isolating genes is usually done with simple PCR. These days you almost never isolate a gene without knowing it's sequence and then it's a matter of designing PCR primers that are unique (fairly straightforward). You can check that the appropriate gene is isolated by simple gel electrophoresis and once it's inserted into the vector (which makes amplification in an organism very easy) you can once again sequence.

Isolating the protein product after expression is a little bit more difficult but is well understood. Almost always you will use one or more liquid chromatography methods. You can append small "tag" sequences to the gene to create an amino-acid sequence that specifically binds to certain metals (HHH binds Nickel) or an antibody which can be immobilized on a solid "column". You can then flow a slurry of the bacteria that expressed the protein across the column and everything will flow past except your protein of interest. The protein can then be removed off the column with a different specific buffer. There are many variations depending on your protein of interest and application needs, but this is the general approach. In cases where tags can't be added due to activity needs or cost requirements, proteins can be isolated similarly based on their size, charge, and hydrophobicity. If you perform those separations in various orders and be clever about it, you can get very pure product.

Finally, confirming the final identity and activity of the enzyme might require mass spec and activity assays (if enzyme) where you can compare reaction rate of product vs. total amount of protein added to get an idea of the fraction of protein that is active.

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