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I've recently been learning bits and pieces of starch metabolism, and am wondering if anyone could explain exactly what is going on when alpha or beta amylase "cleaves" the 1-4 glycosidic bonds?

I understand that amylase, a water molecule and a starch chain are inputs of the reaction, and somehow a 1-4 bond gets cleaved producing 2 smaller glucose polymers; one of the new terminal residues gains a hydrogen and the other gains a hydroxide. In alpha amylase it cleaves any 1-4 bond along the glucose chain, while beta amylase cleaves the second bond in from the terminus yielding a maltose.

I also understand that, as catalysts, they speed up a reaction that already will occur. What I would like to know is what properties of these proteins allow them to do this?

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An article on amylase on Proteopedia.org states:

The mechanism involved includes catalyzing substrate hydrolysis by a double replacement mechanism, forming a covalent glycosyl-enzyme intermediate and hydrolyzed through oxocarbenium ion-like transition states. One of the carboxylic acids in the active site acts as the catalytic nucleophile during the formation of the intermediate. A second carboxylic acid operates as the acid/base catalyst, supporting the stabilization of the transition states during the hydrolysis.

So the enzyme uses a carboxylate nucleophile to form a covalent intermediate, which then gets hydrolyzed by water.

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