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I've searched the internet for a definition of what the different shells in enzymes/proteins are and haven't found a good answer. This Nature article, which is about an enzyme that catalyzes Kemp Elimination, has the sentence:

The Val12Met and Phe77Ile mutations, also in the second shell, cause changes to internal cavities that cause a slight rotation in the backbone at Trp50 that also favors the alternative configuration.

What is the second shell, and how many different shells are there?

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The authors mention three residues that they consider "second shell", 12, 77, and 102. They are shown in the figure below in green:

enter image description here

Residues Trp50 and Glu101 are directly involved in ligand binding, shown as space-filling in yellow. As you can see, they are part of the central beta barrel. The residues in green are also part of the beta barrel, but the side chains are not involved in ligand binding because they point to the outside. For that reason, the effect of mutating these residues on ligand binding must be indirect by repositioning the residues that point inside.

What is the second shell, and how many different shells are there?

This is not a well-defined concept. The term is more common for hydration of ions (first shell of water molecules makes direct contact, second shell contact first shell and so on), or in the context of coordination spheres (https://en.wikipedia.org/wiki/Coordination_sphere). When researcher write papers, they come up with descriptions that fit their structures. A famous example is to describe polymerase active site secondary structure as hands, with a palm, thumb and fingers, see e.g. https://marlin-prod.literatumonline.com/cms/attachment/c9a1e727-5b4b-45ce-81c8-8a24c1aea2ef/gr1.jpg. Some of these descriptions become common (such as the beta-barrel mentioned above) and some do not (my research advisor once described an enzyme as hat-shaped - it did not stick, maybe because hats come in many different shapes and forms, and barrels are more defined).

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