Is there an framework/formalism which defines computational models based on proteins other than Adleman's DNA model or this work by Cherry and Qian?
Membrane Computing is a model that is based on the possibility or not of movement of molecules through membranes; also on possible reactions of these molecules inside a membrane compartment. While this is not specifically talking about proteins, in reality some of these molecules and the channels through which they pass would be proteins.
Here is an overview written by the creator of this model.
Another model possibly involving proteins is Computing by Observing which models the way that a person conducting an experiment extracts information from what she/he sees. The observed process may very well model protein interactions.