Are there protein-based computational models? Is there a framework/formalism that defines computational models based on proteins other than Adleman's DNA model or this work by Cherry and Qian?

Edit 2020 (related models/ideas based on DNA):
DNA allows to store large amount of data, it seems to me that the combination of DNA-based memory methods and technologies like CRISPR as computational workers might be an interesting and relevant venue.

DNA-sequencing-based readers
DNA sequencing is the most direct way to extract information from DNA-based recording devices. Sanger sequencing can provide low-throughput but high-accuracy sequences of ~800 bp. Nucleotide polymorphism frequencies across a population at specific DNA addresses can also be determined from Sanger chromatograms88. Alternatively, NGS can determine the sequence of DNA addresses at a much larger scale, and progress in this arena14 has enabled analysis of many recent recording devices. Short-read sequencing-by-synthesis (from Illumina) can currently provide the highest throughput and read quality, albeit with a maximum read length of ~600 bp89. For DNA addresses with longer lengths (for example, large recombinase-targeted loci87,90), long-read sequencing technologies such as single-molecule real-time sequencing (SMRT; from Pacific Biosciences) or nanopore sequencing (from Oxford Nanopore Technologies) are necessary. Although long-read sequencing modalities currently have a relatively lower throughput and lower quality than more mature short-read NGS platforms, portable instruments such as the MinION nanopore sequencer offer exciting real-time readout of DNA data storage91.

Sheth, Ravi U., and Harris H. Wang. &quot;DNA-based memory devices for recording cellular events.&quot; Nature Reviews Genetics 19.11 (2018): 718-732.
As mentioned in the answers, $\pi$-calculus and bioambinet calculus are relevant here as well:

An Abstract Machine for the Stochastic Bioambient Calculus by Andrew Phillip
The π-calculus as an Abstraction for Biomolecular Systems by Aviv Regev &amp; Ehud Shapiro
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.