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. "DNA-based memory devices for recording cellular events." Nature Reviews Genetics 19.11 (2018): 718-732.
As mentioned in the answers, $\pi$-calculus and bioambinet calculus are relevant here as well: