All proteins have neither DNA nor RNA, that is why they are called “proteins” they are made up of a chain of amino acids coded for by DNA in the form of a messenger RNA. The Prion protein has just as much “DNA” and “RNA” as any other gene product, in that a prion protein is coded for by a DNA gene using an RNA intermediary.
The “prion problem” is that they are misfolded, and can apparently, when ingested, make native proteins misfold as well.
When you say “In the Shannon model, prions would be incoherent in the channel because they have no discernible message” that would also be true of any protein, or any misfolded protein if you assume the “message” of any functional protein is that it is able to perform its function.
In other words the very first thing I read was a mess that made no sense to anybody who understand the science of molecular biology; but no worries, I don't think the target audience either desires or possesses such knowledge.
“Prions — typified as non-autonomous protein noise/mutation contributing to successful/failed communication (protein crystallization). Prions are protein molecules that have neither DNA nor RNA. Currently, prions are the suspected cause of bovine spongiform encephalopathy — Mad Cow Disease. In the Shannon model, prions would be incoherent in the channel because they have no discernible message; that is, neither DNA nor RNA. Thus the prion would lead to channel or decoding malfunctions.”
I take responsibility for the sloppy wording about prions.
That section of our book tried to address the inevitable "laundry list" of anomalies when discussing information theory applied to molecular biology.
The point was that the prion contains no message (DNA, RNA) and is not autonomous to the molecular machine. Under the Shannon model, it would a type of "noise."
In sum, when the prion is introduced into the communication - it causes malfunctions either in the channel or decoding, e.g. blocking the successful communication of beneficial messages perhaps as you say by causing other proteins to misfold.
The emphasis was on the communication (information theory) of the message (DNA, RNA.) Prions have no message and are not part of the communication, they are like meteorites pounding a radio transmitter/receiver. They are made of the same 'stuff' but they don't belong there.