As a mathematician interested in biology I am very curious about informed answers to this Biology StackExchange question. Here I add mine, with the understanding that it is in no way complete and it might be very biased or ignorant as concerns biology.
We may understand a cell as a chemical and physics based program which runs itself. The cell is a computational device in the sense that the outcomes of its activity are computable functions of its inputs (a reasonable hypothesis), but more than that: at the cell level there is no distinction between the computer, the program which runs on the computer, input and output data AND the execution of the program. All is at the same level, i.e. every abstraction is embodied in a concrete chemical or physical thing.
This concreteness part adds, I believe to the difficulty, because the usual thinking in computer science is all about structuring abstractions, while in biology everything is ultimately at only one level: real, physics and chemistry embodied.
This is a claim which needs strong supporting evidence. Being a matter of principle, it cannot be proved rigorously, but it might be given a rigorous support by constructing simple proofs of principle models.
There are many computing models which are inspired by chemistry, therefore in return they can be seen as such proof of principles.
There are Chemical Reaction Networks and Petri Nets models which are more like structuring tools than real embodied models of computation, because they don’t consider the structure of molecules (they are just nodes in a graph), nor the way chemical reactions happen (they are edges in a graph). They are very useful tools though and the description given here is very much simplified.
There is the CHAM (chemical abstract machine), G. Berry and G. Boudol. The chemical abstract machine. Theoretical Computer Science, 96(1):217–248, 1992. In this model states of the machine (imagine: a cell) “are chemical solutions where floating molecules can interact according to reaction rules” (cite from the abstract). In this model “solution” means a multiset of molecules, reaction rules are between molecules and they do not apply inside molecules. This is a limitation of the model because the structure of the molecule is not as important as the number of molecules in a species.
Another very interesting model is the Algorithmic Chemistry of Fontana and Buss. The main idea is that chemistry and computation are basically the same. The reasoning goes as following. There are two pillars of the rigorous notion of computation: the Turing Machine (well known) and Church’s lambda calculus. Lambda calculus is less known outside computer science, but is a formalism which may be more helpful to chemists, or biologists even, than the Turing machine. Fontana and Buss propose that lambda calculus is a kind of chemistry, in the sense that the its basic operations, namely abstraction and application, can be given chemical analogies. Molecules are like mathematical functions, abstractions are like reaction sites and applications are like chemical reactions.
The Algorithmic Chemistry is almost as closer as possible to be (proof of principle) answer to the question.
Finally I mention chemlambda, or the Chemical concrete machine, which is like Algorithmic Chemistry, but it is far more concrete. Molecules are graphs, applications and abstractions are molecules, chemical reactions are graph rewrites.
What is very interesting in all these models, in my opinion, is that they suggest that answering to the question “Can we understand a cell as an organic computational device?” is somehow relevant to the Computer Science question “How to design an asynchronous, decentralized Internet?”.