This is an interesting question. I have always though that computational models are the most interesting description of genetic mechanisms (DNA primary sequence, epigenetics, transcriptional control, etc.) and singaling networks (RNA, proteins, metabolic networks, etc). They abstract away complexity and give us tools to make powerfull predictions. What you are suggesting here is that biology is more than a universal Turing machine. What makes you believe so?
The statment "biology is a Turing machine" does not make any sense to me, so I won't discuss whether it is true or not. Just pursue the analogy of DNA as a program for a few steps more and you will see how it goes. If DNA is a program what would be the computer? Ribosome? The universe? What would be the outcome? I guess the protein. But then the result of the computation is dependent on hundreds of factors outside of both the DNA and the ribosome. The "result" immediately gets distorted by whatever physics applies to it, the "instructions" might have different effects on the "computer" depending on position in the strand and on hundreds of external factors etc. What insights does this metaphor give you beside endless confusion?
The "computational" models that have been successfully used to model the DNA have nothing to do with treating it as a description of a computation, they are only computational in the sense that they use mathematics and computing for making predictions about biological phenomena. Those are mostly probabilistic models that treat the DNA as a string of discrete symbols, and encode the physical and chemical complexities associated with this approach in various ways, one good example is Hidden Markov Models, where the probabilities might encode physical or chemical knowledge:
