The question is not really what does the computer depend on, but what does _the program_ depend on. The boolean algebra that computers are an implementation of is very simple and can be implemented in a wide variety of ways and regardless which one you choose, programs can be compiled and ran using the new architecture obtained in this way. So you could hypothetically build a biological computer that would realize ANDs, ORs etc., in the end providing the same instruction set as our current PC, and the unchanged Linux source code (example from the parent post) could be compiled an ran on it. In other words, while our computers depend on laws of physics in some way, those laws are not part of the computational model.
This is different with the DNA. There is no middle layer like Boolean algebra to abstract-out the device that does the computation. The protein that will be the result of connecting the amino acids specified by the DNA, it's form and function, is very highly dependent directly on physical laws in a very complicated way - if you simulate protein folding (and consider how hard this is in the first place), you can see how much the outcome will vary when you for example change the value of some physical constant by a small amount. Then all those proteins start interacting with each other in highly complicated ways, also dependent on a wide variety of physical laws and on the outside environment, and if you consider DNA a program, those physical laws are parts of the computational model, of course if the concept of a computational model makes any sense when studying non-man-made artefacts. That's roughly why applying computer science metaphors to the DNA always sounds a bit ridiculous to me.
Of course, it is a different question whether we can find a computational model that would explain to us the working of a healthy, fully-developed human brain, I think it is worth mentioning as it is easy to confuse those two questions.