The hash function is used because it is irreversible, easily checkable, small in size, and probably some other things I'm forgetting. Even if you could, for example, encode the transactions as a polypeptide (chain of amino acids), and then made folding the polypeptide into a protein the proof-of-work, such a proof-of-work could not be checked without redoing the entire computation.
I'm no biochemist, so I'm not sure about this, but I think you could do that. Folding proteins is about finding states with low energy (because in nature, proteins occur in the state that has the lowest energy). So you could encode the information in a polypeptide, have some energy limit that you have to fit in order to generate the block and fold the polypeptide until you find a state with low enough energy. Verifying what energy does a certain state have should be relatively fast.
On the other hand, such computations probably don't have any value to biochemists, so this would just replace one useless computation with another.