This paper explores a possibility of constructing massively parallel molecular computing systems using molecular electronic devices called enzyme transistors. The enzyme transistor is, in a sense, an artificial catalyst which selects a specific substrate molecule and transforms it into a specific product. Using this primitive function, various active continuous media for signal transfer/processing can be realized. Prominent examples discussed in this paper are: (i) Turing pattern formation and (ii) excitable wave propagation in a two-dimensional enzyme transistor array. This paper demonstrates the potential of enzyme transistors for creating reaction-diffusion dynamics that performs useful computations in a massively parallel fashion.