Are there ‘biologically computing agents’ capable to compute Turing uncomputable functions? It is perhaps tempting to dismiss this question with a negative answer. Quite the opposite, for the first time in the literature on molecular computing we contend that the answer is not theoretically negative. Our results will be formulated in the language of membrane computing (P systems). Some mathematical results presented here are interesting in themselves. In contrast with most speed-up methods which are based on non-determinism, our results rest upon some universality results proved for deterministic P systems. These results will be used for building “accelerated P systems”. In contrast with the case of Turing machines, acceleration is a part of the hardware (not a quality of the environment) and it is realised either by decreasing the size of “reactors” or by speeding-up the communication channels. Consequently, two acceleration postulates of biological inspiration are introduced; each of them poses specific questions to biology. Finally, in a more speculative part of the paper, we will deal with Turing non-computability activity of the brain and possible forms of (extraterrestrial) intelligence.