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The Trp operon has 2 methods to check for presence of tryptophan, one indicating the absence of free Trp and the other the absence of charged Trp tRNA. If both methods showed absence of Trp there is tryptophan synthesis. However, the synthesis of tryptophan involves 5 enzymes to be synthesized, and usually we find all the amino acids in a protein such as an enzyme. Therefore, there is high probability that Trp is found in these 5 enzymes. Where does that Trp come from if there is neither free tryptophan nor charged tRNA for Trp?

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You're right that if there was really a complete lack of Trp, then the cell wouldn't be able to make the enzymes. But the cell doesn't have to wait until [Trp] drops all the way to zero -- it can induce expression of the operon when the concentration gets sufficiently low (in practice there's some continous relationship between [Trp] and expression). Note that there's always going to be a little bit of Trp around as there are constantly proteins being degraded (and resynthesized).

But you're on to something -- in fact there is some bias towards lower usage of a particular amino acid in enzymes that are responsible for synthesizing that amino acid. Here's the paper: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1839009/ although the effect isn't that high for Tryptophan.

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