Negative gene regulation is when something (a protein/transcription factor) binds to the regulatory sequences of a gene (either a promoter or operator) and represses that gene's expression (i.e. makes it so RNA is not produced). When the gene's not repressed, it's being used as a template for RNA synthesis and so there will be a lot of that RNA in the cell, which will then in turn bind to ribosomes and the RNA will be translated into protein. When expression is repressed, no new RNA will be made by that gene, so the level of that RNA in the cell will go down (this depends on how stable the individual RNA molecules are) and in turn the protein level will go down (this depends on the protein's stability). RNAs and proteins that are more stable with kick around in the cell longer, so the effect of the negative gene regulation will take longer to become apparent. The more unstable a protein or RNA molecule is, the faster the effect of negative regulation will become apparent.
Operons are groups of genes and their associated regulatory sequences that are expressed as a unit (they're referred to as "polycistronic" meaning that multiple proteins are encoded by one RNA molecule; this almost never happens in eukaryotes). Most known operons have a role in either synthesizing or using certain molecules the cell needs to survive. To conserve energy, however, they'll only be turned on when their activity is required by/useful for the cell.
The two most well-characterized operons are probably the lac operon and the trp operon.
The lac operon is responsible for transporting lactose into bacteria and then subsequently metabolizing it for energy. As such, it's only useful for the cell to make the proteins encoded in the lac operon when lactose is available (otherwise, the cell would be wasting energy to make proteins it has no use for). The lacI gene on the operon is always active and produces a protein called the lac repressor. In the absence of lactose, the lac repressor binds that lac operator and shuts off expression of the other genes in the operon (lacZ, lacY and lacA). When lactose is around, some of it spontaneously converts to allolactose; allolactose then binds to the lac repressor, changing its shape and preventing it to bind to the lac operator, so expression of lacZ, Y and A are activated. This is, therefore, a positive feedback loop.
The trp operon, on the other hand, works by negative feedback inhibition. This operon is responsible for synthesis of tryptophan (an amino acid), and as such, is only useful when the cell lacks tryptophan. In the trp operon, the trpR gene encodes the trp repressor. The gene is always active and the trp repressor proteins form tetramers. If present, tryptophan binds to the tetramer, making it able to bind DNA and turn off expression of the operon. When tryptophan is missing, the tetramers do not bind DNA, so the operon is expressed.
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