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Living cells are characterized by small populations of key molecular components that have large stochastic noise associated with them. Various gene network motifs exist within cells that help reduce these stochastic fluctuations. A common such motif is an auto-regulatory gene network where the protein expressed from the gene inhibits its own transcription. Here the transcription rate of the gene is given as some function of the number of protein molecules present in the cell. We refer to this function as the transcriptional response of the gene network. We develop analytical formulas that relate the stochastic fluctuations in protein numbers with the functional form of the transcriptional response. This is done by first approximating the transcriptional response by a polynomial and then using recently developed moment closure techniques to solve for the statistical moments of the protein population. We show that the protein noise level in these auto-regulatory gene networks is related to the stability of the network and increasing (decreasing) stability leads to attenuation (magnification) of protein noise. Using the above formulas we also investigate the transcriptional response of a specific gene network in lambda phage and show that this network is especially effective at reducing stochastic fluctuations in protein levels.