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Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of chemical reactions and the small populations of proteins, mRNAs present inside cells. These fluctuations are usually reported in terms of the first and second order statistical moments of the protein populations. If the birth-death rates of the mRNAs or the proteins are nonlinear, then the dynamics of these moments generally do not form a closed system of differential equations, in the sense that their time-derivatives depends on moments of order higher than two. Recent work has developed techniques to obtain the two lowest-order moments by closing their dynamics, which involves approximating the higher order moments as nonlinear functions of the two lowest ones. This paper uses these moment closure techniques to quantify noise in several gene regulatory networks. In gene expression mechanisms in which a protein inhibits its own transcription, the resulting negative feedback reduces stochastic variations in the protein populations. Often the protein itself is not active and combines with itself to form an active multimer, which them inhibits the transcription. We demonstrate that this more sophisticated form of negative feedback (using multimerization) is more effective in suppressing noise. We also consider a two-gene cascade activation network in which the protein expressed by one gene activates another gene to express a second protein. Analysis shows that the stochastic fluctuations in the population of the activated protein increases with the degree of multimerization in the activating protein.