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Molecular noises in gene networks come from intrinsic fluctuations, transmitted noise from upstream genes, and the global noise affecting all genes. Knowledge of molecular noise filtering in gene networks is crucial to understand the signal processing in gene networks and to design noise-tolerant gene circuits for synthetic biology. A nonlinear stochastic dynamic model is proposed in describing a gene network under intrinsic molecular fluctuations and extrinsic molecular noises. The stochastic molecular-noise-processing scheme of gene regulatory networks for attenuating these molecular noises is investigated from the nonlinear robust stabilization and filtering perspective. In order to improve the robust stability and noise filtering, a robust gene circuit design for gene networks is proposed based on the nonlinear robust stochastic stabilization and filtering scheme, which needs to solve a nonlinear Hamilton-Jacobi inequality. However, in order to avoid solving these complicated nonlinear stabilization and filtering problems, a fuzzy approximation method is employed to interpolate several linear stochastic gene networks at different operation points via fuzzy bases to approximate the nonlinear stochastic gene network. In this situation, the method of linear matrix inequality technique could be employed to simplify the gene circuit design problems to improve robust stability and molecular-noise-filtering ability of gene networks to overcome intrinsic molecular fluctuations and extrinsic molecular noises.