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By assuming the Dirichlet boundary conditions, aiming to estimate the mRNA and protein concentrations via available network measurements. Firstly, sufficient F\u2010T stability conditions for the filtering error system have been investigated via constructing an appropriate Lyapunov\u2013Krasovskii functional (LKF) and using several integral inequalities and (reciprocally) convex technique simultaneously. These conditions are delay\u2010dependent and reaction\u2010diffusion\u2010dependent and can be checked by MATLAB toolbox. Furthermore, a method is proposed to design an <jats:italic>r<\/jats:italic>\u2010order F\u2010T state observer, and the explicit expressions of observer gains are given. 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