This paper tackles the problem of recovering time-varying gene networks from a series of undersampled and noisy observations. Gene regulatory networks evolve over time in response to functional requirements in the cell and environmental conditions. Collected genetic profiles from dynamic biological processes, such as cell development, cancer progression and treatment recovery, underlie genetic interactions that rewire over the course of time. We formulate the problem of estimating time-varying networks in a state-space framework. We show that, due to the small number of measurements, the system is unobservable; thus making the application of the standard Kalman filter ineffective. We remedy the problem by performing simultaneous compression and state estimation. The sparsity property of gene regulatory networks is incorporated as a constraint in the Kalman filter, leading to a compressed Kalman estimate and reducing the number of required observations for effective tracking of the network. Moreover, we improve the estimation accuracy by taking into account the entire sample set for each time instant estimate of the network through a forward-backward smoothing procedure. The proposed constrained and smoothed Kalman filter is shown to yield good tracking results for varying small and medium-size networks.