Reservoir computing (RC) is a computational framework for neural network based information processing. Little work, however, has been conducted on adapting the structure of the neural reservoir. In this paper, we propose a developmental approach to structural self-organization in reservoir computing. More specifically, a recurrent spiking neural network is adopted for building up the reservoir, whose synaptic and structural plasticity are regulated by a gene regulatory network (GRN). Meanwhile, the expression dynamics of the GRN is directly influenced by the activity of the neurons in the reservoir. We term this proposed model as GRN-regulated self-organizing RC (GRN-SO-RC). Contrary to a randomly initialized and fixed structure used in most existing RC models, the structure of the reservoir in the GRN-SO-RC model is self-organized to adapt to the specific task using the GRN-based mechanism. To evaluate the proposed model, experiments have been conducted on several benchmark problems widely used in RC models, such as memory capacity and nonlinear auto-regressive moving average. In addition, we apply the GRN-SO-RC model to solving complex real-world problems, including speech recognition and human action recognition. Our experimental results on both the benchmark and real-world problems demonstrate that the GRN-SO-RC model is effective and robust in solving different types of problems.