Before revision:  We first show that starting from a full Delbr\"uck-Gillespie-process description of the gene regulation network, two much simpler stochastic biochemical kinetic systems can be deduced, respectively, in the limit of either the stochastic gene-state switching or transcription factor copy-number fluctuation being dominant among other sources of noise.
After revision:  We first show that starting from a full chemical-master-equation description we derive a model in which the protein copy-number, for a given gene state, follow a deterministic mean-field description while the protein synthesis rates fluctuate due to stochastic gene-state switching or transcription factor copy-number fluctuation being dominant among other sources of noise.