Predicting gene expression from DNA sequence remains a major goal in the field of gene regulation. A challenge to this goal is the connectivity of the network, whose role in altering gene expression remains unclear. Here, we study a common autoregulatory network motif, the negative single-input module, to explore the regulatory properties inherited from the motif. Using stochastic simulations and a synthetic biology approach in E. coli, we find that the TF gene and its target genes have inherent asymmetry in regulation, even when their promoters are identical; the TF gene being more repressed than its targets. The magnitude of asymmetry depends on network features such as network size and TF binding affinities. Intriguingly, asymmetry disappears when the growth rate is too fast or too slow and is most significant for typical growth conditions. These results highlight the importance of accounting for network architecture in quantitative models of gene expression.