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The mathematical study of biological networks has become an important research field. In particular, the study of time lags or delays has played a significant role in the dynamics of gene network models. In principle, the influence of such delays can result in oscillatory mRNA and protein expressions. I will describe a gene network model and its dynamical behavior close to a Hopf bifurcation. This will be accomplished by using perturbation methods and a center manifold reduction on the system. The final outcome results in closed form expressions for the limit cycle amplitude and frequency of oscillation. These results provide analytical evidence that delays can drive oscillations in gene activity.