We conduct numerical analysis of the 2-dimensional discrete-time gene expression model originally introduced by Andrecut and Kauffman (Phys. Lett. A 367: 281–287, 2007). In contrast to the previous studies, we analyze the dynamics with different reaction rates α1 and α2 for each of the two genes under consideration. We explore bifurcation diagrams for the model with α1 varying in a wide range and α2 fixed. We detect chaotic dynamics...