Evolutionary dynamics of a two-gene model for chemical reactions, corresponding to gene expression and regulation, has been studied in detail. Bifurcation analysis has been carried out to understand behavior of steady state solutions leading to chaotic evolution for different cases. Numerical simulations have been performed and measurable quantities like Lyapunov exponents, topological entropies and correlation dimensions have been calculated for certain sets of parameter values. These measures explain complexity and chaotic nature of evolution. © 2019 Forum-Editrice Universitaria Udinese SRL. All Rights Reserved.