Gene expression is controlled by networks of regulatory proteins that interact specifically with external signals and DNA regulatory sequences. These interactions force the network components to co-evolve so as to continually maintain function. Yet, existing models of evolution mostly focus on isolated genetic elements. In contrast, we study the essential process by which regulatory networks grow: the duplication and subsequent specialization of network components. We synthesize a biophysical model of molecular interactions with the evolutionary framework to find the conditions and pathways by which new regulatory functions emerge. We show that specialization of new network components is usually slow, but can be drastically accelerated in the presence of regulatory crosstalk and mutations that promote promiscuous interactions between network components.Gene networks evolve by transcription factor (TF) duplication and divergence of their binding site specificities, but little is known about the global constraints at play. Here, the authors study the coevolution of TFs and binding sites using a biophysical-evolutionary approach, and show that the emerging complex fitness landscapes strongly influence regulatory evolution with a role for crosstalk.