Background Movement of populations on fitness landscapes has been a problem of interest for a long time. While the subject has been extensively developed theoretically, reconciliation of the theoretical work with recent experimental data has not yet happened. In this work, we develop a computational framework and study evolution of the simplest transcription network between a single regulator, R and a single target protein, T. Results Through our simulations, we track evolution of this transcription network and comment on its dynamics and statistics of this movement. Significantly, we report that there exists a critical parameter which controls the ability of a network to reach the global fitness peak on the landscape. This parameter is the fraction of all permissible values of a biochemical parameter that can be accessed from its current value via a single mutation. Conclusions Overall, through this work, we aim to present a general framework for analysis of movement of populations (and particularly regulatory networks) on landscapes. Electronic supplementary material The online version of this article (10.1186/s12918-019-0704-0) contains supplementary material, which is available to authorized users.