The theoretical approach used in this study, originally developed to describe gene regulation by the lambda bacteriophage repressor CI (Ackers et al., 1982), relies on statistical thermodynamics assumptions to model initiation of transcription. (a) In this framework, each DNA-binding protein is assigned a binding energy (Ei) to an arbitrary stretch of DNA. Given a set of DNA-binding proteins (a generic RNAP-like and a generic repressor-like TF, in this case) and a promoter sequence, a Boltzmann weight can be assigned to any configuration of these TFs on the promoter. By assigning a Boltzmann weight to all configurations, one can calculate the probability of finding the system in a configuration that leads to the initiation of transcription. (b) When considering only the binding of a single protein to DNA (for example ‘RNAP’ only), if mutations have a negative effect on protein-DNA binding, the model predicts negative epistasis between them in terms of expression. This prediction arises from the non-linear relationship between binding energy and gene expression pon (dotted line). In this illustration, we show a relative change in binding energy compared to the sequence with highest possible binding, in kT. (c) By generalizing the properties of the relationship between binding and gene expression, we conclude that the sign of epistasis depends only on the sign of individual mutation effects (p1 and p2) upon binding. When both ‘RNAP’ and ‘repressor’ are present in the system, epistasis depends on the ‘repressor’ concentration and the magnitude of single mutation effects on ‘RNAP’ and ‘repressor’ binding (d,e,f,g). (d) One point mutation negatively affects only ‘RNAP’ binding, while the other only ‘repressor’ binding. (e) Under such circumstances, the system shows no epistasis at low ‘repressor’ concentrations, but is in positive epistasis when ‘repressor’ concentration increases. Finally, at very high repressor concentrations, epistasis approaches 0. (f) Point mutations negatively affect both ‘RNAP’ and ‘repressor’ binding. (g) Under such conditions, epistasis changes the sign from negative to positive as repressor concentration increases.