We explore minimalist models of transcription in which we take into account that a cis-regulatory sequence is embedded in, and interacts with, a complex genome. The classical Hill equation is the simplest way to represent a transcriptional response. However, it may overlook the fact that a transcription factor (TF) establishes specific and nonspecific nonfunctional interactions with chromatin. Classical papers have shown that nonfunctional binding (not leading to transcription) may influence gene expression. We examine how the presence of additional binding sites for a TF, besides those on the gene(s) of interest, affect the shape and parameters of the transcriptional response. We consider two conditions: at equilibrium and at steady-state. In many cases the TF level is determined by the position of the cell within a spatial or temporal gradient. We show that such gradients can be adjusted by evolutionary selection to compensate for the alteration of the gene transcription response by the presence of nonfunctional binding sites. Finally, we analyse how the transcriptional response is affected by a decrease in TF concentration, as in cases of haploinsufficiency. We show that the nonlinearity of the transcriptional response as a function of [TF] exacerbates the effect of a decrease in the latter, at least for weakly expressed TFs. Although decades of work on TFs have led to the impression that almost everything is known about the control of gene expression, we show that even the simplest models of transcription control have not delivered all their secrets yet.