A challenge facing biology is to develop quantitative, predictive models of gene regulation. Eukaryotic promoters contain transcription factor binding sites of differing affinity and accessibility, but we understand little about how these variables combine to generate a fine-tuned, quantitative transcriptional response. Here we used the PHO5 promoter in budding yeast to quantify the relationship between transcription factor input and gene expression output, termed the gene-regulation function (GRF). A model that captures variable interactions between transcription factors, nucleosomes and the promoter faithfully reproduced the observed quantitative changes in the GRF that occur upon altering the affinity of transcription factor binding sites, and implicates nucleosome-modulated accessibility of transcription factor binding sites in increasing the diversity of gene expression profiles. This work establishes a quantitative framework that can be applied to predict GRFs of other eukaryotic genes.