(A) Example of additive vs. multiplicative effects on expression of hypothetical gene X, mathematical formulation of the combined response factor, and illustration of how the value of the combined response factor (c value) reflects whether a combined gene expression response is sub-additive, additive, multiplicative, or super-multiplicative. (B) Schematic of signal response experiments in MCF-7 cells. Briefly, we treated MCF-7 cells with three different dosages of retinoic acid, TGF-β, or both signals for 72 hr, then performed bulk RNA-seq and ATAC-seq at the endpoint. We show the number of differentially expressed genes and peaks for each dose of each condition as well as the overlap between the sets of differentially expressed genes and differential peaks. (C) Five example genes representing sub-additive to super-multiplicative combined transcriptional responses, where we show each gene’s transcripts per million (TPM) value for each replicate after single or combined signal treatments. Horizontal gray bars show the average TPM value, and error bars represent the 80% confidence interval of the estimated underlying Gaussian distribution of each dosage and condition (see Materials and methods for parameter estimation details). (D) Illustrated definition of master set of upregulated genes. (E) Frequency of each type of combined response behavior for each dosage in the master set of genes. (F) Simulated, observed, and residual histograms of c value distributions for the medium and high doses. In the simulated mixture model, we randomly simulated combined responses to be either additive or multiplicative based on the relative frequency of additive vs. multiplicative combined transcriptional responses that we observed at each dose in 1E. Annotated percentages at broken bars represent the fraction of c values in the tail beyond the limits of the x axis of the graph. *For all c value analyses, 14 genes with a control TPM of zero were removed from the master set of genes, as they end up misleadingly having c values of exactly 0 regardless of the effects of retinoic acid and TGF-β.