Stochastic Modeling of the Relationship between Copy Number and Gene Expression Based on Transcriptional Logic

DNA copy number alterations (CNAs) can cause genetic diseases, and studies have revealed a relationship between CNAs and gene expression; however, the manner in which CNAs relate to gene expression and what regulatory mechanisms underlying the relationship remain unclear. In many instances, real data have revealed a nonlinear relationship between copy number and gene expression. In this paper, queueing theory is used to model this relationship, with the basic structural parameters involving transcription factor (TF) arrival and departure rates. A key finding is that the ratio of TF arrival rate to TF departure rate is critical: small and large ratios corresponding to nonlinear and linear relationships, respectively. Indeed, copy number amplifications do not necessarily lead to expression increases: when one of the regulatory TFs exists in a small amount, copy number gains can cause a down regulation. Using the concept of mutual information, we show that the TF with minimum activation probability can have maximum dependence in regulation: a TF in small amount could result in a nonlinear copy-number-gene-expression relationship and play a major role in regulation. The expectation-maximization algorithm is used to estimate the ratio of TF arrival rate to TF departure rate. The theoretical results are illustrated via simulations.