Influence of gene copy number on self-regulated gene expression

Using an analytically solvable stochastic model, we study the properties of a simple genetic circuit consisting of multiple copies of a self-regulating gene. We analyse how the variation in gene copy number and the mutations changing the auto-regulation strength affect the steady-state distribution of protein concentration. We predict that one-reporter assay, an experimental method where the extrinsic noise level is inferred from the comparison of expression variance of a single and duplicated reporter gene, may give an incorrect estimation of the extrinsic noise contribution when applied to self-regulating genes. We also show that an imperfect duplication of an auto-activated gene, changing the regulation strength of one of the copies, may lead to a hybrid, binary+graded response of these genes to external signal. The analysis of relative changes in mean gene expression before and after duplication suggests that evolutionary accumulation of gene duplications may, at a given mean burst size, non-trivially depend on the inherent noisiness of a given gene, quantified by the inverse of the maximal mean frequency of bursts. Moreover, we find that the dependence of gene expression noise on gene copy number and auto-regulation strength may qualitatively differ, e.g. in monotonicity, depending on whether the noise is measured by Fano factor or coefficient of variation. Thus, experimentally-based hypotheses linking gene expression noise and evolutionary optimisation in the context of gene copy number variation may be ambiguous as they are dependent on the particular function chosen to quantify noise.