  Using an analytically solvable stochastic model, we study the properties of a
simple genetic circuit consisting of multiple copies of an 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
non-trivially depend on the inherent noisiness of a given gene, quantified by
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 may be ambiguous as they are
dependent on the particular function chosen to quantify noise.
