I am reading Elowitz et al. (2002). When talking about intrinsic noise, i.e. noise due to microscopic events that govern which reactions occur and in what order, it is stated that:
Models of stochastic gene expression predict that intrinsic noise should increase as the amount of transcript decreases.
Noise, defined in the article as the standard deviation divided by the mean, can be divided in intrinsic (due to microscopic events during transcription) and extrinsic (due to localization, state and concentration of other cellular components, e.g. polymerases).
I think the intrinsic noise should increase at low level of transcript amount because the average (mean) amount will be lower but deviation decrease is smaller than mean decrease. I don't know, however, if this is the actual reason (and why deviation decrease should be smaller than mean decrease) or if deviation decreases with the mean and there's another more likely explanation.
- Elowitz, M. B., Levine, A. J., Siggia, E. D. & Swain, P. S. Stochastic gene expression in a single cell. Science 297, 1183–1186 (2002).