# Why models of stochastic gene expression predict that intrinsic noise should increase as the amount of transcript decrease

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.

1. Elowitz, M. B., Levine, A. J., Siggia, E. D. & Swain, P. S. Stochastic gene expression in a single cell. Science 297, 1183–1186 (2002).

Reading Kaufmann and van Oudenaarden (2007), it seems to validate the first alternative (using results from the Central Limit Theorem):

Although biochemical fluctuations influence all stages of gene expression, those involving molecules in extremely low abundance are expected from a statistical standpoint to be larger in magnitude and therefore to contribute disproportionately to the overall variation between cells.

1. Kaufmann, B. B. & van Oudenaarden, A. Stochastic gene expression: from single molecules to the proteome. Curr. Opin. Genet. Dev. 17, 107–112 (2007).

The inverse relationship between variance and mean is a consequence of the stochastic nature of gene expression, which can be modeled as a discrete birth-death process. One of the simplest models assumes that the gene is expressed constitutively. Then, using the chemical master equation, mRNA production can be shown to evolve as a discrete Poisson process. The steady-state distribution is Poisson. For the Poisson distribution, coefficient of variation ($$\sigma^2/\mu^2$$) is equal to $$1/\mu$$. Further, if protein is translated in bursts from mRNA (translation burst), protein noise can be written as

$$$$\label{eq1}\tag{1} \eta^2_P = \frac{\sigma_P^2}{\mu_P^2} = \frac{B}{\mu_P},$$$$

where $$\eta_P^2, \sigma_P^2,\mu_P,$$ and B are the coefficient of variation, variance, mean and burst size for protein. From \eqref{eq1}, the inverse relationship is evident.

Gene expression models can be made more detailed by assuming different states for the promoter. The promoter switches between these states, and each of these states can have different rates of transcription. For these models, $$\eta_P^2$$ can again be shown to follow an inverse relationship with $$\mu_P^2$$.

Reference:

Paulsson, Johan. "Models of stochastic gene expression." Physics of life reviews 2.2 (2005): 157-175.