Abstract
Gene expression is influenced by extrinsic noise (involving a fluctuating environment of cellular processes) and intrinsic noise (referring to fluctuations within a cell under constant environment). We study the standard model of gene expression including an (in-)active gene, mRNA and protein. Gene expression is regulated in the sense that the protein feeds back and either represses (negative feedback) or enhances (positive feedback) its production at the stage of transcription. While it is well-known that negative (positive) feedback reduces (increases) intrinsic noise, we give a precise result on the resulting fluctuations in protein numbers. The technique we use is an extension of the Langevin approximation and is an application of a central limit theorem under stochastic averaging for Markov jump processes (Kang et al. in Ann Appl Probab 24:721–759, 2014). We find that (under our scaling and in equilibrium), negative feedback leads to a reduction in the Fano factor of at most 2, while the noise under positive feedback is potentially unbounded. The fit with simulations is very good and improves on known approximations.
Highlights
It is widely accepted that gene expression is a stochastic process
While negative feedback is known to reduce noise under auto-regulated gene expression, we improve on the quantification of this effect, i.e. our results account for all possible sources of noise due to gene activation, mRNA fluctuations and the protein processes itself
We provide the same quantification of noise for positive feedback, where noise is increased
Summary
It is widely accepted that gene expression is a stochastic process. The reason is that a single cell is a system with only one or two copies of each gene and of the order tens for mRNA molecules (Swain et al 2002; Elowitz et al 2002; Raj and van Oudenaarden 2008). This stochasticity can even be observed directly by single-cell measurements such as flow cytometry and fluorescence microscopy, which show the inherent fluctuations of protein numbers arising from cell to cell (Li and Xie 2011). Ensemble averages eliminate intrinsic noise, while single-cell measurements over time can be thought of having a constant environment, eliminating extrinsic noise (Singh and Soltani 2013; Singh 2014)
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