Summary of a paper. Paulsson, J. (2004). Summing up the noise in gene networks. Nature, 427(6973), 415–418. https://doi.org/10.1038/nature02257 (original). Rights: © publisher — not reproduced. This is a short account of it in our own words; the work itself is not reproduced here.
Summing up the noise in gene networks¶
What this covers¶
How to compare noise in gene expression across different experiments. It speaks to the physics of stochastic gene networks and single-cell biology, at the point where several independent GFP studies had just produced measurements that looked hard to reconcile with one another.
The question¶
By the early 2000s several groups had measured cell-to-cell variability in gene expression using GFP reporters, each targeting a different stage of the central dogma: plasmid replication, transcriptional autorepression, and chromosomal expression in bacteria, yeast and two simultaneously-tagged genes in E. coli. The studies used different models, different noise measures and different biological systems, and seemed to point to different conclusions — for instance, about whether autorepression suppresses noise, where noise actually originates, and whether eukaryotes differ fundamentally from prokaryotes in how noise arises. Paulsson's question is whether these apparently conflicting pictures are really describing different phenomena, or whether they are special cases of one underlying relationship that the separate analyses obscured.
The approach¶
Paulsson models a generic two-species system — a fluctuating "environment" species \(X_1\) (for example mRNA or an upstream regulator) driving the births and deaths of a second species \(X_2\) (for example protein or plasmid copies) as a birth-and-death Markov process. Applying the \(\Omega\)-expansion, whose first two orders recover the macroscopic rate equations and a linear-noise (fluctuation-dissipation) approximation around a stable steady state, he derives a single compact equation for the relative variance of \(n_2\). It separates into an intrinsic term set by the average copy number of \(X_2\) and a logarithmic gain \(H_{22}\) (how strongly \(X_2\)'s own production and elimination respond to its own abundance — a scale-free elasticity borrowed from metabolic control analysis), and an extrinsic term built from \(X_1\)'s own noise, a ratio of logarithmic gains \(H_{21}/H_{22}\) that measures how much a change in \(X_1\) is transmitted to \(X_2\) relative to how strongly \(X_2\) corrects itself, and a time-averaging factor that depends on how the relaxation time of \(X_2\) compares with the correlation time of \(X_1\). Because the logarithmic gains are properties of the reaction rates themselves, they can often be read directly off the proposed kinetic model for a given system, which lets the same equation be reinterpreted for plasmid replication, transcription, translation or autorepression simply by substituting the appropriate gains.
What it found¶
Applying this single relation to the published studies, Paulsson argues that much of the apparent disagreement dissolves. Autorepression of plasmid replication and of transcription both reduce noise for the same two reasons in the model — faster normalized adjustment toward steady state suppresses intrinsic fluctuations, and reduced susceptibility to the upstream environment suppresses extrinsic fluctuations — which reconciles the replication and transcription experiments under one mechanism despite their different biology. For the transcriptional autorepression data, the predicted gain ratio (\(H_{21}/H_{22}\approx1/4\) under autorepression versus \(1\) without it) matches the observed roughly twofold drop in relative standard deviation. For chromosomal expression in Bacillus subtilis, a burst-size model with no free parameters predicts relative variance \(\approx 1+b\), where \(b\) is the average number of proteins made per transcript, and this fits the measured translation-dependence of the noise. Across systems, fully induced genes show relative standard deviations as low as 15% in both a prokaryote (B. subtilis) and a eukaryote (S. cerevisiae), and Paulsson argues the differences reported between studies trace mostly to differences in network design — which rate constants and upstream inputs were varied — rather than to a fundamental difference between how noise arises in prokaryotic and eukaryotic systems.
Limits and context¶
The unifying equation is a linear, first-order approximation valid near a stable fixed point; it is not built to capture genuinely nonlinear or dynamically disordered regimes, such as unstable self-replication without a template. Paulsson is explicit that some comparisons remain unsettled: the non-monotonic relation between transcription rate and noise reported for S. cerevisiae could reflect extra mechanisms specific to eukaryotes, such as chromatin remodelling or transcriptional reinitiation, but he states plainly that alternative explanations involving random association and dissociation of activators or repressors are equally consistent with the same data, and that "there are at this point no reasons to favour one interpretation over another." He also cautions that GFP is an imperfect reporter of single-cell plasmid copy number, limiting what the replication study can establish about mechanism, and that normalized noise measures can make the same underlying data look either monotonic or non-monotonic depending on which quantity is held fixed. He argues against the claim that eukaryotic and prokaryotic gene expression differ in kind, holding that no principled difference has actually been demonstrated by the data assembled so far, and calls for further experiments — including real-time tracking rather than population snapshots — to distinguish between the remaining candidate explanations.
Citation¶
Paulsson, J. (2004). Summing up the noise in gene networks. Nature, 427(6973), 415–418. https://doi.org/10.1038/nature02257. Available via Nature (subscription/institutional access) or from the library catalogue entry for this source.