Careful accounting of extrinsic noise in protein expression reveals correlations among its sources

In order to grow and replicate, living cells must express a diverse array of proteins, but the process by which proteins are made includes a great deal of inherent randomness. Understanding this randomness—whether it arises from the discrete stochastic nature of chemical reactivity (“intrinsic” noise), or from cell-to-cell variability in the concentrations of molecules involved in gene expression, or from the timings of important cell-cycle events like DNA replication and cell division (“extrinsic” noise)—remains a challenge. In this article we analyze a model of gene expression that accounts for several extrinsic sources of noise, including those associated with chromosomal replication, cell division, and variability in the numbers of RNA polymerase, ribonuclease E, and ribosomes. We then attempt to fit our model to a large proteomics and transcriptomics data set and find that only through the introduction of a few key correlations among the extrinsic noise sources can we accurately recapitulate the experimental data. These include significant correlations between the rate of mRNA degradation (mediated by ribonuclease E) and the rates of both transcription (RNA polymerase) and translation (ribosomes) and, strikingly, an anticorrelation between the transcription and the translation rates themselves.

Figure 1

The central dogma of molecular biology for a replicating chromosome. Replication forks form at the origin of replication and proceed along both sides of the chromosome until they meet at the terminus. Each gene, depending on its location, gets copied at its own gene replication time $t_r$; prior to $t_r$ a single copy exists, and afterward two copies exist. The gene can be transcribed into mRNA, which in turn can be translated to form proteins.

Figure 2

(a) Analytical and simulated mRNA and protein statistics for a “median” E. coli gene as a function of the gene loci. Circles represent statistics calculated from 5000 simulated cell cycles. At the start of each cycle, the mRNA and protein copy numbers were drawn from a binomial distribution based on the final counts in the previous cycle. The simulated counts were normalized to account for cell growth and the fact that cell ages are exponentially distributed. The lines represent analytical results evaluated according to Eqs. (A47). (b) Simulated mRNA and protein traces with $\chi =0.5$. The thin black lines indicate five individual cell cycles, while the thick black lines indicate the mean of 5000 cell cycles. Red areas indicate $\pm 1$ SD.

Figure 3

Protein noise broken down by contributing source for the “median” gene assuming either that (a) all extrinsic noise sources act independently or (b) extrinsic noise sources exhibit correlations among themselves. Note that in (b), bars indicate the median noise contributions calculated using the top $0.5\%$ of the sampled correlation matrices, while the error bars indicate their respective median absolute deviations (see Sec. 4 for details).

Figure 4

Fitting our model to the Taniguchi et al. data set assuming that either (a) all extrinsic noise sources act independently (i.e., $\rho =\mathbb{1}$) or (b) extrinsic noise sources exhibit correlations among themselves (see Sec. 4). Black points represent experimental data from [2], while open circles represent the best fits from our model [colored by the squared fitting error, $\mathrm{\Delta }\left(\rho \right)$; see Eq. (C1)]. Note that in (b), open circles represent median values of $E\left[p\right]$, $\mathrm{Var}\left[p\right]/E{\left[p\right]}^2$, and $\mathrm{\Delta }\left(\rho \right)$, calculated using the top $0.5\%$ of the sampled correlation matrices (see Sec. 4 for details). Also noted is the mean squared fitting error, $\langle \mathrm{\Delta }\left(\mathbb{1}\right)\rangle$, when extrinsic noise sources are assumed to act independently, as well as the median mean squared fitting error for the best-performing matrices.

Figure 5

(a) Marginal distributions of off-diagonal terms in sampled correlation matrices. Blue areas indicate the distributions of all 50 000 matrices; red areas, the distributions of the best-performing matrices [those among the $0.5\%$ with the lowest associated $\langle \mathrm{\Delta }\left(\rho \right)\rangle$ values]. (b) Scatterplots of $\rho (k_t,k_r)$, $\rho (k_t,k_d)$, and $\rho (k_r,k_d)$ with respect to each other for the best-performing matrices.

Figure 6

Comparison of fit kinetic parameters using our model versus the earlier gamma distribution model [34]. (a) Comparison of fit transcription rates. (b) Comparison of fit translation rates. (c) Comparison of fit mRNA degradation rates. Points are colored by mean protein expression level. Diagonal red lines indicate perfect agreement. In all cases, the plotted “ext. noise fit” values represent each gene's median transcription, translation, and mRNA degradation values obtained by performing the fits described in Eq. (C1) using each of our best-performing correlation matrices, with the respective median absolute deviation shown by the vertical black lines. The plotted Gamma Fit values represent the analogous fits performed using Equation (5) rather than our $\rho$-dependent extrinsic noise model.

Figure 7

Comparison of experimental [2] and fit mean mRNA copy numbers. Points are colored by mean protein expression level and represent the median mean mRNA copy numbers computed over our bestperforming matrices, with their associated median absolute deviations shown by the black vertical lines. The solid red line indicates the line of perfect agreement, while the dashed red line indicates the line of perfect agreement if the Taniguchi mRNA counts had been scaled to 2400 total mRNA's per cell.