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. 2018 Oct 5;14(10):e1006386.
doi: 10.1371/journal.pcbi.1006386. eCollection 2018 Oct.

Noise Propagation in an Integrated Model of Bacterial Gene Expression and Growth

Free PMC article

Noise Propagation in an Integrated Model of Bacterial Gene Expression and Growth

Istvan T Kleijn et al. PLoS Comput Biol. .
Free PMC article


In bacterial cells, gene expression, metabolism, and growth are highly interdependent and tightly coordinated. As a result, stochastic fluctuations in expression levels and instantaneous growth rate show intricate cross-correlations. These correlations are shaped by feedback loops, trade-offs and constraints acting at the cellular level; therefore a quantitative understanding requires an integrated approach. To that end, we here present a mathematical model describing a cell that contains multiple proteins that are each expressed stochastically and jointly limit the growth rate. Conversely, metabolism and growth affect protein synthesis and dilution. Thus, expression noise originating in one gene propagates to metabolism, growth, and the expression of all other genes. Nevertheless, under a small-noise approximation many statistical quantities can be calculated analytically. We identify several routes of noise propagation, illustrate their origins and scaling, and establish important connections between noise propagation and the field of metabolic control analysis. We then present a many-protein model containing >1000 proteins parameterized by previously measured abundance data and demonstrate that the predicted cross-correlations between gene expression and growth rate are in broad agreement with published measurements.

Conflict of interest statement

The authors have declared that no competing interests exist.


Fig 1
Fig 1. Integrated model of stochastic gene expression and cell growth.
The cell contains many protein species, with proteome mass fractions ϕi that sum to 1. Mass fractions are increased by protein synthesis but diluted by growth. The synthesis rate πi of each species i is modulated by a noise source Ni. The instantaneous growth rate μ reflects the total rate of protein synthesis. Proteins affect metabolism and thus the deterministic growth rate μd(ϕ), as quantified by growth-control coefficients Ciμ. A fraction fi of the total metabolic flux is allotted to the synthesis of protein i. The inherent noise in the expression of each gene reverberates through the cell, affecting cell growth and the expression of every other gene.
Fig 2
Fig 2. Limitations of the operational definition of in- and extrinsic expression noise.
(A) Extrinsic noise is measured by the covariance between the expression levels of two identical reporter systems R and G. This presupposes that the intrinsic noise NR of system R affects concentration ϕR but not ϕG (orange outline), so that the covariance between ϕR and ϕG quantifies the contribution of extrinsic sources Next,i. (B) But in our model, NR affects the growth rate and thus the dilution of ϕG. This adds a negative term to the covariance, which no longer measures just the extrinsic noise.
Fig 3
Fig 3. Noise modes in a toy model containing only two protein species, X and Y.
(A) Analytical solution for the cross-correlation between protein Y’s proteome fraction ϕY and growth rate μ (gray curve), verified by simulations (gray diamonds, details in S1 Text, p. 9). The contributing noise modes are indicated (colored curves). (B) Same as (A), but for the synthesis rate πY. The cross-correlation functions are linear combinations of three classes of functions, called Ai(τ), Bi(τ), and Si(τ) (see S1 Text, equations (47)–(49) for their definitions). In panels (A) and (B), noise modes that are proportional to just one of these functions are annotated accordingly. (C)–(F) Noise propagation routes underlying the noise modes. The control mode and the autogenic mode arise from noise source NY alone. Both noise sources NX and NY contribute to the dilution and transmission modes, but only the contribution of NX is illustrated in Fig (D) and (F). Parameters for (A) and (B): CYμ=0.25; ϕ0,Y = 0.33; mean growth rate μ0 = 1 h−1; noise sources of NY and NX have amplitudes θY = 0.5 and θX = 0.5 and reversion rates βY = βX = 4μ0.
Fig 4
Fig 4. Expression–growth cross-correlations in the many-protein model.
(A) Cartoon of the noise propagation network. (B) Monod curve describing the mean growth rate as a function of lac expression. Black dots indicate the operon mass fractions and growth rate used to calculate the cross-correlations in (D)-(F). (C) Noise distribution of the proteome (gray cloud) taken from Ref. [51], and the values chosen for proteins on the lac operon (black dots). Green dashed lines are guides for the eye. (D)–(F) Experimental [5] (top panels) and theoretical (middle and bottom panels) cross-correlations for three growth conditions. Proteome fraction–growth and production–growth cross-correlations are plotted as solid and dashed black lines, respectively. As in Fig 3A and 3B, colored lines show the contributing noise modes.
Fig 5
Fig 5. Deceptive concentration–growth cross-correlations.
(A) Positive Pearson correlation despite a negative operon GCC, due to a dominant autogenic mode. Same parameters as Fig 4F, but with COμ=-0.035. (B) Negative Pearson correlation despite a positive operon GCC, due to noisy GFP expression. Same parameters as Fig 4F, but with operon noise much smaller than GFP noise (see Materials and methods).

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LHJK was supported by the NWO (Nederlandse Organisatie voor Wetenschappelijk Onderzoek, (Grant 022.005.023). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.