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. 2021 May;18(178):20210274.
doi: 10.1098/rsif.2021.0274. Epub 2021 May 26.

Coordination of gene expression noise with cell size: analytical results for agent-based models of growing cell populations

Philipp Thomas  1 , Vahid Shahrezaei  1
Affiliations

Coordination of gene expression noise with cell size: analytical results for agent-based models of growing cell populations

Philipp Thomas et al. J R Soc Interface. 2021 May.

Abstract

The chemical master equation and the Gillespie algorithm are widely used to model the reaction kinetics inside living cells. It is thereby assumed that cell growth and division can be modelled through effective dilution reactions and extrinsic noise sources. We here re-examine these paradigms through developing an analytical agent-based framework of growing and dividing cells accompanied by an exact simulation algorithm, which allows us to quantify the dynamics of virtually any intracellular reaction network affected by stochastic cell size control and division noise. We find that the solution of the chemical master equation-including static extrinsic noise-exactly agrees with the agent-based formulation when the network under study exhibits stochastic concentration homeostasis, a novel condition that generalizes concentration homeostasis in deterministic systems to higher order moments and distributions. We illustrate stochastic concentration homeostasis for a range of common gene expression networks. When this condition is not met, we demonstrate by extending the linear noise approximation to agent-based models that the dependence of gene expression noise on cell size can qualitatively deviate from the chemical master equation. Surprisingly, the total noise of the agent-based approach can still be well approximated by extrinsic noise models.

Keywords: agent-based modelling; chemical master equation; single-cell analysis; stochastic gene expression.

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Figures

Figure 1.
Figure 1.
Modelling approaches for cell size dependence of gene expression. (a) The effective dilution model describes cells at constant size with intracellular reactions coupled to effective dilution reactions. (b) The extrinsic noise model incorporates static cell size variability as a source of extrinsic noise coupled with effective dilution models. (c) The agent-based approach models intracellular reactions occurring across a growing and dividing cell population without the need for effective dilution reactions.
Figure 2.
Figure 2.
Distributions of CME and agent-based models agree for reaction networks with stochastic concentration homeostasis. (a–c) The EDM (solid lines, analytical solution [53,54]) agrees with agent-based simulations (shaded areas) for a range of gene expression models. Panels show (a) bursty transcription [53] with transcription rate proportional to cell size, (b) bursty translation [54], and (c) bursty transcription and translation [54] with geometrically distributed bursts ms whose average is proportional to cell size s (see main text for details). (d) mRNA distributions simulated using the ABM (shaded areas) are shown for cells of sizes s = s0 (red), s = 1.5s0 (orange) and s = 2s0 (green), which agree with the effective dilution model (dots, Poisson distribution). (e) Simulated protein distributions (shaded areas) disagree with the effective dilution model (solid lines, solution in [55]). (f) Absolute error (ℓ1) of the effective dilution model as a function of cell size for mRNA (teal) and protein (red) distributions. ABM simulations were obtained using the First-Division Algorithm (box 1) assuming an adder model (a = 1) and parameters k0 = 10, kdm = 9, ktl = 100, α = 1. Cell cycle noise assumes gamma distribution φ~(Δ) with unit mean and CVφ[Δ] = 0.1, while division noise assumes symmetric beta distribution with CVπ[θ]=0.01.
Figure 3.
Figure 3.
Comparing the statistics of the effective dilution and agent-based models. (a) Simple model of mRNA transcription and protein translation transcriptional size-scaling (2.13). (b) Mean mRNA (top) and protein levels (bottom) agree with the EDM (solid grey lines) and ABM simulations (blue dots). (c) mRNA statistics display unit Fano factor indicating Poisson statistics in agreement with EDM. (d) ABM simulations (dots, box 1) display non-monotonic cell size scaling of protein noise, which are predicted by the agent-based theory (solid red) but not by the EDM (solid grey). Parameters are k0 = 10, kdm = 10, ktl = 100, α = 1. Cell size control parameters are as in figure 2.
Figure 4.
Figure 4.
Effect of noise in cell size control and division on gene expression noise in single cells. (a–d) Protein noise as a function of cell size s for various noise levels in added size CV[Δ]. For comparison, the prediction without cell cycle noise (dashed black line, equation (3.11)) and the cell size distributions (shaded grey) are shown. (e–h) Same as (a–d) but with division noise affecting the inherited size fraction CV[θ]. Analytical predictions (solid lines, equation (3.13a) with (3.11) and (3.13b)) and ABM simulations (dots) using the First-Division Algorithm (box 1) are shown. Gene expression model and all other parameters are as given in figure 3. Added size Δ assumes a gamma distribution with unit mean and CV[Δ] = 0.01 (e,f) while division errors θ followed a symmetric beta distribution with CV[θ] = 0.01 (a–d).
Figure 5.
Figure 5.
The extrinsic noise model approximates gene expression noise with size control and division errors. (a) Scaling of mRNA concentration noise with mean concentrations for various noise levels in added size CV[Δ] and partition noise CV[θ] when the transcription rate k0 is varied (top). Corresponding scaling is shown for mRNA numbers (bottom). Analytical predictions of the ENM (solid lines, equation (2.8) with (2.10) and (2.12)) and ABM simulations using the First-Division Algorithm (dots, box 1) are shown. The inset shows the relative error in CV of the EDM compared to ABM simulations [100%×(CVENM/CVABM−1)]. (b) Scaling of protein noise with mean protein concentration (top) and numbers (bottom) when the translation rate ktl is varied. (c) Same as (b) but varying the transcription rate ktl. For comparison, the ABM predictions without cell cycle noise are shown (dashed black lines, equation (3.16)) and the error bounds of 2% predicted by the theory (solid grey). See caption of figure 3 for the remaining parameters.
Figure 6.
Figure 6.
Retrospective averages of birth size. Matched asymptotic expansions of EΠ[s0|s] (equation (3.13b), lines) for the adder size control (a = 1, s0 = 1) and agent-based simulations (dots) for (a) varying size control noise CV[Δ] and (b) division errors CV[θ].
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