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. 2019 Jan 24;9(1):474.
doi: 10.1038/s41598-018-35927-x.

Intrinsic and extrinsic noise of gene expression in lineage trees

Philipp Thomas  1
Affiliations

Intrinsic and extrinsic noise of gene expression in lineage trees

Philipp Thomas. Sci Rep. .

Abstract

Cell-to-cell heterogeneity is driven by stochasticity in intracellular reactions and the population dynamics. While these sources are usually studied separately, we develop an agent-based framework that accounts for both factors while tracking every single cell of a growing population. Apart from the common intrinsic variability, the framework also predicts extrinsic noise without the need to introduce fluctuating rate constants. Instead, extrinsic fluctuations are explained by cell cycle fluctuations and differences in cell age. We provide explicit formulas to quantify mean molecule numbers, intrinsic and extrinsic noise statistics in two-colour experiments. We find that these statistics differ significantly depending on the experimental setup used to observe the cells. We illustrate this fact using (i) averages over an isolated cell lineage tracked over many generations as observed in the mother machine, (ii) population snapshots with known cell ages as recorded in time-lapse microscopy, and (iii) snapshots with unknown cell ages as measured from static images or flow cytometry. Applying the method to models of stochastic gene expression and feedback regulation elucidates that isolated lineages, as compared to snapshot data, can significantly overestimate the mean number of molecules, overestimate extrinsic noise but underestimate intrinsic noise and have qualitatively different sensitivities to cell cycle fluctuations.

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Conflict of interest statement

The author declares no competing interests.

Figures

Figure 1
Figure 1
Single-cell statistics of cell populations and isolated lineages. (a) Distributions of the molecule numbers x measured in a cell of age τ across a lineage tree from time-lapse movies. The population dynamics is described by the age distribution Π(τ) and the distribution of interdivision times ρ(τ). Gene expression dynamics is described by the distribution Π(x|τ) for cells of age τ (open grey circles). Accordingly, the distributions are Π(x|0) at cell birth (open black circles) and Eρ[Π(x|τd)] at cell division (filled black circles). The final state of the population (red dots) describes a snapshot with distributed cell ages EΠ[Π(x|τ)] as obtained from static images or flow cytometry. (b) The corresponding statistics for a lineage following an isolated cell over time as observed in the mother machine.
Figure 2
Figure 2
Agent-based model of clonal population dynamics with stochastic gene expression and cell cycle variability. (a) Illustration of a growing population as a branching process with stochastic interdivision times. Each cell expresses two identical but non-interacting reporters (green and red) that are partitioned randomly at cell division. Red and green cells express more molecules of either reporter, which indicates intrinsic variability between cells. Yellow cells express similar levels of reporter molecules, but their levels vary over the cell cycle resulting in extrinsic variability. A snapshot of the population (blue dashed box) quantifies the cell-to-cell variability across the population. A lineage (blue path) quantifies variability over time and tracks an isolated cell over successive cell divisions by randomly selecting one of the daughter cells. (b) Simulated trajectories of cell age and stochastic protein expression of two identical reporters on a branched tree. Line colour indicates reporter expression in the same cell. (c) Cell age and reporter expression of an isolated cell lineage. (d) Comparison of simulated distributions of lineages and population snapshots. Exemplary simulations of the reactions (29) assume k0 = 10, km = 1, ks = 10 for each reporter and lognormal-distributed interdivision times with unit mean and standard deviation.
Figure 3
Figure 3
Intrinsic and extrinsic noise propagation over the cell cycle. (a) Total noise as a function of cell age τ with gamma (top) and log-normal-distributed (bottom) interdivision times. Population snapshot statistics (solid) are compared to lineages (dashed lines). (b) Intrinsic noise peaks as a function of cell age and increases with cell cycle fluctuations in populations but not in lineages. (c) Extrinsic noise is lower in the population than in lineages. Parameters are k0 = 1, b = 100 and interdivision time distributions assume unit mean.
Figure 4
Figure 4
Statistics of population snapshots and isolated lineages for cells of unknown age. (a) Mean protein number as a function of the cell cycle variations CVφ[τd] in lineages (dashed) and snapshots (solid lines). For lineages, the mean protein number increases with cell cycle variability and is independent of the interdivision time distribution. In snapshots, the mean decreases with cell cycle variability with a rate that depends on higher moments of the distribution. The predictions for gamma- and log-normal distributed interdivision times are shown. (b) Sensitivity of intrinsic and extrinsic noise sources to cell cycle fluctuations. Intrinsic noise (red lines) increases in lineages but decreases in snapshots consistent with the dependence of the respective means shown in (a). The transmitted cell cycle noise (blue lines) shows a similar dependence on cell-cycle variability in lineages and snapshots for the gamma-distribution, but is lower in snapshots for the log-normal distribution. (c) Total noise (black lines) broken down into individual noise components for the gamma-distribution. Transmitted cell cycle noise and the uncertainty due to distributed cell ages (purple lines) contribute to the total extrinsic noise (teal). (d) The corresponding decomposition for the log-normal distribution. Parameters are k0 = 10 and b = 10 and Eφ[τd] = 1.
Figure 5
Figure 5
Noise decomposition of a negative feedback circuit. Sensitivities to cell cycle noise CVϕ2[τd] of mean, intrinsic and extrinsic noise contributions are shown for weak (yellow, K = 200), moderate (blue, K = 100) and strong feedback (red, K = 50). Predictions by the linear noise approximation (solid lines) are in good qualitative agreement with stochastic simulations (dots). (a) In lineages, the mean mRNA number always decreases with cell cycle variability while this is not true in snapshots for moderate to high feedback. (b) In contrast, protein levels always increase in lineages but decrease in snapshots. (c) The corresponding intrinsic noise profiles of mRNAs typically increase with cell cycle noise except in snapshots with strong feedback. (d) Intrinsic noise of proteins always increases with cell cycle noise in snapshots but not in lineages. (e,f) Total extrinsic noise increases with cell cycle variability for mRNAs and proteins. However, strong feedback may significantly reduce extrinsic noise in snapshots compared to lineages. Deviations between the approximation (lines) and the simulations (dots) are most pronounced for strong feedback. Parameters are k0 = 10, km = 1, ks = 10 and log-normal distributed interdivision times with unit-mean.
Figure 6
Figure 6
Feedback strategies for noise suppression in lineages and populations. Intrinsic and extrinsic noise statistics of negative autoregulatory feedback circuit are shown as a function of K, the inverse feedback strength, for three different levels of cell cycle noise CVϕ2[τd]=0.1 (red), 0.25 (blue) and 0.75 (yellow). (a) Intrinsic noise exhibits a minimum as a function of the repression strength both in lineage (dashed) and in snapshot statistics (solid lines). The predictions obtained using the linear noise approximation (lines) are in good agreement with exact stochastic simulations using the First-Division Algorithm (dots for population, open circles for lineages). (b) Optimal feedback strength (1/K) to minimise intrinsic noise is shown. The feedback strength increases with cell cycle noise in lineages but decreases in population snapshots. (c) The transmitted cell cycle noise shows a minimum in dependence of the repression strength both in lineage (dashed) and in snapshot statistics (solid lines). (d) The optimal feedback strength to minimise transmitted cell cycle noise decreases with interdivision time noise both in lineages and to a lesser extent in the population. Parameters as in Fig. 5.
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