Fig 1.
Three discrete growth phases in metabolic pathways.
(A) A metabolic pathway consists of enzymes A and B that produce and consume the intracellular metabolite M, respectively. (B) Mathematical models predict three growth phases for combinations of metabolite production rate and demand δ. Color bar indicates normalized growth rate. Colored spots correspond to the colored lines in panel C. The black line represents the effect of experimental conditions changing extracellular lactose concentrations in E. coli causing intracellular lactose concentration changes because of LacY activity. Results are for two models of toxicity: metabolite buildup (left) and permease proton symport (right). (C) Mathematical models predict that growth is maximized below a threshold production rate (dashed line between cyan and yellow) past which no steady state exists. Increasing the rate of metabolite production (V+) translates the rate curves upward. When the rate is beyond the dashed line, there is a runaway buildup of metabolite and consequent toxic effects. Results are for two models of toxicity: metabolite buildup (left) and permease proton symport (toxic byproduct; right). The toxic byproduct model has two variables; we plot only the rate of toxic byproduct buildup for simplicity because it crosses the threshold at a lower V+.
Fig 2.
Predicted population growth rates for varying substrate import rates in simulated metabolic pathway models.
(A) Average single cell growth rates predicted with a stochastic model (N = 10,000 trajectories) with substrate toxicity. Intracellular metabolite production rates are controlled by changing the transcription rate of Enzyme A (ktA). The discrepancy with experimentally measured growth rates at high metabolite concentrations (Fig 3a) are discussed in Methods. (B) Correlations between single-cell molecule numbers and growth rates reveal mechanisms of growth arrest in stochastic simulations. Positive correlations imply that the molecule species improves growth, while negative correlations imply the opposite. The black dashed line indicates the parameter value predicted to give peak growth rate. Correlations where all trajectories enter growth arrest (making model predictions irrelevant) are not shown. (C) Predicted population densities (color bar) of cellular growth rate and metabolite from stochastic simulations in the low toxicity case. All trajectories in these three cases start with low metabolite concentrations. As the simulations proceed, trajectories exceeding a threshold concentration of metabolite transition from the low-metabolite subpopulation to the high-metabolite subpopulation. The predicted growth rate of the high-metabolite subpopulation decreases as metabolite continues to build up.
Table 1.
Reactions, propensities and parameter values used for stochastic simulations.
Fig 3.
Population and individual cell fitness of E. coli (lacI− B REL606) in varying growth conditions.
(A) Mean ± SEM (N = 3) growth rate (p < 10−6 for no trend in lactose concentrations > 1 mg/ml) and fraction of propidium iodide-stained (PI+) cells (p = 0.0051 for no PI+ trend in lactose concentrations > 1 mg/ml) at various lactose concentrations. Dashed lines indicate quadratic regression models, which fit significantly better than do linear models (see text for details). PI+ fractions at low lactose concentrations are shown in S4c Fig. (B) Mean ± SEM (N = 3) expression of GFP at various lactose concentrations (p = 0.00016 for no trend). CV, coefficient of variation (p < 10−6 for no trend). Dashed lines indicate fits of statistical models used as a guide to the eye (fitted model is: ). (C) PI-stained E. coli grown in a microfluidic device perfused with the indicated concentration of lactose (mg/ml). Note the patchy distribution of fast growing (low GFP) and slow- or non-growing (high GFP) cells at 50 mg/ml lactose. Brightfield alone is shown below. PI staining identifies dead cells and appears red or yellow, depending on the amount of GFP in the same cell. Dark spots are silicone support structures.
Fig 4.
Protection from antibiotics in growth-arrest-prone media.
(A) Survival curves of bacteria grown at indicated lactose concentrations in ampicillin-treated cultures (100 μg/ml). (B) Culture conditions favoring growth arrest enhance the presence of antibiotic tolerant cells after 20 h treatment of 32 μg/ml doxycycline (blue bars; ANOVA p = 0.003) or 100 μg/ml ampicillin (yellow bars; ANOVA p = 0.004). Survival ratios are normalized by cell densities of untreated cultures in corresponding conditions. N = 3 for each condition and the survival curve points, reporting mean ± SEM. In the final timepoint of the survival curve for 0.1 mg/ml lactose, two of the replicates were below the level of detection; the remaining replicate value is plotted.
Fig 5.
A framework for population growth dynamics in the presence of metabolic risk.
(A). Growth conditions, gene expression, replication, and regulatory factors determine timescales for switching between different types of growth. On a characteristic timescale τ1, cells stochastically switch from balanced growth (g) to a condition of rapidly changing growth rate (ĝ) and escape on a timescale of τ-1. Growth arrest arises from the growth shift state on a timescale of τ2. Escape from growth arrest permits cells to resume growth on a timescale of τ-2, or die on a timescale of τ3. (B) In the limit of large τ1 or small τ-1, populations have classical balanced growth. (C) In the limit of small τ1 and τ2 with large τ-1 and τ-2, the metastable population model holds.