Model (A+V) | Model (S+V) | |
Division rate | | |
Variability | In the aging rate | In the growth rate |
Variations | (Figure 1) | (Figure 2) |
Comments | Analytic result: Theorems 2.4 and 2.5 | - |
Recent biological studies draw attention to the question of variability between cells. We refer to the study of Kiviet et al. published in 2014 [
Citation: |
Figure 1. Model (A+V). $CV_{\rho_\alpha } \leadsto \lambda_{B,\rho_\alpha }$ defined by (11) for $\rho_\alpha (v) = \alpha ^{-1}\rho\big(\alpha ^{-1}(v- (1-\alpha )\bar v)\big)$ (the baseline density $\rho$ is a Gaussian density with mean $\bar v = 1$ and standard deviation $0.7$ truncated on $[0,2]$ and different division rates) $\gamma(a,v) = vB(a)$ with $B(a) = (a-1)^\beta \mathbf{1}_{\{a\geq1\}}$, $\beta \in \{0, 0.25, 0.5, 0.75, 1, 2, \ldots, 7\}$. Reference (all cells age at rate $\bar v = 1$): point of null abscissa and y-coordinate $\lambda_{B,\bar v = 1}$ defined by (10)
Figure 2. Model (S+V). Division rate $\gamma(x,v) = vxB(x)$ with $B(x) = (x-1)^2\mathbf{1}_{\{x\geq1\}}$. Estimated curve $CV_{\rho_\alpha } \leadsto \lambda_{B,\rho_\alpha }$ using estimator (16) (mean and 95% confidence interval based on $M=50$ Monte Carlo continuous time trees). Reference (all cells grow at a rate $\bar v=1$): $\lambda_{B,\bar v} = \bar v = 1$
Figure 3. Model (S+V). Standard deviation of two estimators of the Malthus parameter as $T$ increases (based on $M=50$ Monte Carlo continuous time trees simulated up to time $T$), for $\rho_{\alpha = 0.3}$ and division rate $\gamma(x,v) = vxB(x)$ with $B(x) = (x-1)^2 \mathbf{1}_{\{x\geq 1\}}$. Blue lower curve: estimation by (16) via the biomass. Green upper curve: estimation by (33) via the number of cells.
Table 1.
Variations of the Malthus parameter compared to the reference value when introducing variability between cells, for an experimentally realistic division rate. Note:
Model (A+V) | Model (S+V) | |
Division rate | | |
Variability | In the aging rate | In the growth rate |
Variations | (Figure 1) | (Figure 2) |
Comments | Analytic result: Theorems 2.4 and 2.5 | - |
Table 2.
Model (S+V). Division rate
10.5 | 11 | 11.25 | 11.5 | 11.75 | 12 | 12.25 | 12.5 | 13 | |
95% CI | [0.9974, 0.9999] | [0.9923, 0.9954] | [0.9841, 0.9893] | [0.9717, 0.9789] | [0.9583, 0.9656] | [0.9397, 0.9505] | [0.9178, 0.9312] | [0.8920, 0.9036] | [0.8650, 0.8794] |
Table 3.
Model (S+V). Division rate
10.5 | 11 | 11.25 | 11.5 | 11.75 | 12 | 12.25 | 12.5 | 13 | |
95% CI | [0.9975, 0.9995] | [0.9918, 0.9952] | [0.9833, 0.9876] | [0.9705, 0.9763] | [0.9554, 0.9628] | [0.9332, 0.9426] | [0.9113, 0.9214] | [0.8820, 0.8945] | [0.8489, 0.8655] |
Table 4.
Model (S+V). Division rate
10.5 | 11 | 11.25 | 11.5 | 11.75 | 12 | 12.25 | 12.5 | 13 | |
95% CI | [0.9972, 0.9996] | [0.9932, 0.9963] | [0.9855, 0.9906] | [0.9755, 0.9824] | [0.9634, 0.9706] | [0.9472, 0.9545] | [0.9263, 0.9372] | [0.9018, 0.9166] | [0.8743, 0.8925] |
Table 5.
Model (S+V). Division rate
17.5 | 18 | 18.25 | 18.5 | 18.75 | 19 | 19.25 | 19.5 | 20 | |
95% CI | [0.6086, 0.6138] | [0.6066, 0.6115] | [0.6017, 0.6071] | [0.5945, 0.6010] | [0.5838, 0.5942] | [0.5752, 0.5861] | [0.5607, 0.5702] | [0.5438, 0.5578] | [0.5270, 0.5413] |
Table 6.
Model (S+V). Division rate
10.5 | 10.75 | 11 | 11.25 | 11.5 | 11.75 | 12 | 12.25 | 12.5 | |
95% CI | [0.9982, 1.0006] | [0.9949, 0.9996] | [0.9894, 0.9966] | [0.9861, 0.9933] | [0.9784, 0.9878] | [0.9688, 0.9794] | [0.9588, 0.9715] | [0.9466, 0.9590] | [0.9317, 0.9470] |
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Model (A+V).
Model (S+V). Division rate
Model (S+V). Standard deviation of two estimators of the Malthus parameter as