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# How does variability in cell aging and growth rates influence the Malthus parameter?

• Recent biological studies draw attention to the question of variability between cells. We refer to the study of Kiviet et al. published in 2014 . A cell in a controlled culture grows at a constant rate $v>0$, but this rate can differ from one individual to another. The biological question we address here states as follows. How does individual variability in the growth rate influence the growth speed of the population? The growth speed of the population is measured by the Malthus parameter we define thereafter, also called in the literature fitness. Even if the variability in the growth rate among cells is small, with a distribution of coefficient of variation around 10%, and even if its influence on the Malthus parameter would be still smaller, such an influence may become determinant since it characterises the exponential growth speed of the population.

Mathematics Subject Classification: Primary: 35Q92, 47A75; Secondary: 60J80, 92D25.

 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: $\lambda_{\gamma,\rho} \searrow$ means that $\lambda_{\gamma,\rho} < \lambda_{\gamma,\bar v}$ for a non-degenerated probability distribution $\rho(\cdot)$ with mean $\bar v$ (truncated Gaussian), and so on

 Model (A+V) Model (S+V) Division rate $\gamma$ $v(a-1)^2 \mathbf{1}_{\{a\geq 1\}}$ $vx (x-1)^2 \mathbf{1}_{\{x\geq 1\}}$ Variability In the aging rate In the growth rate Variations $\lambda_{\gamma,\rho} \searrow$ (Figure 1) $\lambda_{\gamma,\rho} \searrow$ (Figure 2) Comments Analytic result: Theorems 2.4 and 2.5 -

Table 2.  Model (S+V). Division rate $\gamma(x,v) = vxB(x)$ with $B(x) = (x-1)^2 \mathbf{1}_{\{x \geq 1\}}$. Estimation of the Malthus parameter $\lambda_{B,\rho_\alpha}$ (mean and 95% confidence interval based on $M=50$ Monte Carlo continuous time trees simulated up to time $T$) with respect to the coefficient of variation of the growth rates density $\rho_\alpha$ with mean $\bar v = 1$. Reference (all cells grow at a rate $\bar v=1$): $\lambda_{B,\bar v} = 1$

 $\boldsymbol{CV_{\rho_\alpha} = 5\%}$ $\boldsymbol{CV_{\rho_\alpha} = 10\%}$ $\boldsymbol{CV_{\rho_\alpha} = 15\%}$ $\boldsymbol{CV_{\rho_\alpha} = 20\%}$ $\boldsymbol{CV_{\rho_\alpha} = 25\%}$ $\boldsymbol{CV_{\rho_\alpha} = 30\%}$ $\boldsymbol{CV_{\rho_\alpha} = 35\%}$ $\boldsymbol{CV_{\rho_\alpha} = 40\%}$ $\boldsymbol{CV_{\rho_\alpha} = 45\%}$ $\boldsymbol{T}$ 10.5 11 11.25 11.5 11.75 12 12.25 12.5 13 $\boldsymbol{\underset{{\rm (Min.}\leq\cdot\leq {\rm Max.)}}{{\rm Mean}}}\boldsymbol{|\partial \mathcal{T}_T|}$ $\underset{(42~358\leq\cdot\leq52~147)}{46~837}$ $\underset{(57~254\leq\cdot\leq87~282)}{73~100}$ $\underset{(53~615\leq\cdot\leq116~052)}{90~027}$ $\underset{(68~946\leq\cdot\leq128~379)}{98~270}$ $\underset{(71~884\leq\cdot\leq157~032)}{107~305}$ $\underset{(63~409\leq\cdot\leq200~860)}{120~102}$ $\underset{(52~116\leq\cdot\leq172~328)}{104~628}$ $\underset{(28~171\leq\cdot\leq192~021)}{117~208}$ $\underset{(39~238\leq\cdot\leq238~181)}{114~180}$ $\boldsymbol{\underset{{\rm (sd.)}}{{\rm Mean}}} \, \boldsymbol{\widehat \lambda_T}$ $\underset{(0.0006)}{0.9985}$ $\underset{(0.0009)}{0.9938}$ $\underset{(0.0014)}{0.9867}$ $\underset{(0.0018)}{0.9757}$ $\underset{(0.0019)}{0.9617}$ $\underset{(0.0027)}{0.9450}$ $\underset{(0.0035)}{0.9245}$ $\underset{(0.0030)}{0.8985}$ $\underset{(0.0039)}{0.8722}$ 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 $\gamma(x,v) = vxB(x)$ with $B(x) = (x-1)^8 \mathbf{1}_{\{x \geq 1\}}$. Estimation of the Malthus parameter $\lambda_{B,\rho_\alpha}$ (mean and 95% confidence interval based on $M=50$ Monte Carlo continuous time trees simulated up to time $T$) with respect to the coefficient of variation of the growth rates density $\rho_\alpha$ with mean $\bar v = 1$. Reference (all cells grow at a rate $\bar v=1$): $\lambda_{B,\bar v} = 1$

 $\boldsymbol{CV_{\rho_\alpha} = 5\%}$ $\boldsymbol{CV_{\rho_\alpha} = 10\%}$ $\boldsymbol{CV_{\rho_\alpha} = 15\%}$ $\boldsymbol{CV_{\rho_\alpha} = 20\%}$ $\boldsymbol{CV_{\rho_\alpha} = 25\%}$ $\boldsymbol{CV_{\rho_\alpha} = 30\%}$ $\boldsymbol{CV_{\rho_\alpha} = 35\%}$ $\boldsymbol{CV_{\rho_\alpha} = 40\%}$ $\boldsymbol{CV_{\rho_\alpha} = 45\%}$ $\boldsymbol{T}$ 10.5 11 11.25 11.5 11.75 12 12.25 12.5 13 $\boldsymbol{\underset{{\rm (Min.}\leq\cdot\leq {\rm Max.)}}{{\rm Mean}}}\boldsymbol{|\partial \mathcal{T}_T|}$ $\underset{(41~357\leq\cdot\leq53~270)}{47~160}$ $\underset{(61~758\leq\cdot\leq84~191)}{73~670}$ $\underset{(66~499\leq\cdot\leq118~486)}{86~410}$ $\underset{(61~924\leq\cdot\leq127~299)}{95~230}$ $\underset{(53~902\leq\cdot\leq156~868)}{104~480}$ $\underset{(53~156\leq\cdot\leq145~125)}{107~540}$ $\underset{(50~784\leq\cdot\leq162~615)}{100~480}$ $\underset{(42~533\leq\cdot\leq192~984)}{90~440}$ $\underset{(22~600\leq\cdot\leq200~034)}{102~880}$ $\boldsymbol{\underset{{\rm (sd.)}}{{\rm Mean}}} \, \boldsymbol{\widehat \lambda_T}$ $\underset{(0.0005)}{0.9984}$ $\underset{(0.0009)}{0.9934}$ $\underset{(0.0012)}{0.9855}$ $\underset{(0.0015)}{0.9732}$ $\underset{(0.0019)}{0.9589}$ $\underset{(0.0023)}{0.9384}$ $\underset{(0.0025)}{0.9166}$ $\underset{(0.0036)}{0.8890}$ $\underset{(0.0044)}{0.8597}$ 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 $\gamma(x,v) = vxB(x)$ with $B(x) = (x-1)^2 \mathbf{1}_{\{x \geq 1\}}$. Asymmetric division (a cell of size $x$ splits into two cells of size $ux$ and $(1-u)x$ for $u$ uniformly drawn on $[0.1,0.9]$). Estimation of the Malthus parameter $\lambda_{B,\rho_\alpha}$ (mean and 95% confidence interval based on $M=50$ Monte Carlo continuous time trees simulated up to time $T$) with respect to the coefficient of variation of the growth rates density $\rho_\alpha$ with mean $\bar v = 1$. Reference (all cells grow at a rate $\bar v=1$): $\lambda_{B,\bar v} = 1$

 $\boldsymbol{CV_{\rho_\alpha} = 5\%}$ $\boldsymbol{CV_{\rho_\alpha} = 10\%}$ $\boldsymbol{CV_{\rho_\alpha} = 15\%}$ $\boldsymbol{CV_{\rho_\alpha} = 20\%}$ $\boldsymbol{CV_{\rho_\alpha} = 25\%}$ $\boldsymbol{CV_{\rho_\alpha} = 30\%}$ $\boldsymbol{CV_{\rho_\alpha} = 35\%}$ $\boldsymbol{CV_{\rho_\alpha} = 40\%}$ $\boldsymbol{CV_{\rho_\alpha} = 45\%}$ $\boldsymbol{T}$ 10.5 11 11.25 11.5 11.75 12 12.25 12.5 13 $\boldsymbol{\underset{{\rm (Min.}\leq\cdot\leq {\rm Max.)}}{{\rm Mean}}}\boldsymbol{|\partial \mathcal{T}_T|}$ $\underset{(49~880\leq\cdot\leq59~486)}{53~590}$ $\underset{(69~343\leq\cdot\leq97~237)}{85~310}$ $\underset{(82~182\leq\cdot\leq129~410)}{101~350}$ $\underset{(86~751\leq\cdot\leq154~226)}{121~570}$ $\underset{(84~620\leq\cdot\leq234~613)}{129~770}$ $\underset{(67~334\leq\cdot\leq222~004)}{135~650}$ $\underset{(50~493\leq\cdot\leq234~646)}{141~660}$ $\underset{(23~530\leq\cdot\leq243~023)}{140~170}$ $\underset{(18~187\leq\cdot\leq359~824)}{154~120}$ $\boldsymbol{\underset{{\rm (sd.)}}{{\rm Mean}}} \, \boldsymbol{\widehat \lambda_T}$ $\underset{(0.0006)}{0.9987}$ $\underset{(0.0008)}{0.9948}$ $\underset{(0.0014)}{0.9880}$ $\underset{(0.0016)}{0.9783}$ $\underset{(0.0019)}{0.9665}$ $\underset{(0.0021)}{0.9511}$ $\underset{(0.0026)}{0.9322}$ $\underset{(0.0038)}{0.9099}$ $\underset{(0.0039)}{0.8836}$ 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 $\gamma(x,v) = v B(x)$ with $B(x) = (x-1)^2 \mathbf{1}_{\{x \geq 1\}}$. Estimation of the Malthus parameter $\lambda_{B,\rho_\alpha}$ (mean and 95% confidence interval based on $M=50$ Monte Carlo continuous time trees simulated up to time $T$) with respect to the coefficient of variation of the growth rates density $\rho_\alpha$ with mean $\bar v = 1$. Reference (all cells grow at a rate $\bar v=1$): $\lambda_{B,\bar v} \approx 0.6130$ (over 50 continuous time trees simulated up to time 17.25, sd. 0.0016). Among the 50 realisations, 95% lie between $0.6098$ and $0.6161$. The mean-size of the 50 trees is 46 353 (the smallest tree counts $30~553$ cells and the largest $70~914$)

 $\boldsymbol{CV_{\rho_\alpha} = 5\%}$ $\boldsymbol{CV_{\rho_\alpha} = 10\%}$ $\boldsymbol{CV_{\rho_\alpha} = 15\%}$ $\boldsymbol{CV_{\rho_\alpha} = 20\%}$ $\boldsymbol{CV_{\rho_\alpha} = 25\%}$ $\boldsymbol{CV_{\rho_\alpha} = 30\%}$ $\boldsymbol{CV_{\rho_\alpha} = 35\%}$ $\boldsymbol{CV_{\rho_\alpha} = 40\%}$ $\boldsymbol{CV_{\rho_\alpha} = 45\%}$ $\boldsymbol{T}$ 17.5 18 18.25 18.5 18.75 19 19.25 19.5 20 $\boldsymbol{\underset{{\rm (Min.}\leq\cdot\leq {\rm Max.)}}{{\rm Mean}}}\boldsymbol{|\partial \mathcal{T}_T|}$ $\underset{(38~138\leq\cdot\leq71~168)}{55~219}$ $\underset{(43~802\leq\cdot\leq95~071)}{67~760}$ $\underset{(40~296\leq\cdot\leq113~904)}{75~748}$ $\underset{(32~035\leq\cdot\leq119~198)}{76~084}$ $\underset{(29~071\leq\cdot\leq131~343)}{73~931}$ $\underset{(30~940\leq\cdot\leq141~046)}{76~074}$ $\underset{(28~704\leq\cdot\leq118~295)}{69~719}$ $\underset{(10~488\leq\cdot\leq120~506)}{57~913}$ $\underset{(3~017\leq\cdot\leq190~355)}{62~582}$ $\boldsymbol{\underset{{\rm (sd.)}}{{\rm Mean}}} \, \boldsymbol{\widehat \lambda_T}$ $\underset{(0.0014)}{0.6116}$ $\underset{(0.0014)}{0.6090}$ $\underset{(0.0015)}{0.6043}$ $\underset{(0.0018)}{0.5976}$ $\underset{(0.0021)}{0.5893}$ $\underset{(0.0023)}{0.5788}$ $\underset{(0.0025)}{0.5658}$ $\underset{(0.0033)}{0.5513}$ $\underset{(0.0038)}{0.5348}$ 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 $\gamma(x,v) = B(x)$ with $B(x) = (x-1)^2 \mathbf{1}_{\{x \geq 1\}}$. Estimation of the Malthus parameter $\lambda_{B,\rho_\alpha}$ (mean and 95% confidence interval based on $M=50$ Monte Carlo continuous time trees simulated up to time $T$) with respect to the coefficient of variation of the growth rates density $\rho_\alpha$ with mean $\bar v = 1$. Reference (all cells grow at a rate $\bar v=1$): $\lambda_{B,\bar v} = 1$

 $\boldsymbol{CV_{\rho_\alpha} = 5\%}$ $\boldsymbol{CV_{\rho_\alpha} = 10\%}$ $\boldsymbol{CV_{\rho_\alpha} = 15\%}$ $\boldsymbol{CV_{\rho_\alpha} = 20\%}$ $\boldsymbol{CV_{\rho_\alpha} = 25\%}$ $\boldsymbol{CV_{\rho_\alpha} = 30\%}$ $\boldsymbol{CV_{\rho_\alpha} = 35\%}$ $\boldsymbol{CV_{\rho_\alpha} = 40\%}$ $\boldsymbol{CV_{\rho_\alpha} = 45\%}$ $\boldsymbol{T}$ 10.5 10.75 11 11.25 11.5 11.75 12 12.25 12.5 $\boldsymbol{\underset{{\rm (Min.}\leq\cdot\leq {\rm Max.)}}{{\rm Mean}}}\boldsymbol{|\partial \mathcal{T}_T|}$ $\underset{(35~256\leq\cdot\leq42~659)}{39~660}$ $\underset{(38~675\leq\cdot\leq60~374)}{49~520}$ $\underset{(47~384\leq\cdot\leq82~371)}{61~150}$ $\underset{(48~639\leq\cdot\leq111~048)}{79~470}$ $\underset{(50~665\leq\cdot\leq139~785)}{92~490}$ $\underset{(60~083\leq\cdot\leq171~387)}{109~600}$ $\underset{(48~810\leq\cdot\leq231~667)}{124~320}$ $\underset{(45~032\leq\cdot\leq239~816)}{143~760}$ $\underset{(43~934\leq\cdot\leq287~633)}{146~400}$ $\boldsymbol{\underset{{\rm (sd.)}}{{\rm Mean}}} \, \boldsymbol{\widehat \lambda_T}$ $\underset{(0.0006)}{0.9993}$ $\underset{(0.0013)}{0.9974}$ $\underset{(0.0016)}{0.9937}$ $\underset{(0.0019)}{0.9893}$ $\underset{(0.0022)}{0.9827}$ $\underset{(0.0027)}{0.9743}$ $\underset{(0.0034)}{0.9644}$ $\underset{(0.0030)}{0.9530}$ $\underset{(0.0041)}{0.9400}$ 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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