June  2017, 10(2): 481-512. doi: 10.3934/krm.2017019

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

Université Paris-Dauphine, PSL Research University, CNRS, UMR [7534], CEREMADE, 75016 Paris, France

Received  February 2016 Revised  June 2016 Published  November 2016

Recent biological studies draw attention to the question of variability between cells. We refer to the study of Kiviet et al. published in 2014 [15]. 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.

Citation: AdélaÏde Olivier. How does variability in cell aging and growth rates influence the Malthus parameter?. Kinetic and Related Models, 2017, 10 (2) : 481-512. doi: 10.3934/krm.2017019
References:
[1]

A. Amir, Cell size regulation in bacteria, Physical Review Letters, 112 (2014), 208102.  doi: 10.1103/PhysRevLett.112.208102.

[2]

V. BansayeJ.-F. DelmasL. Marsalle and V. C. Tran, Limit theorems for Markov processes indexed by continuous time Galton-Watson trees, The Annals of Applied Probability, 21 (2011), 2263-2314.  doi: 10.1214/10-AAP757.

[3]

F. BillyJ. ClairambaultO. FercoqS. GaubertT. LepoutreT. Ouillon and S. Saito, Synchronisation and control of proliferation in cycling cell population models with age structure, Mathematics and Computers in Simulation, 96 (2014), 66-94.  doi: 10.1016/j.matcom.2012.03.005.

[4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. 
[5]

V. CalvezM. Doumic and P. Gabriel, Self-similarity in a general aggregation-fragmentation problem. Application to fitness analysis, Journal de mathématiques pures et appliquées, 98 (2012), 1-27.  doi: 10.1016/j.matpur.2012.01.004.

[6]

F. Campillo, N. Champagnat and C. Fritsch, On the Variations of the Principal Eigenvalue and the Probability of Survival with Respect to a Parameter in Growth-Fragmentation-Death Models arXiv: 1601. 02516.

[7]

J. ClairambaultP. Michel and B. Perthame, Circadian rhythm and tumour growth, Comptes Rendus Mathematique de l'Académie des Sciences Paris, 342 (2006), 17-22.  doi: 10.1016/j.crma.2005.10.029.

[8]

B. Cloez, Limit Theorems for some Branching Measure-Valued Processes arXiv: 1106. 0660.

[9] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1990. 
[10]

M. Doumic, Analysis of a population model structured by the cells molecular content, Mathematical Modelling of Natural Phenomena, 2 (2007), 121-152.  doi: 10.1051/mmnp:2007006.

[11]

M. Doumic and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Mathematical Models and Methods in Applied Sciences, 20 (2010), 757-783.  doi: 10.1142/S021820251000443X.

[12]

M. DoumicM. HoffmannN. Krell and L. Robert, Statistical estimation of a growth-fragmentation model observed on a genealogical tree, Bernoulli, 21 (2015), 1760-1799.  doi: 10.3150/14-BEJ623.

[13]

S. Gaubert and T. Lepoutre, Discrete limit and monotonicity properties of the Floquet eigenvalue in an age structured cell division cycle model, Journal of Mathematical Biology, 71 (2015), 1663-1703.  doi: 10.1007/s00285-015-0874-3.

[14]

J. Guyon, Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging, The Annals of Applied Probability, 17 (2007), 1538-1569.  doi: 10.1214/105051607000000195.

[15]

D. J. KivietP. NgheN. WalkerS. BoulineauV. Sunderlikova and S. J. Tans, Stochasticity of metabolism and growth at the single-cell level, Nature, 514 (2014), 376-379.  doi: 10.1038/nature13582.

[16]

J. L. Lebowitz and S. I. Rubinow, A theory for the age and generation time distribution of a microbial population, Journal of Mathematical Biology, 1 (1974), 17-36.  doi: 10.1007/BF02339486.

[17]

A. G. MarrR. J. Harvey and W. C. Trentini, Growth and division of Escherichia coli, Journal of Bacteriology, 91 (1966), 2388-2389. 

[18]

J. A. J. Metz and O. Diekmann, Formulating models for structured populations, In The dynamics of physiologically structured populations (Amsterdam, 1983), Lecture Notes in Biomathematics, 68 (1986), 78-135. doi: 10.1007/978-3-662-13159-6_3.

[19]

P. Michel, Optimal proliferation rate in a cell division model, Mathematical Modelling of Natural Phenomena, 1 (2006), 23-44.  doi: 10.1051/mmnp:2008002.

[20]

S. MischlerB. Perthame and L. Ryzhik, Stability in a nonlinear population maturation model, Mathematical Models and Methods in Applied Sciences, 12 (2002), 1751-1772.  doi: 10.1142/S021820250200232X.

[21]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.

[22]

A. Olivier, Statistical Analysis of Growth-Fragmentation Models Ph. D thesis, University Paris-Dauphine, 2015. Available from: https://hal.archives-ouvertes.fr/tel-01235239/document

[23]

M. OsellaE. Nugent and M. Cosentino Lagomarsino, Concerted control of Escherichia coli cell division, PNAS, 111 (2014), 3431-3435. 

[24]

B. Perthame, Transport Equations Arising In Biology, Birckhäuser Frontiers in mathematics edition, 2007.

[25]

L. Robert, M. Hoffmann, N. Krell, S. Aymerich, J. Robert and M. Doumic, Division Control in Escherichia Coli is Based on a Size-Sensing Rather than a Timing Mechanism BMC Biology, 2014.

[26]

M. Rotenberg, Transport theory for growing cell populations, Journal of Theoretical Biology, 103 (1983), 181-199.  doi: 10.1016/0022-5193(83)90024-3.

[27]

M. SchaechterJ. P. WilliamsonJ. R. Hood Jun and A. L. Koch, Growth, Cell and Nuclear Divisions in some Bacteria, Microbiology, 29 (1962), 421-434.  doi: 10.1099/00221287-29-3-421.

[28]

I. SoiferL. RobertN. Barkai and A. Amir, Single-cell analysis of growth in budding yeast and bacteria reveals a common size regulation strategy, Current Biology, 26 (2016), 356-361.  doi: 10.1016/j.cub.2015.11.067.

[29]

S. Taheri-AraghiS. BraddeJ. T. SaulsN. S. HillP. A. LevinJ. PaulssonM. Vergassola and S. Jun, Cell-size control and homeostasis in bacteria, Current Biology, 25 (2015), 385-391.  doi: 10.1016/j.cub.2014.12.009.

show all references

References:
[1]

A. Amir, Cell size regulation in bacteria, Physical Review Letters, 112 (2014), 208102.  doi: 10.1103/PhysRevLett.112.208102.

[2]

V. BansayeJ.-F. DelmasL. Marsalle and V. C. Tran, Limit theorems for Markov processes indexed by continuous time Galton-Watson trees, The Annals of Applied Probability, 21 (2011), 2263-2314.  doi: 10.1214/10-AAP757.

[3]

F. BillyJ. ClairambaultO. FercoqS. GaubertT. LepoutreT. Ouillon and S. Saito, Synchronisation and control of proliferation in cycling cell population models with age structure, Mathematics and Computers in Simulation, 96 (2014), 66-94.  doi: 10.1016/j.matcom.2012.03.005.

[4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. 
[5]

V. CalvezM. Doumic and P. Gabriel, Self-similarity in a general aggregation-fragmentation problem. Application to fitness analysis, Journal de mathématiques pures et appliquées, 98 (2012), 1-27.  doi: 10.1016/j.matpur.2012.01.004.

[6]

F. Campillo, N. Champagnat and C. Fritsch, On the Variations of the Principal Eigenvalue and the Probability of Survival with Respect to a Parameter in Growth-Fragmentation-Death Models arXiv: 1601. 02516.

[7]

J. ClairambaultP. Michel and B. Perthame, Circadian rhythm and tumour growth, Comptes Rendus Mathematique de l'Académie des Sciences Paris, 342 (2006), 17-22.  doi: 10.1016/j.crma.2005.10.029.

[8]

B. Cloez, Limit Theorems for some Branching Measure-Valued Processes arXiv: 1106. 0660.

[9] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1990. 
[10]

M. Doumic, Analysis of a population model structured by the cells molecular content, Mathematical Modelling of Natural Phenomena, 2 (2007), 121-152.  doi: 10.1051/mmnp:2007006.

[11]

M. Doumic and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Mathematical Models and Methods in Applied Sciences, 20 (2010), 757-783.  doi: 10.1142/S021820251000443X.

[12]

M. DoumicM. HoffmannN. Krell and L. Robert, Statistical estimation of a growth-fragmentation model observed on a genealogical tree, Bernoulli, 21 (2015), 1760-1799.  doi: 10.3150/14-BEJ623.

[13]

S. Gaubert and T. Lepoutre, Discrete limit and monotonicity properties of the Floquet eigenvalue in an age structured cell division cycle model, Journal of Mathematical Biology, 71 (2015), 1663-1703.  doi: 10.1007/s00285-015-0874-3.

[14]

J. Guyon, Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging, The Annals of Applied Probability, 17 (2007), 1538-1569.  doi: 10.1214/105051607000000195.

[15]

D. J. KivietP. NgheN. WalkerS. BoulineauV. Sunderlikova and S. J. Tans, Stochasticity of metabolism and growth at the single-cell level, Nature, 514 (2014), 376-379.  doi: 10.1038/nature13582.

[16]

J. L. Lebowitz and S. I. Rubinow, A theory for the age and generation time distribution of a microbial population, Journal of Mathematical Biology, 1 (1974), 17-36.  doi: 10.1007/BF02339486.

[17]

A. G. MarrR. J. Harvey and W. C. Trentini, Growth and division of Escherichia coli, Journal of Bacteriology, 91 (1966), 2388-2389. 

[18]

J. A. J. Metz and O. Diekmann, Formulating models for structured populations, In The dynamics of physiologically structured populations (Amsterdam, 1983), Lecture Notes in Biomathematics, 68 (1986), 78-135. doi: 10.1007/978-3-662-13159-6_3.

[19]

P. Michel, Optimal proliferation rate in a cell division model, Mathematical Modelling of Natural Phenomena, 1 (2006), 23-44.  doi: 10.1051/mmnp:2008002.

[20]

S. MischlerB. Perthame and L. Ryzhik, Stability in a nonlinear population maturation model, Mathematical Models and Methods in Applied Sciences, 12 (2002), 1751-1772.  doi: 10.1142/S021820250200232X.

[21]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.

[22]

A. Olivier, Statistical Analysis of Growth-Fragmentation Models Ph. D thesis, University Paris-Dauphine, 2015. Available from: https://hal.archives-ouvertes.fr/tel-01235239/document

[23]

M. OsellaE. Nugent and M. Cosentino Lagomarsino, Concerted control of Escherichia coli cell division, PNAS, 111 (2014), 3431-3435. 

[24]

B. Perthame, Transport Equations Arising In Biology, Birckhäuser Frontiers in mathematics edition, 2007.

[25]

L. Robert, M. Hoffmann, N. Krell, S. Aymerich, J. Robert and M. Doumic, Division Control in Escherichia Coli is Based on a Size-Sensing Rather than a Timing Mechanism BMC Biology, 2014.

[26]

M. Rotenberg, Transport theory for growing cell populations, Journal of Theoretical Biology, 103 (1983), 181-199.  doi: 10.1016/0022-5193(83)90024-3.

[27]

M. SchaechterJ. P. WilliamsonJ. R. Hood Jun and A. L. Koch, Growth, Cell and Nuclear Divisions in some Bacteria, Microbiology, 29 (1962), 421-434.  doi: 10.1099/00221287-29-3-421.

[28]

I. SoiferL. RobertN. Barkai and A. Amir, Single-cell analysis of growth in budding yeast and bacteria reveals a common size regulation strategy, Current Biology, 26 (2016), 356-361.  doi: 10.1016/j.cub.2015.11.067.

[29]

S. Taheri-AraghiS. BraddeJ. T. SaulsN. S. HillP. A. LevinJ. PaulssonM. Vergassola and S. Jun, Cell-size control and homeostasis in bacteria, Current Biology, 25 (2015), 385-391.  doi: 10.1016/j.cub.2014.12.009.

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 $\gamma$ $v(a-1)^2 \mathbf{1}_{\{a\geq 1\}}$ $vx (x-1)^2 \mathbf{1}_{\{x\geq 1\}}$
VariabilityIn the aging rateIn 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-
Model (A+V) Model (S+V)
Division rate $\gamma$ $v(a-1)^2 \mathbf{1}_{\{a\geq 1\}}$ $vx (x-1)^2 \mathbf{1}_{\{x\geq 1\}}$
VariabilityIn the aging rateIn 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]
$\boldsymbol{CV_{\rho_\alpha} = 5\%}$ $\boldsymbol{CV_{\rho_\alpha} = 10\%}$ $\boldsymbol{CV_{\rho_\alpha} = 15\%}$ $\boldsymbol{CV_{\rho_\alpha} = 20\%}$ $\boldsymbol{CV_{\rho_\alpha} = 25\%}$ $\boldsymbol{CV_{\rho_\alpha} = 30\%}$ $\boldsymbol{CV_{\rho_\alpha} = 35\%}$ $\boldsymbol{CV_{\rho_\alpha} = 40\%}$ $\boldsymbol{CV_{\rho_\alpha} = 45\%}$
$\boldsymbol{T}$ 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]
$\boldsymbol{CV_{\rho_\alpha} = 5\%}$ $\boldsymbol{CV_{\rho_\alpha} = 10\%}$ $\boldsymbol{CV_{\rho_\alpha} = 15\%}$ $\boldsymbol{CV_{\rho_\alpha} = 20\%}$ $\boldsymbol{CV_{\rho_\alpha} = 25\%}$ $\boldsymbol{CV_{\rho_\alpha} = 30\%}$ $\boldsymbol{CV_{\rho_\alpha} = 35\%}$ $\boldsymbol{CV_{\rho_\alpha} = 40\%}$ $\boldsymbol{CV_{\rho_\alpha} = 45\%}$
$\boldsymbol{T}$ 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]
$\boldsymbol{CV_{\rho_\alpha} = 5\%}$ $\boldsymbol{CV_{\rho_\alpha} = 10\%}$ $\boldsymbol{CV_{\rho_\alpha} = 15\%}$ $\boldsymbol{CV_{\rho_\alpha} = 20\%}$ $\boldsymbol{CV_{\rho_\alpha} = 25\%}$ $\boldsymbol{CV_{\rho_\alpha} = 30\%}$ $\boldsymbol{CV_{\rho_\alpha} = 35\%}$ $\boldsymbol{CV_{\rho_\alpha} = 40\%}$ $\boldsymbol{CV_{\rho_\alpha} = 45\%}$
$\boldsymbol{T}$ 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]
$\boldsymbol{CV_{\rho_\alpha} = 5\%}$ $\boldsymbol{CV_{\rho_\alpha} = 10\%}$ $\boldsymbol{CV_{\rho_\alpha} = 15\%}$ $\boldsymbol{CV_{\rho_\alpha} = 20\%}$ $\boldsymbol{CV_{\rho_\alpha} = 25\%}$ $\boldsymbol{CV_{\rho_\alpha} = 30\%}$ $\boldsymbol{CV_{\rho_\alpha} = 35\%}$ $\boldsymbol{CV_{\rho_\alpha} = 40\%}$ $\boldsymbol{CV_{\rho_\alpha} = 45\%}$
$\boldsymbol{T}$ 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]
$\boldsymbol{CV_{\rho_\alpha} = 5\%}$ $\boldsymbol{CV_{\rho_\alpha} = 10\%}$ $\boldsymbol{CV_{\rho_\alpha} = 15\%}$ $\boldsymbol{CV_{\rho_\alpha} = 20\%}$ $\boldsymbol{CV_{\rho_\alpha} = 25\%}$ $\boldsymbol{CV_{\rho_\alpha} = 30\%}$ $\boldsymbol{CV_{\rho_\alpha} = 35\%}$ $\boldsymbol{CV_{\rho_\alpha} = 40\%}$ $\boldsymbol{CV_{\rho_\alpha} = 45\%}$
$\boldsymbol{T}$ 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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