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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 |
Recent biological studies draw attention to the question of variability between cells. We refer to the study of Kiviet et al. published in 2014 [
References:
[1] |
A. Amir,
Cell size regulation in bacteria, Physical Review Letters, 112 (2014), 208102.
doi: 10.1103/PhysRevLett.112.208102. |
[2] |
V. Bansaye, J.-F. Delmas, L. 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. Billy, J. Clairambault, O. Fercoq, S. Gaubert, T. Lepoutre, T. 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. Calvez, M. 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. Clairambault, P. 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. Doumic, M. Hoffmann, N. 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. Kiviet, P. Nghe, N. Walker, S. Boulineau, V. 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. Marr, R. 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. Mischler, B. 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. Osella, E. 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. Schaechter, J. P. Williamson, J. 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. Soifer, L. Robert, N. 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-Araghi, S. Bradde, J. T. Sauls, N. S. Hill, P. A. Levin, J. Paulsson, M. 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. Bansaye, J.-F. Delmas, L. 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. Billy, J. Clairambault, O. Fercoq, S. Gaubert, T. Lepoutre, T. 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. Calvez, M. 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. Clairambault, P. 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. Doumic, M. Hoffmann, N. 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. Kiviet, P. Nghe, N. Walker, S. Boulineau, V. 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. Marr, R. 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. Mischler, B. 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. Osella, E. 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. Schaechter, J. P. Williamson, J. 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. Soifer, L. Robert, N. 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-Araghi, S. Bradde, J. T. Sauls, N. S. Hill, P. A. Levin, J. Paulsson, M. 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. |



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 | - |
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 | - |
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] |
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] |
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] |
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] |
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] |
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] |
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] |
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] |
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] |
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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