March  2019, 14(1): 149-171. doi: 10.3934/nhm.2019008

Steady distribution of the incremental model for bacteria proliferation

1. 

Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 45 Avenue des États-Unis, 78035 Versailles cedex, France

2. 

Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Sorbonne université, 4 place Jussieu, 75005 Paris, France

* Corresponding author: hugo.martin@sorbonne-universite.fr

Received  March 2018 Published  January 2019

Fund Project: P. Gabriel has been supported by the ANR project KIBORD, ANR-13-BS01-0004, funded by the French Ministry of Research. H. Martin has been supported by the ERC Starting Grant SKIPPERAD (number 306321)

We study the mathematical properties of a model of cell division structured by two variables – the size and the size increment – in the case of a linear growth rate and a self-similar fragmentation kernel. We first show that one can construct a solution to the related two dimensional eigenproblem associated to the eigenvalue $ 1 $ from a solution of a certain one dimensional fixed point problem. Then we prove the existence and uniqueness of this fixed point in the appropriate $ {\rm{L}} ^1 $ weighted space under general hypotheses on the division rate. Knowing such an eigenfunction proves useful as a first step in studying the long time asymptotic behaviour of the Cauchy problem.

Citation: Pierre Gabriel, Hugo Martin. Steady distribution of the incremental model for bacteria proliferation. Networks & Heterogeneous Media, 2019, 14 (1) : 149-171. doi: 10.3934/nhm.2019008
References:
[1]

G. I. Bell and E. C. Anderson, Cell growth and division: Ⅰ. a mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophysical Journal, 7 (1967), 329-351.

[2]

E. Bernard, M. Doumic and P. Gabriel, Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts, Kinetic Related Models, to appear, arXiv: 1609.03846.

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.

[4]

M. J. Cáceres, J. A. Cañizo and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, J. Math. Pures Appl. (9), 96 (2011), 334–362. doi: 10.1016/j.matpur.2011.01.003.

[5]

B. de Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149-153. doi: 10.1007/BF01162028.

[6]

M. Doumic, Analysis of a population model structured by the cells molecular content, Math. Model. Nat. Phenom., 2 (2007), 121-152. doi: 10.1051/mmnp:2007006.

[7]

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.

[8]

M. Doumic Jauffret and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783. doi: 10.1142/S021820251000443X.

[9]

Y. Du, Order structure and topological methods in nonlinear partial differential equations, Vol. 1. Maximum principles and applications, Series in Partial Differential Equations and Applications. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. Maximum principles and applications.

[10]

A. J. HallG. C. Wake and P. W. Gandar, Steady size distributions for cells in one-dimensional plant tissues, J. Math. Biol., 30 (1991), 101-123. doi: 10.1007/BF00160330.

[11]

M. Lerch, Sur un point de la théorie des fonctions géné ratrices d'Abel, Acta Math., 27 (1903), 339-351. doi: 10.1007/BF02421315.

[12]

J. A. J. Metz and O. Diekmann, editors, The Dynamics of Physiologically Structured Populations, volume 68 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.

[13]

P. MichelS. Mischler and B. Perthame, General entropy equations for structured population models and scattering, C. R., Math., Acad. Sci. Paris, 338 (2004), 697-702. doi: 10.1016/j.crma.2004.03.006.

[14]

P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl. (9), 84 (2005), 1235–1260. doi: 10.1016/j.matpur.2005.04.001.

[15]

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

[16]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33 (2016), 849-898. doi: 10.1016/j.anihpc.2015.01.007.

[17]

A. Olivier, How does variability in cells aging and growth rates influence the malthus parameter?, Kinetic and Related Models, 10 (2017), 481-512. doi: 10.3934/krm.2017019.

[18]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.

[19]

B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, J. Differential Equations, 210 (2005), 155-177. doi: 10.1016/j.jde.2004.10.018.

[20]

J. D. F. Richard L. Burden, Numerical Analysis, Boston, MA: PWS Publishing Company; London: ITP International Thomson Publishing, 5th ed. edition, 1993.

[21]

M. Rotenberg, Transport theory for growing cell populations, J. Theoret. Biol., 103 (1983), 181-199. doi: 10.1016/0022-5193(83)90024-3.

[22]

J. T. SaulsD. Li and S. Jun, Adder and a coarse-grained approach to cell size homeostasis in bacteria, Current Opinion in Cell Biology, 38 (2016), 38-44. doi: 10.1016/j.ceb.2016.02.004.

[23]

H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, New York-Heidelberg, 1974.

[24]

J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. doi: 10.2307/1934533.

[25]

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.

[26]

G. F. Webb, Dynamics of populations structured by internal variables, Math. Z., 189 (1985), 319-335. doi: 10.1007/BF01164156.

[27]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, volume 89 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1985.

[28]

G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763. doi: 10.1090/S0002-9947-1987-0902796-7.

show all references

References:
[1]

G. I. Bell and E. C. Anderson, Cell growth and division: Ⅰ. a mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophysical Journal, 7 (1967), 329-351.

[2]

E. Bernard, M. Doumic and P. Gabriel, Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts, Kinetic Related Models, to appear, arXiv: 1609.03846.

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.

[4]

M. J. Cáceres, J. A. Cañizo and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, J. Math. Pures Appl. (9), 96 (2011), 334–362. doi: 10.1016/j.matpur.2011.01.003.

[5]

B. de Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149-153. doi: 10.1007/BF01162028.

[6]

M. Doumic, Analysis of a population model structured by the cells molecular content, Math. Model. Nat. Phenom., 2 (2007), 121-152. doi: 10.1051/mmnp:2007006.

[7]

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.

[8]

M. Doumic Jauffret and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783. doi: 10.1142/S021820251000443X.

[9]

Y. Du, Order structure and topological methods in nonlinear partial differential equations, Vol. 1. Maximum principles and applications, Series in Partial Differential Equations and Applications. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. Maximum principles and applications.

[10]

A. J. HallG. C. Wake and P. W. Gandar, Steady size distributions for cells in one-dimensional plant tissues, J. Math. Biol., 30 (1991), 101-123. doi: 10.1007/BF00160330.

[11]

M. Lerch, Sur un point de la théorie des fonctions géné ratrices d'Abel, Acta Math., 27 (1903), 339-351. doi: 10.1007/BF02421315.

[12]

J. A. J. Metz and O. Diekmann, editors, The Dynamics of Physiologically Structured Populations, volume 68 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.

[13]

P. MichelS. Mischler and B. Perthame, General entropy equations for structured population models and scattering, C. R., Math., Acad. Sci. Paris, 338 (2004), 697-702. doi: 10.1016/j.crma.2004.03.006.

[14]

P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl. (9), 84 (2005), 1235–1260. doi: 10.1016/j.matpur.2005.04.001.

[15]

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

[16]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33 (2016), 849-898. doi: 10.1016/j.anihpc.2015.01.007.

[17]

A. Olivier, How does variability in cells aging and growth rates influence the malthus parameter?, Kinetic and Related Models, 10 (2017), 481-512. doi: 10.3934/krm.2017019.

[18]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.

[19]

B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, J. Differential Equations, 210 (2005), 155-177. doi: 10.1016/j.jde.2004.10.018.

[20]

J. D. F. Richard L. Burden, Numerical Analysis, Boston, MA: PWS Publishing Company; London: ITP International Thomson Publishing, 5th ed. edition, 1993.

[21]

M. Rotenberg, Transport theory for growing cell populations, J. Theoret. Biol., 103 (1983), 181-199. doi: 10.1016/0022-5193(83)90024-3.

[22]

J. T. SaulsD. Li and S. Jun, Adder and a coarse-grained approach to cell size homeostasis in bacteria, Current Opinion in Cell Biology, 38 (2016), 38-44. doi: 10.1016/j.ceb.2016.02.004.

[23]

H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, New York-Heidelberg, 1974.

[24]

J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. doi: 10.2307/1934533.

[25]

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.

[26]

G. F. Webb, Dynamics of populations structured by internal variables, Math. Z., 189 (1985), 319-335. doi: 10.1007/BF01164156.

[27]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, volume 89 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1985.

[28]

G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763. doi: 10.1090/S0002-9947-1987-0902796-7.

Figure 1.  schematic representation of the variables on an E. coli bacterium
Figure 2.  Left: simulation of the function $ f $ by the power method with $ B(a) = \frac{2}{1+a}{1}_{\{1\leq a\}} $ and $ \mu(z) = 2\delta_{\frac{1}{2}}(z). $ Right: level set of the density $ N(a,x) $ obtained from this function $ f. $ Straight line: the set $ \{x = a+1\}. $
Figure 3.  Domain of the model, with respect to the choice of variables to describe the bacterium. Grey: domain where the bacteria densities may be positive. Arrows: transport. Left: size increment/size. Right: size increment/birth size. Dashed: location of cells of size $x_1$
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