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.   Google Scholar

[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. Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.   Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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. Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.   Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

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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