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May  2019, 24(5): 2365-2381. doi: 10.3934/dcdsb.2019099

Applications of stochastic semigroups to cell cycle models

Katarzyna Pichór 1, and  Ryszard Rudnicki 2,,

1. 

Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland
2. 

Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland

* Corresponding author: Ryszard Rudnicki

Received  December 2017 Revised  January 2019 Published  May 2019 Early access  March 2019

Fund Project: This research was partially supported by the National Science Centre (Poland) Grant No. 2017/27/B/ST1/00100.

We consider a generational and continuous-time two-phase model of the cell cycle. The first model is given by a stochastic operator, and the second by a piecewise deterministic Markov process. In the second case we also introduce a stochastic semigroup which describes the evolution of densities of the process. We study long-time behaviour of these models. In particular we prove theorems on asymptotic stability and sweeping. We also show the relations between both models.

Keywords: Cell cycle, positive linear operator, stochastic semigroup, asymptotic stability, Markov process.
Mathematics Subject Classification: Primary: 47D06; Secondary: 60J75 92C37.
Citation: Katarzyna Pichór, Ryszard Rudnicki. Applications of stochastic semigroups to cell cycle models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2365-2381. doi: 10.3934/dcdsb.2019099
References:
[1]

M. Adimy, F. Crauste, M. L. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math., 70 (2010), 1611-1633.  doi: 10.1137/080742713.  Google Scholar

[2]

M. Adimy, F. Crauste and C. Marquet, Asymptotic behavior and stability switch for a mature-immature model of cell differentiation, Nonlinear Analysis: Real World, 11 (2010), 2913-2929.  doi: 10.1016/j.nonrwa.2009.11.001.  Google Scholar

[3]

B. Alberts, A. Johnson, J. Lewis, D. Morgan, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, Taylor & Francis Group, Abingdon UK, 2014. Google Scholar

[4]

J. Banasiak, K. Pichór and R. Rudnicki, Asynchronous exponential growth of a general structured population model, Acta Appl. Math., 119 (2012), 149-166.  doi: 10.1007/s10440-011-9666-y.  Google Scholar

[5]

F. J. Burns and I. F. Tannock, On the existence of a $G_0$ phase in the cell cycle, Cell Tissue Kinet., 3 (1970), 321-334.  doi: 10.1111/j.1365-2184.1970.tb00340.x.  Google Scholar

[6]

R. Crabb, M. C. Mackey and A. Rey, Propagating fronts, chaos and multistability in a cell replication model, Chaos, 6 (1996), 477-492.  doi: 10.1063/1.166195.  Google Scholar

[7]

F. Crauste, I. Demin, O. Gandrillon and V. Volpert, Mathematical study of feedback control roles and relevance in stress erythropoiesis, J.Theor. Biology, 263 (2010), 303-316.  doi: 10.1016/j.jtbi.2009.12.026.  Google Scholar

[8]

O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biology, 19 (1984), 227-248.  doi: 10.1007/BF00277748.  Google Scholar

[9]

S. Fischer, P. Kurbatova and N. Bessonov, Modeling erythroblastic islands: Using a hybrid model to assess the function of central macrophage, J. Theor. Biology, 298 (2012), 92-106.  doi: 10.1016/j.jtbi.2012.01.002.  Google Scholar

[10]

H. Gacki and A. Lasota, Markov operators defined by Volterra type integrals with advanced argument, Ann. Polon. Math., 51 (1990), 155-166.  doi: 10.4064/ap-51-1-155-166.  Google Scholar

[11]

M. Gyllenberg and H. J. A. M. Heijmans, An abstract delay-differential equation modelling size dependent cell growth and division, SIAM J. Math. Anal., 18 (1987), 74-88.  doi: 10.1137/0518006.  Google Scholar

[12] P. C. L. John (ed.), The Cell Cycle, Cambridge University Press, London, 1981.   Google Scholar
[13]

A. Lasota and M. C. Mackey, Globally asymptotic properties of proliferating cell populations, J. Math. Biol., 19 (1984), 43-62.  doi: 10.1007/BF00275930.  Google Scholar

[14]

A. Lasota and M. C. Mackey, Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, Ⅱ edition, Springer Applied Mathematical Sciences, 97, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[15]

A. Lasota, M. C. Mackey and J. Tyrcha, The statistical dynamics of recurrent biological events, J. Math. Biol., 30 (1992), 775-800.  doi: 10.1007/BF00176455.  Google Scholar

[16]

J. L. Lebowitz and S. L. Rubinow, A theory for the age and generation time distribution of microbial population, J. Math. Biol., 1 (1974), 17-36.  doi: 10.1007/BF02339486.  Google Scholar

[17]

K. Łoskot and R. Rudnicki, Sweeping of some integral operators, Bull. Pol. Ac.: Math., 37 (1989), 229-235.   Google Scholar

[18]

T. Luzyanina, D. Roose and G. Bocharov, Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data, J. Math. Biology, 59 (2009), 581-603.  doi: 10.1007/s00285-008-0244-5.  Google Scholar

[19]

M.C. Mackey and R. Rudnicki, Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol., 33 (1994), 89-109.  doi: 10.1007/BF00160175.  Google Scholar

[20]

M. C. Mackey and M. Tyran-Kamińska, Dynamics and density evolution in piecewise deterministic growth processes, Ann. Polon. Math., 94 (2008), 111-129.  doi: 10.4064/ap94-2-2.  Google Scholar

[21] D. O. Morgan, The Cell Cycle: Principles of Control, New Science Press, London, 2007.   Google Scholar
[22]

K. Pichór, Asymptotic behaviour of a structured population model, Mathematical and Computer Modelling, 57 (2013), 1240-1249.  doi: 10.1016/j.mcm.2012.10.027.  Google Scholar

[23]

K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685.  doi: 10.1006/jmaa.2000.6968.  Google Scholar

[24]

K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic operators and semigroups, J. Math. Anal. Appl., 436 (2016), 305-321.  doi: 10.1016/j.jmaa.2015.12.009.  Google Scholar

[25]

K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic semigroups and applications, Stochastics and Dynamics, 18 (2018), 1850001, 18 pp. doi: 10.1142/S0219493718500016.  Google Scholar

[26]

K. Pichór and R. Rudnicki, Stability of stochastic semigroups and applications to Stein's neuronal model, Discrete Contin. Dyn. Syst. B, 23 (2018), 377-385.  doi: 10.3934/dcdsb.2018026.  Google Scholar

[27]

L. Pujo-Menjouet, Blood cell dynamics: Half of a century of modelling, Math. Model. Nat. Phenom., 11 (2016), 92-115.  doi: 10.1051/mmnp/201611106.  Google Scholar

[28]

L. Pujo-Menjouet and R. Rudnicki, Global stability of cellular populations with unequal division, Canad. Appl. Math. Quart., 8 (2000), 185-202.  doi: 10.1216/camq/1032375042.  Google Scholar

[29]

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

[30]

S. I. Rubinow, A maturity time representation for cell populations, Biophy. J., 8 (1968), 1055-1073.  doi: 10.1016/S0006-3495(68)86539-7.  Google Scholar

[31]

R. Rudnicki, Stability in L1 of some integral operators, Integr Equat. Oper. Th., 24 (1996), 320-327.  doi: 10.1007/BF01204604.  Google Scholar

[32]

R. Rudnicki, Stochastic operators and semigroups and their applications in physics and biology, in J. Banasiak, M. Mokhtar-Kharroubi (eds.), Evolutionary Equations with Applications in Natural Sciences, Lecture Notes in Mathematics Springer, Heidelberg, 2126 (2015), 255–318. doi: 10.1007/978-3-319-11322-7_6.  Google Scholar

[33]

R. Rudnicki and K. Pichór, Markov semigroups and stability of the cell maturation distribution, J. Biol. Systems, 8 (2000), 69-94.   Google Scholar

[34]

R. Rudnicki and M. Tyran-Kamińska, Piecewise Deterministic Processes in Biological Models, SpringerBriefs in Applied Sciences and Technology, Mathematical Methods, Springer, Cham, Switzerland, 2017. doi: 10.1007/978-3-319-61295-9.  Google Scholar

[35]

J. A. Smith and L. Martin, Do cells cycle?, Proc. Natl. Acad. Sci. U.S.A., 70 (1973), 1263-1267.   Google Scholar

[36]

J. J. Tyson and K. B. Hannsgen, Cell growth and division: A deterministic/probabilistic model of the cell cycle, J. Math. Biol., 23 (1986), 231-246.  doi: 10.1007/BF00276959.  Google Scholar

[37]

J. Tyrcha, Asymptotic stability in a generalized probabilistic/deterministic model of the cell cycle, J. Math. Biology, 26 (1988), 465-475.  doi: 10.1007/BF00276374.  Google Scholar

[38]

H. von Foerster, Some remarks on changing populations, in The Kinetics of Cellular Proliferation, 382–407, Ed. F. Stohlman, Grune and Stratton, New York, 1959. Google Scholar

show all references

References:
[1]

M. Adimy, F. Crauste, M. L. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math., 70 (2010), 1611-1633.  doi: 10.1137/080742713.  Google Scholar

[2]

M. Adimy, F. Crauste and C. Marquet, Asymptotic behavior and stability switch for a mature-immature model of cell differentiation, Nonlinear Analysis: Real World, 11 (2010), 2913-2929.  doi: 10.1016/j.nonrwa.2009.11.001.  Google Scholar

[3]

B. Alberts, A. Johnson, J. Lewis, D. Morgan, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, Taylor & Francis Group, Abingdon UK, 2014. Google Scholar

[4]

J. Banasiak, K. Pichór and R. Rudnicki, Asynchronous exponential growth of a general structured population model, Acta Appl. Math., 119 (2012), 149-166.  doi: 10.1007/s10440-011-9666-y.  Google Scholar

[5]

F. J. Burns and I. F. Tannock, On the existence of a $G_0$ phase in the cell cycle, Cell Tissue Kinet., 3 (1970), 321-334.  doi: 10.1111/j.1365-2184.1970.tb00340.x.  Google Scholar

[6]

R. Crabb, M. C. Mackey and A. Rey, Propagating fronts, chaos and multistability in a cell replication model, Chaos, 6 (1996), 477-492.  doi: 10.1063/1.166195.  Google Scholar

[7]

F. Crauste, I. Demin, O. Gandrillon and V. Volpert, Mathematical study of feedback control roles and relevance in stress erythropoiesis, J.Theor. Biology, 263 (2010), 303-316.  doi: 10.1016/j.jtbi.2009.12.026.  Google Scholar

[8]

O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biology, 19 (1984), 227-248.  doi: 10.1007/BF00277748.  Google Scholar

[9]

S. Fischer, P. Kurbatova and N. Bessonov, Modeling erythroblastic islands: Using a hybrid model to assess the function of central macrophage, J. Theor. Biology, 298 (2012), 92-106.  doi: 10.1016/j.jtbi.2012.01.002.  Google Scholar

[10]

H. Gacki and A. Lasota, Markov operators defined by Volterra type integrals with advanced argument, Ann. Polon. Math., 51 (1990), 155-166.  doi: 10.4064/ap-51-1-155-166.  Google Scholar

[11]

M. Gyllenberg and H. J. A. M. Heijmans, An abstract delay-differential equation modelling size dependent cell growth and division, SIAM J. Math. Anal., 18 (1987), 74-88.  doi: 10.1137/0518006.  Google Scholar

[12] P. C. L. John (ed.), The Cell Cycle, Cambridge University Press, London, 1981.   Google Scholar
[13]

A. Lasota and M. C. Mackey, Globally asymptotic properties of proliferating cell populations, J. Math. Biol., 19 (1984), 43-62.  doi: 10.1007/BF00275930.  Google Scholar

[14]

A. Lasota and M. C. Mackey, Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, Ⅱ edition, Springer Applied Mathematical Sciences, 97, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[15]

A. Lasota, M. C. Mackey and J. Tyrcha, The statistical dynamics of recurrent biological events, J. Math. Biol., 30 (1992), 775-800.  doi: 10.1007/BF00176455.  Google Scholar

[16]

J. L. Lebowitz and S. L. Rubinow, A theory for the age and generation time distribution of microbial population, J. Math. Biol., 1 (1974), 17-36.  doi: 10.1007/BF02339486.  Google Scholar

[17]

K. Łoskot and R. Rudnicki, Sweeping of some integral operators, Bull. Pol. Ac.: Math., 37 (1989), 229-235.   Google Scholar

[18]

T. Luzyanina, D. Roose and G. Bocharov, Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data, J. Math. Biology, 59 (2009), 581-603.  doi: 10.1007/s00285-008-0244-5.  Google Scholar

[19]

M.C. Mackey and R. Rudnicki, Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol., 33 (1994), 89-109.  doi: 10.1007/BF00160175.  Google Scholar

[20]

M. C. Mackey and M. Tyran-Kamińska, Dynamics and density evolution in piecewise deterministic growth processes, Ann. Polon. Math., 94 (2008), 111-129.  doi: 10.4064/ap94-2-2.  Google Scholar

[21] D. O. Morgan, The Cell Cycle: Principles of Control, New Science Press, London, 2007.   Google Scholar
[22]

K. Pichór, Asymptotic behaviour of a structured population model, Mathematical and Computer Modelling, 57 (2013), 1240-1249.  doi: 10.1016/j.mcm.2012.10.027.  Google Scholar

[23]

K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685.  doi: 10.1006/jmaa.2000.6968.  Google Scholar

[24]

K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic operators and semigroups, J. Math. Anal. Appl., 436 (2016), 305-321.  doi: 10.1016/j.jmaa.2015.12.009.  Google Scholar

[25]

K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic semigroups and applications, Stochastics and Dynamics, 18 (2018), 1850001, 18 pp. doi: 10.1142/S0219493718500016.  Google Scholar

[26]

K. Pichór and R. Rudnicki, Stability of stochastic semigroups and applications to Stein's neuronal model, Discrete Contin. Dyn. Syst. B, 23 (2018), 377-385.  doi: 10.3934/dcdsb.2018026.  Google Scholar

[27]

L. Pujo-Menjouet, Blood cell dynamics: Half of a century of modelling, Math. Model. Nat. Phenom., 11 (2016), 92-115.  doi: 10.1051/mmnp/201611106.  Google Scholar

[28]

L. Pujo-Menjouet and R. Rudnicki, Global stability of cellular populations with unequal division, Canad. Appl. Math. Quart., 8 (2000), 185-202.  doi: 10.1216/camq/1032375042.  Google Scholar

[29]

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

[30]

S. I. Rubinow, A maturity time representation for cell populations, Biophy. J., 8 (1968), 1055-1073.  doi: 10.1016/S0006-3495(68)86539-7.  Google Scholar

[31]

R. Rudnicki, Stability in L1 of some integral operators, Integr Equat. Oper. Th., 24 (1996), 320-327.  doi: 10.1007/BF01204604.  Google Scholar

[32]

R. Rudnicki, Stochastic operators and semigroups and their applications in physics and biology, in J. Banasiak, M. Mokhtar-Kharroubi (eds.), Evolutionary Equations with Applications in Natural Sciences, Lecture Notes in Mathematics Springer, Heidelberg, 2126 (2015), 255–318. doi: 10.1007/978-3-319-11322-7_6.  Google Scholar

[33]

R. Rudnicki and K. Pichór, Markov semigroups and stability of the cell maturation distribution, J. Biol. Systems, 8 (2000), 69-94.   Google Scholar

[34]

R. Rudnicki and M. Tyran-Kamińska, Piecewise Deterministic Processes in Biological Models, SpringerBriefs in Applied Sciences and Technology, Mathematical Methods, Springer, Cham, Switzerland, 2017. doi: 10.1007/978-3-319-61295-9.  Google Scholar

[35]

J. A. Smith and L. Martin, Do cells cycle?, Proc. Natl. Acad. Sci. U.S.A., 70 (1973), 1263-1267.   Google Scholar

[36]

J. J. Tyson and K. B. Hannsgen, Cell growth and division: A deterministic/probabilistic model of the cell cycle, J. Math. Biol., 23 (1986), 231-246.  doi: 10.1007/BF00276959.  Google Scholar

[37]

J. Tyrcha, Asymptotic stability in a generalized probabilistic/deterministic model of the cell cycle, J. Math. Biology, 26 (1988), 465-475.  doi: 10.1007/BF00276374.  Google Scholar

[38]

H. von Foerster, Some remarks on changing populations, in The Kinetics of Cellular Proliferation, 382–407, Ed. F. Stohlman, Grune and Stratton, New York, 1959. Google Scholar

Figure 1.  Evolution of maturity of a mother cell: (1) – resting phase; (2) – proliferating phase and a daughter cell: (3) – resting phase; (4) – proliferating phase
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Figure 2.  The set $ X $
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