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

Applications of stochastic semigroups to cell cycle models

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.

Citation: Katarzyna Pichór, Ryszard Rudnicki. Applications of stochastic semigroups to cell cycle models. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2365-2381. doi: 10.3934/dcdsb.2019099
References:
[1]

M. AdimyF. CrausteM. 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.

[2]

M. AdimyF. 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.

[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.

[4]

J. BanasiakK. 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.

[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.

[6]

R. CrabbM. 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.

[7]

F. CrausteI. DeminO. 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.

[8]

O. DiekmannH. 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.

[9]

S. FischerP. 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.

[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.

[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.

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

[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.

[15]

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

[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.

[17]

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

[18]

T. LuzyaninaD. 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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[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.

[33]

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

[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.

[35]

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

[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.

[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.

[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.

show all references

References:
[1]

M. AdimyF. CrausteM. 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.

[2]

M. AdimyF. 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.

[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.

[4]

J. BanasiakK. 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.

[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.

[6]

R. CrabbM. 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.

[7]

F. CrausteI. DeminO. 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.

[8]

O. DiekmannH. 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.

[9]

S. FischerP. 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.

[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.

[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.

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

[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.

[15]

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

[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.

[17]

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

[18]

T. LuzyaninaD. 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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[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.

[33]

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

[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.

[35]

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

[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.

[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.

[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.

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
Figure 2.  The set $ X $
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