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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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 |
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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 and 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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. |
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