January  2017, 14(1): 17-30. doi: 10.3934/mbe.2017002

A singular limit for an age structured mutation problem

1. 

Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

2. 

Institute of Mathematics, Technical University of Łódź, Łódź, Poland

Received  October 31, 2015 Accepted  March 10, 2016 Published  October 2016

Fund Project: The paper was presented at the conference Micro and Macro Systems in Life Sciences, B¸edlewo, 8-13 June 2015 and was supported by the statutory grant of the Institute of Mathematics of Łódź University of Technology. Participation of A. F. was sponsored by the organizers of the conference.

The spread of a particular trait in a cell population often is modelled by an appropriate system of ordinary differential equations describing how the sizes of subpopulations of the cells with the same genome change in time. On the other hand, it is recognized that cells have their own vital dynamics and mutations, leading to changes in their genome, mostly occurring during the cell division at the end of its life cycle. In this context, the process is described by a system of McKendrick type equations which resembles a network transport problem. In this paper we show that, under an appropriate scaling of the latter, these two descriptions are asymptotically equivalent.

Citation: Jacek Banasiak, Aleksandra Falkiewicz. A singular limit for an age structured mutation problem. Mathematical Biosciences & Engineering, 2017, 14 (1) : 17-30. doi: 10.3934/mbe.2017002
References:
[1]

H. Amann and J. Escher, Analysis II Birkhäuser, Basel 2008.

[2]

W. Arendt, Resolvent positive operators, Proc. Lond. Math. Soc., 54 (1987), 321-349.  doi: 10.1112/plms/s3-54.2.321.

[3] J. Banasiak and L. Arlotti, Positive Perturbations of Semigroups with Applications, Springer Verlag, London, 2006. 
[4]

J. Banasiak and A. Falkiewicz, Some transport and diffusion processes on networks and their graph realizability Appl. Math. Lett. ,45 (2015), 25-30 doi: 10.1016/j.aml.2015.01.006.

[5]

J. Banasiak, A. Falkiewicz and P. Namayanja, Semigroup approach to diffusion and transport problems on networks Semigroup Forum. [DOI 10.1007/s00233-015-9730-4] doi: 10.1007/s00233-015-9730-4.

[6]

J. BanasiakA. Falkiewicz and P. Namayanja, Asymptotic state lumping in transport and diffusion problems on networks with applications to population problems, Math. Models Methods Appl. Sci., 26 (2016), 215-247.  doi: 10.1142/S0218202516400017.

[7]

J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-05140-6.

[8]

J. Banasiak and M. Moszyński, Dynamics of birth-and-death processes with proliferation -stability and chaos, Discrete Contin. Dyn. Syst., 29 (2011), 67-79.  doi: 10.3934/dcds.2011.29.67.

[9] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511614583.
[10] A. Bobrowski, Convergence of One-parameter Operator Semigroups. In Models of Mathematical Biology and Elsewhere, Cambridge University Press, Cambridge, 2016. 
[11]

A. Bobrowski, On Hille-type approximation of degenerate semigroups of operators, Linear Algebra and its Applications, 511 (2016), 31-53. 

[12]

A. Bobrowski and M. Kimmel, Asymptotic behaviour of an operator exponential related to branching random walk models of DNA repeats, J. Biol. Systems, 7 (1999), 33-43.  doi: 10.1142/S0218339099000048.

[13]

B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356.  doi: 10.1007/s00233-007-9036-2.

[14] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer Verlag, New York, 1999. 
[15]

M. Kimmel and D. N. Stivers, Time-continuous branching walk models of unstable gene amplification, Bull. Math. Biol., 50 (1994), 337-357. 

[16]

M. KimmelA. Świerniak and A. Polański, Infinite-dimensional model of evolution of drug resistance of cancer cells, J. Math. Systems Estimation Control, 8 (1998), 1-16. 

[17]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162.  doi: 10.1007/s00209-004-0695-3.

[18]

J. L. Lebowitz and S. I. Rubinov, A theory for the age and generation time distribution of a microbial population, J. Theor. Biol., 1 (1974), 17-36.  doi: 10.1007/BF02339486.

[19]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719512.

[20]

P. Namayanja, Transport on Network Structures Ph. D thesis, UKZN, 2012.

[21]

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

[22]

A. Świerniak, A. Polański and M. Kimmel, Control problems arising in chemotherapy under evolving drug resistance, Preprints of the 13th World Congress of IFAC 1996, Volume B, 411-416.

[23]

H. T. K. Tse, W. McConnell Weaver and D. Di Carlo, Increased asymmetric and multi-daughter cell division in mechanically confined microenvironments PLoS ONE, 7 (2012), e38986. doi: 10.1371/journal.pone.0038986.

show all references

References:
[1]

H. Amann and J. Escher, Analysis II Birkhäuser, Basel 2008.

[2]

W. Arendt, Resolvent positive operators, Proc. Lond. Math. Soc., 54 (1987), 321-349.  doi: 10.1112/plms/s3-54.2.321.

[3] J. Banasiak and L. Arlotti, Positive Perturbations of Semigroups with Applications, Springer Verlag, London, 2006. 
[4]

J. Banasiak and A. Falkiewicz, Some transport and diffusion processes on networks and their graph realizability Appl. Math. Lett. ,45 (2015), 25-30 doi: 10.1016/j.aml.2015.01.006.

[5]

J. Banasiak, A. Falkiewicz and P. Namayanja, Semigroup approach to diffusion and transport problems on networks Semigroup Forum. [DOI 10.1007/s00233-015-9730-4] doi: 10.1007/s00233-015-9730-4.

[6]

J. BanasiakA. Falkiewicz and P. Namayanja, Asymptotic state lumping in transport and diffusion problems on networks with applications to population problems, Math. Models Methods Appl. Sci., 26 (2016), 215-247.  doi: 10.1142/S0218202516400017.

[7]

J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-05140-6.

[8]

J. Banasiak and M. Moszyński, Dynamics of birth-and-death processes with proliferation -stability and chaos, Discrete Contin. Dyn. Syst., 29 (2011), 67-79.  doi: 10.3934/dcds.2011.29.67.

[9] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511614583.
[10] A. Bobrowski, Convergence of One-parameter Operator Semigroups. In Models of Mathematical Biology and Elsewhere, Cambridge University Press, Cambridge, 2016. 
[11]

A. Bobrowski, On Hille-type approximation of degenerate semigroups of operators, Linear Algebra and its Applications, 511 (2016), 31-53. 

[12]

A. Bobrowski and M. Kimmel, Asymptotic behaviour of an operator exponential related to branching random walk models of DNA repeats, J. Biol. Systems, 7 (1999), 33-43.  doi: 10.1142/S0218339099000048.

[13]

B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356.  doi: 10.1007/s00233-007-9036-2.

[14] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer Verlag, New York, 1999. 
[15]

M. Kimmel and D. N. Stivers, Time-continuous branching walk models of unstable gene amplification, Bull. Math. Biol., 50 (1994), 337-357. 

[16]

M. KimmelA. Świerniak and A. Polański, Infinite-dimensional model of evolution of drug resistance of cancer cells, J. Math. Systems Estimation Control, 8 (1998), 1-16. 

[17]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162.  doi: 10.1007/s00209-004-0695-3.

[18]

J. L. Lebowitz and S. I. Rubinov, A theory for the age and generation time distribution of a microbial population, J. Theor. Biol., 1 (1974), 17-36.  doi: 10.1007/BF02339486.

[19]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719512.

[20]

P. Namayanja, Transport on Network Structures Ph. D thesis, UKZN, 2012.

[21]

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

[22]

A. Świerniak, A. Polański and M. Kimmel, Control problems arising in chemotherapy under evolving drug resistance, Preprints of the 13th World Congress of IFAC 1996, Volume B, 411-416.

[23]

H. T. K. Tse, W. McConnell Weaver and D. Di Carlo, Increased asymmetric and multi-daughter cell division in mechanically confined microenvironments PLoS ONE, 7 (2012), e38986. doi: 10.1371/journal.pone.0038986.

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