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  Mathematical models of cell population dynamics

Werner, B. (2013). Mathematical models of cell population dynamics. PhD Thesis, University, Lübeck.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0026-A6E4-B Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0026-A6E5-9
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 Creators:
Werner, Benjamin1, Author              
Traulsen, Arne1, Referee              
Deutsch, Andreas, Referee
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1Research Group Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Max Planck Society, ou_1445641              

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 Abstract: Cancers result from altered cell proliferation properties, caused by mutations in specific genes. An accumulation of multiple mutations within a cell increases the risk to develop cancer. However, mechanisms evolved to prevent such multiple mutations. One such mechanism is a hierarchically organized tissue structure. At the root of the hierarchy are a few, slow proliferating stem cells. After some cell differentiations all functional cells of a tissue are obtained. In the first two chapters of this thesis, we mathematically and computationally evaluate a multi compartment model that is an abstract representation of such hierarchical tissues. We find analytical expressions for stem cell and non stem cell driven cell populations without further mutations. We show that non stem cell mutations give raise to clonal waves, that travel trough the hierarchy and are lost in the long run. We calculate the average extinction times of such clonal waves. In the third chapter we allow for arbitrary many mutations in hierarchically organized tissues and find exact expressions for the reproductive capacity of cells, highlighting that multiple mutations are strongly suppressed by the hierarchy. In the fourth chapter we turn to a related problem, the evolution of resistance against molecular targeted cancer drugs. We develop a minimalistic mathematical model and compare the predicted dynamics to experimental derived observations. Interestingly we find that resistance can be induced either by mutation or intercellular processes such as phenotypic switching. In the fifth chapter of this thesis, we investigate the shortening of telomeres in detail. The comparison of mathematical results to experimental data reveals interesting properties of stem cell dynamics. We find hints for an increasing stem cell pool size with age, caused by a small number of symmetric stem cell divisions. We also implement disease scenarios and find exact expressions how the patterns of telomere shortening differ for healthy and sick persons. Our model provides a simple explanation for the pronounced increase of telomere shortening in the first years of live, followed by an almost linear decrease for healthy adults. In the final chapter, we implement a method to introduce arbitrary many random mutations into the framework of frequency dependent selection. We show how disadvantageous mutations can reach fixation under a deterministic scenario and discuss possible applications to cancer modeling.

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Language(s): eng - English
 Dates: 2013-09-092013-04-24
 Publication Status: Published in print
 Pages: IV, 155 S.
 Publishing info: Lübeck : University
 Table of Contents: 1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Biological basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Hierarchically organized tissues . . . . . . . . . . . . . . . . . 4
1.2.2 Cancer biology . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Molecular targeted treatment strategies . . . . . . . . . . . . 9
1.2.4 Telomeres and telomerase . . . . . . . . . . . . . . . . . . . . 11
1.3 Stochastic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Gillespie algorithm . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Moran process . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3 Towards deterministic dynamics . . . . . . . . . . . . . . . . . 22
1.4 Deterministic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Replicator equation . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Single mutations in hierarchical tissues 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Mathematical model and results . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 Stem cell driven dynamics . . . . . . . . . . . . . . . . . . . . 31
2.2.2 Non stem cell driven dynamics . . . . . . . . . . . . . . . . . 34
2.2.3 Mutant extinction times . . . . . . . . . . . . . . . . . . . . . 37
2.2.4 Example: Dynamics of PIG-A mutants . . . . . . . . . . . . . 39
2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Multiple mutations in hierarchical tissues 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Mathematical model and results . . . . . . . . . . . . . . . . . . . . . 47
3.2.1 Time continuous dynamics of multiple mutations . . . . . . . 49
3.2.2 Cell reproductive capacity . . . . . . . . . . . . . . . . . . . . 54
3.2.3 Reproductive capacity of neutral mutants . . . . . . . . . . . 55
3.2.4 Number of distinct neutral mutations . . . . . . . . . . . . . 56
3.2.5 Example: clonal diversity in acute lymphoblastic leukemia . . 58
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Resistance evolution 63
4.1 Quasi species equation with time dependent fitness . . . . . . . . . . 64
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Mathematical model and results . . . . . . . . . . . . . . . . . . . . . 71
4.4.1 Analytical approximation for large population size . . . . . . 73
4.4.2 Development of Imatinib resistance in cell culture . . . . . . . 74
4.4.3 Fitting the mathematical model to the experimental data . . 76
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 A mathematical model of telomere shortening 81
5.1 Mathematical model and results . . . . . . . . . . . . . . . . . . . . . 83
5.1.1 Asymmetric cell divisions . . . . . . . . . . . . . . . . . . . . 84
5.1.2 Symmetric cell divisions . . . . . . . . . . . . . . . . . . . . . 89
5.1.3 T-cell mediated stem cell death . . . . . . . . . . . . . . . . . 93
6 Impact of random mutations on population fitness 99
6.1 Random mutant games . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3 Mathematical model and results . . . . . . . . . . . . . . . . . . . . . 103
6.3.1 Games with two types . . . . . . . . . . . . . . . . . . . . . . 103
6.3.2 Games with n types . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.3 Games with equal gains from switching . . . . . . . . . . . . 112
6.3.4 Diploid populations with two alleles . . . . . . . . . . . . . . 115
6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.5 Random mutations and cancer . . . . . . . . . . . . . . . . . . . . . 118
7 Summary and Outlook 121
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Bibliography 125
 Rev. Method: -
 Identifiers: Other: Diss/12615
 Degree: PhD

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