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Population Dynamics with epistatic interactions and its applications to mathematical models of cancer

Bauer, B. (2015). Population Dynamics with epistatic interactions and its applications to mathematical models of cancer. PhD Thesis, Max-Planck-Institut für Evolutionsbiologie, Plön.

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Creators:
Bauer, Benedikt1, Author
Traulsen, Arne1, Referee
Keller, Karsten, Referee
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1Department Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Max Planck Society, ou_1445641

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Abstract: Cells of every organism undergo somatic mutations. Many mutations do not significantly affect the gene’s function, while other mutations impair the gene’s function. Often, this impairment leads to decreased survival chance of the cell, such that the cell dies. If necessary, this cell can be unproblematically be replaced. Sometimes, however, that impairment disturbs the cell’s life cycle in a way that decreases its chance of cell death, or apoptosis, or increases its rate of cell division. Uncontrolled cell proliferation can then lead to the formation of cancer, and ultimately to a tumor. Usually not only one, but a handful of mutations are necessary to affect different safety mechanisms of the cell. Often, genes influence each other, which is called epistasis. Also some oncogenes underlie epistatic interactions. In the Burkitt Lymphoma the hallmark mutation – an IG/MYC translocation – is believed to actually lower the cell’s chance of survival. In concert with other mutations, however, it forms a lymphoma. Basic knowledge about the initiation of cancers where the genes of the cancerous mutations underlie epistatic interactions is rare and difficult to acquire in experimental system. The work described in this thesis is a theoretical analysis of systems with epistatic interactions in cancer initiation. The population dynamics of an abstract system with two different types of mutations between which epistatic interactions exist is analyzed. One type is deleterious by itself, the other one is (nearly) neutral. If the deleterious mutation is accompanied by enough mutations of the other types, the cell has a fitness advantage. We find, amongst others, that the cancer deploying cell lineage has most likely acquired that specific mutation only subsequently to the other, non-deleterious mutations, which inhibit the negative effect of that particular mutation. It is conceivable that epistatic effects could change the order of mutations not only in the survival and proliferation rates of the cell, but also in the mutation rates. Hence, we further pursue the question: “If the deleterious mutation increases the mutation rate for acquiring the necessary, additional mutations, how does this change the probability for the order of mutations?”. We develop a recursive algorithm for the computation of the probability density functions of the different mutational pathways over time. Finally, we develop a model aiming at describing the initiation of Burkitt Lymphoma. Lastly, an outlook is given explaining future research directions based on epistasis in cancer initiation.

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Language(s): eng - English
Dates: 2015-09-142015-09-14
Publication Status: Published in print
Pages: 112 S.
Publishing info: Plön : Max-Planck-Institut für Evolutionsbiologie
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Biological Background . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Epistasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Branching Process and Probability Generating Function . . . 8
1.5 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . 12
2 Cancer Initiation with Epistatic Interactions Between Driver
and Passenger Mutations 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Analytical Results . . . . . . . . . . . . . . . . . . . . 22
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Calculation of Time Distribution and Path Probabilities 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Model and Results . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Two Dimensional Fitness Landscape . . . . . . . . . . 34
3.2.2 Time Distribution . . . . . . . . . . . . . . . . . . . . . 36
3.2.3 Path Probabilities . . . . . . . . . . . . . . . . . . . . . 37
3.2.4 Multiple Mutations in two Dimensions . . . . . . . . . 39
3.2.5 Multi Dimensional Fitness Landscapes . . . . . . . . . 40
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Model for the Initiation of Burkitt Lymphoma 45
4.1 A Model for the Sequence of Cancer Initiating Events in Burkitt
Lymphoma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.1 Materials and Methods . . . . . . . . . . . . . . . . . . 46
4.1.2 Results and Discussion . . . . . . . . . . . . . . . . . . 51
4.2 Timing and Nature of Relapses . . . . . . . . . . . . . . . . . 55
5 Further Research 63
5.1 Branching Process with Frequency Dependent Fitness . . . . . 63
5.2 Epistasis in Spatially Structured Populations . . . . . . . . . . 65
6 Summary 73
7 Appendix 75
7.1 Analytic Expression for the Average Number of Cells without
the Primary Driver Mutation at Generation t . . . . . . . . . 75
7.1.1 Secondary Driver Fitness Advantage is unequal to Zero
- k Secondary Driver Mutations . . . . . . . . . . . . . 75
7.2 Analytic Expression for the Average Number of Cells with the
Primary Driver Mutation at Generation t . . . . . . . . . . . . 76
7.3 Intuitive Description of Equation (11) . . . . . . . . . . . . . . 80
7.4 General Probability Generating Functions . . . . . . . . . . . 82
7.5 Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.6 Single-Path Time Distribution . . . . . . . . . . . . . . . . . . 85
7.7 Implementation of Burkitt Lymphoma Model . . . . . . . . . 88
Bibliography 89
Rev. Method: -
Identifiers: Other: Diss/12642
Degree: PhD

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