Dynamics of Mutant Cells in Hierarchical Organized Tissues

Most tissues in multicellular organisms are maintained by continuous cell renewal processes. However, high turnover of many cells implies a large number of error-prone cell divisions. Hierarchical organized tissue structures with stem cell driven cell differentiation provide one way to prevent the accumulation of mutations, because only few stem cells are long lived. We investigate the deterministic dynamics of cells in such a hierarchical multi compartment model, where each compartment represents a certain stage of cell differentiation. The dynamics of the interacting system is described by ordinary differential equations coupled across compartments. We present analytical solutions for these equations, calculate the corresponding extinction times and compare our results to individual based stochastic simulations. Our general compartment structure can be applied to different tissues, as for example hematopoiesis, the epidermis, or colonic crypts. The solutions provide a description of the average time development of stem cell and non stem cell driven mutants and can be used to illustrate general and specific features of the dynamics of mutant cells in such hierarchically structured populations. We illustrate one possible application of this approach by discussing the origin and dynamics of PIG-A mutant clones that are found in the bloodstream of virtually every healthy adult human. From this it is apparent, that not only the occurrence of a mutant but also the compartment of origin is of importance

A deterministic model for the occurrence and dynamics of multiple mutations in hierarchically organized tissues

We model a general, hierarchically organized tissue by a multi compartment approach, allowing any number of mutations within a cell. We derive closed solutions for the deterministic clonal dynamics and the reproductive capacity of single clones. Our results hold for the average dynamics in a hierarchical tissue characterized by an arbitrary combination of proliferation parameters.Comment: 4 figures, to appear in Royal Society Interfac

Phenotypic heterogeneity in modeling cancer evolution

The unwelcome evolution of malignancy during cancer progression emerges through a selection process in a complex heterogeneous population structure. In the present work, we investigate evolutionary dynamics in a phenotypically heterogeneous population of stem cells (SCs) and their associated progenitors. The fate of a malignant mutation is determined not only by overall stem cell and differentiated cell growth rates but also differentiation and dedifferentiation rates. We investigate the effect of such a complex population structure on the evolution of malignant mutations. We derive exact analytic results for the fixation probability of a mutant arising in each of the subpopulations. The analytic results are in almost perfect agreement with the numerical simulations. Moreover, a condition for evolutionary advantage of a mutant cell versus the wild type population is given in the present study. We also show that microenvironment-induced plasticity in invading mutants leads to more aggressive mutants with higher fixation probability. Our model predicts that decreasing polarity between stem and differentiated cells turnover would raise the survivability of non-plastic mutants; while it would suppress the development of malignancy for plastic mutants. We discuss our model in the context of colorectal/intestinal cancer (at the epithelium). This novel mathematical framework can be applied more generally to a variety of problems concerning selection in heterogeneous populations, in other contexts such as population genetics, and ecology.Comment: 28 pages, 7 figures, 2 table

Mathematical models of cell population dynamics

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.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 12

Effect of Dedifferentiation on Time to Mutation Acquisition in Stem Cell-Driven Cancers

Accumulating evidence suggests that many tumors have a hierarchical organization, with the bulk of the tumor composed of relatively differentiated short-lived progenitor cells that are maintained by a small population of undifferentiated long-lived cancer stem cells. It is unclear, however, whether cancer stem cells originate from normal stem cells or from dedifferentiated progenitor cells. To address this, we mathematically modeled the effect of dedifferentiation on carcinogenesis. We considered a hybrid stochastic-deterministic model of mutation accumulation in both stem cells and progenitors, including dedifferentiation of progenitor cells to a stem cell-like state. We performed exact computer simulations of the emergence of tumor subpopulations with two mutations, and we derived semi-analytical estimates for the waiting time distribution to fixation. Our results suggest that dedifferentiation may play an important role in carcinogenesis, depending on how stem cell homeostasis is maintained. If the stem cell population size is held strictly constant (due to all divisions being asymmetric), we found that dedifferentiation acts like a positive selective force in the stem cell population and thus speeds carcinogenesis. If the stem cell population size is allowed to vary stochastically with density-dependent reproduction rates (allowing both symmetric and asymmetric divisions), we found that dedifferentiation beyond a critical threshold leads to exponential growth of the stem cell population. Thus, dedifferentiation may play a crucial role, the common modeling assumption of constant stem cell population size may not be adequate, and further progress in understanding carcinogenesis demands a more detailed mechanistic understanding of stem cell homeostasis

Cancer modeling: from optimal cell renewal to immunotherapy

Cancer is a disease caused by mutations in normal cells. According to the National Cancer Institute, in 2016, an estimated 1.6 million people were diagnosed and approximately 0.5 million people died from the disease in the United States. There are many factors that shape cancer at the cellular and organismal level, including genetic, immunological, and environmental components. In this thesis, we show how mathematical modeling can be used to provide insight into some of the key mechanisms underlying cancer dynamics. First, we use mathematical modeling to investigate optimal homeostatic cell renewal in tissues such as the small intestine with an emphasis on division patterns and tissue architecture. We find that the division patterns that delay the accumulation of mutations are strictly associated with the population sizes of the tissue. In particular, patterns with long chains of differentiation delay the time to observe a second-hit mutant, which is important given that for many cancers two mutations are enough to initiate a tumor. We also investigated homeostatic cell renewal under a selective pressure and find that hierarchically organized tissues act as suppressors of selection; we find that an architecture with a small number of stem cells and larger pools of transit amplifying cells and mature differentiated cells, together with long chains of differentiation, form a robust evolutionary strategy to delay the time to observe a second-hit mutant when mutations acquire a fitness advantage or disadvantage. We also formulate a model of the immune response to cancer in the presence of costimulatory and inhibitory signals. We demonstrate that the coordination of such signals is crucial to initiate an effective immune response, and while immunotherapy has become a promising cancer treatment over the past decade, these results offer some explanations for why it can fail