On the dynamics of neutral mutations in a mathematical model for a homogeneous stem cell population

Stem cell mutations, associated cancer risk, and consequences for regenerative medicine

Mutation accumulation in stem cells has been associated with cancer risk. However, the presence of numerous mutant clones in healthy tissues has raised the question of what limits cancer initiation. Here, we review recent developments in characterizing mutation accumulation in healthy tissues and compare mutation rates in stem cells during development and adult life with corresponding cancer risk. A certain level of mutagenesis within the stem cell pool might be beneficial to limit the size of malignant clones through competition. This knowledge impacts our understanding of carcinogenesis with potential consequences for the use of stem cells in regenerative medicine.

Open problems in mathematical biology

Biology is data-rich, and it is equally rich in concepts and hypotheses. Part of trying to understand biological processes and systems is therefore to confront our ideas and hypotheses with data using statistical methods to determine the extent to which our hypotheses agree with reality. But doing so in a systematic way is becoming increasingly challenging as our hypotheses become more detailed, and our data becomes more complex. Mathematical methods are therefore gaining in importance across the life- and biomedical sciences. Mathematical models allow us to test our understanding, make testable predictions about future behaviour, and gain insights into how we can control the behaviour of biological systems. It has been argued that mathematical methods can be of great benefit to biologists to make sense of data. But mathematics and mathematicians are set to benefit equally from considering the often bewildering complexity inherent to living systems. Here we present a small selection of open problems and challenges in mathematical biology. We have chosen these open problems because they are of both biological and mathematical interest.

Increased stem cell proliferation in atherosclerosis accelerates clonal hematopoiesis

Clonal hematopoiesis, a condition in which individual hematopoietic stem cell clones generate a disproportionate fraction of blood leukocytes, correlates with higher risk for cardiovascular disease. The mechanisms behind this association are incompletely understood. Here, we show that hematopoietic stem cell division rates are increased in mice and humans with atherosclerosis. Mathematical analysis demonstrates that increased stem cell proliferation expedites somatic evolution and expansion of clones with driver mutations. The experimentally determined division rate elevation in atherosclerosis patients is sufficient to produce a 3.5-fold increased risk of clonal hematopoiesis by age 70. We confirm the accuracy of our theoretical framework in mouse models of atherosclerosis and sleep fragmentation by showing that expansion of competitively transplanted Tet2^-/- cells is accelerated under conditions of chronically elevated hematopoietic activity. Hence, increased hematopoietic stem cell proliferation is an important factor contributing to the association between cardiovascular disease and clonal hematopoiesis.

CELLULAR FEEDBACK NETWORKS AND THEIR RESILIENCE AGAINST MUTATIONS

Many tissues undergo a steady turnover, where cell divisions are on average balanced with cell deaths. Cell fate decisions such as stem cell (SC) differentiations, proliferations, or differentiated cell (DC) deaths, may be controlled by cell populations through cell-to-cell signaling. Here, we examine a class of mathematical models of turnover in SC lineages to understand engineering design principles of control (feedback) loops, that may operate in such systems. By using ordinary differential equations that describe the co-dynamics of SCs and DCs, we study the effect of different types of mutations that interfere with feedback present within cellular networks. For instance, we find that mutants that do not participate in feedback are less dangerous in the sense that they will not rise from low numbers, whereas mutants that do not respond to feedback signals could rise and replace the wild-type population. Additionally, we asked if different feedback networks can have different degrees of resilience against such mutations. We found that all minimal networks, that is networks consisting of exactly one feedback loop that is sufficient for homeostatic stability of the wild-type population, are equally vulnerable. Mutants with a weakened/eliminated feedback parameter might expand from lower numbers and either enter unlimited growth or reach an equilibrium with an increased number of SCs and DCs. Therefore, from an evolutionary viewpoint, it appears advantageous to combine feedback loops, creating redundant feedback networks. Interestingly, from an engineering prospective, not all such redundant systems are equally resilient. For some of them, any mutation that weakens/eliminates one of the loops will lead to a population growth of SCs. For others, the population of SCs can actually shrink as a result of “cutting” one of the loops, thus slowing down further unwanted transformations.

Stem Cell DNA Damage and Genome Mutation in the Context of Aging and Cancer Initiation

Adult stem cells fuel tissue homeostasis and regeneration through their unique ability to self-renew and differentiate into specialized cells. Thus, their DNA provides instructions that impact the tissue as a whole. Since DNA is not an inert molecule, but rather dynamic, interacting with a myriad of chemical and physical factors, it encounters damage from both endogenous and exogenous sources. Damage to DNA introduces deviations from its normal intact structure and, if left unrepaired, may result in a genetic mutation. In turn, mutant genomes of stem and progenitor cells are inherited in cells of the lineage, thus eroding the genetic information that maintains homeostasis of the somatic cell population. Errors arising in stem and progenitor cells will have a substantially larger impact on the tissue in which they reside than errors occurring in postmitotic differentiated cells. Therefore, maintaining the integrity of genomic DNA within our stem cells is essential to protect the instructions necessary for rebuilding healthy tissues during homeostatic renewal. In this review, we will first discuss DNA damage arising in stem cells and cell- and tissue-intrinsic mechanisms that protect against harmful effects of this damage. Secondly, we will examine how erroneous DNA repair and persistent DNA damage in stem and progenitor cells impact stem cells and tissues in the context of cancer initiation and aging. Finally, we will discuss the use of invertebrate and vertebrate model systems to address unanswered questions on the role that DNA damage and mutation may play in aging and precancerous conditions.

Stochastic cell renewal process and lengthening of cell cycle

Evolution of the stem cell population responsible for homeostatic cell renewal processes is analyzed. We assume that this regime is the product of a delicate balance between symmetric divisions that, after each cell cycle, originates a new stem cell or its disappearance (through cell differentiation). This dynamics leads to a monoclonal population, that is for an initial homogeneous set of stem cells, fixation of each clone is equiprobable. In this work we show that if there is an altered stem cell with a longer cell cycle than the rest, the fixation of this altered clone is more likely. We also study the consequeces of the appearance of successive alterations with these characteristics and their fixations. This effect is purely due to inherent characteristics of the cell renewal dynamics and as time goes by it leads to a quiescence state for stem cells owing to the recurrent fixation of such altered cells. Therefore it would contribute to the aging process.

Clonal hematopoiesis of indeterminate potential and its impact on patient trajectories after stem cell transplantation

Clonal hematopoiesis of indeterminate potential (CHIP) is a recently identified process where older patients accumulate distinct subclones defined by recurring somatic mutations in hematopoietic stem cells. CHIP's implications for stem cell transplantation have been harder to identify due to the high degree of mutational heterogeneity that is present within the genetically distinct subclones. In order to gain a better understanding of CHIP and the impact of clonal dynamics on transplantation outcomes, we created a mathematical model of clonal competition dynamics. Our analyses highlight the importance of understanding competition intensity between healthy and mutant clones. Importantly, we highlight the risk that CHIP poses in leading to dominance of precancerous mutant clones and the risk of donor derived leukemia. Furthermore, we estimate the degree of competition intensity and bone marrow niche decline in mice during aging by using our modeling framework. Together, our work highlights the importance of better characterizing the ecological and clonal composition in hematopoietic donor populations at the time of stem cell transplantation.

Long-term treatment-free remission of chronic myeloid leukemia with falling levels of residual leukemic cells

Following the achievement of deep molecular response on tyrosine kinase inhibitors (TKIs), approximately half of patients with chronic myeloid leukemia (CML) can discontinue TKI and remain in treatment-free remission (TFR). The ALLG CML8 study enrolled 40 imatinib-treated patients with undetectable BCR-ABL1 mRNA (approximately MR^4.5). Molecular relapse was defined as detectable BCR-ABL1 on two consecutive tests or any single value >0.1%. With a median follow-up of 8.6 years (range 5.7-11.2 years), 18 patients remain in continuous TFR (45.0%; 95% confidence interval 31.9-63.4%). The latest relapse detected was 27 months after stopping imatinib. No patient progressed to advanced phase. Twenty-two patients met criteria for imatinib re-treatment and all regained undetectable molecular response. Nine patients in long-term TFR were monitored by highly sensitive individualized BCR-ABL1 DNA PCR in a sufficient number of samples to enable more precise quantification of residual leukemia. BCR-ABL1 DNA decreased from a median of MR^5.0 in the first year of TFR to MR^6.1 in the sixth year of TFR. Our results support the long-term safety and remarkable stability of response after imatinib discontinuation in appropriately selected CML patients. Serial high sensitivity testing provides a new and unexpected finding of gradually reducing CML cells in patients in long-term TFR.

A mathematical model of stem cell regeneration with epigenetic state transitions

In this paper, we study a mathematical model of stem cell regeneration with epigenetic state transitions. In the model, the heterogeneity of stem cells is considered through the epigenetic state of each cell, and each epigenetic state defines a subpopulation of stem cells. The dynamics of the subpopulations are modeled by a set of ordinary differential equations in which epigenetic state transition in cell division is given by the transition probability. We present analysis for the existence and linear stability of the equilibrium state. As an example, we apply the model to study the dynamics of state transition in breast cancer stem cells.

Evolutionary dynamics of paroxysmal nocturnal hemoglobinuria

Paroxysmal nocturnal hemoglobinuria (PNH) is an acquired clonal blood disorder characterized by hemolysis and a high risk of thrombosis, that is due to a deficiency in several cell surface proteins that prevent complement activation. Its origin has been traced to a somatic mutation in the PIG-A gene within hematopoietic stem cells (HSC). However, to date the question of how this mutant clone expands in size to contribute significantly to hematopoiesis remains under debate. One hypothesis posits the existence of a selective advantage of PIG-A mutated cells due to an immune mediated attack on normal HSC, but the evidence supporting this hypothesis is inconclusive. An alternative (and simpler) explanation attributes clonal expansion to neutral drift, in which case selection neither favours nor inhibits expansion of PIG-A mutated HSC. Here we examine the implications of the neutral drift model by numerically evolving a Markov chain for the probabilities of all possible outcomes, and investigate the possible occurrence and evolution, within this framework, of multiple independently arising clones within the HSC pool. Predictions of the model agree well with the known incidence of the disease and average age at diagnosis. Notwithstanding the slight difference in clonal expansion rates between our results and those reported in the literature, our model results lead to a relative stability of clone size when averaging multiple cases, in accord with what has been observed in human trials. The probability of a patient harbouring a second clone in the HSC pool was found to be extremely low (~10-8). Thus our results suggest that in clinical cases of PNH where two independent clones of mutant cells are observed, only one of those is likely to have originated in the HSC pool.

The mechanisms leading to expansion of HSC with mutations in the PIG-A gene that leads to the PNH phenotype remains unclear. Data so far suggests there is no intrinsic fitness advantage of the mutant cells compared to normal cells. Assuming neutral drift within the HSC compartment, we determined from first principles the incidence of the disease in a population, the average clone size in patients, the probability of clonal extinction, the likelihood of several separate clones coexisting in the HSC pool, and the expected expansion rate of a mutant clone. Our results are similar to what is observed in clinical practice. We also find that in such a model the probability of multiple PNH clones arising independently in the HSC pool is exceptionally small. This suggests that in clinical cases where more than one distinct clone is observed, all but one of the clones are likely to have emerged in cells that are downstream of the HSC population. We propose that PNH is perhaps the first disease where neutral drift alone may be responsible for clonal expansion leading to a clinical problem.

Optimizing homeostatic cell renewal in hierarchical tissues

In order to maintain homeostasis, mature cells removed from the top compartment of hierarchical tissues have to be replenished by means of differentiation and self-renewal events happening in the more primitive compartments. As each cell division is associated with a risk of mutation, cell division patterns have to be optimized, in order to minimize or delay the risk of malignancy generation. Here we study this optimization problem, focusing on the role of division tree length, that is, the number of layers of cells activated in response to the loss of terminally differentiated cells, which is related to the balance between differentiation and self-renewal events in the compartments. Using both analytical methods and stochastic simulations in a metapopulation-style model, we find that shorter division trees are advantageous if the objective is to minimize the total number of one-hit mutants in the cell population. Longer division trees on the other hand minimize the accumulation of two-hit mutants, which is a more likely evolutionary goal given the key role played by tumor suppressor genes in cancer initiation. While division tree length is the most important property determining mutant accumulation, we also find that increasing the size of primitive compartments helps to delay two-hit mutant generation.

Cells in multicellular organisms are organized hierarchically. A stem cell gives rise to a chain of dividing and progressively differentiating offspring. At the end of this chain (called a lineage) are terminally differentiated cells that perform their function and undergo programmed cell death, to be replaced by new divisions of less differentiated cells. Here we are interested in the design of such lineages. At one extreme, one can imagine that a loss of terminally differentiated cells only results in divisions of cells in close hierarchical proximity to them, giving rise to very short division trees. On the other hand, it is possible that a long chain of increasingly primitive cells gets activated in response to the loss of differentiated cells. We expect that an important type of selection pressure acting upon tissue design is the minimization of mutations that happen in the course of everyday tissue maintenance (homeostasis). For example, tumor suppressor gene inactivation (two consecutive mutations) is an early rate-limiting step in many cancers. Using mathematical and computational methods, we find that the length of division trees is anti-correlated with the likelihood of double mutations, and lengthening the trees may provide an evolutionary advantage to the organism by delaying the onset of cancer.

Clonal dominance and transplantation dynamics in hematopoietic stem cell compartments

Hematopoietic stem cells in mammals are known to reside mostly in the bone marrow, but also transitively passage in small numbers in the blood. Experimental findings have suggested that they exist in a dynamic equilibrium, continuously migrating between these two compartments. Here we construct an individual-based mathematical model of this process, which is parametrised using existing empirical findings from mice. This approach allows us to quantify the amount of migration between the bone marrow niches and the peripheral blood. We use this model to investigate clonal hematopoiesis, which is a significant risk factor for hematologic cancers. We also analyse the engraftment of donor stem cells into non-conditioned and conditioned hosts, quantifying the impact of different treatment scenarios. The simplicity of the model permits a thorough mathematical analysis, providing deeper insights into the dynamics of both the model and of the real-world system. We predict the time taken for mutant clones to expand within a host, as well as chimerism levels that can be expected following transplantation therapy, and the probability that a preconditioned host is reconstituted by donor cells.

Clonal hematopoiesis—where mature myeloid cells in the blood deriving from a single stem cell are over-represented—is a major risk factor for overt hematologic malignancies. To quantify how likely this phenomena is, we combine existing observations with a novel stochastic model and extensive mathematical analysis. This approach allows us to observe the hidden dynamics of the hematopoietic system. We conclude that for a clone to be detectable within the lifetime of a mouse, it requires a selective advantage. I.e. the clonal expansion cannot be explained by neutral drift alone. Furthermore, we use our model to describe the dynamics of hematopoiesis after stem cell transplantation. In agreement with earlier findings, we observe that niche-space saturation decreases engraftment efficiency. We further discuss the implications of our findings for human hematopoiesis where the quantity and role of stem cells is frequently debated.

Mathematical Modeling Links Pregnancy-Associated Changes and Breast Cancer Risk

Recent debate has concentrated on the contribution of bad luck to cancer development. The tight correlation between the number of tissue-specific stem cell divisions and cancer risk of the same tissue suggests that bad luck has an important role to play in tumor development, but the full extent of this contribution remains an open question. Improved understanding of the interplay between extrinsic and intrinsic factors at the molecular level is one promising route to identifying the limits on extrinsic control of tumor initiation, which is highly relevant to cancer prevention. Here, we use a simple mathematical model to show that recent data on the variation in numbers of breast epithelial cells with progenitor features due to pregnancy are sufficient to explain the known protective effect of full-term pregnancy in early adulthood for estrogen receptor-positive (ER⁺) breast cancer later in life. Our work provides a mechanism for this previously ill-understood effect and illuminates the complex influence of extrinsic factors at the molecular level in breast cancer. These findings represent an important contribution to the ongoing research into the role of bad luck in human tumorigenesis. Cancer Res; 77(11); 2800-9. ©2017 AACR.

Model-Based Evaluation of Spontaneous Tumor Regression in Pilocytic Astrocytoma

Pilocytic astrocytoma (PA) is the most common brain tumor in children. This tumor is usually benign and has a good prognosis. Total resection is the treatment of choice and will cure the majority of patients. However, often only partial resection is possible due to the location of the tumor. In that case, spontaneous regression, regrowth, or progression to a more aggressive form have been observed. The dependency between the residual tumor size and spontaneous regression is not understood yet. Therefore, the prognosis is largely unpredictable and there is controversy regarding the management of patients for whom complete resection cannot be achieved. Strategies span from pure observation (wait and see) to combinations of surgery, adjuvant chemotherapy, and radiotherapy. Here, we introduce a mathematical model to investigate the growth and progression behavior of PA. In particular, we propose a Markov chain model incorporating cell proliferation and death as well as mutations. Our model analysis shows that the tumor behavior after partial resection is essentially determined by a risk coefficient γ, which can be deduced from epidemiological data about PA. Our results quantitatively predict the regression probability of a partially resected benign PA given the residual tumor size and lead to the hypothesis that this dependency is linear, implying that removing any amount of tumor mass will improve prognosis. This finding stands in contrast to diffuse malignant glioma where an extent of resection threshold has been experimentally shown, below which no benefit for survival is expected. These results have important implications for future therapeutic studies in PA that should include residual tumor volume as a prognostic factor.

Cellular population dynamics control the robustness of the stem cell niche

Within populations of cells, fate decisions are controlled by an indeterminate combination of cell-intrinsic and cell-extrinsic factors. In the case of stem cells, the stem cell niche is believed to maintain ‘stemness’ through communication and interactions between the stem cells and one or more other cell-types that contribute to the niche conditions. To investigate the robustness of cell fate decisions in the stem cell hierarchy and the role that the niche plays, we introduce simple mathematical models of stem and progenitor cells, their progeny and their interplay in the niche. These models capture the fundamental processes of proliferation and differentiation and allow us to consider alternative possibilities regarding how niche-mediated signalling feedback regulates the niche dynamics. Generalised stability analysis of these stem cell niche systems enables us to describe the stability properties of each model. We find that although the number of feasible states depends on the model, their probabilities of stability in general do not: stem cell–niche models are stable across a wide range of parameters. We demonstrate that niche-mediated feedback increases the number of stable steady states, and show how distinct cell states have distinct branching characteristics. The ecological feedback and interactions mediated by the stem cell niche thus lend (surprisingly) high levels of robustness to the stem and progenitor cell population dynamics. Furthermore, cell–cell interactions are sufficient for populations of stem cells and their progeny to achieve stability and maintain homeostasis. We show that the robustness of the niche – and hence of the stem cell pool in the niche – depends only weakly, if at all, on the complexity of the niche make-up: simple as well as complicated niche systems are capable of supporting robust and stable stem cell dynamics.

Summary: Stem cell niche dynamics are very robust to external and physiological perturbations because proliferation and differentiation are naturally balanced and controlled by the reliance on a shared niche environment.

Clonal expansion of hepatocytes with a selective advantage occurs during all stages of chronic hepatitis B virus infection

Hepatocyte clone size was measured in liver samples of 21 patients in various stages of chronic hepatitis B virus (HBV) infection and from 21 to 76 years of age. Hepatocyte clones containing unique virus-cell DNA junctions formed by the integration of HBV DNA were detected using inverse nested PCR. The maximum hepatocyte clone size tended to increase with age, although there was considerable patient-to-patient variation in each age group. There was an upward trend in maximum clone size with increasing fibrosis, inflammatory activity and with seroconversion from HBV e-antigen (HBeAg)-positive to HBeAg-negative, but these differences did not reach statistical significance. Maximum hepatocyte clone size did not differ between patients with and without a coexisting hepatocellular carcinoma. Thus, large hepatocyte clones containing integrated HBV DNA were detected during all stages of chronic HBV infection. Using laser microdissection, no significant difference in clone size was observed between foci of HBV surface antigen (HBsAg)-positive and HBsAg-negative hepatocytes, suggesting that expression of HBsAg is not a significant factor in clonal expansion. Laser microdissection also revealed that hepatocytes with normal-appearing histology make up a major fraction of the cells undergoing clonal expansion. Thus, preneoplasia does not appear to be a factor in the clonal expansion detected in our assays. Computer simulations suggest that the large hepatocyte clones are not produced by random hepatocyte turnover but have an as-yet-unknown selective advantage that drives increased clonal expansion in the HBV-infected liver.

Dynamics of leukemia stem-like cell extinction in acute promyelocytic leukemia

Many tumors are believed to be maintained by a small number of cancer stem-like cells, where cure is thought to require eradication of this cell population. In this study, we investigated the dynamics of acute promyelocytic leukemia (APL) before and during therapy with regard to disease initiation, progression, and therapeutic response. This investigation used a mathematical model of hematopoiesis and a dataset derived from the North American Intergroup Study INT0129. The known phenotypic constraints of APL could be explained by a combination of differentiation blockade of PML-RARα-positive cells and suppression of normal hematopoiesis. All-trans retinoic acid (ATRA) neutralizes the differentiation block and decreases the proliferation rate of leukemic stem cells in vivo. Prolonged ATRA treatment after chemotherapy can cure patients with APL by eliminating the stem-like cell population over the course of approximately one year. To our knowledge, this study offers the first estimate of the average duration of therapy that is required to eliminate stem-like cancer cells from a human tumor, with the potential for the refinement of treatment strategies to better manage human malignancy.

Interactions and tradeoffs between cell recruitment, proliferation, and differentiation affect CNS regeneration

Regeneration of central nervous system (CNS) lesions requires movement of progenitor cells and production of their differentiated progeny. Although damage to the CNS clearly promotes these two processes, the interplay between these complex events and how it affects a response remains elusive. Here, we use spatial stochastic modeling to show that tradeoffs arise between production and recruitment during regeneration. Proper spatial control of cell cycle timing can mitigate these tradeoffs, maximizing recruitment, improving infiltration into the lesion, and reducing wasteful production outside of it. Feedback regulation of cell lineage dynamics alone however leads to spatial defects in cell recruitment, suggesting a novel, to our knowledge, hypothesis for the aggregation of cells to the periphery of a lesion in multiple sclerosis. Interestingly, stronger chemotaxis does not correct this aggregation and instead, substantial random cell motions near the site of the lesion are required to improve CNS regeneration.

Robustness of differentiation cascades with symmetric stem cell division

Stem cells (SCs) perform the task of maintaining tissue homeostasis by both self-renewal and differentiation. While it has been argued that SCs divide asymmetrically, there is also evidence that SCs undergo symmetric division. Symmetric SC division has been speculated to be key for expanding cell numbers in development and regeneration after injury. However, it might lead to uncontrolled growth and malignancies such as cancer. In order to explore the role of symmetric SC division, we propose a mathematical model of the effect of symmetric SC division on the robustness of a population regulated by a serial differentiation cascade and we show that this may lead to extinction of such population. We examine how the extinction likelihood depends on defining characteristics of the population such as the number of intermediate cell compartments. We show that longer differentiation cascades are more prone to extinction than systems with less intermediate compartments. Furthermore, we have analysed the possibility of mixed symmetric and asymmetric cell division against invasions by mutant invaders in order to find optimal architecture. Our results show that more robust populations are those with unfrequent symmetric behaviour.

Mathematical model of adult stem cell regeneration with cross-talk between genetic and epigenetic regulation

Adult stem cells, which exist throughout the body, multiply by cell division to replenish dying cells or to promote regeneration to repair damaged tissues. To perform these functions during the lifetime of organs or tissues, stem cells need to maintain their populations in a faithful distribution of their epigenetic states, which are susceptible to stochastic fluctuations during each cell division, unexpected injury, and potential genetic mutations that occur during many cell divisions. However, it remains unclear how the three processes of differentiation, proliferation, and apoptosis in regulating stem cells collectively manage these challenging tasks. Here, without considering molecular details, we propose a genetic optimal control model for adult stem cell regeneration that includes the three fundamental processes, along with cell division and adaptation based on differential fitnesses of phenotypes. In the model, stem cells with a distribution of epigenetic states are required to maximize expected performance after each cell division. We show that heterogeneous proliferation that depends on the epigenetic states of stem cells can improve the maintenance of stem cell distributions to create balanced populations. A control strategy during each cell division leads to a feedback mechanism involving heterogeneous proliferation that can accelerate regeneration with less fluctuation in the stem cell population. When mutation is allowed, apoptosis evolves to maximize the performance during homeostasis after multiple cell divisions. The overall results highlight the importance of cross-talk between genetic and epigenetic regulation and the performance objectives during homeostasis in shaping a desirable heterogeneous distribution of stem cells in epigenetic states.

Complete numerical solution of the diffusion equation of random genetic drift

A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size.

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

Cancers are rarely caused by single mutations, but often develop as a result of the combined effects of multiple mutations. For most cells, the number of possible cell divisions is limited because of various biological constraints, such as progressive telomere shortening, cell senescence cascades or a hierarchically organized tissue structure. Thus, the risk of accumulating cells carrying multiple mutations is low. Nonetheless, many diseases are based on the accumulation of such multiple mutations. 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. We show that hierarchically organized tissues strongly suppress cells carrying multiple mutations and derive closed solutions for the expected size and diversity of clonal populations founded by a single mutant within the hierarchy. We discuss the example of childhood acute lymphoblastic leukaemia in detail and find good agreement between our predicted results and recently observed clonal diversities in patients. This result can contribute to the explanation of very diverse mutation profiles observed by whole genome sequencing of many different cancers.