Genetic instability and clonal expansion

Mathematical modeling for mutator phenotype and clonal selection advantage in the risk analysis of lung cancer

Cancer is one of the leading diseases for human mortality. Although substantial research works have been conducted to investigate the initiation and progression of cancer disease, it is still an active debate regarding the function of mutations conferring a clone advantage and the importance of mutator phenotypes caused by the mutation of stability genes. To address this issue further, we develop a mathematical model based on the incidence data of non-small cell lung cancer and small cell lung cancer from the Surveillance Epidemiology and End Results registry in the USA. The key biological parameters have been analyzed to investigate the potential effective measures for inhibiting the risk of lung cancer. Although the first event is the gene mutation that leads to clonal expansion of cells for lung cancer, the simulation results show that the clonal advantage of cancer cells alone is insufficient to cause tumorigenesis. Our analysis suggests that mutations in genes that keep genetic stability are critical in the development of lung cancer. This implies that mutator phenotype is an important indicator for the diagnosis of lung cancer, which can enable early detection and treatment to reduce the risk of lung cancer effectively. Furthermore, the parameter analysis indicates that it would be highly effective to control the risk of lung cancer by inhibiting the transformation rate from the normal cells to mutated cells and the clonal expansion of cells with fewer gene mutations.

Computational tools for assessing gene therapy under branching process models of mutation

Multitype branching processes are ideal for studying the population dynamics of stem cell populations undergoing mutation accumulation over the years following transplant. In such stochastic models, several quantities are of clinical interest as insertional mutagenesis carries the potential threat of leukemogenesis following gene therapy with autologous stem cell transplantation. In this paper, we develop a three-type branching process model describing accumulations of mutations in a population of stem cells distinguished by their ability for long-term self-renewal. Our outcome of interest is the appearance of a double-mutant cell, which carries a high potential for leukemic transformation. In our model, a single-hit mutation carries a slight proliferative advantage over a wild-type stem cells. We compute marginalized transition probabilities that allow us to capture important quantitative aspects of our model, including the probability of observing a double-hit mutant and relevant moments of a single-hit mutation population over time. We thoroughly explore the model behavior numerically, varying birth rates across the initial sizes and populations of wild type stem cells and single-hit mutants, and compare the probability of observing a double-hit mutant under these conditions. We find that increasing the number of single-mutants over wild-type particles initially present has a large effect on the occurrence of a double-mutant, and that it is relatively safe for single-mutants to be quite proliferative, provided the lentiviral gene addition avoids creating single mutants in the original insertion process. Our approach is broadly applicable to an important set of questions in cancer modeling and other population processes involving multiple stages, compartments, or types.

Risk of lung cancer due to external environmental factor and epidemiological data analysis

Lung cancer is a cancer with the fastest growth in the incidence and mortality all over the world, which is an extremely serious threat to human's life and health. Evidences reveal that external environmental factors are the key drivers of lung cancer, such as smoking, radiation exposure and so on. Therefore, it is urgent to explain the mechanism of lung cancer risk due to external environmental factors experimentally and theoretically. However, it is still an open issue regarding how external environment factors affect lung cancer risk. In this paper, we summarize the main mathematical models involved the gene mutations for cancers, and review the application of the models to analyze the mechanism of lung cancer and the risk of lung cancer due to external environmental exposure. In addition, we apply the model described and the epidemiological data to analyze the influence of external environmental factors on lung cancer risk. The result indicates that radiation can cause significantly an increase in the mutation rate of cells, in particular the mutation in stability gene that leads to genomic instability. These studies not only can offer insights into the relationship between external environmental factors and human lung cancer risk, but also can provide theoretical guidance for the prevention and control of lung cancer.

Mathematical modelling the pathway of genomic instability in lung cancer

Genomic instability plays a significant role in lung cancer. Although substantial research has been conducted using both clinical and theoretical studies, it is still a hotly debated issue to whether genomic instability is necessary or whether genomic instability precedes oncogenes activation and tumor suppressor genes inactivation for lung cancer. In response to this issue, we come up with a mathematical model incorporating effects of genomic instability to investigate the genomic instability pathway of human lung cancer. The presented model are applied to match the incidence rate data of lung cancer from the Life Span Study cohort of the atomic bomb survivors in Nagasaki and Hiroshima and the Surveillance Epidemiology and End Results registry in the United States. Model results suggest that genomic instability is necessary in the tumorigenesis of lung cancer, and genomic instability has no significant impact on the net proliferation rate of cells by statistical criteria. By comparing the results of the LSS data to those of the SEER data, we conclude that the genomic instability pathway exhibits a sensitivity to radiation exposure, more intensive in male patients.

Aging, clonal hematopoiesis and preleukemia: not just bad luck?

Chronological human aging is associated with a number of changes in the hematopoietic system, occurring at many levels from stem to mature cells, and the marrow microenvironment as well. This review will focus mainly on the aging of hematopoietic stem and progenitor cells (HSPCs), and on the associated increases in the incidence of hematological malignancies. HSPCs manifest reduced function and acquire molecular changes with chronological aging. Furthermore, while for many years it has been known that the human hematopoietic system becomes increasingly clonal with chronological aging (clonal hematopoiesis), only in the last few years has it become clear that clonal hematopoiesis may result from the accumulation of preleukemic mutations in HSPCs. Such mutations confer a selective advantage that leads to clonal hematopoiesis, and that may occasionally result in the development of leukemia, and define the existence of both preleukemic stem cells, and of 'preleukemia' as a clinical entity. While it is well appreciated that clonal hematopoiesis is very common in the elderly, several questions remain unanswered: why and how does clonal hematopoiesis develop? How is clonal hematopoiesis related to the age-related changes observed in the hematopoietic system? And why do only some individuals with clonal hematopoiesis develop leukemia?

Genetic and non-genetic instability in tumor progression: link between the fitness landscape and the epigenetic landscape of cancer cells

Genetic instability is invoked in explaining the cell phenotype changes that take place during cancer progression. However, the coexistence of a vast diversity of distinct clones, most prominently visible in the form of non-clonal chromosomal aberrations, suggests that Darwinian selection of mutant cells is not operating at maximal efficacy. Conversely, non-genetic instability of cancer cells must also be considered. Such mutation-independent instability of cell states is most prosaically manifest in the phenotypic heterogeneity within clonal cell populations or in the reversible switching between immature "cancer stem cell-like" and more differentiated states. How are genetic and non-genetic instability related to each other? Here, we review basic theoretical foundations and offer a dynamical systems perspective in which cancer is the inevitable pathological manifestation of modes of malfunction that are immanent to the complex gene regulatory network of the genome. We explain in an accessible, qualitative, and permissively simplified manner the mathematical basis for the "epigenetic landscape" and how the latter relates to the better known "fitness landscape." We show that these two classical metaphors have a formal basis. By combining these two landscape concepts, we unite development and somatic evolution as the drivers of the relentless increase in malignancy. Herein, the cancer cells are pushed toward cancer attractors in the evolutionarily unused regions of the epigenetic landscape that encode more and more "dedifferentiated" states as a consequence of both genetic (mutagenic) and non-genetic (regulatory) perturbations-including therapy. This would explain why for the cancer cell, the principle of "What does not kill me makes me stronger" is as much a driving force in tumor progression and development of drug resistance as the simple principle of "survival of the fittest."

Fastest time to cancer by loss of tumor suppressor genes

Genetic instability promotes cancer progression (by increasing the probability of cancerous mutations) as well as hinders it (by imposing a higher cell death rate for cells susceptible to cancerous mutation). With the loss of tumor suppressor gene function known to be responsible for a high percentage of breast and colorectal cancer (and a good fraction of lung cancer and other types as well), it is important to understand how genetic instability can be orchestrated toward carcinogenesis. In this context, this paper gives a complete characterization of the optimal (time-varying) cell mutation rate for the fastest time to a target cancerous cell population through the loss of both copies of a tumor suppressor gene. Similar to the (one-step) oncogene activation model previously analyzed, the optimal mutation rate of the present two-step model changes qualitatively with the convexity of the (mutation rate-dependent) cell death rate. However, the structure of the Hamiltonian for the new model differs significantly and intrinsically from that of the one-step model, and a completely new approach is needed for the solution of the present two-step problem. Considerable insight into the biology of optimal switching (between corner controls) is extracted from numerical results for cases with nonconvex death rates.

A new model of time scheme for progression of colorectal cancer

BACKGROUND

tumourigenesis can be regarded as an evolutionary process, in which the transformation of a normal cell into a tumour cell involves a number of limiting genetic and epigenetic events. To study the progression process, time schemes have been proposed for studying the process of colorectal cancer based on extensive clinical investigations. Moreover, a number of mathematical models have been designed to describe this evolutionary process. These models assumed that the mutation rate of genes is constant during different stages. However, it has been pointed that the subsequent driver mutations appear faster than the previous ones and the cumulative time to have more driver mutations grows with the growing number of gene mutations. Thus it is still a challenge to calculate the time when the first mutation occurs and to determine the influence of tumour size on the mutation rate.

RESULTS

In this work we present a general framework to remedy the shortcoming of existing models. Rather than considering the information of gene mutations based on a population of patients, we for the first time determine the values of the selective advantage of cancer cells and initial mutation rate for individual patients. The averaged values of doubling time and selective advantage coefficient determined by our model are consistent with the predictions made by the published models. Our calculation showed that the values of biological parameters, such as the selective advantage coefficient, initial mutation rate and cell doubling time diversely depend on individuals. Our model has successfully predicted the values of several important parameters in cancer progression, such as the selective advantage coefficient, initial mutation rate and cell doubling time. In addition, experimental data validated our predicted initial mutation rate and cell doubling time.

CONCLUSIONS

The introduced new parameter makes our proposed model more flexible to fix various types of information based on different patients in cancer progression.

Cancer quasispecies and stem-like adaptive aneuploidy

In this paper we develop a theoretical frame to understand self-regulation of aneuploidy rate in cancer and stem cells. This is accomplished building upon quasispecies theory, by leaving its formal mathematical structure intact, but by drastically changing the meaning of its objects. In particular, we propose a novel definition of chromosomal master sequence, as a sequence of physically distinct whole or fragmented chromosomes, whose length is taken to be the sum of the copy numbers of each whole or fragmented chromosome. This fundamental change in the functional objects of quasispecies theory allows us to show that previously measured aneuploidy rates in cancer populations are already close to a formally derived aneuploid error threshold, and that any value of aneuploidy rate larger than the aneuploid error threshold would lead to a loss of fitness of a tumor population. Finally, we make a phenomenological analysis of existing experimental evidence to argue that single clone cancer cells, derived from an aneuploid cancer subpopulation, are capable of self-regulating their aneuploidy rate and of adapting it to distinct environments, namely primary and metastatic microenvironments. We also discuss the potential origin of this self-regulatory ability in the wider context of developmental and comparative biology and we hypothesize the existence of a diversification factor, i.e. a cellular mechanism that regulates adaptation of aneuploidy rates, active in all embryo, adult and cancer stem cells.

The effect of one additional driver mutation on tumor progression

Tumor growth is caused by the acquisition of driver mutations, which enhance the net reproductive rate of cells. Driver mutations may increase cell division, reduce cell death, or allow cells to overcome density-limiting effects. We study the dynamics of tumor growth as one additional driver mutation is acquired. Our models are based on two-type branching processes that terminate in either tumor disappearance or tumor detection. In our first model, both cell types grow exponentially, with a faster rate for cells carrying the additional driver. We find that the additional driver mutation does not affect the survival probability of the lesion, but can substantially reduce the time to reach the detectable size if the lesion is slow growing. In our second model, cells lacking the additional driver cannot exceed a fixed carrying capacity, due to density limitations. In this case, the time to detection depends strongly on this carrying capacity. Our model provides a quantitative framework for studying tumor dynamics during different stages of progression. We observe that early, small lesions need additional drivers, while late stage metastases are only marginally affected by them. These results help to explain why additional driver mutations are typically not detected in fast-growing metastases.

Riparian ecosystems in human cancers

Intratumoral evolution produces extensive genetic heterogeneity in clinical cancers. This is generally attributed to an increased mutation rate that continually produces new genetically defined clonal lineages. Equally important are the interactions between the heritable traits of cancer cells and their microenvironment that produces natural selection favoring some clonal ‘species’ over others. That is, while mutations produce the heritable variation, environmental selection and cellular adaptation govern the strategies (and genotypes) that can proliferate within the tumor ecosystem. Here we ask: What are the dominant evolutionary forces in the cancer ecosystem? We propose that the tumor vascular network is a common and primary cause of intratumoral heterogeneity. Specifically, variations in blood flow result in variability in substrate, such as oxygen, and metabolites, such as acid, that serve as critical, but predictable, environmental selection forces. We examine the evolutionary and ecological consequences of variable blood flow by drawing an analogy to riparian habitats within desert landscapes. We propose that the phenotypic properties of cancer cells will exhibit predictable spatial variation within tumor phenotypes as a result of proximity to blood flow. Just as rivers in the desert create an abrupt shift from the lush, mesic riparian vegetation along the banks to sparser, xeric and dry-adapted plant species in the adjacent drylands, we expect blood vessels within tumors to promote similarly distinct communities of cancer cells that change abruptly with distance from the blood vessel. We propose vascular density and blood flow within a tumor as a primary evolutionary force governing variations in the phenotypic properties of cancer cells thus providing a unifying ecological framework for understanding intratumoral heterogeneity.

Effect of chromosomal instability on the mutation-selection balance in unicellular populations

This paper develops a mathematical model describing the evolutionary dynamics of a unicellular, asexually replicating population exhibiting chromosomal instability. Chromosomal instability is a form of genetic instability characterized by the gain or loss of entire chromosomes during cell division. We assume that the cellular genome is divided into several homologous groups of chromosomes, and that a single functional chromosome per homologous group is required for the cell to have the wild-type fitness. If the fitness is unaffected by the total number of chromosomes in the cell, our model is analytically solvable, and yields a mean fitness at mutation-selection balance that is identical to the mean fitness when there is no chromosomal instability. If this assumption is relaxed and the total number of chromosomes in the cell is not allowed to increase without bound, then chromosomal instability leads to a reduction in mean fitness. The results of this paper provide a useful baseline that can inform both future theoretial and experimental studies of chromosomal instability.

Evolutionary mechanisms and diversity in cancer

The recently introduced genome theory of cancer evolution provides a new framework for evolutionary studies on cancer. In particular, the established relationship between the large number of individual molecular mechanisms and the general evolutionary mechanism of cancer calls upon a change in our strategies that have been based on the characterization of common cancer gene mutations and their defined pathways. To further explain the significance of the genome theory of cancer evolution, a brief review will be presented describing the various attempts to illustrate the evolutionary mechanism of cancer, followed by further analysis of some key components of somatic cell evolution, including the diversity of biological systems, the multiple levels of information systems and control systems, the two phases (the punctuated or discontinuous phase and gradual Darwinian stepwise phase) and dynamic patterns of somatic cell evolution where genome replacement is the driving force. By linking various individual molecular mechanisms to the level of genome population diversity and tumorigenicity, the general mechanism of cancer has been identified as the evolutionary mechanism of cancer, which can be summarized by the following three steps including stress-induced genome instability, population diversity or heterogeneity, and genome-mediated macroevolution. Interestingly, the evolutionary mechanism is equal to the collective aggregate of all individual molecular mechanisms. This relationship explains why most of the known molecular mechanisms can contribute to cancer yet there is no single dominant mechanism for the majority of clinical cases. Despite the fact that each molecular mechanism can serve as a system stress and initiate the evolutionary process, to achieve cancer, multiple cycles of genome-mediated macroevolution are required and are a stochastically determined event. Finally, the potential clinical implications of the evolutionary mechanism of cancer are briefly reviewed.

The evolutionary mechanism of cancer

Identification of the general molecular mechanism of cancer is the Holy Grail of cancer research. Since cancer is believed to be caused by a sequential accumulation of cancer gene mutations, the identification, characterization, and targeting of common genetic alterations and their defined pathways have dominated the field for decades. Despite the impressive data accumulated from studies of gene mutations, epigenetic dysregulation, and pathway alterations, an overwhelming amount of diverse molecular information has offered limited understanding of the general mechanisms of cancer. To solve this paradox, the newly established genome theory is introduced here describing how somatic cells evolve within individual patients. The evolutionary mechanism of cancer is characterized using only three key components of somatic cell evolution that include increased system dynamics induced by stress, elevated genetic and epigenetic heterogeneity, and genome alteration mediated natural selection. Cancer progression represents a macro-evolutionary process where karyotype change or genome replacement plays the key dominant role. Furthermore, the recently identified relationship between the evolutionary mechanism and a large number of diverse individual molecular mechanisms is discussed. The total sum of all the individual molecular mechanisms is equal to the evolutionary mechanism of cancer. Individual molecular mechanisms including all the molecular mechanisms described to date are stochastically selected and unpredictable and are therefore clinically impractical. Recognizing the fundamental importance of the underlying basis of the evolutionary mechanism of cancer mandates the development of new strategies in cancer research.

Evolutionary dynamics of tumor progression with random fitness values

Most human tumors result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Mutations that confer a fitness advantage to the cell are known as driver mutations and are causally related to tumorigenesis. Other mutations, however, do not change the phenotype of the cell or even decrease cellular fitness. While much experimental effort is being devoted to the identification of the functional effects of individual mutations, mathematical modeling of tumor progression generally considers constant fitness increments as mutations are accumulated. In this paper we study a mathematical model of tumor progression with random fitness increments. We analyze a multi-type branching process in which cells accumulate mutations whose fitness effects are chosen from a distribution. We determine the effect of the fitness distribution on the growth kinetics of the tumor. This work contributes to a quantitative understanding of the accumulation of mutations leading to cancer.

Evolution of cell motility in an individual-based model of tumour growth

Tumour invasion is driven by proliferation and importantly migration into the surrounding tissue. Cancer cell motility is also critical in the formation of metastases and is therefore a fundamental issue in cancer research. In this paper we investigate the emergence of cancer cell motility in an evolving tumour population using an individual-based modelling approach. In this model of tumour growth each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. With this model we have investigated the impact of the micro-environment on the emergence of a motile invasive phenotype. The results show that when a motile phenotype emerges the dynamics of the model are radically changed and we observe faster growing tumours exhibiting diffuse morphologies. Further we observe that the emergence of a motile subclone can occur in a wide range of micro-environmental growth conditions. Iterated simulations showed that in identical growth conditions the evolutionary dynamics either converge to a proliferating or migratory phenotype, which suggests that the introduction of cell motility into the model changes the shape of fitness landscape on which the cancer cell population evolves and that it now contains several local maxima. This could have important implications for cancer treatments which focus on the gene level, as our results show that several distinct genotypes and critically distinct phenotypes can emerge and become dominant in the same micro-environment.

Hormesis without cell killing

Stochastic two-stage clonal expansion (TSCE) models of carcinogenesis offer the following clear theoretical explanation for U-shaped cancer dose-response relations. Low doses that kill initiated (premalignant) cells thereby create a protective effect. At higher doses, this effect is overwhelmed by an increase in the net number of initiated cells. The sum of these two effects, from cell killing and cell proliferation, gives a U-shaped or J-shaped dose-response relation. This article shows that exposures that do not kill, repair, or decrease cell populations, but that only hasten transitions that lead to cancer, can also generate U-shaped and J-shaped dose-response relations in a competing-risk (modified TSCE) framework where exposures disproportionately hasten transitions into carcinogenic pathways with relatively long times to tumor. Quantitative modeling of the competing effects of more transitions toward carcinogenesis (risk increasing) and a higher proportion of transitions into the slower pathway (risk reducing) shows that a J-shaped dose-response relation can occur even if exposure increases the number of initiated cells at every positive dose level. This suggests a possible new explanation for hormetic dose-response relations in response to carcinogenic exposures that do not have protective (cell-killing) effects. In addition, the examples presented emphasize the role of time in hormesis: exposures that monotonically increase risks at younger ages may nonetheless produce a U-shaped or J-shaped dose-response relation for lifetime risk of cancer.

Could removing arsenic from tobacco smoke significantly reduce smoker risks of lung cancer?

If a specific biological mechanism could be determined by which a carcinogen increases lung cancer risk, how might this knowledge be used to improve risk assessment? To explore this issue, we assume (perhaps incorrectly) that arsenic in cigarette smoke increases lung cancer risk by hypermethylating the promoter region of gene p16INK4a, leading to a more rapid entry of altered (initiated) cells into a clonal expansion phase. The potential impact on lung cancer of removing arsenic is then quantified using a three-stage version of a multistage clonal expansion (MSCE) model. This refines the usual two-stage clonal expansion (TSCE) model of carcinogenesis by resolving its intermediate or "initiated" cell compartment into two subcompartments, representing experimentally observed "patch" and "field" cells. This refinement allows p16 methylation effects to be represented as speeding transitions of cells from the patch state to the clonally expanding field state. Given these assumptions, removing arsenic might greatly reduce the number of nonsmall cell lung cancer cells (NSCLCs) produced in smokers, by up to two-thirds, depending on the fraction (between 0 and 1) of the smoking-induced increase in the patch-to-field transition rate prevented if arsenic were removed. At present, this fraction is unknown (and could be as low as zero), but the possibility that it could be high (close to 1) cannot be ruled out without further data.

Characterization and quantification of clonal heterogeneity among hematopoietic stem cells: a model-based approach

Hematopoietic stem cells (HSCs) show pronounced heterogeneity in self-renewal and differentiation behavior, which is reflected in their repopulation kinetics. Here, a single-cell-based mathematical model of HSC organization is used to examine the basis of HSC heterogeneity. Our modeling results, which are based on the analysis of limiting dilution competitive repopulation experiments in mice, demonstrate that small quantitative but clonally fixed differences of cellular properties are necessary and sufficient to account for the observed functional heterogeneity. The model predicts, and experimental data validate, that competitive pressures will amplify small clonal differences into large changes in the number of differentiated progeny. We further predict that the repertoire of HSC clones will evolve over time. Last, our results suggest that larger differences in cellular properties have to be assumed to account for genetically determined differences in HSC behavior as observed in different inbred mice strains. The model provides comprehensive systemic and quantitative insights into the clonal heterogeneity among HSCs with potential applications in predicting the behavior of malignant and/or genetically modified cells.

Selective pressures for and against genetic instability in cancer: a time-dependent problem

Genetic instability in cancer is a two-edge sword. It can both increase the rate of cancer progression (by increasing the probability of cancerous mutations) and decrease the rate of cancer growth (by imposing a large death toll on dividing cells). Two of the many selective pressures acting upon a tumour, the need for variability and the need to minimize deleterious mutations, affect the tumour's 'choice' of a stable or unstable 'strategy'. As cancer progresses, the balance of the two pressures will change. In this paper, we examine how the optimal strategy of cancerous cells is shaped by the changing selective pressures. We consider the two most common patterns in multistage carcinogenesis: the activation of an oncogene (a one-step process) and an inactivation of a tumour-suppressor gene (a two-step process). For these, we formulate an optimal control problem for the mutation rate in cancer cells. We then develop a method to find optimal time-dependent strategies. It turns out that for a wide range of parameters, the most successful strategy is to start with a high rate of mutations and then switch to stability. This agrees with the growing biological evidence that genetic instability, prevalent in early cancers, turns into stability later on in the progression. We also identify parameter regimes where it is advantageous to keep stable (or unstable) constantly throughout the growth.