The optimal rate of chromosome loss for the inactivation of tumor suppressor genes in cancer

YY2/BUB3 Axis promotes SAC Hyperactivation and Inhibits Colorectal Cancer Progression via Regulating Chromosomal Instability

Spindle assembly checkpoint (SAC) is a crucial safeguard mechanism of mitosis fidelity that ensures equal division of duplicated chromosomes to the two progeny cells. Impaired SAC can lead to chromosomal instability (CIN), a well-recognized hallmark of cancer that facilitates tumor progression; paradoxically, high CIN levels are associated with better therapeutic response and prognosis. However, the mechanism by which CIN determines tumor cell survival and therapeutic response remains poorly understood. Here, using a cross-omics approach, YY2 is identified as a mitotic regulator that promotes SAC activity by activating the transcription of budding uninhibited by benzimidazole 3 (BUB3), a component of SAC. While both conditions induce CIN, a defect in YY2/SAC activity enhances mitosis and tumor growth. Meanwhile, hyperactivation of SAC mediated by YY2/BUB3 triggers a delay in mitosis and suppresses growth. Furthermore, it is revealed that YY2/BUB3-mediated excessive CIN causes higher cell death rates and drug sensitivity, whereas residual tumor cells that survived DNA damage-based therapy have moderate CIN and increased drug resistance. These results provide insights into the role of SAC activity and CIN levels in influencing tumor cell survival and drug response, as well as suggest a novel anti-tumor therapeutic strategy that combines SAC activity modulators and DNA-damage agents.

The two sides of chromosomal instability: drivers and brakes in cancer

Chromosomal instability (CIN) is a hallmark of cancer and is associated with tumor cell malignancy. CIN triggers a chain reaction in cells leading to chromosomal abnormalities, including deviations from the normal chromosome number or structural changes in chromosomes. CIN arises from errors in DNA replication and chromosome segregation during cell division, leading to the formation of cells with abnormal number and/or structure of chromosomes. Errors in DNA replication result from abnormal replication licensing as well as replication stress, such as double-strand breaks and stalled replication forks; meanwhile, errors in chromosome segregation stem from defects in chromosome segregation machinery, including centrosome amplification, erroneous microtubule-kinetochore attachments, spindle assembly checkpoint, or defective sister chromatids cohesion. In normal cells, CIN is deleterious and is associated with DNA damage, proteotoxic stress, metabolic alteration, cell cycle arrest, and senescence. Paradoxically, despite these negative consequences, CIN is one of the hallmarks of cancer found in over 90% of solid tumors and in blood cancers. Furthermore, CIN could endow tumors with enhanced adaptation capabilities due to increased intratumor heterogeneity, thereby facilitating adaptive resistance to therapies; however, excessive CIN could induce tumor cells death, leading to the "just-right" model for CIN in tumors. Elucidating the complex nature of CIN is crucial for understanding the dynamics of tumorigenesis and for developing effective anti-tumor treatments. This review provides an overview of causes and consequences of CIN, as well as the paradox of CIN, a phenomenon that continues to perplex researchers. Finally, this review explores the potential of CIN-based anti-tumor therapy.

Misaligned Chromosomes are a Major Source of Chromosomal Instability in Breast Cancer

Chromosomal instability (CIN), the persistent reshuffling of chromosomes during mitosis, is a hallmark of human cancers that contributes to tumor heterogeneity and has been implicated in driving metastasis and altering responses to therapy. Though multiple mechanisms can produce CIN, lagging chromosomes generated from abnormal merotelic attachments are the major cause of CIN in a variety of cell lines, and are expected to predominate in cancer. Here, we quantify CIN in breast cancer using a tumor microarray, matched primary and metastatic samples, and patient-derived organoids from primary breast cancer. Surprisingly, misaligned chromosomes are more common than lagging chromosomes and represent a major source of CIN in primary and metastatic tumors. This feature of breast cancers is conserved in a majority of breast cancer cell lines. Importantly, though a portion of misaligned chromosomes align before anaphase onset, the fraction that remain represents the largest source of CIN in these cells. Metastatic breast cancers exhibit higher rates of CIN than matched primary cancers, primarily due to increases in misaligned chromosomes. Whether CIN causes immune activation or evasion is controversial. We find that misaligned chromosomes result in immune-activating micronuclei substantially less frequently than lagging and bridge chromosomes and that breast cancers with greater frequencies of lagging chromosomes and chromosome bridges recruit more stromal tumor-infiltrating lymphocytes. These data indicate misaligned chromosomes represent a major mechanism of CIN in breast cancer and provide support for differential immunostimulatory effects of specific types of CIN.

Significance

We surveyed the single-cell landscape of mitotic defects that generate CIN in primary and metastatic breast cancer and relevant models. Misaligned chromosomes predominate, and are less immunostimulatory than other chromosome segregation errors.

The Mutator Phenotype: Adapting Microbial Evolution to Cancer Biology

The mutator phenotype hypothesis was postulated almost 40 years ago to reconcile the observation that while cancer cells display widespread mutational burden, acquisition of mutations in non-transformed cells is a rare event. Moreover, it also suggested that cancer evolution could be fostered by increased genome instability. Given the evolutionary conservation throughout the tree of life and the genetic tractability of model organisms, yeast and bacterial species pioneered studies to dissect the functions of genes required for genome maintenance (caretaker genes) or for cell growth control (gatekeeper genes). In this review, we first provide an overview of what we learned from model organisms about the roles of these genes and the genome instability that arises as a consequence of their dysregulation. We then discuss our current understanding of how mutator phenotypes shape the evolution of bacteria and yeast species. We end by bringing clinical evidence that lessons learned from single-cell organisms can be applied to tumor evolution.

Chromosome instability in neuroblastoma

Neuroblastoma is a neural crest-derived tumor that accounts for 7-10% of all malignancies in children and ~15% of all childhood cancer-associated mortalities. Approximately 50% of patients are characterized as high-risk (HR) and have an overall survival of <40% at 5 years from diagnosis. HR patients with unfavorable prognosis exhibit several structural copy number variations (CNVs), whereas localized tumors belonging to patients in the low- and intermediate-risk classes, have favorable outcomes and display several numerical CNVs. Taken together these results are indicative of chromosome instability (CIN) in neuroblastoma tumor cells. The present review discusses multiple aspects of CIN including methods of measuring CIN, CIN targeting as a therapeutic strategy in cancer and the effects of CIN in neuroblastoma development and aggressiveness with particular emphasis on the CIN gene signature associated with HR neuroblastoma patients.

New Cell Cycle Inhibitors Target Aneuploidy in Cancer Therapy

Aneuploidy is a hallmark of cancer. Defects in chromosome segregation result in aneuploidy. Multiple pathways are engaged in this process, including errors in kinetochore-microtubule attachments, supernumerary centrosomes, spindle assembly checkpoint (SAC) defects, and chromosome cohesion defects. Although aneuploidy provides an adaptation and proliferative advantage in affected cells, excessive aneuploidy beyond a critical level can be lethal to cancer cells. Given this, enhanced chromosome missegregation is hypothesized to limit survival of aneuploid cancer cells, especially when compared to diploid cells. Based on this concept, proteins and pathways engaged in chromosome segregation are being exploited as candidate therapeutic targets for aneuploid cancers. Agents that induce chromosome missegregation and aneuploidy now exist, including SAC inhibitors, those that alter centrosome fidelity and others that are under active study in preclinical and clinical contexts. This review explores the therapeutic potentials of such new agents, including the benefits of combining them with other antineoplastic agents.

Centrosome aberrations and chromosome instability contribute to tumorigenesis and intra-tumor heterogeneity

Centrosomes serve as the major microtubule organizing centers in cells and thereby contribute to cell shape, polarity, and motility. Also, centrosomes ensure equal chromosome segregation during mitosis. Centrosome aberrations arise when the centrosome cycle is deregulated, or as a result of cytokinesis failure. A long-standing postulate is that centrosome aberrations are involved in the initiation and progression of cancer. However, this notion has been a subject of controversy because until recently the relationship has been correlative. Recently, it was shown that numerical or structural centrosome aberrations can initiate tumors in certain tissues in mice, as well as invasion. Particularly, we will focus on centrosome amplification and chromosome instability as drivers of intra-tumor heterogeneity and their consequences in cancer. We will also discuss briefly the controversies surrounding this theory to highlight the fact that the role of both centrosome amplification and chromosome instability in cancer is highly context-dependent. Further, we will discuss single-cell sequencing as a novel technique to understand intra-tumor heterogeneity and some therapeutic approaches to target chromosome instability.

Determinants and clinical implications of chromosomal instability in cancer

Aberrant chromosomal architecture, ranging from small insertions or deletions to large chromosomal alterations, is one of the most common characteristics of cancer genomes. Chromosomal instability (CIN) underpins much of the intratumoural heterogeneity observed in cancers and drives phenotypic adaptation during tumour evolution. Thus, an urgent need exists to increase our efforts to target CIN as if it were a molecular entity. Indeed, CIN accelerates the development of anticancer drug resistance, often leading to treatment failure and disease recurrence, which limit the effectiveness of most current therapies. Identifying novel strategies to modulate CIN and to exploit the fitness cost associated with aneuploidy in cancer is, therefore, of paramount importance for the successful treatment of cancer. Modern sequencing and analytical methods greatly facilitate the identification and cataloguing of somatic copy-number alterations and offer new possibilities to better exploit the dynamic process of CIN. In this Review, we describe the principles governing CIN propagation in cancer and how CIN might influence sensitivity to immune-checkpoint inhibition, and survey the vulnerabilities associated with CIN that offer potential therapeutic opportunities.

Living in CIN: Mitotic Infidelity and Its Consequences for Tumor Promotion and Suppression

Errors in chromosome segregation during mitosis have been recognized as a hallmark of tumor cells since the late 1800s, resulting in the long-standing hypothesis that mitotic abnormalities drive tumorigenesis. Recent work has shown that mitotic defects can promote tumors, suppress them, or do neither, depending on the rate of chromosome missegregation. Here we discuss the causes of chromosome missegregation, their effects on tumor initiation and progression, and the evidence that increasing the rate of chromosome missegregation may be an effective chemotherapeutic strategy.

Evolution of genetic instability in heterogeneous tumors

Genetic instability is an important characteristic of cancer. While most cancers develop genetic instability at some stage of their progression, sometimes a temporary rise of instability is followed by the return to a relatively stable genome. Neither the reasons for these dynamics, nor, more generally, the role of instability in tumor progression, are well understood. In this paper we develop a class of mathematical models to study the evolutionary competition dynamics among different sub-populations in a heterogeneous tumor. We observe that despite the complexity of this multi-component and multi-process system, there is only a small number of scenarios expected in the context of the evolution of instability. If the penalty incurred by unstable cells (the decrease in the growth due to deleterious mutations) is high compared with the gain (the production rate of advantageous mutations), then instability does not evolve. In the opposite case, instability evolves and comes to dominate the system. In the intermediate parameter regime, instability is generated but later gives way to stable clones. Moreover, the model also informs us of the patterns of instability for cancer lineages corresponding to different stages of progression. It is predicted that mutations causing instability are merely "passengers" in tumors that have undergone only a small number of malignant mutations. Further down the path of carcinogenesis, however, unstable cells are more likely to give rise to the winning clonal wave that takes over the tumor and carries the evolution forward, thus conferring a causal role of the instability in such cases. Further, each individual clonal wave (i.e. cells harboring a fixed number of malignant driver mutations) experiences its own evolutionary history. It can fall under one of three types of temporal behavior: stable throughout, unstable to stable, or unstable throughout. Which scenario is realized depends on the subtle (but predictable) interplay among mutation rates and the death toll associated with the instability. The modeling approach provided here sheds light onto important aspects of the evolutionary dynamics of instability, which may be relevant to treatment scenarios that target instability or damage repair.

Dependence of Human Colorectal Cells Lacking the FBW7 Tumor Suppressor on the Spindle Assembly Checkpoint

FBW7 (F-box and WD repeat domain containing 7), also known as FBXW7 or hCDC4, is a tumor suppressor gene mutated in a broad spectrum of cancer cell types. As a component of the SCF E3 ubiquitin ligase, FBW7 is responsible for specifically recognizing phosphorylated substrates, many important for tumor progression, and targeting them for ubiquitin-mediated degradation. Although the role of FBW7 as a tumor suppressor is well established, less well studied is how FBW7-mutated cancer cells might be targeted for selective killing. To explore this further, we undertook a genome-wide RNAi screen using WT and FBW7 knockout colorectal cell lines and identified the spindle assembly checkpoint (SAC) protein BUBR1, as a candidate synthetic lethal target. We show here that asynchronous FBW7 knockout cells have increased levels of mitotic APC/C substrates and are sensitive to knockdown of not just BUBR1 but BUB1 and MPS1, other known SAC components, suggesting a dependence of these cells on the mitotic checkpoint. Consistent with this dependence, knockdown of BUBR1 in cells lacking FBW7 results in significant cell aneuploidy and increases in p53 levels. The FBW7 substrate cyclin E was necessary for the genetic interaction with BUBR1. In contrast, the establishment of this dependence on the SAC requires the deregulation of multiple substrates of FBW7. Our work suggests that FBW7 knockout cells are vulnerable in their dependence on the mitotic checkpoint and that this may be a good potential target to exploit in FBW7-mutated cancer cells.

Extreme chromosomal instability forecasts improved outcome in ER-negative breast cancer: a prospective validation cohort study from the TACT trial

BACKGROUND

Chromosomal instability (CIN) has been shown to be associated with drug resistance and poor clinical outcome in several cancer types. However, in oestrogen receptor (ER)-negative breast cancer we have previously demonstrated that extreme CIN is associated with improved clinical outcome, consistent with a negative impact of CIN on tumour fitness and growth. The aim of this current study was to validate this finding using previously defined CIN thresholds in a much larger prospective cohort from a randomised, controlled, clinical trial.

PATIENTS AND METHODS

As a surrogate measurement of CIN, dual centromeric fluorescence in situ hybridisation was performed for both chromosomes 2 and 15 on 1173 tumours from the breast cancer TACT trial (CRUK01/001). Each tumour was scored manually and the mean percentage of cells deviating from the modal centromere number was used to define four CIN groups (MCD1-4), where tumours in the MCD4 group were defined as having extreme CIN.

RESULTS

In a multivariate analysis of disease-free survival, with a median follow-up of 91 months, increasing CIN was associated with improved outcome in patients with ER-negative cancer (P trend = 0.03). A similar pattern was seen in ER-negative/HER2-negative cancers (Ptrend = 0.007).

CONCLUSIONS

This prospective validation cohort study further substantiated the association between extreme CIN and improved outcome in ER-negative breast cancers. Identifying such patients with extreme CIN may help distinguish good from poor prognostic groups, and therefore support treatment and risk stratification in this aggressive breast cancer subtype.

Stress-induced mutagenesis and complex adaptation

Because mutations are mostly deleterious, mutation rates should be reduced by natural selection. However, mutations also provide the raw material for adaptation. Therefore, evolutionary theory suggests that the mutation rate must balance between adaptability-the ability to adapt-and adaptedness-the ability to remain adapted. We model an asexual population crossing a fitness valley and analyse the rate of complex adaptation with and without stress-induced mutagenesis (SIM)-the increase of mutation rates in response to stress or maladaptation. We show that SIM increases the rate of complex adaptation without reducing the population mean fitness, thus breaking the evolutionary trade-off between adaptability and adaptedness. Our theoretical results support the hypothesis that SIM promotes adaptation and provide quantitative predictions of the rate of complex adaptation with different mutational strategies.

Can a minimal replicating construct be identified as the embodiment of cancer?

Genomic instability is a hallmark of cancer. Cancer cells that exhibit abnormal chromosomes are characteristic of most advanced tumours, despite the potential threat represented by accumulated genetic damage. Carcinogenesis involves a loss of key components of the genetic and signalling molecular networks; hence some authors have suggested that this is part of a trend of cancer cells to behave as simple, minimal replicators. In this study, we explore this conjecture and suggest that, in the case of cancer, genomic instability has an upper limit that is associated with a minimal cancer cell network. Such a network would include (for a given microenvironment) the basic molecular components that allow cells to replicate and respond to selective pressures. However, it would also exhibit internal fragilities that could be exploited by appropriate therapies targeting the DNA repair machinery. The implications of this hypothesis are discussed.

On the interplay between mathematics and biology: hallmarks toward a new systems biology

This paper proposes a critical analysis of the existing literature on mathematical tools developed toward systems biology approaches and, out of this overview, develops a new approach whose main features can be briefly summarized as follows: derivation of mathematical structures suitable to capture the complexity of biological, hence living, systems, modeling, by appropriate mathematical tools, Darwinian type dynamics, namely mutations followed by selection and evolution. Moreover, multiscale methods to move from genes to cells, and from cells to tissue are analyzed in view of a new systems biology approach.

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.

Chromosomal instability portends superior response of rectal adenocarcinoma to chemoradiation therapy

BACKGROUND

Persistent chromosome segregation errors represent a conspicuous feature of human neoplasms. It is widely accepted that this chromosomal instability is associated with poor prognosis; however, its effect on therapeutic response is a matter of conjecture.

METHODS

Here, the role of chromosome segregation errors in the response of patients with rectal adenocarcinoma to chemoradiation therapy (CRT) was examined. Pretreatment samples from 62 patients were surveyed for evidence of chromosome mis-segregation and mis-segregation frequency was correlated to the pathological response to CRT as determined by the tumor regression grade after surgical resection of irradiated tumors.

RESULTS

Surprisingly, it was found that errors in chromosome segregation predicted enhanced pathological response of rectal adenocarcinoma to CRT (odds ratio, 3.9; P = .02). Furthermore, tumor response inversely correlated with the frequency of cells that exhibited segregation errors during anaphase (correlation coefficient, 0.94; P < .05). Strikingly, elevated chromosome mis-segregation combined with decreased levels of the DNA damage repair protein Mre11 portended a markedly enhanced response (odds ratio, 54.0; P = .008).

CONCLUSIONS

The results of the current study demonstrate that chromosomal instability is a favorable predictor of response to CRT in patients with locally invasive rectal adenocarcinoma. Therefore, the authors propose that downstream structural damage to chromosomes resulting from segregation errors potentiates the effect of DNA-damaging therapies and synergizes with deficiencies in the DNA repair machinery. This work identifies a novel mechanistic marker that foretells treatment response to CRT and suggests that concomitant targeting of whole-chromosome segregation and DNA repair may constitute an effective therapeutic strategy.

Crosstalk between telomere maintenance and radiation effects: A key player in the process of radiation-induced carcinogenesis

It is well established that ionizing radiation induces chromosomal damage, both following direct radiation exposure and via non-targeted (bystander) effects, activating DNA damage repair pathways, of which the proteins are closely linked to telomeric proteins and telomere maintenance. Long-term propagation of this radiation-induced chromosomal damage during cell proliferation results in chromosomal instability. Many studies have shown the link between radiation exposure and radiation-induced changes in oxidative stress and DNA damage repair in both targeted and non-targeted cells. However, the effect of these factors on telomeres, long established as guardians of the genome, still remains to be clarified. In this review, we will focus on what is known about how telomeres are affected by exposure to low- and high-LET ionizing radiation and during proliferation, and will discuss how telomeres may be a key player in the process of radiation-induced carcinogenesis.

Modelling the evolution of genetic instability during tumour progression

The role of genetic instability in driving carcinogenesis remains controversial. Genetic instability should accelerate carcinogenesis by increasing the rate of advantageous driver mutations; however, genetic instability can also potentially retard tumour growth by increasing the rate of deleterious mutation. As such, it is unclear whether genetically unstable clones would tend to be more selectively advantageous than their genetically stable counterparts within a growing tumour. Here, we show the circumstances where genetic instability evolves during tumour progression towards cancer. We employ a Wright-Fisher type model that describes the evolution of tumour subclones. Clones can acquire both advantageous and deleterious mutations, and mutator mutations that increase a cell's intrinsic mutation rate. Within the model, cancers evolve with a mutator phenotype when driver mutations bestow only moderate increases in fitness: very strong or weak selection for driver mutations suppresses the evolution of a mutator phenotype. Genetic instability occurs secondarily to selectively advantageous driver mutations. Deleterious mutations have relatively little effect on the evolution of genetic instability unless selection for additional driver mutations is very weak or if deleterious mutations are very common. Our model provides a framework for studying the evolution of genetic instability in tumour progression. Our analysis highlights the central role of selection in shaping patterns of mutation in carcinogenesis.

Chromosomal instability substantiates poor prognosis in patients with diffuse large B-cell lymphoma

PURPOSE

The specific role of chromosomal instability (CIN) in tumorigenesis has been a matter of conjecture. In part, this is due to the challenge of directly observing chromosome mis-segregation events as well as the inability to distinguish the role of CIN, which consists of increased rates of chromosome mis-segregation, from that of aneuploidy, which is a state of nondiploid chromosome number.

EXPERIMENTAL DESIGN

Here, we examine the contribution of CIN to the prognosis of patients diagnosed with diffuse large B-cell lymphoma (DLBCL) by directly surveying tumor cells, fixed while undergoing anaphase, for evidence of chromosome mis-segregation. Hematoxylin and eosin-stained samples from a cohort of 54 patients were used to examine the relationship between frequencies of chromosome mis-segregation and patient prognosis, overall survival, and response to treatment.

RESULTS

We show that a two-fold increase in the frequency of chromosome mis-segregation led to a 24% decrease in overall survival and 48% decrease in relapse-free survival after treatment. The HR of death in patients with increased chromosome mis-segregation was 2.31 and these patients were more likely to present with higher tumor stage, exhibit tumor bone marrow involvement, and receive a higher International Prognostic Index score.

CONCLUSIONS

Increased rates of chromosome mis-segregation in DLBCL substantiate inferior outcome and poor prognosis. This is likely due to increased heterogeneity of tumor cells leading to a larger predilection for adaptation in response to external pressures such as metastasis and drug treatments. We propose that targeting CIN would yield superior prognosis and improved response to chemotherapeutic drugs.

Cancer chromosomal instability: therapeutic and diagnostic challenges

Chromosomal instability (CIN)-which is a high rate of loss or gain of whole or parts of chromosomes-is a characteristic of most human cancers and a cause of tumour aneuploidy and intra-tumour heterogeneity. CIN is associated with poor patient outcome and drug resistance, which could be mediated by evolutionary adaptation fostered by intra-tumour heterogeneity. In this review, we discuss the clinical consequences of CIN and the challenges inherent to its measurement in tumour specimens. The relationship between CIN and prognosis supports assessment of CIN status in the clinical setting and suggests that stratifying tumours according to levels of CIN could facilitate clinical risk assessment.

A review of spatial computational models for multi-cellular systems, with regard to intestinal crypts and colorectal cancer development

Colon rectal cancers (CRC) are the result of sequences of mutations which lead the intestinal tissue to develop in a carcinoma following a "progression" of observable phenotypes. The actual modeling and simulation of the key biological structures involved in this process is of interest to biologists and physicians and, at the same time, it poses significant challenges from the mathematics and computer science viewpoints. In this report we give an overview of some mathematical models for cell sorting (a basic phenomenon that underlies several dynamical processes in an organism), intestinal crypt dynamics and related problems and open questions. In particular, major attention is devoted to the survey of so-called in-lattice (or grid) models and off-lattice (off-grid) models. The current work is the groundwork for future research on semi-automated hypotheses formation and testing about the behavior of the various actors taking part in the adenoma-carcinoma progression, from regulatory processes to cell-cell signaling pathways.

A review of spatial computational models for multi-cellular systems, with regard to intestinal crypts and colorectal cancer development

Colon rectal cancers (CRC) are the result of sequences of mutations which lead the intestinal tissue to develop in a carcinoma following a "progression" of observable phenotypes. The actual modeling and simulation of the key biological structures involved in this process is of interest to biologists and physicians and, at the same time, it poses significant challenges from the mathematics and computer science viewpoints. In this report we give an overview of some mathematical models for cell sorting (a basic phenomenon that underlies several dynamical processes in an organism), intestinal crypt dynamics and related problems and open questions. In particular, major attention is devoted to the survey of so-called in-lattice (or grid) models and off-lattice (off-grid) models. The current work is the groundwork for future research on semi-automated hypotheses formation and testing about the behavior of the various actors taking part in the adenoma-carcinoma progression, from regulatory processes to cell-cell signaling pathways.

Chromosomal instability and cancer: a complex relationship with therapeutic potential

Chromosomal instability (CIN) is a hallmark of human neoplasms. Despite its widespread prevalence, knowledge of the mechanisms and contributions of CIN in cancer has been elusive. It is now evident that the role of CIN in tumor initiation and growth is more complex than previously thought. Furthermore, distinguishing CIN, which consists of elevated rates of chromosome missegregation, from aneuploidy, which is a state of abnormal chromosome number, is crucial to understanding their respective contributions in cancer. Collectively, experimental evidence suggests that CIN enables tumor adaptation by allowing tumors to constantly sample the aneuploid fitness landscape. This complex relationship, together with the potential to pharmacologically influence chromosome missegregation frequencies in cancer cells, offers previously unrecognized means to limit tumor growth and its response to therapy.

APC and its modifiers in colon cancer

Colon cancer closely follows the paradigm of a single "gatekeeper gene." Mutations inactivating the APC (adenomatous polyposis coli) gene are found in approximately 80% of all human colon tumors and heterozygosity for such mutations produces an autosomal dominant colon cancer predisposition in humans and in murine models. However, this tight association between a single genotype and phenotype belies a complex association of genetic and epigenetic factors that together generate the broad phenotypic spectrum ofboth familial and sporadic colon cancers. In this Chapter, we give a general overview of the structure, function and outstanding issues concerning the role of Apc in human and experimental colon cancer. The availability of increasingly close models for human colon cancer in genetically tractable animal species enables the discovery and eventual molecular identification of genetic modifiers of the Apc-mutant phenotypes, connecting the central role of Apc in colon carcinogenesis to the myriad factors that ultimately determine the course of the disease.

Evolutionary theory of cancer

As Theodosius Dobzhansky famously noted in 1973, "Nothing in biology makes sense except in the light of evolution," and cancer is no exception to this rule. Our understanding of cancer initiation, progression, treatment, and resistance has advanced considerably by regarding cancer as the product of evolutionary processes. Here we review the literature of mathematical models of cancer evolution and provide a synthesis and discussion of the field.

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.

Can loss of apoptosis protect against cancer?

Cells of higher organisms can commit suicide in response to genomic alterations, a process called programmed cell death. Although it is commonly thought that the loss of programmed cell death is required for carcinogenesis, we argue that the situation is more complex and that the loss of programmed cell death can have the converse effect, preventing cancer progression. If the death rate of cancer cells is low, fewer cell divisions are required for the tumor to reach a certain size, resulting in the presence of fewer mutant cells. Therefore, the chances of overcoming potential selective barriers are reduced, rendering the failure of pathogenic progression probable. However, if there is a higher cell death rate, more cell divisions need to occur for the tumor to reach a certain size, resulting in the presence of more mutant cells and in an increased probability of overcoming selective barriers and cancer progression.

Mad2 overexpression promotes aneuploidy and tumorigenesis in mice

Mad2 is an essential component of the spindle checkpoint that blocks activation of Separase and dissolution of sister chromatids until microtubule attachment to kinetochores is complete. We show here that overexpression of Mad2 in transgenic mice leads to a wide variety of neoplasias, appearance of broken chromosomes, anaphase bridges, and whole-chromosome gains and losses, as well as acceleration of myc-induced lymphomagenesis. Moreover, continued overexpression of Mad2 is not required for tumor maintenance, unlike the majority of oncogenes studied to date. These results demonstrate that transient Mad2 overexpression and chromosome instability can be an important stimulus in the initiation and progression of different cancer subtypes.

Spatial stochastic models for cancer initiation and progression

The multistage carcinogenesis hypothesis has been formulated by a number of authors as a stochastic process. However, most previous models assumed "perfect mixing" in the population of cells, and included no information about spatial locations. In this work, we studied the role of spatial dynamics in carcinogenesis. We formulated a 1D spatial generalization of a constant population (Moran) birth-death process, and described the dynamics analytically. We found that in the spatial model, the probability of fixation of advantageous and disadvantageous mutants is lower, and the rate of generation of double-hit mutants (the so-called tunneling rate) is higher, compared to those for the space-free model. This means that the results previously obtained for space-free models give an underestimation for rates of cancer initiation in the case where the first event is the generation of a double-hit mutant, e.g. the inactivation of a tumor-suppressor gene.

Crypt dynamics and colorectal cancer: advances in mathematical modelling

Mathematical modelling forms a key component of systems biology, offering insights that complement and stimulate experimental studies. In this review, we illustrate the role of theoretical models in elucidating the mechanisms involved in normal intestinal crypt dynamics and colorectal cancer. We discuss a range of modelling approaches, including models that describe cell proliferation, migration, differentiation, crypt fission, genetic instability, APC inactivation and tumour heterogeneity. We focus on the model assumptions, limitations and applications, rather than on the technical details. We also present a new stochastic model for stem-cell dynamics, which predicts that, on average, APC inactivation occurs more quickly in the stem-cell pool in the absence of symmetric cell division. This suggests that natural niche succession may protect stem cells against malignant transformation in the gut. Finally, we explain how we aim to gain further understanding of the crypt system and of colorectal carcinogenesis with the aid of multiscale models that cover all levels of organization from the molecular to the whole organ.

Epithelial tissue architecture protects against cancer

We consider the design of colon crypts from the point of view of minimizing the likelihood of generation of cancerous mutations. A stochastic mathematical model (a finite branching process) is developed and fully analyzed. It is found that depending on the mutation rates, different designs are evolutionarily advantageous. If the mutation rates associated with stem cells are a lot higher than the mutation rates of daughter cells, then few stem cells per crypt is the evolutionarily optimal strategy. If the mutation rates of stem cells are of the same order of magnitude or lower than those for daughter cells, then having as many stem cells per crypt as possible is the desirable design. We also found that the optimal evolutionary strategy may work very well to protect the organism from cancer in the young age, but the same strategy becomes detrimental as the organism ages. It pushes the onset of cancer back in time, but it results in an elevated cancer initiation rates as the organism gets older. Our model quantifies the idea that cancer and aging are the two sides of one coin.

Genetic instability and clonal expansion

Inactivation of tumor suppressor genes can lead to clonal expansion. We study the evolutionary dynamics of this process and calculate the probability that inactivation of a tumor suppressor gene is preceded by mutations in genes that confer genetic instability. Unstable cells might have a slower rate of clonal expansion than stable cells because of an increased probability of generating lethal mutations or inducing apoptosis. We show that the different growth rates of genetically stable and unstable cells during clonal expansion represent, in general, only a small disadvantage for genetic instability. The intuitive reason for this conclusion is that robust clonal expansion, where cellular birth rates are significantly greater than death rates, occurs on a much faster time scale than waiting for those mutations that allow clonal expansion. Moreover, in special cases where clonal expansion is very slow, genetically unstable cells have a higher probability to accumulate additional mutations during clonal expansion that confer a selective advantage. Clonal expansion represents a major disadvantage for genetic instability only when inactivation of the tumor suppressor gene leads to a very small increase of the cellular reproductive rate that is cancelled by the increased mortality of unstable cells.

Evolutionary biology of cancer

Cancer is driven by the somatic evolution of cell lineages that have escaped controls on replication and by the population-level evolution of genes that influence cancer risk. We describe here how recent evolutionary ecological studies have elucidated the roles of predation by the immune system and competition among normal and cancerous cells in the somatic evolution of cancer. Recent analyses of the evolution of cancer at the population level show how rapid changes in human environments have augmented cancer risk, how strong selection has frequently led to increased cancer risk as a byproduct, and how anticancer selection has led to tumor-suppression systems, tissue designs that slow somatic evolution, constraints on morphological evolution and even senescence itself. We discuss how applications of the tools of ecology and evolutionary biology are poised to revolutionize our understanding and treatment of this disease.

Population genetics of tumor suppressor genes

Cancer emerges when a single cell receives multiple mutations. For example, the inactivation of both alleles of a tumor suppressor gene (TSG) can imply a net reproductive advantage of the cell and might lead to clonal expansion. In this paper, we calculate the probability as a function of time that a population of cells has generated at least one cell with two inactivated alleles of a TSG. Different kinetic laws hold for small and large populations. The inactivation of the first allele can either be neutral or lead to a selective advantage or disadvantage. The inactivation of the first and of the second allele can occur at equal or different rates. Our calculations provide insights into basic aspects of population genetics determining cancer initiation and progression.

Telomere length heterogeneity and chromosome instability

Chromosome aberrations are the hallmark of cancer cells. Although a few specific chromosome aberrations are frequently detected in some types of cancer, the majority of karyotypic abnormalities tend to differ between different histological types and between individuals with the same type of cancer. Recent work indicates that telomeres may be directly involved in shaping the karyotypes of tumor cells. In particular, the heterogeneity of telomere lengths within cells may have direct influence on the frequency with which chromosomes engage in telomeric fusions and in subsequent breakage-fusion-bridge cycles. Since telomere length distribution among chromosome arms is a polymorphic trait, difference in distributions between individuals may account, at least in part, for the karyotypic differences found among tumors of the same type. Conversely, if single telomere lengths happen to be inherited, the segregation of particularly short telomeres in families may increase the incidence of specific chromosome aberrations during tumor evolution, and perhaps contribute, along with other factors, to cancer pre-disposition.