Stochastic tunnels in evolutionary dynamics

We study a situation that arises in the somatic evolution of cancer. Consider a finite population of replicating cells and a sequence of mutations: type 0 can mutate to type 1, which can mutate to type 2. There is no back mutation. We start with a homogeneous population of type 0. Mutants of type 1 emerge and either become extinct or reach fixation. In both cases, they can generate type 2, which also can become extinct or reach fixation. If mutation rates are small compared to the inverse of the population size, then the stochastic dynamics can be described by transitions between homogeneous populations. A "stochastic tunnel" arises, when the population moves from all 0 to all 2 without ever being all 1. We calculate the exact rate of stochastic tunneling for the case when type 1 is as fit as type 0 or less fit. Type 2 has the highest fitness. We discuss implications for the elimination of tumor suppressor genes and the activation of genetic instability. Although our theory is developed for cancer genetics, stochastic tunnels are general phenomena that could arise in many circumstances.     

Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing

In this paper, we outline the theory of epidemic percolation networks and their use in the analysis of stochastic susceptible-infectious-removed (SIR) epidemic models on undirected contact networks. We then show how the same theory can be used to analyze stochastic SIR models with random and proportionate mixing. The epidemic percolation networks for these models are purely directed because undirected edges disappear in the limit of a large population. In a series of simulations, we show that epidemic percolation networks accurately predict the mean outbreak size and probability and final size of an epidemic for a variety of epidemic models in homogeneous and heterogeneous populations. Finally, we show that epidemic percolation networks can be used to re-derive classical results from several different areas of infectious disease epidemiology. In an Appendix, we show that an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model in a closed population and prove that the distribution of outbreak sizes given the infection of any given node in the SIR model is identical to the distribution of its out-component sizes in the corresponding probability space of epidemic percolation networks. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR models in closed populations.     

Mutation-Selection Balance with Stochastic Selection

Diffusion theory has been used to analyze a model of mutation-selection balance in which the selection process is assumed to be stochastic in time. The limiting outcome of the mutation-stochastic selection process is determined qualitatively by the geometric mean fitnesses of the genotypes, and the conditions for fixation or polymorphism are similar to those that determine the outcome of the mutation-selection process when selection is constant. However, in the case of a completely recessive allele, detailed numerical study of the polymorphism associated with stochastic selection has shown that the average allele frequency maintained is greater than the equilibrium frequency expected when selection is constant, even when the geometric mean fitness of the recessive homozygotes is identical in the stochastic and deterministic models. Thus, allele frequencies in natural populations that are too high to be plausibly explained by a balance between mutation and constant selection can be accounted for if selection is stochastic.     

Linear Model of Colon Cancer Initiation

Cancer results if regulatory mechanisms of cell birth and death are disrupted. Colorectal tumorigenesis is initiated by somatic or inherited mutations in the APC tumor suppressor gene pathway. Several additional genetic hits in other tumor suppressor genes and oncogenes drive the progression from polyps to malignant, invasive cancer. The majority of colorectal cancers present chromosomal instability, CIN, which is caused by mutations in genes that are required to maintain chromosomal stability. A major question in cancer genetics is whether CIN is an early event and thus a driving force of tumor progression. We present a new mathematical model of colon cancer initiation assuming a linear flow from stem cells to differentiated cells to apoptosis. We study the consequences of mutations in different cell types and calculate the conditions for CIN to precede APC inactivation. We find that early emergence of CIN is very likely in colorectal tumorigenesis.     

Linear model of colon cancer initiation

Cancer results if regulatory mechanisms of cell birth and death are disrupted. Colorectal tumorigenesis is initiated by somatic or inherited mutations in the APC tumor suppressor gene pathway. Several additional genetic hits in other tumor suppressor genes and oncogenes drive the progression from polyps to malignant, invasive cancer. The majority of colorectal cancers present chromosomal instability, CIN, which is caused by mutations in genes that are required to maintain chromosomal stability. A major question in cancer genetics is whether CIN is an early event and thus a driving force of tumor progression. We present a new mathematical model of colon cancer initiation assuming a linear flow from stem cells to differentiated cells to apoptosis. We study the consequences of mutations in different cell types and calculate the conditions for CIN to precede APC inactivation. We find that early emergence of CIN is very likely in colorectal tumorigenesis.     

The evolution of tumor metastases during clonal expansion

Cancer is a leading cause of morbidity and mortality in many countries. Solid tumors generally initiate at one particular site called the primary tumor, but eventually disseminate and form new colonies in other organs. The development of such metastases greatly diminishes the potential for a cure of patients and is thought to represent the final stage of the multi-stage progression of human cancer. The concept of early metastatic dissemination, however, postulates that cancer cell spread might arise early during the development of a tumor. It is important to know whether metastases are present at diagnosis since this determines treatment strategies and outcome. In this paper, we design a stochastic mathematical model of the evolution of tumor metastases in an expanding cancer cell population. We calculate the probability of metastasis at a given time during tumor evolution, the expected number of metastatic sites, and the total number of cancer cells as well as metastasized cells. Furthermore, we investigate the effect of drug administration and tumor resection on these quantities and predict the survival time of cancer patients. The model presented in this paper allows us to determine the probability and number of metastases at diagnosis and to identify the optimum treatment strategy to maximally prolong survival of cancer patients.     

A Possible Explanation for the Variable Frequencies of Cancer Stem Cells in Tumors

A controversy surrounds the frequency of cancer stem cells (CSCs) in solid tumors. Initial studies indicated that these cells had a frequency ranging from to of the total cells. Recent studies have shown that this does not always seem to be the case. Some of these studies have indicated a frequency of . In this paper we propose a stochastic model that is able to capture this potential variability in the frequency of CSCs among the various type of tumors. Considerations regarding the heterogeneity of the tumor cells and its consequences are included. Possible effects on conventional treatments in clinical practice are also described. The model results suggest that traditional attempts to combat cancer cells with rapid cycling can be very stimulating for the cancer stem cell populations.     

The different effects of apoptosis and DNA repair on tumorigenesis

Complex multicellular organisms have evolved mechanisms to ensure that individual cells follow their proper developmental and somatic programs. Tumorigenesis, or uncontrolled cellular proliferation, is caused by somatic mutations to those genetic constraints that normally operate within a tissue. Genes involved in DNA repair and apoptosis are particularly instrumental in safeguarding cells against tumorigenesis. In this paper, we introduce a stochastic framework to analyse the somatic evolution of cancer initiation. Within this model, we study how apoptosis and DNA repair can maintain the transient stability of somatic cells and delay the onset of cancer. Focusing on individual cell lineages, we calculate the waiting time before tumorigenesis in the presence of varying degrees of apoptosis and DNA repair. We find that the loss of DNA repair or the loss of apoptosis both hasten tumorigenesis, but in characteristically different ways.     

Chromosome instability and tumor lethality suppression in carcinogenesis

The maintenance and survival of each organism depends on its genome integrity. Alterations of essential genes, or aberrant chromosome number and structure lead to cell death. Paradoxically, cancer cells, especially in solid tumors, contain somatic gene mutations and are chromosome instability (CIN), suggesting a mechanism that cancer cells have acquired to suppress the lethal mutations and/or CIN. Herein we will discuss a tumor lethality suppression concept based on the studies of yeast genetic interactions and transgenic mice. During the early stages of the multistep process of tumorigenesis, incipient cancer cells probably have adopted genetic and epigenetic alterations to tolerate the lethal mutations of other genes that ensue, and to a larger extent CIN. In turn, CIN mediated massive gain and loss of genes provides a wider buffer for further genetic reshuffling, resulting in cancer cell heterogeneity, drug resistance and evasion of oncogene addiction, thus CIN may be both the effector and inducer of tumorigenesis. Accordingly, interfering with tumor lethality suppression could lead to cancer cell death or growth defects. Further validation of the tumor lethality suppression concept would help to elucidate the role of CIN in tumorigenesis, the relationship between CIN and somatic gene mutations, and would impact the design of anticancer drug development.     

Emerging Roles of Ruk/CIN85 in Vesicle‐Mediated Transport,Adhesion, Migration and Malignancy

Ruk/CIN85 is an adaptor protein. Similar to many other proteins of this type, Ruk/CIN85 is known to take part in multiple cellular processes including signal transduction, vesicle‐mediated transport, cytoskeleton remodelling, programmed cell death and viral infection. Recent studies have also revealed the potential importance of Ruk/CIN85 in cancer cell invasiveness. In this review we summarize the various roles of this protein as well as the potential contribution of Ruk/CIN85 to malignancy and the invasiveness of cancer cells. In the last section of the paper we also speculate on the utility of Ruk/CIN85 as a target for novel anti‐cancer therapies.     

CIN and Aneuploidy: Different Concepts,Different Consequences

Chromosomal instability (CIN) and aneuploidy are similar concepts but not synonymous. CIN is the process that leads to chromosome copy number alterations, and aneuploidy is the result. While CIN and resulting aneuploidy often cause growth defects, they are also selected for in cancer cells. Although such contradicting fates may seem paradoxical at first, they can be better understood when CIN and aneuploidy are assessed separately, taking into account the in vitro or in vivo context, the rate of CIN, and severity of the aneuploid karyotype. As CIN can only be measured in living cells, which proves to be technically challenging in vivo, aneuploidy is more frequently quantified. However, CIN rates might be more predictive for tumor outcome than assessing aneuploidy rates alone. In reviewing the literature, we therefore conclude that there is an urgent need for new models in which we can monitor chromosome mis‐segregation and its consequences in vivo. Also see the video abstract here: https://youtu.be/fL3LxZduchg     

Stochastic Modelling of Chromosomal Segregation: Errors Can Introduce Correction

Cell division is a complex process requiring the cell to have many internal checks so that division may proceed and be completed correctly. Failure to divide correctly can have serious consequences, including progression to cancer. During mitosis, chromosomal segregation is one such process that is crucial for successful progression. Accurate segregation of chromosomes during mitosis requires regulation of the interactions between chromosomes and spindle microtubules. If left uncorrected, chromosome attachment errors can cause chromosome segregation defects which have serious effects on cell fates. In early prometaphase, where kinetochores are exposed to multiple microtubules originating from the two poles, there are frequent errors in kinetochore-microtubule attachment. Erroneous attachments are classified into two categories, syntelic and merotelic. In this paper, we consider a stochastic model for a possible function of syntelic and merotelic kinetochores, and we provide theoretical evidence that merotely can contribute to lessening the stochastic noise in the time for completion of the mitotic process in eukaryotic cells.     

Quantifying Clonal and Subclonal Passenger Mutations in Cancer Evolution

The vast majority of mutations in the exome of cancer cells are passengers, which do not affect the reproductive rate of the cell. Passengers can provide important information about the evolutionary history of an individual cancer, and serve as a molecular clock. Passengers can also become targets for immunotherapy or confer resistance to treatment. We study the stochastic expansion of a population of cancer cells describing the growth of primary tumors or metastatic lesions. We first analyze the process by looking forward in time and calculate the fixation probabilities and frequencies of successive passenger mutations ordered by their time of appearance. We compute the likelihood of specific evolutionary trees, thereby informing the phylogenetic reconstruction of cancer evolution in individual patients. Next, we derive results looking backward in time: for a given subclonal mutation we estimate the number of cancer cells that were present at the time when that mutation arose. We derive exact formulas for the expected numbers of subclonal mutations of any frequency. Fitting this formula to cancer sequencing data leads to an estimate for the ratio of birth and death rates of cancer cells during the early stages of clonal expansion.     

Extinction models for cancer stem cell therapy

Cells with stem cell-like properties are now viewed as initiating and sustaining many cancers. This suggests that cancer can be cured by driving these cancer stem cells to extinction. The problem with this strategy is that ordinary stem cells are apt to be killed in the process. This paper sets bounds on the killing differential (difference between death rates of cancer stem cells and normal stem cells) that must exist for the survival of an adequate number of normal stem cells. Our main tools are birth-death Markov chains in continuous time. In this framework, we investigate the extinction times of cancer stem cells and normal stem cells. Application of extreme value theory from mathematical statistics yields an accurate asymptotic distribution and corresponding moments for both extinction times. We compare these distributions for the two cell populations as a function of the killing rates. Perhaps a more telling comparison involves the number of normal stem cells N_H at the extinction time of the cancer stem cells. Conditioning on the asymptotic time to extinction of the cancer stem cells allows us to calculate the asymptotic mean and variance of N_H. The full distribution of N_H can be retrieved by the finite Fourier transform and, in some parameter regimes, by an eigenfunction expansion. Finally, we discuss the impact of quiescence (the resting state) on stem cell dynamics. Quiescence can act as a sanctuary for cancer stem cells and imperils the proposed therapy. We approach the complication of quiescence via multitype branching process models and stochastic simulation. Improvements to the τ-leaping method of stochastic simulation make it a versatile tool in this context. We conclude that the proposed therapy must target quiescent cancer stem cells as well as actively dividing cancer stem cells. The current cancer models demonstrate the virtue of attacking the same quantitative questions from a variety of modeling, mathematical, and computational perspectives.     

Aneuploidy,cell delamination and tumorigenesis in Drosophila epithelia

Chromosomal instability (CIN) is a common feature in human cancer, and highly aneuploid tumors are frequently associated with poor prognosis; however, the molecular and cellular mechanisms underlying CIN-induced tumorigenesis are poorly understood. Here we review recent findings about the role of CIN in driving tumor-like growth and host invasiveness in Drosophila epithelia and discuss the commonalities of CIN-induced tumors with other Drosophila-based cancer models. We also discuss possible scenarios that can account for the participation of CIN in tumorigenesis and propose that, alternatively to the classical role of aneuploidy in promoting the accumulation of mutations in cancer cells, aneuploidy can be a source of stress that may contribute to cancer initiation and/or progression.     

Uncoupled Analysis of Stochastic Reaction Networks in Fluctuating Environments

The dynamics of stochastic reaction networks within cells are inevitably modulated by factors considered extrinsic to the network such as, for instance, the fluctuations in ribosome copy numbers for a gene regulatory network. While several recent studies demonstrate the importance of accounting for such extrinsic components, the resulting models are typically hard to analyze. In this work we develop a general mathematical framework that allows to uncouple the network from its dynamic environment by incorporating only the environment''s effect onto the network into a new model. More technically, we show how such fluctuating extrinsic components (e.g., chemical species) can be marginalized in order to obtain this decoupled model. We derive its corresponding process- and master equations and show how stochastic simulations can be performed. Using several case studies, we demonstrate the significance of the approach.     

Aneuploid colon cancer cells have a robust spindle checkpoint

Colon cancer cells frequently display minisatellite instability (MIN) or chromosome instability (CIN). While MIN is caused by mismatch repair defects, the lesions responsible for CIN are unknown. The observation that CIN cells fail to undergo mitotic arrest following spindle damage suggested that mutations in spindle checkpoint genes may account for CIN. However, here we show that CIN cells do undergo mitotic arrest in response to spindle damage. Although the maximum mitotic index achieved by CIN lines is diminished relative to MIN lines, CIN cells clearly have a robust spindle checkpoint. Consistently, mutations in spindle checkpoint genes are rare in human tumours. In contrast, the adenomatous polyposis coli (APC) gene is frequently mutated in CIN cells. Significantly, we show here that expression of an APC mutant in MIN cells reduces the mitotic index following spindle damage to a level observed in CIN cells, suggesting that APC dysfunction may contribute to CIN.     

Analysis of stochastic strategies in bacterial competence: a master equation approach

Competence is a transiently differentiated state that certain bacterial cells reach when faced with a stressful environment. Entrance into competence can be attributed to the excitability of the dynamics governing the genetic circuit that regulates this cellular behavior. Like many biological behaviors, entrance into competence is a stochastic event. In this case cellular noise is responsible for driving the cell from a vegetative state into competence and back. In this work we present a novel numerical method for the analysis of stochastic biochemical events and use it to study the excitable dynamics responsible for competence in Bacillus subtilis. Starting with a Finite State Projection (FSP) solution of the chemical master equation (CME), we develop efficient numerical tools for accurately computing competence probability. Additionally, we propose a new approach for the sensitivity analysis of stochastic events and utilize it to elucidate the robustness properties of the competence regulatory genetic circuit. We also propose and implement a numerical method to calculate the expected time it takes a cell to return from competence. Although this study is focused on an example of cell-differentiation in Bacillus subtilis, our approach can be applied to a wide range of stochastic phenomena in biological systems.     

An ecological resilience perspective on cancer: Insights from a toy model

In this paper we propose an ecological resilience point of view on cancer. This view is based on the analysis of a simple ODE model for the interactions between cancer and normal cells. The model presents two regimes for tumor growth. In the first, cancer arises due to three reasons: a partial corruption of the functions that avoid the growth of mutated cells, an aggressive phenotype of tumor cells and exposure to external carcinogenic factors. In this case, treatments may be effective if they drive the system to the basin of attraction of the cancer cure state. In the second regime, cancer arises because the repair system is intrinsically corrupted. In this case, the complete cure is not possible since the cancer cure state is no more stable, but tumor recurrence may be delayed if treatment is prolongued. We review three indicators of the resilience of a stable equilibrium, related with size and shape of its basin of attraction: latitude, precariousness and resistance. A novel method to calculate these indicators is proposed. This method is simpler and more efficient than those currently used, and may be easily applied to other population dynamics models. We apply this method to the model and investigate how these indicators behave with parameters changes. Finally, we present some simulations to illustrate how the resilience analysis can be applied to validated models in order to obtain indicators for personalized cancer treatments.