Genetic instability and the quasispecies model

Modeling tumors as complex ecosystems

Many cancers resist therapeutic intervention. This is fundamentally related to intratumor heterogeneity: multiple cell populations, each with different phenotypic signatures, coexist within a tumor and its metastases. Like species in an ecosystem, cancer populations are intertwined in a complex network of ecological interactions. Most mathematical models of tumor ecology, however, cannot account for such phenotypic diversity or predict its consequences. Here, we propose that the generalized Lotka-Volterra model (GLV), a standard tool to describe species-rich ecological communities, provides a suitable framework to model the ecology of heterogeneous tumors. We develop a GLV model of tumor growth and discuss how its emerging properties provide a new understanding of the disease. We discuss potential extensions of the model and their application to phenotypic plasticity, cancer-immune interactions, and metastatic growth. Our work outlines a set of questions and a road map for further research in cancer ecology.

Molecular evolution between chemistry and biology : The interplay of competition, cooperation, and mutation

Biological evolution is reduced to three fundamental processes in the spirit of a minimal model: (i) Competition caused by differential fitness, (ii) cooperation of competitors in the sense of symbiosis, and (iii) variation introduced by mutation understood as error-prone reproduction. The three combinations of two fundamental processes each, ([Formula: see text]) competition and mutation, ([Formula: see text]) cooperation and competition, and ([Formula: see text]) cooperation and mutation, are analyzed. Changes in population dynamics that are induced by bifurcations and threshold phenomena are discussed.

Asymptotic Behavior of Eigen's Quasispecies Model

We study Eigen's quasispecies model in the asymptotic regime where the length of the genotypes goes to [Formula: see text] and the mutation probability goes to 0. A limiting infinite system of differential equations is obtained. We prove convergence of trajectories, as well as convergence of the equilibrium solutions. We give analogous results for a discrete-time version of Eigen's model, which coincides with a model proposed by Moran.

Commonly Occurring Cell Subsets in High-Grade Serous Ovarian Tumors Identified by Single-Cell Mass Cytometry

We have performed an in-depth single-cell phenotypic characterization of high-grade serous ovarian cancer (HGSOC) by multiparametric mass cytometry (CyTOF). Using a CyTOF antibody panel to interrogate features of HGSOC biology, combined with unsupervised computational analysis, we identified noteworthy cell types co-occurring across the tumors. In addition to a dominant cell subset, each tumor harbored rarer cell phenotypes. One such group co-expressed E-cadherin and vimentin (EV), suggesting their potential role in epithelial mesenchymal transition, which was substantiated by pairwise correlation analyses. Furthermore, tumors from patients with poorer outcome had an increased frequency of another rare cell type that co-expressed vimentin, HE4, and cMyc. These poorer-outcome tumors also populated more cell phenotypes, as quantified by Simpson's diversity index. Thus, despite the recognized genomic complexity of the disease, the specific cell phenotypes uncovered here offer a focus for therapeutic intervention and disease monitoring.

Trans-heteroclinic bifurcation: a novel type of catastrophic shift

Global and local bifurcations are extremely important since they govern the transitions between different qualitative regimes in dynamical systems. These transitions or tipping points, which are ubiquitous in nature, can be smooth or catastrophic. Smooth transitions involve a continuous change in the steady state of the system until the bifurcation value is crossed, giving place to a second-order phase transition. Catastrophic transitions involve a discontinuity of the steady state at the bifurcation value, giving place to first-order phase transitions. Examples of catastrophic shifts can be found in ecosystems, climate, economic or social systems. Here we report a new type of global bifurcation responsible for a catastrophic shift. This bifurcation, identified in a family of quasi-species equations and named as trans-heteroclinic bifurcation, involves an exchange of stability between two distant and heteroclinically connected fixed points. Since the two fixed points interchange the stability without colliding, a catastrophic shift takes place. We provide an exhaustive description of this new bifurcation, also detailing the structure of the replication-mutation matrix of the quasi-species equation giving place to this bifurcation. A perturbation analysis is provided around the bifurcation value. At this value the heteroclinic connection is replaced by a line of fixed points in the quasi-species model. But it is shown that, if the replication-mutation matrix satisfies suitable conditions, then, under a small perturbation, the exchange of heteroclinic connections is preserved, except on a tiny range around the bifurcation value whose size is of the order of magnitude of the perturbation. The results presented here can help to understand better novel mechanisms behind catastrophic shifts and contribute to a finer identification of such transitions in theoretical models in evolutionary biology and other dynamical systems.

Theoretical Models of Generalized Quasispecies

Theoretical modeling of quasispecies has progressed in several directions. In this chapter, we review the works of Emmanuel Tannenbaum, who, together with Eugene Shakhnovich at Harvard University and later with colleagues and students at Ben-Gurion University in Beersheva, implemented one of the more useful approaches, by progressively setting up various formulations for the quasispecies model and solving them analytically. Our review will focus on these papers that have explored new models, assumed the relevant mathematical approximations, and proceeded to analytically solve for the steady-state solutions and run stochastic simulations . When applicable, these models were related to real-life problems and situations, including changing environments, presence of chemical mutagens, evolution of cancer and tumor cells , mutations in Escherichia coli, stem cells , chromosomal instability (CIN), propagation of antibiotic drug resistance , dynamics of bacteria with plasmids , DNA proofreading mechanisms, and more.

Cancer evolution: mathematical models and computational inference

Cancer is a somatic evolutionary process characterized by the accumulation of mutations, which contribute to tumor growth, clinical progression, immune escape, and drug resistance development. Evolutionary theory can be used to analyze the dynamics of tumor cell populations and to make inference about the evolutionary history of a tumor from molecular data. We review recent approaches to modeling the evolution of cancer, including population dynamics models of tumor initiation and progression, phylogenetic methods to model the evolutionary relationship between tumor subclones, and probabilistic graphical models to describe dependencies among mutations. Evolutionary modeling helps to understand how tumors arise and will also play an increasingly important prognostic role in predicting disease progression and the outcome of medical interventions, such as targeted therapy.

Repression/depression of conjugative plasmids and their influence on the mutation-selection balance in static environments

We study the effect that conjugation-mediated Horizontal Gene Transfer (HGT) has on the mutation-selection balance of a population in a static environment. We consider a model whereby a population of unicellular organisms, capable of conjugation, comes to mutation-selection balance in the presence of an antibiotic, which induces a first-order death rate constant for genomes that are not resistant. We explicitly take into consideration the repression/de-repression dynamics of the conjugative plasmid, and assume that a de-repressed plasmid remains temporarily de-repressed after copying itself into another cell. We assume that both repression and de-repression are characterized by first-order rate constants and , respectively. We find that conjugation has a deleterious effect on the mean fitness of the population, suggesting that HGT does not provide a selective advantage in a static environment, but is rather only useful for adapting to new environments. This effect can be ameliorated by repression, suggesting that while HGT is not necessarily advantageous for a population in a static environment, its deleterious effect on the mean fitness can be negated via repression. Therefore, it is likely that HGT is much more advantageous in a dynamic landscape. Furthermore, in the limiting case of a vanishing spontaneous de-repression rate constant, we find that the fraction of conjugators in the population undergoes a phase transition as a function of population density. Below a critical population density, the fraction of conjugators is zero, while above this critical population density the fraction of conjugators rises continuously to one. Our model for conjugation-mediated HGT is related to models of infectious disease dynamics, where the conjugators play the role of the infected (I) class, and the non-conjugators play the role of the susceptible (S) class.

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.

A systems approach defining constraints of the genome architecture on lineage selection and evolvability during somatic cancer evolution

Most clinically distinguishable malignant tumors are characterized by specific mutations, specific patterns of chromosomal rearrangements and a predominant mechanism of genetic instability but it remains unsolved whether modifications of cancer genomes can be explained solely by mutations and selection through the cancer microenvironment.

It has been suggested that internal dynamics of genomic modifications as opposed to the external evolutionary forces have a significant and complex impact on Darwinian species evolution. A similar situation can be expected for somatic cancer evolution as molecular key mechanisms encountered in species evolution also constitute prevalent mutation mechanisms in human cancers. This assumption is developed into a systems approach of carcinogenesis which focuses on possible inner constraints of the genome architecture on lineage selection during somatic cancer evolution. The proposed systems approach can be considered an analogy to the concept of evolvability in species evolution.

The principal hypothesis is that permissive or restrictive effects of the genome architecture on lineage selection during somatic cancer evolution exist and have a measurable impact. The systems approach postulates three classes of lineage selection effects of the genome architecture on somatic cancer evolution: i) effects mediated by changes of fitness of cells of cancer lineage, ii) effects mediated by changes of mutation probabilities and iii) effects mediated by changes of gene designation and physical and functional genome redundancy. Physical genome redundancy is the copy number of identical genetic sequences. Functional genome redundancy of a gene or a regulatory element is defined as the number of different genetic elements, regardless of copy number, coding for the same specific biological function within a cancer cell. Complex interactions of the genome architecture on lineage selection may be expected when modifications of the genome architecture have multiple and possibly opposed effects which manifest themselves at disparate times and progression stages.

Dissection of putative mechanisms mediating constraints exerted by the genome architecture on somatic cancer evolution may provide an algorithm for understanding and predicting as well as modifying somatic cancer evolution in individual patients.

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.

A race between tumor immunoescape and genome maintenance selects for optimum levels of (epi)genetic instability

The human immune system functions to provide continuous body-wide surveillance to detect and eliminate foreign agents such as bacteria and viruses as well as the body's own cells that undergo malignant transformation. To counteract this surveillance, tumor cells evolve mechanisms to evade elimination by the immune system; this tumor immunoescape leads to continuous tumor expansion, albeit potentially with a different composition of the tumor cell population ("immunoediting"). Tumor immunoescape and immunoediting are products of an evolutionary process and are hence driven by mutation and selection. Higher mutation rates allow cells to more rapidly acquire new phenotypes that help evade the immune system, but also harbor the risk of an inability to maintain essential genome structure and functions, thereby leading to an error catastrophe. In this paper, we designed a novel mathematical framework, based upon the quasispecies model, to study the effects of tumor immunoediting and the evolution of (epi)genetic instability on the abundance of tumor and immune system cells. We found that there exists an optimum number of tumor variants and an optimum magnitude of mutation rates that maximize tumor progression despite an active immune response. Our findings provide insights into the dynamics of tumorigenesis during immune system attacks and help guide the choice of treatment strategies that best inhibit diverse tumor cell populations.

Quasispecies Dynamics of RNA Viruses

RNA viruses, such as human immunodeficiency virus, hepatitis C virus, influenza virus, and poliovirus replicate with very high mutation rates and exhibit very high genetic diversity. The extremely high genetic diversity of RNA virus populations originates that they replicate as complex mutant spectra known as viral quasispecies. The quasispecies dynamics of RNA viruses are closely related to viral pathogenesis and disease, and antiviral treatment strategies. Over the past several decades, the quasispecies concept has been expanded to provide an adequate framework to explain complex behavior of RNA virus populations. Recently, the quasispecies concept has been used to study other complex biological systems, such as tumor cells, bacteria, and prions. Here, we focus on some questions regarding viral and theoretical quasispecies concepts, as well as more practical aspects connected to pathogenesis and resistance to antiviral treatments. A better knowledge of virus diversification and evolution may be critical in preventing and treating the spread of pathogenic viruses.

Quasispecies as a matter of fact: viruses and beyond

We review the origins of the quasispecies concept and its relevance for RNA virus evolution, viral pathogenesis and antiviral treatment strategies. We emphasize a critical point of quasispecies that refers to genome collectivities as the unit of selection, and establish parallels between RNA viruses and some cellular systems such as bacteria and tumor cells. We refer also to tantalizing new observations that suggest quasispecies behavior in prions, perhaps as a result of the same quantum-mechanical indeterminations that underlie protein conformation and error-prone replication in genetic systems. If substantiated, these observations with prions could lead to new research on the structure-function relationship of non-nucleic acid biological molecules.

Stochastic dynamics of leukemic cells under an intermittent targeted therapy

The evolutionary dynamics of cancerous cell populations in a model of Chronic Myeloid Leukemia (CML) is investigated in the presence of an intermittent targeted therapy. Cancer development and progression is modeled by simulating the stochastic evolution of initially healthy cells which can experience genetic mutations and modify their reproductive behavior, becoming leukemic clones. Front line therapy for the treatment of patients affected by CML is based on the administration of tyrosine kinase inhibitors, namely imatinib (Gleevec) or, more recently, dasatinib or nilotinib. Despite the fact that they represent the first example of a successful molecular targeted therapy, the development of resistance to these drugs is observed in a proportion of patients, especially those in advanced stages. In this study, we simulate an imatinib-like treatment of CML by modifying the fitness and the death rate of cancerous cells and describe the several scenarios in the evolutionary dynamics of white blood cells as a consequence of the efficacy of the different modeled therapies. The patient response to the therapy is investigated by simulating a drug administration following a continuous or pulsed time scheduling. A permanent disappearance of leukemic clones is achieved with a continuous therapy. This theoretical behavior is in a good agreement with that observed in previous clinical investigations. However, these findings demonstrate that an intermittent therapy could represent a valid alternative in patients with high risk of toxicity. A suitable tuned pulsed therapy can also reduce the probability of developing resistance.

Mutant spectra in virus behavior

RNA viruses replicate as complex mutant spectra, also termed ‘mutant clouds’, known as viral quasispecies. While this is a widely observed viral population structure, it is less known that a number of biologically relevant features of this important group of viral pathogens depend on (or are strongly influenced by) the complexity and composition of mutant spectra. Among them, fitness increase or decrease depending on intrapopulation complementation or interference, selection triggered by memory genomes, pathogenic potential of viruses, disease evolution and the response to antiviral treatments. Quasispecies represent the recognition of complex behavior in viruses, and it is an oversimplification to equate such a population structure with the classic polymorphism of population biology. Darwinian principles acting on genome collectivities that replicate with high error rates provide a unique population structure prone to flexible and largely unpredictable behavior.

Unfinished stories on viral quasispecies and Darwinian views of evolution

Experimental evidence that RNA virus populations consist of distributions of mutant genomes, termed quasispecies, was first published 31 years ago. This work provided the earliest experimental support for a theory to explain a system that replicated with limited fidelity and to understand the self-organization and adaptability of early life forms on Earth. High mutation rates and quasispecies dynamics of RNA viruses are intimately related to both viral disease and antiviral treatment strategies. Moreover, the quasispecies concept is being applied to other biological systems such as cancer research in which cellular mutant spectra can be also detected. This review addresses some of the unanswered questions regarding viral and theoretical quasispecies concepts as well as more practical aspects concerning resistance to antiviral treatments and pathogenesis.

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.

Clonal origin and evolution of a transmissible cancer

The transmissible agent causing canine transmissible venereal tumor (CTVT) is thought to be the tumor cell itself. To test this hypothesis, we analyzed genetic markers including major histocompatibility (MHC) genes, microsatellites, and mitochondrial DNA (mtDNA) in naturally occurring tumors and matched blood samples. In each case, the tumor is genetically distinct from its host. Moreover, tumors collected from 40 dogs in 5 continents are derived from a single neoplastic clone that has diverged into two subclades. Phylogenetic analyses indicate that CTVT most likely originated from a wolf or an East Asian breed of dog between 200 and 2500 years ago. Although CTVT is highly aneuploid, it has a remarkably stable genotype. During progressive growth, CTVT downmodulates MHC antigen expression. Our findings have implications for understanding genome instability in cancer, natural transplantation of allografts, and the capacity of a somatic cell to evolve into a transmissible parasite.