The linear process of somatic evolution

Fixation times on directed graphs

Computing the rate of evolution in spatially structured populations is difficult. A key quantity is the fixation time of a single mutant with relative reproduction rate r which invades a population of residents. We say that the fixation time is "fast" if it is at most a polynomial function in terms of the population size N. Here we study fixation times of advantageous mutants (r > 1) and neutral mutants (r = 1) on directed graphs, which are those graphs that have at least some one-way connections. We obtain three main results. First, we prove that for any directed graph the fixation time is fast, provided that r is sufficiently large. Second, we construct an efficient algorithm that gives an upper bound for the fixation time for any graph and any r ≥ 1. Third, we identify a broad class of directed graphs with fast fixation times for any r ≥ 1. This class includes previously studied amplifiers of selection, such as Superstars and Metafunnels. We also show that on some graphs the fixation time is not a monotonically declining function of r; in particular, neutral fixation can occur faster than fixation for small selective advantages.

The speed of neutral evolution on graphs

The speed of evolution on structured populations is crucial for biological and social systems. The likelihood of invasion is key for evolutionary stability. But it makes little sense if it takes long. It is far from known what population structure slows down evolution. We investigate the absorption time of a single neutral mutant for all the 112 non-isomorphic undirected graphs of size 6. We find that about three-quarters of the graphs have an absorption time close to that of the complete graph, less than one-third are accelerators, and more than two-thirds are decelerators. Surprisingly, determining whether a graph has a long absorption time is too complicated to be captured by the joint degree distribution. Via the largest sojourn time, we find that echo-chamber-like graphs, which consist of two homogeneous graphs connected by few sparse links, are likely to slow down absorption. These results are robust for large graphs, mutation patterns as well as evolutionary processes. This work serves as a benchmark for timing evolution with complex interactions, and fosters the understanding of polarization in opinion formation.

How mutation accumulation depends on the structure of the cell lineage tree

All the cells of a multicellular organism are the product of cell divisions that trace out a single binary tree, the so-called cell lineage tree. Because cell divisions are accompanied by replication errors, the shape of the cell lineage tree is a key determinant of how somatic evolution, which can potentially lead to cancer, proceeds. Carcinogenesis requires the accumulation of a certain number of driver mutations. By mapping the accumulation of mutations into a graph theoretical problem, we present an exact numerical method to calculate the probability of collecting a given number of mutations and show that for low mutation rates it can be approximated with a simple analytical formula, which depends only on the distribution of the lineage lengths, and is dominated by the longest lineages. Our results are crucial in understanding how natural selection can shape the cell lineage trees of multicellular organisms and curtail somatic evolution.

The dark side of stemness - the role of hematopoietic stem cells in development of blood malignancies

Hematopoietic stem cells (HSCs) produce all blood cells throughout the life of the organism. However, the high self-renewal and longevity of HSCs predispose them to accumulate mutations. The acquired mutations drive preleukemic clonal hematopoiesis, which is frequent among elderly people. The preleukemic state, although often asymptomatic, increases the risk of blood cancers. Nevertheless, the direct role of preleukemic HSCs is well-evidenced in adult myeloid leukemia (AML), while their contribution to other hematopoietic malignancies remains less understood. Here, we review the evidence supporting the role of preleukemic HSCs in different types of blood cancers, as well as present the alternative models of malignant evolution. Finally, we discuss the clinical importance of preleukemic HSCs in choosing the therapeutic strategies and provide the perspective on further studies on biology of preleukemic HSCs.

Adult stem cell activity in naked mole rats for long-term tissue maintenance

The naked mole rat (NMR), Heterocephalus glaber, the longest-living rodent, provides a unique opportunity to explore how evolution has shaped adult stem cell (ASC) activity and tissue function with increasing lifespan. Using cumulative BrdU labelling and a quantitative imaging approach to track intestinal ASCs (Lgr5+) in their native in vivo state, we find an expanded pool of Lgr5+ cells in NMRs, and these cells specifically at the crypt base (Lgr5+CBC) exhibit slower division rates compared to those in short-lived mice but have a similar turnover as human LGR5+CBC cells. Instead of entering quiescence (G0), NMR Lgr5+CBC cells reduce their division rates by prolonging arrest in the G1 and/or G2 phases of the cell cycle. Moreover, we also observe a higher proportion of differentiated cells in NMRs that confer enhanced protection and function to the intestinal mucosa which is able to detect any chemical imbalance in the luminal environment efficiently, triggering a robust pro-apoptotic, anti-proliferative response within the stem/progenitor cell zone.

Spatial exclusion leads to "tug-of-war" ecological dynamics between competing species within microchannels

Competition is ubiquitous in microbial communities, shaping both their spatial and temporal structure and composition. Classical minimal models of competition, such as the Moran model, have been employed in ecology and evolutionary biology to understand the role of fixation and invasion in the maintenance of population diversity. Informed by recent experimental studies of cellular competition in confined spaces, we extend the Moran model to incorporate mechanical interactions between cells that divide within the limited space of a one-dimensional open microchannel. The model characterizes the skewed collective growth of the cells dividing within the channel, causing cells to be expelled at the channel ends. The results of this spatial exclusion model differ significantly from those of its classical well-mixed counterpart. The mean time to fixation of a species is greatly accelerated, scaling logarithmically, rather than algebraically, with the system size, and fixation/extinction probability sharply depends on the species' initial fractional abundance. By contrast, successful takeovers by invasive species, whether through mutation or immigration, are substantially less likely than in the Moran model. We also find that the spatial exclusion tends to attenuate the effects of fitness differences on the fixation times and probabilities. We find that these effects arise from the combination of the quasi-neutral "tug-of-war" diffusion dynamics of the inter-species boundary around an unstable equipoise point and the quasi-deterministic avalanche dynamics away from the fixed point. These results, which can be tested in microfluidic monolayer devices, have implications for the maintenance of species diversity in dense bacterial and cellular ecosystems where spatial exclusion is central to the competition, such as in organized biofilms or intestinal crypts.

Optimization Therapy by Coupling Intermittent Androgen Suppression with Impulsive Chemotherapy for a Prostate Cancer Model

Intermittent androgen suppression in the prostate cancer is often relapsed by the increasing of prostate specific antigen level during the on-treatment. Historically, chemotherapy has had a limited role in the treatment of prostate cancer. However, new agents are showing promise in patients with advanced disease. Intermittent androgen suppression plus chemotherapy in pulsed pattern has become an indispensable clinical scheme for prostate cancer, which is presented to describe the transformation mechanism for three kinds of cancer cells in this paper. The model is then extended to include the residual effect of chemotherapy which suppresses the cancer cells production, thereby preventing the relapse. The optimal controls represent the efficiencies of both intermittent androgen suppression and chemotherapy in suppressing relapse of prostate cancer. Based on an optimal algorithm, numerical simulations are implemented not only to show the optimal durations of on- and off-treatment and chemotherapy dosages but also to present the effectiveness of different strategies in inhibiting the relapse for three types of patients. Results reveal that the optimal intermittent androgen suppression scheme with alterable treatment cycles is pivotal for type I and II patients, in part because it can greatly reduce the on-treatment time and degrade the level of prostate specific antigen. Furthermore, optimal hybrid schedule even averts the relapse of prostate cancer for type II and III patients. Finally, comparing the prostate specific antigen under intermittent androgen suppression schedule with residual effect of chemotherapy to one without residual effect of chemotherapy demonstrates the validity of both our model and algorithms in lessening the prostate specific antigen and decreasing the chemotherapy dosages.

Spectral dynamics of guided edge removals and identifying transient amplifiers for death-Birth updating

The paper deals with two interrelated topics: (1) identifying transient amplifiers in an iterative process, and (2) analyzing the process by its spectral dynamics, which is the change in the graph spectra by edge manipulation. Transient amplifiers are networks representing population structures which shift the balance between natural selection and random drift. Thus, amplifiers are highly relevant for understanding the relationships between spatial structures and evolutionary dynamics. We study an iterative procedure to identify transient amplifiers for death-Birth updating. The algorithm starts with a regular input graph and iteratively removes edges until desired structures are achieved. Thus, a sequence of candidate graphs is obtained. The edge removals are guided by quantities derived from the sequence of candidate graphs. Moreover, we are interested in the Laplacian spectra of the candidate graphs and analyze the iterative process by its spectral dynamics. The results show that although transient amplifiers for death-Birth updating are generally rare, a substantial number of them can be obtained by the proposed procedure. The graphs identified share structural properties and have some similarity to dumbbell and barbell graphs. We analyze amplification properties of these graphs and also two more families of bell-like graphs and show that further transient amplifiers for death-Birth updating can be found. Finally, it is demonstrated that the spectral dynamics possesses characteristic features useful for deducing links between structural and spectral properties. These feature can also be taken for distinguishing transient amplifiers among evolutionary graphs in general.

Categorizing update mechanisms for graph-structured metapopulations

The structure of a population strongly influences its evolutionary dynamics. In various settings ranging from biology to social systems, individuals tend to interact more often with those present in their proximity and rarely with those far away. A common approach to model the structure of a population is evolutionary graph theory. In this framework, each graph node is occupied by a reproducing individual. The links connect these individuals to their neighbours. The offspring can be placed on neighbouring nodes, replacing the neighbours-or the progeny of its neighbours can replace a node during the course of ongoing evolutionary dynamics. Extending this theory by replacing single individuals with subpopulations at nodes yields a graph-structured metapopulation. The dynamics between the different local subpopulations is set by an update mechanism. There are many such update mechanisms. Here, we classify update mechanisms for structured metapopulations, which allows to find commonalities between past work and illustrate directions for further research and current gaps of investigation.

Spatial biology of cancer evolution

The natural history of cancers can be understood through the lens of evolution given that the driving forces of cancer development are mutation and selection of fitter clones. Cancer growth and progression are spatial processes that involve the breakdown of normal tissue organization, invasion and metastasis. For these reasons, spatial patterns are an integral part of histological tumour grading and staging as they measure the progression from normal to malignant disease. Furthermore, tumour cells are part of an ecosystem of tumour cells and their surrounding tumour microenvironment. A range of new spatial genomic, transcriptomic and proteomic technologies offers new avenues for the study of cancer evolution with great molecular and spatial detail. These methods enable precise characterizations of the tumour microenvironment, cellular interactions therein and micro-anatomical structures. In conjunction with spatial genomics, it emerges that tumours and microenvironments co-evolve, which helps explain observable patterns of heterogeneity and offers new routes for therapeutic interventions.

Suppressors of fixation can increase average fitness beyond amplifiers of selection

Evolutionary dynamics on graphs has remarkable features: For example, it has been shown that amplifiers of selection exist that-compared to an unstructured population-increase the fixation probability of advantageous mutations, while they decrease the fixation probability of disadvantageous mutations. So far, the theoretical literature has focused on the case of a single mutant entering a graph-structured population, asking how the graph affects the probability that a mutant takes over a population and the time until this typically happens. For continuously evolving systems, the more relevant case is that mutants constantly arise in an evolving population. Typically, such mutations occur with a small probability during reproduction events. We thus focus on the low mutation rate limit. The probability distribution for the fitness in this process converges to a steady state at long times. Intuitively, amplifiers of selection are expected to increase the population's mean fitness in the steady state. Similarly, suppressors of selection are expected to decrease the population's mean fitness in the steady state. However, we show that another set of graphs, called suppressors of fixation, can attain the highest population mean fitness. The key reason behind this is their ability to efficiently reject deleterious mutants. This illustrates the importance of the deleterious mutant regime for the long-term evolutionary dynamics, something that seems to have been overlooked in the literature so far.

The role of spatial structures of tissues in cancer initiation dynamics

It is widely believed that biological tissues evolved to lower the risks of cancer development. One of the specific ways to minimize the chances of tumor formation comes from proper spatial organization of tissues. However, the microscopic mechanisms of underlying processes remain not fully understood. We present a theoretical investigation on the role of spatial structures in cancer initiation dynamics. In our approach, the dynamics of single mutation fixations are analyzed using analytical calculations and computer simulations by mapping them to Moran processes on graphs with different connectivity that mimic various spatial structures. It is found that while the fixation probability is not affected by modifying the spatial structures of the tissues, the fixation times can change dramatically. The slowest dynamics is observed in "quasi-one-dimensional" structures, while the fastest dynamics is observed in "quasi-three-dimensional" structures. Theoretical calculations also suggest that there is a critical value of the degree of graph connectivity, which mimics the spatial dimension of the tissue structure, above which the spatial structure of the tissue has no effect on the mutation fixation dynamics. An effective discrete-state stochastic model of cancer initiation is utilized to explain our theoretical results and predictions. Our theoretical analysis clarifies some important aspects on the role of the tissue spatial structures in the cancer initiation processes.

Branching structure of genealogies in spatially growing populations and its implications for population genetics inference

Spatial models where growth is limited to the population edge have been instrumental to understanding the population dynamics and the clone size distribution in growing cellular populations, such as microbial colonies and avascular tumours. A complete characterization of the coalescence process generated by spatial growth is still lacking, limiting our ability to apply classic population genetics inference to spatially growing populations. Here, we start filling this gap by investigating the statistical properties of the cell lineages generated by the two dimensional Eden model, leveraging their physical analogy with directed polymers. Our analysis provides quantitative estimates for population measurements that can easily be assessed via sequencing, such as the average number of segregating sites and the clone size distribution of a subsample of the population. Our results not only reveal remarkable features of the genealogies generated during growth, but also highlight new properties that can be misinterpreted as signs of selection if non-spatial models are inappropriately applied.

Using an Unsupervised Clustering Model to Detect the Early Spread of SARS-CoV-2 Worldwide

Deciphering the population structure of SARS-CoV-2 is critical to inform public health management and reduce the risk of future dissemination. With the continuous accruing of SARS-CoV-2 genomes worldwide, discovering an effective way to group these genomes is critical for organizing the landscape of the population structure of the virus. Taking advantage of recently published state-of-the-art machine learning algorithms, we used an unsupervised deep learning clustering algorithm to group a total of 16,873 SARS-CoV-2 genomes. Using single nucleotide polymorphisms as input features, we identified six major subtypes of SARS-CoV-2. The proportions of the clusters across the continents revealed distinct geographical distributions. Comprehensive analysis indicated that both genetic factors and human migration factors shaped the specific geographical distribution of the population structure. This study provides a different approach using clustering methods to study the population structure of a never-seen-before and fast-growing species such as SARS-CoV-2. Moreover, clustering techniques can be used for further studies of local population structures of the proliferating virus.

Trade-off between reducing mutational accumulation and increasing commitment to differentiation determines tissue organization

Species-specific differences control cancer risk across orders of magnitude variation in body size and lifespan, e.g., by varying the copy numbers of tumor suppressor genes. It is unclear, however, how different tissues within an organism can control somatic evolution despite being subject to markedly different constraints, but sharing the same genome. Hierarchical differentiation, characteristic of self-renewing tissues, can restrain somatic evolution both by limiting divisional load, thereby reducing mutation accumulation, and by increasing cells’ commitment to differentiation, which can “wash out” mutants. Here, we explore the organization of hierarchical tissues that have evolved to limit their lifetime incidence of cancer. Estimating the likelihood of cancer in the presence of mutations that enhance self-proliferation, we demonstrate that a trade-off exists between mutation accumulation and the strength of washing out. Our results explain differences in the organization of widely different hierarchical tissues, such as colon and blood.

The observation that tissues that undergo more stem cell divisions are less prone to develop cancer presents a paradox as these tissues should have more opportunity to accumulate cancer-causing mutations. Here, the authors present a solution to the paradox by showing how hierarchical tissues can maintain low cancer incidence by balancing mutation accumulation and the cells’ commitment to differentiation.

DNA in extracellular vesicles: from evolution to its current application in health and disease

Extracellular vesicle (EV) secretion is a highly conserved evolutionary trait in all organisms in the three domains of life. The packaging and release of EVs appears to be a bulk-flow process which takes place mainly under extreme conditions. EVs participate in horizontal gene transfer, which supports the survival of prokaryotic and eukaryotic microbes. In higher eukaryotes, almost all cells secrete a heterogeneous population of EVs loaded with various biomolecules. EV secretion is typically higher in cancer microenvironments, promoting tumor progression and metastasis. EVs are now recognized as additional mediators of autocrine and paracrine communication in health and disease. In this context, proteins and RNAs have been studied the most, but extracellular vesicle DNA (EV-DNA) has started to gain in importance in the last few years. In this review, we summarize new findings related to the loading mechanism(s), localization, and post-shedding function of EV-DNA. We also discuss the feasibility of using EV-DNA as a biomarker when performing a liquid biopsy, at the same time emphasizing the lack of data from clinical trials in this regard. Finally, we outline the potential of EV-DNA uptake and its interaction with the host genome as a promising tool for understanding the mechanisms of cancer evolution.

Protecting DNA in membrane vesicles seems to be a conserved phenomenon for the horizontal genetic flux between prokaryotes and lower eukaryotes.

Capturing and analyzing this vesicular DNA enables quick and non-invasive monitoring of natural ecosystems.

Cancer-derived extracellular vesicles containing DNA open up novel directions in cell-to-cell communication and therefore disease monitoring.

Complex and fluctuating conditions of the tumor microenvironment, mimicking natural ecosystems, could favor EV-DNA release, mediating tumor multi-clonal evolution and providing survival benefits.

Population genetics in microchannels

Many microbial populations proliferate in small channels. In such environments, reproducing cells organize in parallel lanes. Reproducing cells shift these lanes, potentially expelling other cells from the channel. In this paper, we combine theory and experiments to understand how these dynamics affects the diversity of a microbial population. We theoretically predict that genetic diversity is quickly lost along lanes of cells. Our experiments confirm that a population of proliferating Escherichia coli in a microchannel organizes into lanes of genetically identical cells within a few generations. Our findings elucidate the effect of lane formation on populations evolution, with potential applications ranging from microbial ecology in soil to dynamics of epithelial tissues in higher organisms.

Spatial constraints, such as rigid barriers, affect the dynamics of cell populations, potentially altering the course of natural evolution. In this paper, we investigate the population genetics of Escherichia coli proliferating in microchannels with open ends. Our analysis is based on a population model, in which reproducing cells shift entire lanes of cells toward the open ends of the channel. The model predicts that diversity is lost very rapidly within lanes but at a much slower pace among lanes. As a consequence, two mixed, neutral E. coli strains competing in a microchannel must organize into an ordered regular stripe pattern in the course of a few generations. These predictions are in quantitative agreement with our experiments. We also demonstrate that random mutations appearing in the middle of the channel are much more likely to reach fixation than those occurring elsewhere. Our results illustrate fundamental mechanisms of microbial evolution in spatially confined space.

Constrained optimization of divisional load in hierarchically organized tissues during homeostasis

It has been hypothesized that the structure of tissues and the hierarchy of differentiation from stem cell to terminally differentiated cell play a significant role in reducing the incidence of cancer in that tissue. One specific mechanism by which this risk can be reduced is by minimizing the number of divisions—and hence the mutational risk—that cells accumulate as they divide to maintain tissue homeostasis. Here, we investigate a mathematical model of cell division in a hierarchical tissue, calculating and minimizing the divisional load while constraining parameters such that homeostasis is maintained. We show that the minimal divisional load is achieved by binary division trees with progenitor cells incapable of self-renewal. Contrary to the protection hypothesis, we find that an increased stem cell turnover can lead to lower divisional load. Furthermore, we find that the optimal tissue structure depends on the time horizon of the duration of homeostasis, with faster stem cell division favoured in short-lived organisms and more progenitor compartments favoured in longer-lived organisms.

Temporal order of mutations influences cancer initiation dynamics

Cancer is a set of genetic diseases that are driven by mutations. It was recently discovered that the temporal order of genetic mutations affects the cancer evolution and even the nature of the decease itself. The mechanistic origin of these observations, however, remain not well understood. Here we present a theoretical model for cancer initiation dynamics that allows us to quantify the impact of the temporal order of mutations. In our approach, the cancer initiation process is viewed as a set of stochastic transitions between discrete states defined by the different numbers of mutated cells. Using a first-passage analysis, probabilities and times before the cancer initiation are explicitly evaluated for two alternative sequences of two mutations. It is found that the probability of cancer initiation is determined only by the first mutation, while the dynamics depends on both mutations. In addition, it is shown that the acquisition of a mutation with higher fitness before mutation with lower fitness increases the probability of the tumor formation but delays the cancer initiation. Theoretical results are explained using effective free-energy landscapes.

Fast and strong amplifiers of natural selection

Selection and random drift determine the probability that novel mutations fixate in a population. Population structure is known to affect the dynamics of the evolutionary process. Amplifiers of selection are population structures that increase the fixation probability of beneficial mutants compared to well-mixed populations. Over the past 15 years, extensive research has produced remarkable structures called strong amplifiers which guarantee that every beneficial mutation fixates with high probability. But strong amplification has come at the cost of considerably delaying the fixation event, which can slow down the overall rate of evolution. However, the precise relationship between fixation probability and time has remained elusive. Here we characterize the slowdown effect of strong amplification. First, we prove that all strong amplifiers must delay the fixation event at least to some extent. Second, we construct strong amplifiers that delay the fixation event only marginally as compared to the well-mixed populations. Our results thus establish a tight relationship between fixation probability and time: Strong amplification always comes at a cost of a slowdown, but more than a marginal slowdown is not needed.

Spectral analysis of transient amplifiers for death-birth updating constructed from regular graphs

A central question of evolutionary dynamics on graphs is whether or not a mutation introduced in a population of residents survives and eventually even spreads to the whole population, or becomes extinct. The outcome naturally depends on the fitness of the mutant and the rules by which mutants and residents may propagate on the network, but arguably the most determining factor is the network structure. Some structured networks are transient amplifiers. They increase for a certain fitness range the fixation probability of beneficial mutations as compared to a well-mixed population. We study a perturbation method for identifying transient amplifiers for death–birth updating. The method involves calculating the coalescence times of random walks on graphs and finding the vertex with the largest remeeting time. If the graph is perturbed by removing an edge from this vertex, there is a certain likelihood that the resulting perturbed graph is a transient amplifier. We test all pairwise nonisomorphic regular graphs up to a certain order and thus cover the whole structural range expressible by these graphs. For cubic and quartic regular graphs we find a sufficiently large number of transient amplifiers. For these networks we carry out a spectral analysis and show that the graphs from which transient amplifiers can be constructed share certain structural properties. Identifying spectral and structural properties may promote finding and designing such networks.

Cooperation of partially-transformed clones: an invisible force behind the early stages of carcinogenesis

1Most tumours exhibit significant heterogeneity and are best described as communities of cellular populations competing for resources. Growing experimental evidence also suggests, however, that cooperation between cancer clones is important as well for the maintenance of tumour heterogeneity and tumour progression. However, a role for cell communication during the earliest steps in oncogenesis is not well characterised despite its vital importance in normal tissue and clinically manifest tumours. Here, we present a simple analytical model and stochastic lattice-based simulations to study how the interaction between the mutational process and cell-to-cell communication in three-dimensional tissue architecture might contribute to shape early oncogenesis. We show that non-cell-autonomous mechanisms of carcinogenesis could support and accelerate pre-cancerous clonal expansion through the cooperation of different, non- or partially- transformed mutants. We predict the existence of a 'cell-autonomous time-horizon', a time before which cooperation between cell-to-cell communication and DNA mutations might be one of the most fundamental forces shaping the early stages of oncogenesis. The understanding of this process could shed new light on the mechanisms leading to clinically manifest cancers.

Fixation probabilities in graph-structured populations under weak selection

A population's spatial structure affects the rate of genetic change and the outcome of natural selection. These effects can be modeled mathematically using the Birth-death process on graphs. Individuals occupy the vertices of a weighted graph, and reproduce into neighboring vertices based on fitness. A key quantity is the probability that a mutant type will sweep to fixation, as a function of the mutant's fitness. Graphs that increase the fixation probability of beneficial mutations, and decrease that of deleterious mutations, are said to amplify selection. However, fixation probabilities are difficult to compute for an arbitrary graph. Here we derive an expression for the fixation probability, of a weakly-selected mutation, in terms of the time for two lineages to coalesce. This expression enables weak-selection fixation probabilities to be computed, for an arbitrary weighted graph, in polynomial time. Applying this method, we explore the range of possible effects of graph structure on natural selection, genetic drift, and the balance between the two. Using exhaustive analysis of small graphs and a genetic search algorithm, we identify families of graphs with striking effects on fixation probability, and we analyze these families mathematically. Our work reveals the nuanced effects of graph structure on natural selection and neutral drift. In particular, we show how these notions depend critically on the process by which mutations arise.

The impact of random microenvironmental fluctuations on tumor control probability

The tumor control probability (TCP) is a metric used to calculate the probability of controlling or eradicating tumors through radiotherapy. Cancer cells vary in their response to radiation, and although many factors are involved, the tumor microenvironment is a crucial one that determines radiation efficacy. The tumor microenvironment plays a significant role in cancer initiation and propagation, as well as in treatment outcome. We have developed stochastic formulations to study the impact of arbitrary microenvironmental fluctuations on TCP and extinction probability (EP), which is defined as the probability of cancer cells removal in the absence of treatment. Since the derivation of analytical solutions are not possible for complicated cases, we employ a modified Gillespie algorithm to analyze TCP and EP, considering the random variations in cellular proliferation and death rates. Our results show that increasing the standard deviation in kinetic rates initially enhances the probability of tumor eradication. However, if the EP does not reach a probability of 1, the increase in the standard deviation subsequently has a negative impact on probability of cancer cells removal, decreasing the EP over time. The greatest effect on EP has been observed when both birth and death rates are being randomly modified and are anticorrelated. In addition, similar results are observed for TCP, where radiotherapy is included, indicating that increasing the standard deviation in kinetic rates at first enhances the probability of tumor eradication. But, it has a negative impact on treatment effectiveness if the TCP does not reach a probability of 1.

The evolution of metapopulation dynamics and the number of stem cells in intestinal crypts and other tissue structures in multicellular bodies

Carcinogenesis is a process of somatic evolution. Previous models of stem and transient amplifying cells in epithelial proliferating units like colonic crypts showed that intermediate numbers of stem cells in a crypt should optimally prevent progression to cancer. If a stem cell population is too small, it is easy for a mutator mutation to drift to fixation. If it is too large, it is easy for selection to drive cell fitness enhancing carcinogenic mutations to fixation. Here, we show that a multiscale microsimulation, that captures both within-crypt and between-crypt evolutionary dynamics, leads to a different conclusion. Epithelial tissues are metapopulations of crypts. We measured time to initiation of a neoplasm, implemented as inactivation of both alleles of a tumor suppressor gene. In our model, time to initiation is dependent on the spread of mutator clones in the crypts. The proportion of selectively beneficial and deleterious mutations in somatic cells is unknown and so was explored with a parameter. When the majority of non-neutral mutations are deleterious, the fitness of mutator clones tends to decline. When crypts are maintained by few stem cells, intercrypt competition tends to remove crypts with fixed mutators. When there are many stem cells within a crypt, there is virtually no crypt turnover, but mutator clones are suppressed by within-crypt competition. If the majority of non-neutral mutations are beneficial to the clone, then these results are reversed and intermediate-sized crypts provide the most protection against initiation. These results highlight the need to understand the dynamics of turnover and the mechanisms that control homeostasis, both at the level of stem cells within proliferative units and at the tissue level of competing proliferative units. Determining the distribution of fitness effects of somatic mutations will also be crucial to understanding the dynamics of tumor initiation and progression.

Emerging Frontiers in the Study of Molecular Evolution

A collection of the editors of Journal of Molecular Evolution have gotten together to pose a set of key challenges and future directions for the field of molecular evolution. Topics include challenges and new directions in prebiotic chemistry and the RNA world, reconstruction of early cellular genomes and proteins, macromolecular and functional evolution, evolutionary cell biology, genome evolution, molecular evolutionary ecology, viral phylodynamics, theoretical population genomics, somatic cell molecular evolution, and directed evolution. While our effort is not meant to be exhaustive, it reflects research questions and problems in the field of molecular evolution that are exciting to our editors.

Signalling architectures can prevent cancer evolution

Cooperation between cells in multicellular organisms is preserved by an active regulation of growth through the control of cell division. Molecular signals used by cells for tissue growth are usually present during developmental stages, angiogenesis, wound healing and other processes. In this context, the use of molecular signals triggering cell division is a puzzle, because any molecule inducing and aiding growth can be exploited by a cancer cell, disrupting cellular cooperation. A significant difference is that normal cells in a multicellular organism have evolved in competition between high-level organisms to be altruistic, being able to send signals even if it is to their detriment. Conversely, cancer cells evolve their abuse over the cancer’s lifespan by out-competing their neighbours. A successful mutation leading to cancer must evolve to be adaptive, enabling a cancer cell to send a signal that results in higher chances to be selected. Using a mathematical model of such molecular signalling mechanism, this paper argues that a signal mechanism would be effective against abuse by cancer if it affects the cell that generates the signal as well as neighbouring cells that would receive a benefit without any cost, resulting in a selective disadvantage for a cancer signalling cell. We find that such molecular signalling mechanisms normally operate in cells as exemplified by growth factors. In scenarios of global and local competition between cells, we calculate how this process affects the fixation probability of a mutant cell generating such a signal, and find that this process can play a key role in limiting the emergence of cancer.

A compartment size-dependent selective threshold limits mutation accumulation in hierarchical tissues

Renewed tissues of multicellular organism accumulate mutations that lead to aging and cancer. To mitigate these effects, self-renewing tissues produce cells along differentiation hierarchies, which have been shown to suppress somatic evolution both by limiting the number of cell divisions, and thus reducing mutational load, and by differentiation “washing out” mutations. Our analytical results reveal the existence of a third mechanism: a compartment size-dependent threshold in proliferative advantage, below which mutations cannot persist, but are rapidly expelled from the tissue by differentiation. In sufficiently small compartments, the resulting selective barrier can greatly slow down somatic evolution and reduce the risk of cancer by preventing the accumulation of mutations even if even they confer substantial proliferative advantage.

Cancer is a genetic disease fueled by somatic evolution. Hierarchical tissue organization can slow somatic evolution by two qualitatively different mechanisms: by cell differentiation along the hierarchy “washing out” harmful mutations and by limiting the number of cell divisions required to maintain a tissue. Here we explore the effects of compartment size on somatic evolution in hierarchical tissues by considering cell number regulation that acts on cell division rates such that the number of cells in the tissue has the tendency to return to its desired homeostatic value. Introducing mutants with a proliferative advantage, we demonstrate the existence of a third fundamental mechanism by which hierarchically organized tissues are able to slow down somatic evolution. We show that tissue size regulation leads to the emergence of a threshold proliferative advantage, below which mutants cannot persist. We find that the most significant determinant of the threshold selective advantage is compartment size, with the threshold being higher the smaller the compartment. Our results demonstrate that, in sufficiently small compartments, even mutations that confer substantial proliferative advantage cannot persist, but are expelled from the tissue by differentiation along the hierarchy. The resulting selective barrier can significantly slow down somatic evolution and reduce the risk of cancer by limiting the accumulation of mutations that increase the proliferation of cells.

Revisiting the tumorigenesis timeline with a data-driven generative model

Recently, investigators have shown that only a few driver gene mutational events appear to be needed for cancer to occur. However, the reason that some mutational events precede others in the same cancer and the explanation for tissue-specific differences in this timing, remain mysterious. We here combine mathematical modeling with epidemiologic studies and sequencing data to address these questions. We suggest that the first driver event in cancers generally occurred at early ages and provide estimates for the fitness of different types of drivers during tumor evolution, showing how they vary with the tissue of origin.

Cancer is driven by the sequential accumulation of genetic and epigenetic changes in oncogenes and tumor suppressor genes. The timing of these events is not well understood. Moreover, it is currently unknown why the same driver gene change appears as an early event in some cancer types and as a later event, or not at all, in others. These questions have become even more topical with the recent progress brought by genome-wide sequencing studies of cancer. Focusing on mutational events, we provide a mathematical model of the full process of tumor evolution that includes different types of fitness advantages for driver genes and carrying-capacity considerations. The model is able to recapitulate a substantial proportion of the observed cancer incidence in several cancer types (colorectal, pancreatic, and leukemia) and inherited conditions (Lynch and familial adenomatous polyposis), by changing only 2 tissue-specific parameters: the number of stem cells in a tissue and its cell division frequency. The model sheds light on the evolutionary dynamics of cancer by suggesting a generalized early onset of tumorigenesis followed by slow mutational waves, in contrast to previous conclusions. Formulas and estimates are provided for the fitness increases induced by driver mutations, often much larger than previously described, and highly tissue dependent. Our results suggest a mechanistic explanation for why the selective fitness advantage introduced by specific driver genes is tissue dependent.

Change in mutation frequency at a TP53 hotspot during culture of ENU-mutagenised human lymphoblastoid cells

Mutations in oncogenes or tumour suppressor genes cause increases in cell growth capacity. In some cases, fully malignant cancer cells develop after additional mutations occur in initially mutated cells. In such instances, the risk of cancer would increase in response to growth of these initially mutated cells. To ascertain whether such a situation might occur in cultured cells, three independent cultures of human lymphoblastoid GM00130 cells were treated with N-ethyl-N-nitrosourea to induce mutations, and the cells were maintained for 12 weeks. Mutant frequencies and spectra of the cells at the MspI and HaeIII restriction sites located at codons 247-250 of the TP53 gene were examined. Mutant frequencies at both sites in the gene exhibited a declining trend during cell culture and reached background levels after 12 weeks; this was also supported by mutation spectra findings. These results indicate that the mutations detected under our assay conditions are disadvantageous to cell growth.

Replicative cellular age distributions in compartmentalized tissues

The cellular age distribution of hierarchically organized tissues can reveal important insights into the dynamics of cell differentiation and self-renewal and associated cancer risks. Here, we examine the effect of progenitor compartments with varying differentiation and self-renewal capacities on the resulting observable distributions of replicative cellular ages. We find that strongly amplifying progenitor compartments, i.e. compartments with high self-renewal capacities, substantially broaden the age distributions which become skewed towards younger cells with a long tail of few old cells. For several of these strongly amplifying compartments, the age distribution becomes virtually independent of the influx from the stem cell compartment. By contrast, if tissues are organized into many downstream compartments with low self-renewal capacity, the shape of the replicative cell distribution in more differentiated compartments is dominated by stem cell dynamics with little added variation. In the limiting case of a strict binary differentiation tree without self-renewal, the shape of the output distribution becomes indistinguishable from that of the input distribution. Our results suggest that a comparison of cellular age distributions between healthy and cancerous tissues may inform about dynamical changes within the hierarchical tissue structure, i.e. an acquired increased self-renewal capacity in certain tumours. Furthermore, we compare our theoretical results to telomere length distributions in granulocyte populations of 10 healthy individuals across different ages, highlighting that our theoretical expectations agree with experimental observations.

The invasion of de-differentiating cancer cells into hierarchical tissues

Many fast renewing tissues are characterized by a hierarchical cellular architecture, with tissue specific stem cells at the root of the cellular hierarchy, differentiating into a whole range of specialized cells. There is increasing evidence that tumors are structured in a very similar way, mirroring the hierarchical structure of the host tissue. In some tissues, differentiated cells can also revert to the stem cell phenotype, which increases the risk that mutant cells lead to long lasting clones in the tissue. However, it is unclear under which circumstances de-differentiating cells will invade a tissue. To address this, we developed mathematical models to investigate how de-differentiation is selected as an adaptive mechanism in the context of cellular hierarchies. We derive thresholds for which de-differentiation is expected to emerge, and it is shown that the selection of de-differentiation is a result of the combination of the properties of cellular hierarchy and de-differentiation patterns. Our results suggest that de-differentiation is most likely to be favored provided stem cells having the largest effective self-renewal rate. Moreover, jumpwise de-differentiation provides a wider range of favorable conditions than stepwise de-differentiation. Finally, the effect of de-differentiation on the redistribution of self-renewal and differentiation probabilities also greatly influences the selection for de-differentiation.

How can a tissue such as the blood system or the skin, which constantly produces a huge number of cells, avoids that errors accumulate in the cells over time? Such tissues are typically organized in cellular hierarchies, which induce a directional relation between different stages of cellular differentiation, minimizing the risk of retention of mutations. However, recent evidence also shows that some differentiated cells can de-differentiate into the stem cell phenotype. Why does de-differentiation arise in some tumors, but not in others? We developed a mathematical model to study the growth competition between de-differentiating mutant cell populations and non de-differentiating resident cell population. Our results suggest that the invasion of de-differentiation is jointly influenced by the cellular hierarchy (e.g. number of cell compartments, inherent cell division pattern) and the de-differentiation pattern, i.e. how exactly cells acquire their stem-cell like properties.

Exploring and mapping the universe of evolutionary graphs identifies structural properties affecting fixation probability and time

Population structure can be modeled by evolutionary graphs, which can have a substantial influence on the fate of mutants. Individuals are located on the nodes of these graphs, competing to take over the graph via the links. Applications for this framework range from the ecology of river systems and cancer initiation in colonic crypts to biotechnological search for optimal mutations. In all these applications, both the probability of fixation and the associated time are of interest. We study this problem for all undirected and unweighted graphs up to a certain size. We devise a genetic algorithm to find graphs with high or low fixation probability and short or long fixation time and study their structure searching for common themes. Our work unravels structural properties that maximize or minimize fixation probability and time, which allows us to contribute to a first map of the universe of evolutionary graphs.

Modeling Cell Dynamics in Colon and Intestinal Crypts: The Significance of Central Stem Cells in Tumorigenesis

Colon and intestinal crypts have been widely chosen to study cell dynamics because of their fairly simple structures. In the colon and intestinal crypts, stem cells (SCs) are located at very bottom of the crypt, fully differentiated cells (FDs) are located in the top of the crypt, and transit-amplifying cells (TAs) are in the middle of the crypt between FDs and SCs. Recently, it has been discovered that there are two types of stem cells in the intestinal crypts: central stem cells (CeSCs) and border stem cells. To investigate dynamics of mutants in colon and intestinal crypts, we develop a four-compartmental stochastic model, which includes two SC compartments, and TAs and FDs compartments. We calculate the probability of the progeny of marked or mutant cells located at each of these compartments taking over the entire crypt or being washed out from the crypt. We found that the progeny of CeSCs will take over the entire crypt with a probability close to one. Interestingly, the progeny of advantageous mutant TAs and FDs will be washed out faster than disadvantageous mutants. Saliently, the model predicts that the time that the progeny of wild-type central stem cells will take over the mouse intestinal crypt is around 60 days, which is in perfect agreement with an experimental observation.

Computational methods for birth-death processes

Many important stochastic counting models can be written as general birth-death processes (BDPs). BDPs are continuous-time Markov chains on the non-negative integers in which only jumps to adjacent states are allowed. BDPs can be used to easily parameterize a rich variety of probability distributions on the non-negative integers, and straightforward conditions guarantee that these distributions are proper. BDPs also provide a mechanistic interpretation - birth and death of actual particles or organisms - that has proven useful in evolution, ecology, physics, and chemistry. Although the theoretical properties of general BDPs are well understood, traditionally statistical work on BDPs has been limited to the simple linear (Kendall) process. Aside from a few simple cases, it remains impossible to find analytic expressions for the likelihood of a discretely-observed BDP, and computational difficulties have hindered development of tools for statistical inference. But the gap between BDP theory and practical methods for estimation has narrowed in recent years. There are now robust methods for evaluating likelihoods for realizations of BDPs: finite-time transition, first passage, equilibrium probabilities, and distributions of summary statistics that arise commonly in applications. Recent work has also exploited the connection between continuously- and discretely-observed BDPs to derive EM algorithms for maximum likelihood estimation. Likelihood-based inference for previously intractable BDPs is much easier than previously thought and regression approaches analogous to Poisson regression are straightforward to derive. In this review, we outline the basic mathematical theory for BDPs and demonstrate new tools for statistical inference using data from BDPs.

A Simple 3-Parameter Model for Cancer Incidences

We propose a simple 3-parameter model that provides very good fits for incidence curves of 18 common solid cancers even when variations due to different locations, races, or periods are taken into account. From a data perspective, we use model selection (Akaike information criterion) to show that this model, which is based on the Weibull distribution, outperforms other simple models like the Gamma distribution. From a modeling perspective, the Weibull distribution can be justified as modeling the accumulation of driver events, which establishes a link to stem cell division based cancer development models and a connection to a recursion formula for intrinsic cancer risk published by Wu et al. For the recursion formula a closed form solution is given, which will help to simplify future analyses. Additionally, we perform a sensitivity analysis for the parameters, showing that two of the three parameters can vary over several orders of magnitude. However, the shape parameter of the Weibull distribution, which corresponds to the number of driver mutations required for cancer onset, can be robustly estimated from epidemiological data.

Reconstruction of kidney renal clear cell carcinoma evolution across pathological stages

Although numerous studies on kidney renal clear cell carcinoma (KIRC) were carried out, the dynamic process of tumor formation was not clear yet. Inadequate attention was paid on the evolutionary paths among somatic mutations and their clinical implications. As the tumor initiation and evolution of KIRC were primarily associated with SNVs, we reconstructed an evolutionary process of KIRC using cross-sectional SNVs in different pathological stages. KIRC driver genes appeared early in the evolutionary tree, and the genes with moderate mutation frequency showed a pattern of stage-by-stage expansion. Although the individual gene mutations were not necessarily associated with survival outcome, the evolutionary paths such as VHL-PBRM1 and FMN2-PCLO could indicate stage-specific prognosis. Our results suggested that, besides mutation frequency, the evolutionary relationship among the mutated genes could facilitate to identify novel drivers and biomarkers for clinical utility.

Optimizing homeostatic cell renewal in hierarchical tissues

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

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

Phenotypic heterogeneity in modeling cancer evolution

The unwelcome evolution of malignancy during cancer progression emerges through a selection process in a complex heterogeneous population structure. In the present work, we investigate evolutionary dynamics in a phenotypically heterogeneous population of stem cells (SCs) and their associated progenitors. The fate of a malignant mutation is determined not only by overall stem cell and non-stem cell growth rates but also differentiation and dedifferentiation rates. We investigate the effect of such a complex population structure on the evolution of malignant mutations. We derive exactly calculated results for the fixation probability of a mutant arising in each of the subpopulations. The exactly calculated results are in almost perfect agreement with the numerical simulations. Moreover, a condition for evolutionary advantage of a mutant cell versus the wild type population is given in the present study. We also show that microenvironment-induced plasticity in invading mutants leads to more aggressive mutants with higher fixation probability. Our model predicts that decreasing polarity between stem and non-stem cells’ turnover would raise the survivability of non-plastic mutants; while it would suppress the development of malignancy for plastic mutants. The derived results are novel and general with potential applications in nature; we discuss our model in the context of colorectal/intestinal cancer (at the epithelium). However, the model clearly needs to be validated through appropriate experimental data. This novel mathematical framework can be applied more generally to a variety of problems concerning selection in heterogeneous populations, in other contexts such as population genetics, and ecology.

Effects of motion in structured populations

In evolutionary processes, population structure has a substantial effect on natural selection. Here, we analyse how motion of individuals affects constant selection in structured populations. Motion is relevant because it leads to changes in the distribution of types as mutations march towards fixation or extinction. We describe motion as the swapping of individuals on graphs, and more generally as the shuffling of individuals between reproductive updates. Beginning with a one-dimensional graph, the cycle, we prove that motion suppresses natural selection for death-birth (DB) updating or for any process that combines birth-death (BD) and DB updating. If the rule is purely BD updating, no change in fixation probability appears in the presence of motion. We further investigate how motion affects evolution on the square lattice and weighted graphs. In the case of weighted graphs, we find that motion can be either an amplifier or a suppressor of natural selection. In some cases, whether it is one or the other can be a function of the relative reproductive rate, indicating that motion is a subtle and complex attribute of evolving populations. As a first step towards understanding less restricted types of motion in evolutionary graph theory, we consider a similar rule on dynamic graphs induced by a spatial flow and find qualitatively similar results, indicating that continuous motion also suppresses natural selection.

The role of backward cell migration in two-hit mutants' production in the stem cell niche

It has been discovered that there are two stem cell groups in the intestinal crypts: central stem cells (CeSCs), which are at the very bottom of the crypt, and border stem cells (BSCs), which are located between CeSCs and transit amplifying cells (TAs). Moreover, backward cell migration from BSCs to CeSCs has been observed. Recently, a bi-compartmental stochastic model, which includes CeSCs and BSCs, has been developed to investigate the probability of two-hit mutant production in the stem cell niche. In this project, we improve this stochastic model by adding the probability of backward cell migration to the model. The model suggests that the probability of two-hit mutant production increases when the frequency of backward cell migration increases. Furthermore, a small non-zero probability of backward cell migration leads to the largest range of optimal values for the frequency of symmetric divisions and the portion of divisions at each stem cell compartment in terms of delaying 2-hit mutant production. Moreover, the probability of two-hit mutant production is more sensitive to the probability of symmetric divisions than to the rate of backward cell migrations. The highest probability of two-hit mutant production corresponds to the case when all stem cell’s divisions are asymmetric.

Evolutionary dynamics on any population structure

Evolution occurs in populations of reproducing individuals. The structure of a population can affect which traits evolve. Understanding evolutionary game dynamics in structured populations remains difficult. Mathematical results are known for special structures in which all individuals have the same number of neighbours. The general case, in which the number of neighbours can vary, has remained open. For arbitrary selection intensity, the problem is in a computational complexity class that suggests there is no efficient algorithm. Whether a simple solution for weak selection exists has remained unanswered. Here we provide a solution for weak selection that applies to any graph or network. Our method relies on calculating the coalescence times of random walks. We evaluate large numbers of diverse population structures for their propensity to favour cooperation. We study how small changes in population structure-graph surgery-affect evolutionary outcomes. We find that cooperation flourishes most in societies that are based on strong pairwise ties.