Spatial stochastic models for cancer initiation and progression

Graph-structured populations elucidate the role of deleterious mutations in long-term evolution

Birth-death models are used to understand the interplay of genetic drift and natural selection. While well-mixed populations remain unaffected by the order of birth and death and where selection acts, evolutionary outcomes in spatially structured populations are affected by these choices. We show that the choice of individual moving to vacant sites-parent or offspring-controls the initial mutant placement on a graph and hence alters its fixation probability. Moving parent individuals introduces, to our knowledge, previously unexplored update rules and fixation categories for heterogeneous graphs. We identify a class of graphs, amplifiers of fixation, where fixation probability is larger than in well-mixed populations, regardless of the mutant fitness. Under death-Birth parent moving, the star graph is an amplifier of fixation, with a non-zero fixation probability for deleterious mutants, in contrast to very large well-mixed populations. Most Erdős-Rényi graphs of size 8 are amplifiers of fixation under death-Birth parent moving, but suppressors of fixation under Birth-death offspring moving. Surprisingly, amplifiers of fixation attain lower fitness in long-term evolution, despite favouring beneficial mutants, while suppressors of fixation attain higher fitness. These counterintuitive findings are explained by the fate of deleterious mutations and highlight the crucial role of deleterious mutants for adaptive evolution.

High-density sampling reveals volume growth in human tumours

In growing cell populations such as tumours, mutations can serve as markers that allow tracking the past evolution from current samples. The genomic analyses of bulk samples and samples from multiple regions have shed light on the evolutionary forces acting on tumours. However, little is known empirically on the spatio-temporal dynamics of tumour evolution. Here, we leverage published data from resected hepatocellular carcinomas, each with several hundred samples taken in two and three dimensions. Using spatial metrics of evolution, we find that tumour cells grow predominantly uniformly within the tumour volume instead of at the surface. We determine how mutations and cells are dispersed throughout the tumour and how cell death contributes to the overall tumour growth. Our methods shed light on the early evolution of tumours in vivo and can be applied to high-resolution data in the emerging field of spatial biology.

Impact of Resistance on Therapeutic Design: A Moran Model of Cancer Growth

Resistance of cancers to treatments, such as chemotherapy, largely arise due to cell mutations. These mutations allow cells to resist apoptosis and inevitably lead to recurrence and often progression to more aggressive cancer forms. Sustained-low dose therapies are being considered as an alternative over maximum tolerated dose treatments, whereby a smaller drug dosage is given over a longer period of time. However, understanding the impact that the presence of treatment-resistant clones may have on these new treatment modalities is crucial to validating them as a therapeutic avenue. In this study, a Moran process is used to capture stochastic mutations arising in cancer cells, inferring treatment resistance. The model is used to predict the probability of cancer recurrence given varying treatment modalities. The simulations predict that sustained-low dose therapies would be virtually ineffective for a cancer with a non-negligible probability of developing a sub-clone with resistance tendencies. Furthermore, calibrating the model to in vivo measurements for breast cancer treatment with Herceptin, the model suggests that standard treatment regimens are ineffective in this mouse model. Using a simple Moran model, it is possible to explore the likelihood of treatment success given a non-negligible probability of treatment resistant mutations and suggest more robust therapeutic schedules.

Amplifiers of selection for the Moran process with both Birth-death and death-Birth updating

Populations evolve by accumulating advantageous mutations. Every population has some spatial structure that can be modeled by an underlying network. The network then influences the probability that new advantageous mutations fixate. Amplifiers of selection are networks that increase the fixation probability of advantageous mutants, as compared to the unstructured fully-connected network. Whether or not a network is an amplifier depends on the choice of the random process that governs the evolutionary dynamics. Two popular choices are Moran process with Birth-death updating and Moran process with death-Birth updating. Interestingly, while some networks are amplifiers under Birth-death updating and other networks are amplifiers under death-Birth updating, so far no spatial structures have been found that function as an amplifier under both types of updating simultaneously. In this work, we identify networks that act as amplifiers of selection under both versions of the Moran process. The amplifiers are robust, modular, and increase fixation probability for any mutant fitness advantage in a range r ∈ (1, 1.2). To complement this positive result, we also prove that for certain quantities closely related to fixation probability, it is impossible to improve them simultaneously for both versions of the Moran process. Together, our results highlight how the two versions of the Moran process differ and what they have in common.

Evolutionary dynamics of mutants that modify population structure

Natural selection is usually studied between mutants that differ in reproductive rate, but are subject to the same population structure. Here we explore how natural selection acts on mutants that have the same reproductive rate, but different population structures. In our framework, population structure is given by a graph that specifies where offspring can disperse. The invading mutant disperses offspring on a different graph than the resident wild-type. We find that more densely connected dispersal graphs tend to increase the invader's fixation probability, but the exact relationship between structure and fixation probability is subtle. We present three main results. First, we prove that if both invader and resident are on complete dispersal graphs, then removing a single edge in the invader's dispersal graph reduces its fixation probability. Second, we show that for certain island models higher invader's connectivity increases its fixation probability, but the magnitude of the effect depends on the exact layout of the connections. Third, we show that for lattices the effect of different connectivity is comparable to that of different fitness: for large population size, the invader's fixation probability is either constant or exponentially small, depending on whether it is more or less connected than the resident.

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.

Counterintuitive properties of evolutionary measures: A stochastic process study in cyclic population structures with periodic environments

Local environmental interactions are a major factor in determining the success of a new mutant in structured populations. Spatial variations in the concentration of genotype-specific resources change the fitness of competing strategies locally and thus can drastically change the outcome of evolutionary processes in unintuitive ways. The question is how such local environmental variations in network population structures change the condition for selection and fixation probability of an advantageous (or deleterious) mutant. We consider linear graph structures and focus on the case where resources have a spatial periodic pattern. This is the simplest model with two parameters, length scale and fitness scales, representing heterogeneity. We calculate fixation probability and fixation times for a constant population birth-death process as fitness heterogeneity and period vary. Fixation probability is affected by not only the level of fitness heterogeneity but also spatial scale of resources variations set by period of distribution T. We identify conditions for which a previously a deleterious mutant (in a uniform environment) becomes beneficial as fitness heterogeneity is increased. We observe cases where the fixation probability of both mutant and resident types are more than their neutral value, 1/N, simultaneously. This coincides with exponential increase in time to fixation which points to potential coexistence of resident and mutant types. Finally, we discuss the effect of the 'fitness shift' where the fitness function of two types has a phase difference. We observe significant increases (or decreases) in the fixation probability of the mutant as a result of such phase shift.

A spatial mutation model with increasing mutation rates

We consider a spatial model of cancer in which cells are points on the d-dimensional torus $\mathcal{T}=[0,L]^d$ , and each cell with $k-1$ mutations acquires a kth mutation at rate $\mu_k$ . We assume that the mutation rates $\mu_k$ are increasing, and we find the asymptotic waiting time for the first cell to acquire k mutations as the torus volume tends to infinity. This paper generalizes results on waiting for $k\geq 3$ mutations in Foo et al. (2020), which considered the case in which all of the mutation rates $\mu_k$ are the same. In addition, we find the limiting distribution of the spatial distances between mutations for certain values of the mutation rates.

Testing for Phylogenetic Signal in Single-Cell RNA-Seq Data

Phylogenetic methods are emerging as a useful tool to understand cancer evolutionary dynamics, including tumor structure, heterogeneity, and progression. Most currently used approaches utilize either bulk whole genome sequencing or single-cell DNA sequencing and are based on calling copy number alterations and single nucleotide variants (SNVs). Single-cell RNA sequencing (scRNA-seq) is commonly applied to explore differential gene expression of cancer cells throughout tumor progression. The method exacerbates the single-cell sequencing problem of low yield per cell with uneven expression levels. This accounts for low and uneven sequencing coverage and makes SNV detection and phylogenetic analysis challenging. In this article, we demonstrate for the first time that scRNA-seq data contain sufficient evolutionary signal and can also be utilized in phylogenetic analyses. We explore and compare results of such analyses based on both expression levels and SNVs called from scRNA-seq data. Both techniques are shown to be useful for reconstructing phylogenetic relationships between cells, reflecting the clonal composition of a tumor. Both standardized expression values and SNVs appear to be equally capable of reconstructing a similar pattern of phylogenetic relationship. This pattern is stable even when phylogenetic uncertainty is taken in account. Our results open up a new direction of somatic phylogenetics based on scRNA-seq data. Further research is required to refine and improve these approaches to capture the full picture of somatic evolutionary dynamics in cancer.

Laws of Spatially Structured Population Dynamics on a Lattice

We consider spatial population dynamics on a lattice, following a type of a contact (birth–death) stochastic process. We show that simple mathematical approximations for the density of cells can be obtained in a variety of scenarios. In the case of a homogeneous cell population, we derive the cellular density for a two-dimensional (2D) spatial lattice with an arbitrary number of neighbors, including the von Neumann, Moore, and hexagonal lattice.are not allowed in the abstract, please move them to the main text, please remember to check the order of references in the full text after revision. We then turn our attention to evolutionary dynamics, where mutant cells of different properties can be generated. For disadvantageous mutants, we derive an approximation for the equilibrium density representing the selection–mutation balance. For neutral and advantageous mutants, we show that simple scaling (power) laws for the numbers of mutants in expanding populations hold in 2D and 3D, under both flat (planar) and range population expansion. These models have relevance for studies in ecology and evolutionary biology, as well as biomedical applications including the dynamics of drug-resistant mutants in cancer and bacterial biofilms.

OPTIMAL CONTROL OF GENETIC DIVERSITY IN THE MORAN MODEL WITH POPULATION GROWTH

In the Moran model of drift and selection of a mutant allele with population growth, instead of examining the consequences of pre-specified selection and population growth, the coexistence of the wild allele and the mutant allele becomes the maximization of the expected sojourn time in a given set. The process is controlled by the additional mortality of the mutant and by population growth. This makes it possible to retroactively assign fitness values as functions of the constraints, thus guiding a conservation policy or the achievement of a wishful proportion of mutants. This also gives the optimal conditions that have allowed an observed coexistence.

The network structure affects the fixation probability when it couples to the birth-death dynamics in finite population

The study of evolutionary dynamics on graphs is an interesting topic for researchers in various fields of science and mathematics. In systems with finite population, different model dynamics are distinguished by their effects on two important quantities: fixation probability and fixation time. The isothermal theorem declares that the fixation probability is the same for a wide range of graphs and it only depends on the population size. This has also been proved for more complex graphs that are called complex networks. In this work, we propose a model that couples the population dynamics to the network structure and show that in this case, the isothermal theorem is being violated. In our model the death rate of a mutant depends on its number of neighbors, and neutral drift holds only in the average. We investigate the fixation probability behavior in terms of the complexity parameter, such as the scale-free exponent for the scale-free network and the rewiring probability for the small-world network.

In this work, we examine an evolutionary model that considers the effect of competition between the mutated individuals for acquiring more resources. This competition has an effect on the death rate of mutants. The model purposes that the death rate of each mutant depends on the number of its neighbors, while the average death rate in the population is equal to one. The birth rate for all individuals is assumed to be the same and equal to one. This situation is called here the ‘neutral drift in average’. We study the dynamics of the model on complex networks to take into account the non-uniformity of the environment. The results show the fixation probability differs from the Moran model. For the construction of this model, we were biologically motivated by the avascular tumour, which consists of a population of normal and cancer cells. The cancer cells likely need more oxygen than normal cells. There is a competition between cells for consuming oxygen, and cancer cells are far more sensitive to the amount of oxygen in the environment than normal cells. This means the death rate of a cancer cell grows by increasing the number of its neighbors.

Emergent spatiotemporal population dynamics with cell-length control of synthetic microbial consortia

The increased complexity of synthetic microbial biocircuits highlights the need for distributed cell functionality due to concomitant increases in metabolic and regulatory burdens imposed on single-strain topologies. Distributed systems, however, introduce additional challenges since consortium composition and spatiotemporal dynamics of constituent strains must be robustly controlled to achieve desired circuit behaviors. Here, we address these challenges with a modeling-based investigation of emergent spatiotemporal population dynamics using cell-length control in monolayer, two-strain bacterial consortia. We demonstrate that with dynamic control of a strain's division length, nematic cell alignment in close-packed monolayers can be destabilized. We find that this destabilization confers an emergent, competitive advantage to smaller-length strains-but by mechanisms that differ depending on the spatial patterns of the population. We used complementary modeling approaches to elucidate underlying mechanisms: an agent-based model to simulate detailed mechanical and signaling interactions between the competing strains, and a reductive, stochastic lattice model to represent cell-cell interactions with a single rotational parameter. Our modeling suggests that spatial strain-fraction oscillations can be generated when cell-length control is coupled to quorum-sensing signaling in negative feedback topologies. Our research employs novel methods of population control and points the way to programming strain fraction dynamics in consortial synthetic biology.

Geometrically regulating evolutionary dynamics in biofilms

The theoretical understanding of evolutionary dynamics in spatially structured populations often relies on nonspatial models. Biofilms are among such populations where a more accurate understanding is of theoretical interest and can reveal new solutions to existing challenges. Here, we studied how the geometry of the environment affects the evolutionary dynamics of expanding populations, using the Eden model. Our results show that fluctuations of subpopulations during range expansion in two- and three-dimensional environments are not Brownian. Furthermore, we found that the substrate's geometry interferes with the evolutionary dynamics of populations that grow upon it. Inspired by these findings, we propose a periodically wedged pattern on surfaces prone to develop biofilms. On such patterned surfaces, natural selection becomes less effective and beneficial mutants would have a harder time establishing. Additionally, this modification accelerates genetic drift and leads to less diverse biofilms. Both interventions are highly desired for biofilms.

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.

Engineering in Medicine To Address the Challenge of Cancer Drug Resistance: From Micro- and Nanotechnologies to Computational and Mathematical Modeling

Drug resistance has profoundly limited the success of cancer treatment, driving relapse, metastasis, and mortality. Nearly all anticancer drugs and even novel immunotherapies, which recalibrate the immune system for tumor recognition and destruction, have succumbed to resistance development. Engineers have emerged across mechanical, physical, chemical, mathematical, and biological disciplines to address the challenge of drug resistance using a combination of interdisciplinary tools and skill sets. This review explores the developing, complex, and under-recognized role of engineering in medicine to address the multitude of challenges in cancer drug resistance. Looking through the "lens" of intrinsic, extrinsic, and drug-induced resistance (also referred to as "tolerance"), we will discuss three specific areas where active innovation is driving novel treatment paradigms: (1) nanotechnology, which has revolutionized drug delivery in desmoplastic tissues, harnessing physiochemical characteristics to destroy tumors through photothermal therapy and rationally designed nanostructures to circumvent cancer immunotherapy failures, (2) bioengineered tumor models, which have benefitted from microfluidics and mechanical engineering, creating a paradigm shift in physiologically relevant environments to predict clinical refractoriness and enabling platforms for screening drug combinations to thwart resistance at the individual patient level, and (3) computational and mathematical modeling, which blends in silico simulations with molecular and evolutionary principles to map mutational patterns and model interactions between cells that promote resistance. On the basis that engineering in medicine has resulted in discoveries in resistance biology and successfully translated to clinical strategies that improve outcomes, we suggest the proliferation of multidisciplinary science that embraces engineering.

Beyond Deterministic Models in Drug Discovery and Development

The model-informed drug discovery and development paradigm is now well established among the pharmaceutical industry and regulatory agencies. This success has been mainly due to the ability of pharmacometrics to bring together different modeling strategies, such as population pharmacokinetics/pharmacodynamics (PK/PD) and systems biology/pharmacology. However, there are promising quantitative approaches that are still seldom used by pharmacometricians and that deserve consideration. One such case is the stochastic modeling approach, which can be important when modeling small populations because random events can have a huge impact on these systems. In this review, we aim to raise awareness of stochastic models and how to combine them with existing modeling techniques, with the ultimate goal of making future drug-disease models more versatile and realistic.

Mutant Evolution in Spatially Structured and Fragmented Expanding Populations

Mutant evolution in spatially structured systems is important for a range of biological systems, but aspects of it still require further elucidation. Adding to previous work, we provide a simple derivation of growth laws that characterize the number of mutants of different relative fitness in expanding populations in spatial models of different dimensionalities. These laws are universal and independent of "microscopic" modeling details. We further study the accumulation of mutants and find that, with advantageous and neutral mutants, more of them are present in spatially structured, compared to well-mixed colonies of the same size. The behavior of disadvantageous mutants is subtle: if they are disadvantageous through a reduction in division rates, the result is the same, and it is the opposite if the disadvantage is due to a death rate increase. Finally, we show that in all cases, the same results are observed in fragmented, nonspatial patch models. This suggests that the patterns observed are the consequence of population fragmentation, and not spatial restrictions per se We provide an intuitive explanation for the complex dependence of disadvantageous mutant evolution on spatial restriction, which relies on desynchronized dynamics in different locations/patches, and plays out differently depending on whether the disadvantage is due to a lower division rate or a higher death rate. Implications for specific biological systems, such as the evolution of drug-resistant cell mutants in cancer or bacterial biofilms, are discussed.

Limits on amplifiers of natural selection under death-Birth updating

The fixation probability of a single mutant invading a population of residents is among the most widely-studied quantities in evolutionary dynamics. Amplifiers of natural selection are population structures that increase the fixation probability of advantageous mutants, compared to well-mixed populations. Extensive studies have shown that many amplifiers exist for the Birth-death Moran process, some of them substantially increasing the fixation probability or even guaranteeing fixation in the limit of large population size. On the other hand, no amplifiers are known for the death-Birth Moran process, and computer-assisted exhaustive searches have failed to discover amplification. In this work we resolve this disparity, by showing that any amplification under death-Birth updating is necessarily bounded and transient. Our boundedness result states that even if a population structure does amplify selection, the resulting fixation probability is close to that of the well-mixed population. Our transience result states that for any population structure there exists a threshold r^⋆ such that the population structure ceases to amplify selection if the mutant fitness advantage r is larger than r^⋆. Finally, we also extend the above results to δ-death-Birth updating, which is a combination of Birth-death and death-Birth updating. On the positive side, we identify population structures that maintain amplification for a wide range of values r and δ. These results demonstrate that amplification of natural selection depends on the specific mechanisms of the evolutionary process.

Extensive literature exists on amplifiers of natural selection for the Birth-death Moran process, but no amplifiers are known for the death-Birth Moran process. Here we show that if amplifiers exist under death-Birth updating, they must be bounded and transient. Boundedness implies weak amplification, and transience implies amplification for only a limited range of the mutant fitness advantage. These results demonstrate that amplification depends on the specific mechanisms of the evolutionary process.

Elucidating the correlations between cancer initiation times and lifetime cancer risks

Cancer is a genetic disease that results from accumulation of unfavorable mutations. As soon as genetic and epigenetic modifications associated with these mutations become strong enough, the uncontrolled tumor cell growth is initiated, eventually spreading through healthy tissues. Clarifying the dynamics of cancer initiation is thus critically important for understanding the molecular mechanisms of tumorigenesis. Here we present a new theoretical method to evaluate the dynamic processes associated with the cancer initiation. It is based on a discrete-state stochastic description of the formation of tumors as a fixation of cancerous mutations in tissues. Using a first-passage analysis the probabilities for the cancer to appear and the times before it happens, which are viewed as fixation probabilities and fixation times, respectively, are explicitly calculated. It is predicted that the slowest cancer initiation dynamics is observed for neutral mutations, while it is fast for both advantageous and, surprisingly, disadvantageous mutations. The method is applied for estimating the cancer initiation times from experimentally available lifetime cancer risks for different types of cancer. It is found that the higher probability of the cancer to occur does not necessary lead to the faster times of starting the cancer. Our theoretical analysis helps to clarify microscopic aspects of cancer initiation processes.

Moran Model of Spatial Alignment in Microbial Colonies

We describe a spatial Moran model that captures mechanical interactions and directional growth in spatially extended populations. The model is analytically tractable and completely solvable under a mean-field approximation and can elucidate the mechanisms that drive the formation of population-level patterns. As an example we model a population of E. coli growing in a rectangular microfluidic trap. We show that spatial patterns can arise as a result of a tug-of-war between boundary effects and growth rate modulations due to cell-cell interactions: Cells align parallel to the long side of the trap when boundary effects dominate. However, when cell-cell interactions exceed a critical value, cells align orthogonally to the trap's long side. This modeling approach and analysis can be extended to directionally-growing cells in a variety of domains to provide insight into how local and global interactions shape collective behavior.

Computational modelling of resistance and associated treatment response heterogeneity in metastatic cancers

Metastatic cancer patients invariably develop treatment resistance. Different levels of resistance lead to observed heterogeneity in treatment response. The main goal was to evaluate treatment response heterogeneity with a computation model simulating the dynamics of drug-sensitive and drug-resistant cells. Model parameters included proliferation, drug-induced death, transition and proportion of intrinsically resistant cells. The model was benchmarked with imaging metrics extracted from 39 metastatic prostate cancer patients who had ¹⁸F-NaF-PET/CT scans performed at baseline and at three cycles into chemotherapy or hormonal therapy. Two initial model assumptions were evaluated: considering only inter-patient heterogeneity and both inter-patient and intra-patient heterogeneity in the proportion of intrinsically resistant cells. The correlation between the median proportion of intrinsically resistant cells and baseline patient-level imaging metrics was assessed with Spearman's rank correlation coefficient. The impact of model parameters on simulated treatment response was evaluated with a sensitivity study. Treatment response after periods of six, nine, and 12 months was predicted with the model. The median predicted range of response for patients treated with both therapies was compared with a Wilcoxon rank sum test. For each patient, the time was calculated when the proportion of disease with a non-favourable response outperformed a favourable response. By taking into account inter-patient and intra-patient heterogeneity in the proportion of intrinsically resistant cells, the model performed significantly better ([Formula: see text]) than by taking into account only inter-patient heterogeneity ([Formula: see text]). The median proportion of intrinsically resistant cells showed a moderate correlation (ρ  =  0.55) with mean patient-level uptake, and a low correlation (ρ  =  0.36) with the dispersion of mean metastasis-level uptake in a patient. The sensitivity study showed a strong impact of the proportion of intrinsically resistant cells on model behaviour after three cycles of therapy. The difference in the median range of response (MRR) was not significant between cohorts at any time point (p   >  0.15). The median time when the proportion of disease with a non-favourable response outperformed the favourable response was eight months, for both cohorts. The model provides an insight into inter-patient and intra-patient heterogeneity in the evolution of treatment resistance.

Population structure determines the tradeoff between fixation probability and fixation time

The rate of biological evolution depends on the fixation probability and on the fixation time of new mutants. Intensive research has focused on identifying population structures that augment the fixation probability of advantageous mutants. But these amplifiers of natural selection typically increase fixation time. Here we study population structures that achieve a tradeoff between fixation probability and time. First, we show that no amplifiers can have an asymptotically lower absorption time than the well-mixed population. Then we design population structures that substantially augment the fixation probability with just a minor increase in fixation time. Finally, we show that those structures enable higher effective rate of evolution than the well-mixed population provided that the rate of generating advantageous mutants is relatively low. Our work sheds light on how population structure affects the rate of evolution. Moreover, our structures could be useful for lab-based, medical, or industrial applications of evolutionary optimization.

An extended Moran process that captures the struggle for fitness

When a new type of individual appears in a stable population, the newcomer is typically not advantageous. Due to stochasticity, the new type can grow in numbers, but the newcomers can only become advantageous if they manage to change the environment in such a way that they increase their fitness. This dynamics is observed in several situations in which a relatively stable population is invaded by an alternative strategy, for instance the evolution of cooperation among bacteria, the invasion of cancer in a multicellular organism and the evolution of ideas that contradict social norms. These examples also show that, by generating different versions of itself, the new type increases the probability of winning the struggle for fitness. Our model captures the imposed cooperation whereby the first generation of newcomers dies while changing the environment such that the next generations become more advantageous.

Emerging functional markers for cancer stem cell-based therapies: Understanding signaling networks for targeting metastasis

Metastasis is one of the most challenging issues in cancer patient management, and effective therapies to specifically target disease progression are missing, emphasizing the urgent need for developing novel anti-metastatic therapeutics. Cancer stem cells (CSCs) gained fast attention as a minor population of highly malignant cells within liquid and solid tumors that are responsible for tumor onset, self-renewal, resistance to radio- and chemotherapies, and evasion of immune surveillance accelerating recurrence and metastasis. Recent progress in the identification of their phenotypic and molecular characteristics and interactions with the tumor microenvironment provides great potential for the development of CSC-based targeted therapies and radical improvement in metastasis prevention and cancer patient prognosis. Here, we report on newly uncovered signaling mechanisms controlling CSC's aggressiveness and treatment resistance, and CSC-specific agents and molecular therapeutics, some of which are currently under investigation in clinical trials, gearing towards decisive functional CSC intrinsic or surface markers. One special research focus rests upon subverted regulatory pathways such as insulin-like growth factor 1 receptor signaling and its interactors in metastasis-initiating cell populations directly related to the gain of stem cell- and EMT-associated properties, as well as key components of the E2F transcription factor network regulating metastatic progression, microenvironmental changes, and chemoresistance. In addition, the study provides insight into systems biology tools to establish complex molecular relationships behind the emergence of aggressive phenotypes from high-throughput data that rely on network-based analysis and their use to investigate immune escape mechanisms or predict clinical outcome-relevant CSC receptor signaling signatures. We further propose that customized vector technologies could drastically enhance systemic drug delivery to target sites, and summarize recent progress and remaining challenges. This review integrates available knowledge on CSC biology, computational modeling approaches, molecular targeting strategies, and delivery techniques to envision future clinical therapies designed to conquer metastasis-initiating cells.

Construction of arbitrarily strong amplifiers of natural selection using evolutionary graph theory

Because of the intrinsic randomness of the evolutionary process, a mutant with a fitness advantage has some chance to be selected but no certainty. Any experiment that searches for advantageous mutants will lose many of them due to random drift. It is therefore of great interest to find population structures that improve the odds of advantageous mutants. Such structures are called amplifiers of natural selection: they increase the probability that advantageous mutants are selected. Arbitrarily strong amplifiers guarantee the selection of advantageous mutants, even for very small fitness advantage. Despite intensive research over the past decade, arbitrarily strong amplifiers have remained rare. Here we show how to construct a large variety of them. Our amplifiers are so simple that they could be useful in biotechnology, when optimizing biological molecules, or as a diagnostic tool, when searching for faster dividing cells or viruses. They could also occur in natural population structures.

Modeling of Interactions between Cancer Stem Cells and their Microenvironment: Predicting Clinical Response

Mathematical models of cancer stem cells are useful in translational cancer research for facilitating the understanding of tumor growth dynamics and for predicting treatment response and resistance to combined targeted therapies. In this chapter, we describe appealing aspects of different methods used in mathematical oncology and discuss compelling questions in oncology that can be addressed with these modeling techniques. We describe a simplified version of a model of the breast cancer stem cell niche, illustrate the visualization of the model, and apply stochastic simulation to generate full distributions and average trajectories of cell type populations over time. We further discuss the advent of single-cell data in studying cancer stem cell heterogeneity and how these data can be integrated with modeling to advance understanding of the dynamics of invasive and proliferative populations during cancer progression and response to therapy.

The effect of spatial randomness on the average fixation time of mutants

The mean conditional fixation time of a mutant is an important measure of stochastic population dynamics, widely studied in ecology and evolution. Here, we investigate the effect of spatial randomness on the mean conditional fixation time of mutants in a constant population of cells, N. Specifically, we assume that fitness values of wild type cells and mutants at different locations come from given probability distributions and do not change in time. We study spatial arrangements of cells on regular graphs with different degrees, from the circle to the complete graph, and vary assumptions on the fitness probability distributions. Some examples include: identical probability distributions for wild types and mutants; cases when only one of the cell types has random fitness values while the other has deterministic fitness; and cases where the mutants are advantaged or disadvantaged. Using analytical calculations and stochastic numerical simulations, we find that randomness has a strong impact on fixation time. In the case of complete graphs, randomness accelerates mutant fixation for all population sizes, and in the case of circular graphs, randomness delays mutant fixation for N larger than a threshold value (for small values of N, different behaviors are observed depending on the fitness distribution functions). These results emphasize fundamental differences in population dynamics under different assumptions on cell connectedness. They are explained by the existence of randomly occurring “dead zones” that can significantly delay fixation on networks with low connectivity; and by the existence of randomly occurring “lucky zones” that can facilitate fixation on networks of high connectivity. Results for death-birth and birth-death formulations of the Moran process, as well as for the (haploid) Wright Fisher model are presented.

We study the influence of randomness on evolutionary dynamics, assuming that a newly arising mutant may experience a different set of environments compared to the wild type. We calculate the mean conditional fixation time of the mutant under different assumptions on spatial interactions, and show that randomness has a strong impact on the fixation time. In particular, it delays the fixation of mutants on 1D circles and accelerates it on complete graphs (the so called mass action, or complete mixing, model). This result holds for advantageous, disadvantageous, and neutral (on average) mutants. The reason for this pattern is quite intuitive: in a rigid, 1D structure, randomness can by chance put a “roadblock” and disrupt mutant spread, causing significant delay. In higher dimensions, there are many ways for a mutant to spread, and it is difficult to block all of them by chance; on the other hand, randomness can enhance fixation by providing an “easier” path. The effects of a random environment are important in biological models such as bacterial growth or cancer initiation/progression.

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.

Genotype by random environmental interactions gives an advantage to non-favored minor alleles

Fixation probability, the probability that the frequency of a newly arising mutation in a population will eventually reach unity, is a fundamental quantity in evolutionary genetics. Here we use a number of models (several versions of the Moran model and the haploid Wright-Fisher model) to examine fixation probabilities for a constant size population where the fitness is a random function of both allelic state and spatial position, despite neither allele being favored on average. The concept of fitness varying with respect to both genotype and environment is important in models of cancer initiation and progression, bacterial dynamics, and drug resistance. Under our model spatial heterogeneity redefines the notion of neutrality for a newly arising mutation, as such mutations fix at a higher rate than that predicted under neutrality. The increased fixation probability appears to be due to rare alleles having an advantage. The magnitude of this effect can be large, and is an increasing function of the spatial variance and skew in fitness. The effect is largest when the fitness values of the mutants and wild types are anti-correlated across environments. We discuss results for both a spatial ring geometry of cells (such as that of a colonic crypt), a 2D lattice and a mass-action (complete graph) arrangement.

An exactly solvable, spatial model of mutation accumulation in cancer

One of the hallmarks of cancer is the accumulation of driver mutations which increase the net reproductive rate of cancer cells and allow them to spread. This process has been studied in mathematical models of well mixed populations, and in computer simulations of three-dimensional spatial models. But the computational complexity of these more realistic, spatial models makes it difficult to simulate realistically large and clinically detectable solid tumours. Here we describe an exactly solvable mathematical model of a tumour featuring replication, mutation and local migration of cancer cells. The model predicts a quasi-exponential growth of large tumours, even if different fragments of the tumour grow sub-exponentially due to nutrient and space limitations. The model reproduces clinically observed tumour growth times using biologically plausible rates for cell birth, death, and migration rates. We also show that the expected number of accumulated driver mutations increases exponentially in time if the average fitness gain per driver is constant, and that it reaches a plateau if the gains decrease over time. We discuss the realism of the underlying assumptions and possible extensions of the model.

Fixation Probabilities for Any Configuration of Two Strategies on Regular Graphs

Population structure and spatial heterogeneity are integral components of evolutionary dynamics, in general, and of evolution of cooperation, in particular. Structure can promote the emergence of cooperation in some populations and suppress it in others. Here, we provide results for weak selection to favor cooperation on regular graphs for any configuration, meaning any arrangement of cooperators and defectors. Our results extend previous work on fixation probabilities of rare mutants. We find that for any configuration cooperation is never favored for birth-death (BD) updating. In contrast, for death-birth (DB) updating, we derive a simple, computationally tractable formula for weak selection to favor cooperation when starting from any configuration containing any number of cooperators. This formula elucidates two important features: (i) the takeover of cooperation can be enhanced by the strategic placement of cooperators and (ii) adding more cooperators to a configuration can sometimes suppress the evolution of cooperation. These findings give a formal account for how selection acts on all transient states that appear in evolutionary trajectories. They also inform the strategic design of initial states in social networks to maximally promote cooperation. We also derive general results that characterize the interaction of any two strategies, not only cooperation and defection.

Spatial Measures of Genetic Heterogeneity During Carcinogenesis

In this work we explore the temporal dynamics of spatial heterogeneity during the process of tumorigenesis from healthy tissue. We utilize a spatial stochastic model of mutation accumulation and clonal expansion in a structured tissue to describe this process. Under a two-step tumorigenesis model, we first derive estimates of a non-spatial measure of diversity: Simpson's Index, which is the probability that two individuals sampled at random from the population are identical, in the premalignant population. We next analyze two new measures of spatial population heterogeneity. In particular we study the typical length scale of genetic heterogeneity during the carcinogenesis process and estimate the extent of a surrounding premalignant clone given a clinical observation of a premalignant point biopsy. This evolutionary framework contributes to a growing literature focused on developing a better understanding of the spatial population dynamics of cancer initiation and progression.

The role of cell location and spatial gradients in the evolutionary dynamics of colon and intestinal crypts

BACKGROUND

Colon and intestinal crypts serve as an important model system for adult stem cell proliferation and differentiation. We develop a spatial stochastic model to study the rate of somatic evolution in a normal crypt, focusing on the production of two-hit mutants that inactivate a tumor suppressor gene. We investigate the effect of cell division pattern along the crypt on mutant production, assuming that the division rate of each cell depends on its location.

RESULTS

We find that higher probability of division at the bottom of the crypt, where the stem cells are located, leads to a higher rate of double-hit mutant production. The optimal case for delaying mutations occurs when most of the cell divisions happen at the top of the crypt. We further consider an optimization problem where the "evolutionary" penalty for double-hit mutant generation is complemented with a "functional" penalty that assures that fully differentiated cells at the top of the crypt cannot divide.

CONCLUSION

The trade-off between the two types of objectives leads to the selection of an intermediate division pattern, where the cells in the middle of the crypt divide with the highest rate. This matches the pattern of cell divisions obtained experimentally in murine crypts.

REVIEWERS

This article was reviewed by David Axelrod (nominated by an Editorial Board member, Marek Kimmel), Yang Kuang and Anna Marciniak-Czochra. For the full reviews, please go to the Reviewers' comments section.

Structural symmetry in evolutionary games

In evolutionary game theory, an important measure of a mutant trait (strategy) is its ability to invade and take over an otherwise-monomorphic population. Typically, one quantifies the success of a mutant strategy via the probability that a randomly occurring mutant will fixate in the population. However, in a structured population, this fixation probability may depend on where the mutant arises. Moreover, the fixation probability is just one quantity by which one can measure the success of a mutant; fixation time, for instance, is another. We define a notion of homogeneity for evolutionary games that captures what it means for two single-mutant states, i.e. two configurations of a single mutant in an otherwise-monomorphic population, to be 'evolutionarily equivalent' in the sense that all measures of evolutionary success are the same for both configurations. Using asymmetric games, we argue that the term 'homogeneous' should apply to the evolutionary process as a whole rather than to just the population structure. For evolutionary matrix games in graph-structured populations, we give precise conditions under which the resulting process is homogeneous. Finally, we show that asymmetric matrix games can be reduced to symmetric games if the population structure possesses a sufficient degree of symmetry.

Spatial Moran models, II: cancer initiation in spatially structured tissue

We study the accumulation and spread of advantageous mutations in a spatial stochastic model of cancer initiation on a lattice. The parameters of this general model can be tuned to study a variety of cancer types and genetic progression pathways. This investigation contributes to an understanding of how the selective advantage of cancer cells together with the rates of mutations driving cancer, impact the process and timing of carcinogenesis. These results can be used to give insights into tumor heterogeneity and the "cancer field effect," the observation that a malignancy is often surrounded by cells that have undergone premalignant transformation.

Modeling Invasion Dynamics with Spatial Random-Fitness Due to Micro-Environment

Numerous experimental studies have demonstrated that the microenvironment is a key regulator influencing the proliferative and migrative potentials of species. Spatial and temporal disturbances lead to adverse and hazardous microenvironments for cellular systems that is reflected in the phenotypic heterogeneity within the system. In this paper, we study the effect of microenvironment on the invasive capability of species, or mutants, on structured grids (in particular, square lattices) under the influence of site-dependent random proliferation in addition to a migration potential. We discuss both continuous and discrete fitness distributions. Our results suggest that the invasion probability is negatively correlated with the variance of fitness distribution of mutants (for both advantageous and neutral mutants) in the absence of migration of both types of cells. A similar behaviour is observed even in the presence of a random fitness distribution of host cells in the system with neutral fitness rate. In the case of a bimodal distribution, we observe zero invasion probability until the system reaches a (specific) proportion of advantageous phenotypes. Also, we find that the migrative potential amplifies the invasion probability as the variance of fitness of mutants increases in the system, which is the exact opposite in the absence of migration. Our computational framework captures the harsh microenvironmental conditions through quenched random fitness distributions and migration of cells, and our analysis shows that they play an important role in the invasion dynamics of several biological systems such as bacterial micro-habitats, epithelial dysplasia, and metastasis. We believe that our results may lead to more experimental studies, which can in turn provide further insights into the role and impact of heterogeneous environments on invasion dynamics.

A spatial model predicts that dispersal and cell turnover limit intratumour heterogeneity

Most cancers in humans are large, measuring centimetres in diameter, and composed of many billions of cells. An equivalent mass of normal cells would be highly heterogeneous as a result of the mutations that occur during each cell division. What is remarkable about cancers is that virtually every neoplastic cell within a large tumour often contains the same core set of genetic alterations, with heterogeneity confined to mutations that emerge late during tumour growth. How such alterations expand within the spatially constrained three-dimensional architecture of a tumour, and come to dominate a large, pre-existing lesion, has been unclear. Here we describe a model for tumour evolution that shows how short-range dispersal and cell turnover can account for rapid cell mixing inside the tumour. We show that even a small selective advantage of a single cell within a large tumour allows the descendants of that cell to replace the precursor mass in a clinically relevant time frame. We also demonstrate that the same mechanisms can be responsible for the rapid onset of resistance to chemotherapy. Our model not only provides insights into spatial and temporal aspects of tumour growth, but also suggests that targeting short-range cellular migratory activity could have marked effects on tumour growth rates.

The role of the bi-compartmental stem cell niche in delaying cancer

In recent years, by using modern imaging techniques, scientists have found evidence of collaboration between different types of stem cells (SCs), and proposed a bi-compartmental organization of the SC niche. Here we create a class of stochastic models to simulate the dynamics of such a heterogeneous SC niche. We consider two SC groups: the border compartment, S1, is in direct contact with transit-amplifying (TA) cells, and the central compartment, S2, is hierarchically upstream from S1. The S1 SCs differentiate or divide asymmetrically when the tissue needs TA cells. Both groups proliferate when the tissue requires SCs (thus maintaining homeostasis). There is an influx of S2 cells into the border compartment, either by migration, or by proliferation. We examine this model in the context of double-hit mutant generation, which is a rate-limiting step in the development of many cancers. We discover that this type of a cooperative pattern in the stem niche with two compartments leads to a significantly smaller rate of double-hit mutant production compared with a homogeneous, one-compartmental SC niche. Furthermore, the minimum probability of double-hit mutant generation corresponds to purely symmetric division of SCs, consistent with the literature. Finally, the optimal architecture (which minimizes the rate of double-hit mutant production) requires a large proliferation rate of S1 cells along with a small, but non-zero, proliferation rate of S2 cells. This result is remarkably similar to the niche structure described recently by several authors, where one of the two SC compartments was found more actively engaged in tissue homeostasis and turnover, while the other was characterized by higher levels of quiescence (but contributed strongly to injury recovery). Both numerical and analytical results are presented.

The duality of spatial death-birth and birth-death processes and limitations of the isothermal theorem

Evolutionary models on graphs, as an extension of the Moran process, have two major implementations: birth-death (BD) models (or the invasion process) and death-birth (DB) models (or voter models). The isothermal theorem states that the fixation probability of mutants in a large group of graph structures (known as isothermal graphs, which include regular graphs) coincides with that for the mixed population. This result has been proved by Lieberman et al. (2005 Nature 433, 312-316. (doi:10.1038/nature03204)) in the case of BD processes, where mutants differ from the wild-types by their birth rate (and not by their death rate). In this paper, we discuss to what extent the isothermal theorem can be formulated for DB processes, proving that it only holds for mutants that differ from the wild-type by their death rate (and not by their birth rate). For more general BD and DB processes with arbitrary birth and death rates of mutants, we show that the fixation probabilities of mutants are different from those obtained in the mass-action populations. We focus on spatial lattices and show that the difference between BD and DB processes on one- and two-dimensional lattices is non-small even for large population sizes. We support these results with a generating function approach that can be generalized to arbitrary graph structures. Finally, we discuss several biological applications of the results.

Spatial Moran Models I. Stochastic Tunneling in the Neutral Case

We consider a multistage cancer model in which cells are arranged in a d-dimensional integer lattice. Starting with all wild-type cells, we prove results about the distribution of the first time when two neutral mutations have accumulated in some cell in dimensions d ≥ 2, extending work done by Komarova [12] for d = 1.

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.

Multifocality and recurrence risk: a quantitative model of field cancerization

Primary tumors often emerge within genetically altered fields of premalignant cells that appear histologically normal but have a high chance of progression to malignancy. Clinical observations have suggested that these premalignant fields pose high risks for emergence of recurrent tumors if left behind after surgical removal of the primary tumor. In this work, we develop a spatio-temporal stochastic model of epithelial carcinogenesis, combining cellular dynamics with a general framework for multi-stage genetic progression to cancer. Using the model, we investigate how various properties of the premalignant fields depend on microscopic cellular properties of the tissue. In particular, we provide analytic results for the size-distribution of the histologically undetectable premalignant fields at the time of diagnosis, and investigate how the extent and the geometry of these fields depend upon key groups of parameters associated with the tissue and genetic pathways. We also derive analytical results for the relative risks of local vs. distant secondary tumors for different parameter regimes, a critical aspect for the optimal choice of post-operative therapy in carcinoma patients. This study contributes to a growing literature seeking to obtain a quantitative understanding of the spatial dynamics in cancer initiation.

Spatial interactions and cooperation can change the speed of evolution of complex phenotypes

Complex traits arise from the interactions among multiple gene products. In the case where the complex phenotype is separated from the wild type by a fitness valley or a fitness plateau, the generation of a complex phenotype may take a very long evolutionary time. Interestingly, the rate of evolution depends in nontrivial ways on various properties of the underlying stochastic process, such as the spatial organization of the population and social interactions among cells. Here we review some of our recent work that investigates these phenomena in asexual populations. The role of spatial constraints is quite complex: there are realistic cases where spatial constrains can accelerate or delay evolution, or even influence it in a nonmonotonic fashion, where evolution works fastest for intermediate-range constraints. Social interactions among cells can be studied in the context of the division-of-labor games. Under a range of circumstances, cooperation among cells can lead to a relatively fast creation of a complex phenotype as an emerging (distributed) property. If we further assume the presence of cheaters, we observe the emergence of a fully mutated population of cells possessing the complex phenotype. Applications of these ideas to cancer initiation and biofilm formation in bacteria are discussed.

Complex role of space in the crossing of fitness valleys by asexual populations

The evolution of complex traits requires the accumulation of multiple mutations, which can be disadvantageous, neutral or advantageous relative to the wild-type. We study two spatial (two-dimensional) models of fitness valley crossing (the constant-population Moran process and the non-constant-population contact process), varying the number of loci involved and the degree of mixing. We find that spatial interactions accelerate the crossing of fitness valleys in the Moran process in the context of neutral and disadvantageous intermediate mutants because of the formation of mutant islands that increase the lifespan of mutant lineages. By contrast, in the contact process, spatial structure can accelerate or delay the emergence of the complex trait, and there can even be an optimal degree of mixing that maximizes the rate of evolution. For advantageous intermediate mutants, spatial interactions always delay the evolution of complex traits, in both the Moran and contact processes. The role of the mutant islands here is the opposite: instead of protecting, they constrict the growth of mutants. We conclude that the laws of population growth can be crucial for the effect of spatial interactions on the rate of evolution, and we relate the two processes explored here to different biological situations.

Effect of dedifferentiation on time to mutation acquisition in stem cell-driven cancers

Accumulating evidence suggests that many tumors have a hierarchical organization, with the bulk of the tumor composed of relatively differentiated short-lived progenitor cells that are maintained by a small population of undifferentiated long-lived cancer stem cells. It is unclear, however, whether cancer stem cells originate from normal stem cells or from dedifferentiated progenitor cells. To address this, we mathematically modeled the effect of dedifferentiation on carcinogenesis. We considered a hybrid stochastic-deterministic model of mutation accumulation in both stem cells and progenitors, including dedifferentiation of progenitor cells to a stem cell-like state. We performed exact computer simulations of the emergence of tumor subpopulations with two mutations, and we derived semi-analytical estimates for the waiting time distribution to fixation. Our results suggest that dedifferentiation may play an important role in carcinogenesis, depending on how stem cell homeostasis is maintained. If the stem cell population size is held strictly constant (due to all divisions being asymmetric), we found that dedifferentiation acts like a positive selective force in the stem cell population and thus speeds carcinogenesis. If the stem cell population size is allowed to vary stochastically with density-dependent reproduction rates (allowing both symmetric and asymmetric divisions), we found that dedifferentiation beyond a critical threshold leads to exponential growth of the stem cell population. Thus, dedifferentiation may play a crucial role, the common modeling assumption of constant stem cell population size may not be adequate, and further progress in understanding carcinogenesis demands a more detailed mechanistic understanding of stem cell homeostasis.

Recent evidence suggests that, like many normal tissues, many cancers are maintained by a small population of immortal stem cells that divide indefinitely to produce many differentiated cells. Cancer stem cells may come directly from mutation of normal stem cells, but this route demands high mutation rates, because there are few normal stem cells. There are, however, many differentiated cells, and mutations can cause such cells to “dedifferentiate” into a stem-like state. We used mathematical modeling to study the effects of dedifferentiation on the time to cancer onset. We found that the effect of dedifferentiation depends critically on how stem cell numbers are controlled by the body. If homeostasis is very tight (due to all divisions being asymmetric), then dedifferentiation has little effect, but if homeostatic control is looser (allowing both symmetric and asymmetric divisions), then dedifferentiation can dramatically hasten cancer onset and lead to exponential growth of the cancer stem cell population. Our results suggest that dedifferentiation may be a very important factor in cancer and that more study of dedifferentiation and stem cell control is necessary to understand and prevent cancer onset.

Spatial invasion dynamics on random and unstructured meshes: implications for heterogeneous tumor populations

In this work we discuss a spatial evolutionary model for a heterogeneous cancer cell population. We consider the gain-of-function mutations that not only change the fitness potential of the mutant phenotypes against normal background cells but may also increase the relative motility of the mutant cells. The spatial modeling is implemented as a stochastic evolutionary system on a structured grid (a lattice, with random neighborhoods, which is not necessarily bi-directional) or on a two-dimensional unstructured mesh, i.e. a bi-directional graph with random numbers of neighbors. We present a computational approach to investigate the fixation probability of mutants in these spatial models. Additionally, we examine the effect of the migration potential on the spatial dynamics of mutants on unstructured meshes. Our results suggest that the probability of fixation is negatively correlated with the width of the distribution of the neighborhood size. Also, the fixation probability increases given a migration potential for mutants. We find that the fixation probability (of advantaged, disadvantaged and neutral mutants) on unstructured meshes is relatively smaller than the corresponding results on regular grids. More importantly, in the case of neutral mutants the introduction of a migration potential has a critical effect on the fixation probability and increases this by orders of magnitude. Further, we examine the effect of boundaries and as intuitively expected, the fixation probability is smaller on the boundary of regular grids when compared to its value in the bulk. Based on these computational results, we speculate on possible better therapeutic strategies that may delay tumor progression to some extent.