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

Evolutionary graph theory beyond single mutation dynamics: on how network-structured populations cross fitness landscapes

Spatially resolved datasets are revolutionizing knowledge in molecular biology, yet are under-utilized for questions in evolutionary biology. To gain insight from these large-scale datasets of spatial organization, we need mathematical representations and modeling techniques that can both capture their complexity, but also allow for mathematical tractability. Evolutionary graph theory utilizes the mathematical representation of networks as a proxy for heterogeneous population structure and has started to reshape our understanding of how spatial structure can direct evolutionary dynamics. However, previous results are derived for the case of a single new mutation appearing in the population and the role of network structure in shaping fitness landscape crossing is still poorly understood. Here we study how network-structured populations cross fitness landscapes and show that even a simple extension to a two-mutational landscape can exhibit complex evolutionary dynamics that cannot be predicted using previous single-mutation results. We show how our results can be intuitively understood through the lens of how the two main evolutionary properties of a network, the amplification and acceleration factors, change the expected fate of the intermediate mutant in the population and further discuss how to link these models to spatially resolved datasets of cellular organization.

Foraging-induced craniofacial plasticity is associated with an early, robust and dynamic transcriptional response

Phenotypic plasticity is the ability of a single genotype to vary its phenotype in response to the environment. Plasticity of the skeletal system in response to mechanical input is widely studied, but the timing of its transcriptional regulation is not well understood. Here, we used the cichlid feeding apparatus to examine the transcriptional dynamics of skeletal plasticity over time. Using three closely related species that vary in their ability to remodel bone and a panel of 11 genes, including well-studied skeletal differentiation markers and newly characterized environmentally sensitive genes, we examined plasticity at one, two, four and eight weeks following the onset of alternate foraging challenges. We found that the plastic species exhibited environment-specific bursts in gene expression beginning at one week, followed by a sharp decline in levels, while the species with more limited plasticity exhibited consistently low levels of gene expression. This trend held across nearly all genes, suggesting that it is a hallmark of the larger plasticity regulatory network. We conclude that plasticity of the cichlid feeding apparatus is not the result of slowly accumulating gene expression difference over time, but rather is stimulated by early bursts of environment-specific gene expression followed by a return to homeostatic levels.

Clonal interactions in cancer: integrating quantitative models with experimental and clinical data

Tumors consist of different genotypically distinct subpopulations-or subclones-of cells. These subclones can influence neighboring clones in a process called "clonal interaction." Conventionally, research on driver mutations in cancer has focused on their cell-autonomous effects that lead to an increase in fitness of the cells containing the driver. Recently, with the advent of improved experimental and computational technologies for investigating tumor heterogeneity and clonal dynamics, new studies have shown the importance of clonal interactions in cancer initiation, progression, and metastasis. In this review we provide an overview of clonal interactions in cancer, discussing key discoveries from a diverse range of approaches to cancer biology research. We discuss common types of clonal interactions, such as cooperation and competition, its mechanisms, and the overall effect on tumorigenesis, with important implications for tumor heterogeneity, resistance to treatment, and tumor suppression. Quantitative models-in coordination with cell culture and animal model experiments-have played a vital role in investigating the nature of clonal interactions and the complex clonal dynamics they generate. We present mathematical and computational models that can be used to represent clonal interactions and provide examples of the roles they have played in identifying and quantifying the strength of clonal interactions in experimental systems. Clonal interactions have proved difficult to observe in clinical data; however, several very recent quantitative approaches enable their detection. We conclude by discussing ways in which researchers can further integrate quantitative methods with experimental and clinical data to elucidate the critical-and often surprising-roles of clonal interactions in human cancers.

Stochastic Fluctuations Drive Non-genetic Evolution of Proliferation in Clonal Cancer Cell Populations

Evolutionary dynamics allows us to understand many changes happening in a broad variety of biological systems, ranging from individuals to complete ecosystems. It is also behind a number of remarkable organizational changes that happen during the natural history of cancers. These reflect tumour heterogeneity, which is present at all cellular levels, including the genome, proteome and phenome, shaping its development and interrelation with its environment. An intriguing observation in different cohorts of oncological patients is that tumours exhibit an increased proliferation as the disease progresses, while the timescales involved are apparently too short for the fixation of sufficient driver mutations to promote explosive growth. Here, we discuss how phenotypic plasticity, emerging from a single genotype, may play a key role and provide a ground for a continuous acceleration of the proliferation rate of clonal populations with time. We address this question by combining the analysis of real-time growth of non-small-cell lung carcinoma cells (N-H460) together with stochastic and deterministic mathematical models that capture proliferation trait heterogeneity in clonal populations to elucidate the contribution of phenotypic transitions on tumour growth dynamics.

The role of evolutionary game theory in spatial and non-spatial models of the survival of cooperation in cancer: a review

Evolutionary game theory (EGT) is a branch of mathematics which considers populations of individuals interacting with each other to receive pay-offs. An individual’s pay-off is dependent on the strategy of its opponent(s) as well as on its own, and the higher its pay-off, the higher its reproductive fitness. Its offspring generally inherit its interaction strategy, subject to random mutation. Over time, the composition of the population shifts as different strategies spread or are driven extinct. In the last 25 years there has been a flood of interest in applying EGT to cancer modelling, with the aim of explaining how cancerous mutations spread through healthy tissue and how intercellular cooperation persists in tumour-cell populations. This review traces this body of work from theoretical analyses of well-mixed infinite populations through to more realistic spatial models of the development of cooperation between epithelial cells. We also consider work in which EGT has been used to make experimental predictions about the evolution of cancer, and discuss work that remains to be done before EGT can make large-scale contributions to clinical treatment and patient outcomes.

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.

Discovering and genotyping genomic structural variations by yeast genome synthesis and inducible evolution

Genomic structural variations (SVs) promote the evolution of Saccharomyces cerevisiae, and play an important role in phenotypic diversities. Yeast genomic structures can be remodeled by design and bottom-up synthesis. The synthesis of yeast genome creates novel copy number variations (CNVs) and SVs and develops new strategies to discover gene functions. Further, an inducible evolution system SCRaMbLE, consisted of 3,932 loxPsym sites, was incorporated on synthetic yeast genome. SCRaMbLE enables genomic rearrangements at will and rapidly generates chromosomal number variations, and massive SVs under customized conditions. The impacts of genetic variations on phenotypes can be revealed by genome analysis and chromosome restructuring. Yeast genome synthesis and SCRaMbLE provide a new research paradigm to explore the genotypic mechanisms of phenotype diversities, and can be used to improve biological traits and optimize industrial chassis.

Universal scaling laws rule explosive growth in human cancers

Most physical and other natural systems are complex entities composed of a large number of interacting individual elements. It is a surprising fact that they often obey the so-called scaling laws relating an observable quantity with a measure of the size of the system. Here we describe the discovery of universal superlinear metabolic scaling laws in human cancers. This dependence underpins increasing tumour aggressiveness, due to evolutionary dynamics, which leads to an explosive growth as the disease progresses. We validated this dynamic using longitudinal volumetric data of different histologies from large cohorts of cancer patients. To explain our observations we put forward increasingly-complex biologically-inspired mathematical models that captured the key processes governing tumor growth. Our models predicted that the emergence of superlinear allometric scaling laws is an inherently three-dimensional phenomenon. Moreover, the scaling laws thereby identified allowed us to define a set of metabolic metrics with prognostic value, thus providing added clinical utility to the base findings.

A Bacterial Tower of Babel: Quorum-Sensing Signaling Diversity and Its Evolution

Quorum sensing is a process in which bacteria secrete and sense a diffusible molecule, thereby enabling bacterial groups to coordinate their behavior in a density-dependent manner. Quorum sensing has evolved multiple times independently, utilizing different molecular pathways and signaling molecules. A common theme among many quorum-sensing families is their wide range of signaling diversity-different variants within a family code for different signal molecules with a cognate receptor specific to each variant. This pattern of vast allelic polymorphism raises several questions-How do different signaling variants interact with one another? How is this diversity maintained? And how did it come to exist in the first place? Here we argue that social interactions between signaling variants can explain the emergence and persistence of signaling diversity throughout evolution. Finally, we extend the discussion to include cases where multiple diverse systems work in concert in a single bacterium.

Structuring biofilm communities living in pesticide contaminated water

The wide use of pesticides in agriculture expose microbiota to stressful conditions that require the development of survival strategies. The bacterial response to many pollutants has not been elucidated in detail, as well as the evolutionary processes that occur to build adapted communities. The purpose of this study was to evaluate the bacterial population structure and adaptation strategies in planktonic and biofilm communities in limited environments, as tanks containing water used for washing herbicide containers. This biodiversity, with high percentage of nonculturable microorganisms, was characterized based on habitat and abiotic parameters using molecular and bioinformatics tools. According to water and wastewater standards, the physicochemical conditions of the tank water were inadequate for survival of the identified bacteria, which had to develop survival strategies in this hostile environment. The biodiversity decreased in the transition from planktonic to biofilm samples, indicating a possible association between genetic drift and selection of individuals that survive under stressful conditions, such as heating in water and the presence of chlorine, fluorine and agrochemicals over a six-month period. The abundance of Enterobacter, Acinetobacter and Pseudomonas in biofilms from water tanks was linked to essential processes, deduced from the genes attributed to these taxonomic units, and related to biofilm formation, structure and membrane transport, quorum sensing and xenobiotic degradation. These characteristics were randomly combined and fixed in the biofilm community. Thus, communities of biofilm bacteria obtained under these environmental conditions serve as interesting models for studying herbicide biodegradation kinetics and the prospects of consortia suitable for use in bioremediation in reservoirs containing herbicide-contaminated wastewater, as biofilters containing biofilm communities capable of degrading herbicides.

Host-microbiome coevolution can promote cooperation in a rock-paper-scissors dynamics

Cooperation is a fundamental behaviour observed in all forms of life. The evolution of cooperation has been widely studied, but almost all theories focused on the cooperating individual and its genes. We suggest a different approach, taking into account the microbes carried by the interacting individuals. Accumulating evidence reveals that microbes can affect their host's well-being and behaviour, yet hosts can evolve mechanisms to resist the manipulations of their microbes. We thus propose that coevolution of microbes with their hosts may favour microbes that induce their host to cooperate. Using computational modelling, we show that microbe-induced cooperation can evolve and be maintained in a wide range of conditions, including when facing hosts' resistance to the microbial effect. We find that host-microbe coevolution leads the population to a rock-paper-scissors dynamics that enables maintenance of cooperation in a polymorphic state. Our results suggest a mechanism for the evolution and maintenance of cooperation that may be relevant to a wide variety of organisms, including cases that are difficult to explain by current theories. This study provides a new perspective on the coevolution of hosts and their microbiome, emphasizing the potential role of microbes in shaping their host's behaviour.

Combination Therapy and the Evolution of Resistance: The Theoretical Merits of Synergism and Antagonism in Cancer

The search for effective combination therapies for cancer has focused heavily on synergistic combinations because they exhibit enhanced therapeutic efficacy at lower doses. Although synergism is intuitively attractive, therapeutic success often depends on whether drug resistance develops. The impact of synergistic combinations (vs. antagonistic or additive combinations) on the process of drug-resistance evolution has not been investigated. In this study, we use a simplified computational model of cancer cell numbers in a population of drug-sensitive, singly-resistant, and fully-resistant cells to simulate the dynamics of resistance evolution in the presence of two-drug combinations. When we compared combination therapies administered at the same combination of effective doses, simulations showed synergistic combinations most effective at delaying onset of resistance. Paradoxically, when the therapies were compared using dose combinations with equal initial efficacy, antagonistic combinations were most successful at suppressing expansion of resistant subclones. These findings suggest that, although synergistic combinations could suppress resistance through early decimation of cell numbers (making them "proefficacy" strategies), they are inherently fragile toward the development of single resistance. In contrast, antagonistic combinations suppressed the clonal expansion of singly-resistant cells, making them "antiresistance" strategies. The distinction between synergism and antagonism was intrinsically connected to the distinction between offensive and defensive strategies, where offensive strategies inflicted early casualties and defensive strategies established protection against anticipated future threats. Our findings question the exclusive focus on synergistic combinations and motivate further consideration of nonsynergistic combinations for cancer therapy.Significance: Computational simulations show that if different combination therapies have similar initial efficacy in cancers, then nonsynergistic drug combinations are more likely than synergistic drug combinations to provide a long-term defense against the evolution of therapeutic resistance. Cancer Res; 78(9); 2419-31. ©2018 AACR.

Key issues review: evolution on rugged adaptive landscapes

Adaptive landscapes represent a mapping between genotype and fitness. Rugged adaptive landscapes contain two or more adaptive peaks: allele combinations with higher fitness than any of their neighbors in the genetic space. How do populations evolve on such rugged landscapes? Evolutionary biologists have struggled with this question since it was first introduced in the 1930s by Sewall Wright. Discoveries in the fields of genetics and biochemistry inspired various mathematical models of adaptive landscapes. The development of landscape models led to numerous theoretical studies analyzing evolution on rugged landscapes under different biological conditions. The large body of theoretical work suggests that adaptive landscapes are major determinants of the progress and outcome of evolutionary processes. Recent technological advances in molecular biology and microbiology allow experimenters to measure adaptive values of large sets of allele combinations and construct empirical adaptive landscapes for the first time. Such empirical landscapes have already been generated in bacteria, yeast, viruses, and fungi, and are contributing to new insights about evolution on adaptive landscapes. In this Key Issues Review we will: (i) introduce the concept of adaptive landscapes; (ii) review the major theoretical studies of evolution on rugged landscapes; (iii) review some of the recently obtained empirical adaptive landscapes; (iv) discuss recent mathematical and statistical analyses motivated by empirical adaptive landscapes, as well as provide the reader with instructions and source code to implement simulations of evolution on adaptive landscapes; and (v) discuss possible future directions for this exciting field.

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.

Homeostasis Back and Forth: An Ecoevolutionary Perspective of Cancer

The role of genetic mutations in cancer is indisputable: They are a key source of tumor heterogeneity and drive its evolution to malignancy. But, the success of these new mutant cells relies on their ability to disrupt the homeostasis that characterizes healthy tissues. Mutated clones unable to break free from intrinsic and extrinsic homeostatic controls will fail to establish a tumor. Here, we will discuss, through the lens of mathematical and computational modeling, why an evolutionary view of cancer needs to be complemented by an ecological perspective to understand why cancer cells invade and subsequently transform their environment during progression. Importantly, this ecological perspective needs to account for tissue homeostasis in the organs that tumors invade, because they perturb the normal regulatory dynamics of these tissues, often coopting them for its own gain. Furthermore, given our current lack of success in treating advanced metastatic cancers through tumor-centric therapeutic strategies, we propose that treatments that aim to restore homeostasis could become a promising venue of clinical research. This ecoevolutionary view of cancer requires mechanistic mathematical models to both integrate clinical with biological data from different scales but also to detangle the dynamic feedback between the tumor and its environment. Importantly, for these models to be useful, they need to embrace a higher degree of complexity than many mathematical modelers are traditionally comfortable with.

Plant Community Richness Mediates Inhibitory Interactions and Resource Competition between Streptomyces and Fusarium Populations in the Rhizosphere

Plant community characteristics impact rhizosphere Streptomyces nutrient competition and antagonistic capacities. However, the effects of Streptomyces on, and their responses to, coexisting microorganisms as a function of plant host or plant species richness have received little attention. In this work, we characterized antagonistic activities and nutrient use among Streptomyces and Fusarium from the rhizosphere of Andropogon gerardii (Ag) and Lespedeza capitata (Lc) plants growing in communities of 1 (monoculture) or 16 (polyculture) plant species. Streptomyces from monoculture were more antagonistic against Fusarium than those from polyculture. In contrast, Fusarium isolates from polyculture had greater inhibitory capacities against Streptomyces than isolates from monoculture. Although Fusarium isolates had on average greater niche widths, the collection of Streptomyces isolates in total used a greater diversity of nutrients for growth. Plant richness, but not plant host, influenced the potential for resource competition between the two taxa. Fusarium isolates had greater niche overlap with Streptomyces in monoculture than polyculture, suggesting greater potential for Fusarium to competitively challenge Streptomyces in monoculture plant communities. In contrast, Streptomyces had greater niche overlap with Fusarium in polyculture than monoculture, suggesting that Fusarium experiences greater resource competition with Streptomyces in polyculture than monoculture. These patterns of competitive and inhibitory phenotypes among Streptomyces and Fusarium populations are consistent with selection for Fusarium-antagonistic Streptomyces populations in the presence of strong Fusarium resource competition in plant monocultures. Similarly, these results suggest selection for Streptomyces-inhibitory Fusarium populations in the presence of strong Streptomyces resource competition in more diverse plant communities. Thus, landscape-scale variation in plant species richness may be critical to mediating the coevolutionary dynamics and selective trajectories for inhibitory and nutrient use phenotypes among Streptomyces and Fusarium populations in soil, with significant implications for microbial community functional characteristics.

With a little help from my friends: cooperation can accelerate the rate of adaptive valley crossing

Background

Natural selection favors changes that lead to genotypes possessing high fitness. A conflict arises when several mutations are required for adaptation, but each mutation is separately deleterious. The process of a population evolving from a genotype encoding for a local fitness maximum to a higher fitness genotype is termed an adaptive peak shift.

Results

Here we suggest cooperative behavior as a factor that can facilitate adaptive peak shifts. We model cooperation in a public goods scenario, wherein each individual contributes resources that are later equally redistributed among all cooperating individuals. We use mathematical modeling and stochastic simulations to study the effect of cooperation on peak shifts in both panmictic and structured populations. Our results show that cooperation can substantially affect the rate of complex adaptation. Furthermore, we show that cooperation increases the population diversity throughout the peak shift process, thus increasing the robustness of the population to sudden environmental changes.

Conclusions

We provide a new explanation to adaptive valley crossing in natural populations and suggest that the long term evolution of a species depends on its social behavior.

Electronic supplementary material

The online version of this article (doi:10.1186/s12862-017-0983-2) contains supplementary material, which is available to authorized users.

The Effectiveness of Reward and Punishment in Spatial Social Games

Punishment and reward are usually regarded as two potential mechanisms to explain the evolution of cooperation especially among multiple participators. However, the performance of these two scenarios in spatial environment needs to be discussed. To figure out this issue, we resort to the [Formula: see text]-player Iterated Snowdrift Dilemma (ISD) game and Iterated Prisoner’s Dilemma (IPD) game. More importantly, the evolution of punishment and reward in social network-structured populations has not been formally addressed. The numerical results show the equilibrium cooperation frequency can be influenced by cost-to-benefit ratio [Formula: see text], the punishment-to-benefit ratio [Formula: see text] and the reward-to-benefit ratio [Formula: see text]. And one intriguing observation is that under the same situation, the punishment is more effective than reward to the population. Then we further probe the effectiveness of neighborhood relationship to the cooperation, which is reflected by the random rewired probability [Formula: see text]. From the distribution of the four roles of the participator we can find that individuals can cooperate easily when they have close relationship. The results of this paper may be helpful to understand the cooperation in complex project or among industry–university–research cooperation project.

A population genetics perspective on the determinants of intra-tumor heterogeneity

Cancer results from the acquisition of somatic alterations in a microevolutionary process that typically occurs over many years, much of which is occult. Understanding the evolutionary dynamics that are operative at different stages of progression in individual tumors might inform the earlier detection, diagnosis, and treatment of cancer. Although these processes cannot be directly observed, the resultant spatiotemporal patterns of genetic variation amongst tumor cells encode their evolutionary histories. Such intra-tumor heterogeneity is pervasive not only at the genomic level, but also at the transcriptomic, phenotypic, and cellular levels. Given the implications for precision medicine, the accurate quantification of heterogeneity within and between tumors has become a major focus of current research. In this review, we provide a population genetics perspective on the determinants of intra-tumor heterogeneity and approaches to quantify genetic diversity. We summarize evidence for different modes of evolution based on recent cancer genome sequencing studies and discuss emerging evolutionary strategies to therapeutically exploit tumor heterogeneity. This article is part of a Special Issue entitled: Evolutionary principles - heterogeneity in cancer?, edited by Dr. Robert A. Gatenby.

Small groups and long memories promote cooperation

Complex social behaviors lie at the heart of many of the challenges facing evolutionary biology, sociology, economics, and beyond. For evolutionary biologists the question is often how group behaviors such as collective action, or decision making that accounts for memories of past experience, can emerge and persist in an evolving system. Evolutionary game theory provides a framework for formalizing these questions and admitting them to rigorous study. Here we develop such a framework to study the evolution of sustained collective action in multi-player public-goods games, in which players have arbitrarily long memories of prior rounds of play and can react to their experience in an arbitrary way. We construct a coordinate system for memory-m strategies in iterated n-player games that permits us to characterize all cooperative strategies that resist invasion by any mutant strategy, and stabilize cooperative behavior. We show that, especially when groups are small, longer-memory strategies make cooperation easier to evolve, by increasing the number of ways to stabilize cooperation. We also explore the co-evolution of behavior and memory. We find that even when memory has a cost, longer-memory strategies often evolve, which in turn drives the evolution of cooperation, even when the benefits for cooperation are low.

Collective behaviour, uncertainty and environmental change

A central aspect of cultural evolutionary theory concerns how human groups respond to environmental change. Although we are painting with a broad brush, it is fair to say that prior to the twenty-first century, adaptation often happened gradually over multiple human generations, through a combination of individual and social learning, cumulative cultural evolution and demographic shifts. The result was a generally resilient and sustainable population. In the twenty-first century, however, considerable change happens within small portions of a human generation, on a vastly larger range of geographical and population scales and involving a greater degree of horizontal learning. As a way of gauging the complexity of societal response to environmental change in a globalized future, we discuss several theoretical tools for understanding how human groups adapt to uncertainty. We use our analysis to estimate the limits of predictability of future societal change, in the belief that knowing when to hedge bets is better than relying on a false sense of predictability.

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.