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We show here that the set of the integral solutions of a nonlocal differential inclusion is dense in the set of the solution set of the corresponding relaxed differential inclusion. We further define a notion of limit solution and show that the set of limit solutions is closed and is the closure of the set of integral solutions. An illustrative example is provided.

Coexistence of multiple strains of a pathogen in a host population can present significant challenges to vaccine development or treatment efficacy. Here we discuss a novel mechanism that can increase rates of long-lived strain polymorphism, rooted in the presence of social structure in a host population. We show that social preference of interaction, in conjunction with differences in immunity between host subgroups, can exert varying selection pressure on pathogen strains, creating a balancing mechanism that supports stable viral coexistence, independent of other known mechanisms. We use population genetic models to study rates of pathogen heterozygosity as a function of population size, host population composition, mutant strain fitness differences and host social preferences of interaction. We also show that even small periodic epochs of host population stratification can lead to elevated strain coexistence. These results are robust to varying social preferences of interaction, overall differences in strain fitnesses, and spatial heterogeneity in host population composition. Our results highlight the role of host population social stratification in increasing rates of pathogen strain diversity, with effects that should be considered when designing policies or treatments with a long-term view of curbing pathogen evolution., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2024 Li et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

Many evolving ecosystems have spatial structures that can be conceptualized as networks, with nodes representing individuals or homogeneous subpopulations and links the patterns of spread between them. Prior models of evolution on networks do not take ecological niche differences and eco-evolutionary interplay into account. Here, we combine a resource competition model with evolutionary graph theory to study how heterogeneous topological structure shapes evolutionary dynamics under global frequency-dependent ecological interactions. We find that the addition of ecological competition for resources can produce a reversal of roles between amplifier and suppressor networks for deleterious mutants entering the population. We show that this effect is a nonlinear function of ecological niche overlap and discuss intuition for the observed dynamics using simulations and analytical approximations. We use these theoretical results together with spatial representations from imaging data to show that, for ductal carcinoma, where tumor growth is highly spatially constrained, with cells confined to a tree-like network of ducts, the topological structure can lead to higher rates of deleterious mutant hitchhiking with metabolic driver mutations, compared to tumors characterized by different spatial topologies., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2024 Devadhasan et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

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., Competing Interests: Conflicts of interest. The author(s) declare no conflicts of interest., (© The Author(s) 2024. Published by Oxford University Press on behalf of The Genetics Society of America.)

How the spatial arrangement of a population shapes its evolutionary dynamics has been of long-standing interest in population genetics. Most previous studies assume a small number of demes or symmetrical structures that, most often, act as well-mixed populations. Other studies use network theory to study more heterogeneous spatial structures, however they usually assume small, regular networks, or strong constraints on the strength of selection considered. Here we build network generation algorithms, conduct evolutionary simulations and derive general analytic approximations for probabilities of fixation in populations with complex spatial structure. We build a unifying evolutionary theory across network families and derive the relevant selective parameter, which is a combination of network statistics, predictive of evolutionary dynamics. We also illustrate how to link this theory with novel datasets of spatial organization and use recent imaging data to build the cellular spatial networks of the stem cell niches of the bone marrow. Across a wide variety of parameters, we find these networks to be strong suppressors of selection, delaying mutation accumulation in this tissue. We also find that decreases in stem cell population size also decrease the suppression strength of the tissue spatial structure., (© 2024. The Author(s).)

To design population topologies that can accelerate rates of solution discovery in directed evolution problems or for evolutionary optimization applications, we must first systematically understand how population structure shapes evolutionary outcome. Using the mathematical formalism of evolutionary graph theory, recent studies have shown how to topologically build networks of population interaction that increase probabilities of fixation of beneficial mutations, at the expense, however, of longer fixation times, which can slow down rates of evolution, under elevated mutation rate. Here we find that moving beyond dyadic interactions in population graphs is fundamental to explain the trade-offs between probabilities and times to fixation of new mutants in the population. We show that higher-order motifs, and in particular three-node structures, allow the tuning of times to fixation, without changes in probabilities of fixation. This gives a near-continuous control over achieving solutions that allow for a wide range of times to fixation. We apply our algorithms and analytic results to two evolutionary optimization problems and show that the rate of solution discovery can be tuned near continuously by adjusting the higher-order topology of the population. We show that the effects of population structure on the rate of evolution critically depend on the optimization landscape and find that decelerators, with longer times to fixation of new mutants, are able to reach the optimal solutions faster than accelerators in complex solution spaces. Our results highlight that no one population topology fits all optimization applications, and we provide analytic and computational tools that allow for the design of networks suitable for each specific task., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2024 Kuo, Carja. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

Circulating tumor DNA (ctDNA) monitoring, while sufficiently advanced to reflect tumor evolution in real time and inform cancer diagnosis, treatment, and prognosis, mainly relies on DNA that originates from cell death via apoptosis or necrosis. In solid tumors, chemotherapy and immune infiltration can induce spatially variable rates of cell death, with the potential to bias and distort the clonal composition of ctDNA. Using a stochastic evolutionary model of boundary-driven growth, we study how elevated cell death on the edge of a tumor can simultaneously impact driver mutation accumulation and the representation of tumor clones and mutation detectability in ctDNA. We describe conditions in which invasive clones are over-represented in ctDNA, clonal diversity can appear elevated in the blood, and spatial bias in shedding can inflate subclonal variant allele frequencies (VAFs). Additionally, we find that tumors that are mostly quiescent can display similar biases but are far less detectable, and the extent of perceptible spatial bias strongly depends on sequence detection limits. Overall, we show that spatially structured shedding might cause liquid biopsies to provide highly biased profiles of tumor state. While this may enable more sensitive detection of expanding clones, it could also increase the risk of targeting a subclonal variant for treatment. Our results indicate that the effects and clinical consequences of spatially variable cell death on ctDNA composition present an important area for future work.

Circulating tumor DNA (ctDNA) monitoring, while sufficiently advanced to reflect tumor evolution in real time and inform on cancer diagnosis, treatment, and prognosis, mainly relies on DNA that originates from cell death via apoptosis or necrosis. In solid tumors, chemotherapy and immune infiltration can induce spatially variable rates of cell death, with the potential to bias and distort the clonal composition of ctDNA. Using a stochastic evolutionary model of boundary-driven growth, we study how elevated cell death on the edge of a tumor can simultaneously impact driver mutation accumulation and the representation of tumor clones and mutation detectability in ctDNA. We describe conditions in which invasive clones end up over-represented in ctDNA, clonal diversity can appear elevated in the blood, and spatial bias in shedding can inflate subclonal variant allele frequencies (VAFs). Additionally, we find that tumors that are mostly quiescent can display similar biases, but are far less detectable, and the extent of perceptible spatial bias strongly depends on sequence detection limits. Overall, we show that spatially structured shedding might cause liquid biopsies to provide highly biased profiles of tumor state. While this may enable more sensitive detection of expanding clones, it could also increase the risk of targeting a subclonal variant for treatment. Our results indicate that the effects and clinical consequences of spatially variable cell death on ctDNA composition present an important area for future work.

The cell cycle, over which cells grow and divide, is a fundamental process of life. Its dysregulation has devastating consequences, including cancer 1-3 . The cell cycle is driven by precise regulation of proteins in time and space, which creates variability between individual proliferating cells. To our knowledge, no systematic investigations of such cell-to-cell proteomic variability exist. Here we present a comprehensive, spatiotemporal map of human proteomic heterogeneity by integrating proteomics at subcellular resolution with single-cell transcriptomics and precise temporal measurements of individual cells in the cell cycle. We show that around one-fifth of the human proteome displays cell-to-cell variability, identify hundreds of proteins with previously unknown associations with mitosis and the cell cycle, and provide evidence that several of these proteins have oncogenic functions. Our results show that cell cycle progression explains less than half of all cell-to-cell variability, and that most cycling proteins are regulated post-translationally, rather than by transcriptomic cycling. These proteins are disproportionately phosphorylated by kinases that regulate cell fate, whereas non-cycling proteins that vary between cells are more likely to be modified by kinases that regulate metabolism. This spatially resolved proteomic map of the cell cycle is integrated into the Human Protein Atlas and will serve as a resource for accelerating molecular studies of the human cell cycle and cell proliferation.

Mutations can occur throughout the virus genome and may be beneficial, neutral or deleterious. We are interested in mutations that yield a C next to a G, producing CpG sites. CpG sites are rare in eukaryotic and viral genomes. For the eukaryotes, it is thought that CpG sites are rare because they are prone to mutation when methylated. In viruses, we know less about why CpG sites are rare. A previous study in HIV suggested that CpG-creating transition mutations are more costly than similar non-CpG-creating mutations. To determine if this is the case in other viruses, we analyzed the allele frequencies of CpG-creating and non-CpG-creating mutations across various strains, subtypes, and genes of viruses using existing data obtained from Genbank, HIV Databases, and Virus Pathogen Resource. Our results suggest that CpG sites are indeed costly for most viruses. By understanding the cost of CpG sites, we can obtain further insights into the evolution and adaptation of viruses., (© The Author(s) 2020.)

Cultural processes, as well as the selection pressures experienced by individuals in a population over time and space, are fundamentally stochastic. Phenotypic variability, together with imperfect phenotypic transmission between parents and offspring, has been previously shown to play an important role in evolutionary rescue and (epi)genetic adaptation of populations to fluctuating temporal environmental pressures. This type of evolutionary bet-hedging does not confer a direct benefit to a single individual, but can instead increase the adaptability of the whole lineage. Here we develop a population-genetic model to explore cultural response strategies to temporally changing selection, as well as the role of local population structure, as exemplified by heterogeneity in the contact network between individuals, in shaping evolutionary dynamics. We use this model to study the evolutionary advantage of cultural bet-hedging, modeling the evolution of a variable cultural trait starting from one copy in a population of individuals with a fixed cultural strategy. We find that the probability of fixation of a cultural bet-hedger is a non-monotonic function of the probability of cultural memory between generations. Moreover, this probability increases for networks of higher mean degree but decreases with increasing heterogeneity of the contact network, tilting the balance of forces toward drift and against selection. These results shed light on the interplay of temporal and spatial stochasticity in shaping cultural evolutionary dynamics and suggest that partly-heritable cultural phenotypic variability may constitute an important evolutionary bet-hedging strategy in response to changing selection pressures., (Copyright © 2019 Elsevier Inc. All rights reserved.)

Environmental variation is commonplace, but unpredictable. Populations that encounter a deleterious environment can sometimes avoid extinction by rapid evolutionary adaptation. Phenotypic variability, whereby a single genotype can express multiple different phenotypes, might play an important role in rescuing such populations from extinction. This type of evolutionary bet-hedging need not confer a direct benefit to a single individual, but it may increase the chance of long-term survival of a lineage. Here, we develop a population genetic model to explore how partly heritable phenotypic variability influences the probability of evolutionary rescue and the mean duration of population persistence in changing environments. We find that the probability of population persistence depends nonmonotonically on the degree of phenotypic heritability between generations: some heritability can help avert extinction, but too much heritability removes any benefit of phenotypic variability. Partly heritable phenotypic variation is particularly advantageous when it extends the persistence time of a declining population and thereby increases the chance of rescue via beneficial mutations at linked loci. We discuss the implications of these results in the context of therapies designed to eradicate populations of pathogens or aberrant cellular lineages., (Copyright © 2019 Carja and Plotkin.)

Recombination in HIV-1 is well documented, but its importance in the low-diversity setting of within-host diversification is less understood. Here we develop a novel computational tool (RAPR (Recombination Analysis PRogram)) to enable a detailed view of in vivo viral recombination during early infection, and we apply it to near-full-length HIV-1 genome sequences from longitudinal samples. Recombinant genomes rapidly replace transmitted/founder (T/F) lineages, with a median half-time of 27 days, increasing the genetic complexity of the viral population. We identify recombination hot and cold spots that differ from those observed in inter-subtype recombinants. Furthermore, RAPR analysis of longitudinal samples from an individual with well-characterized neutralizing antibody responses shows that recombination helps carry forward resistance-conferring mutations in the diversifying quasispecies. These findings provide insight into molecular mechanisms by which viral recombination contributes to HIV-1 persistence and immunopathogenesis and have implications for studies of HIV transmission and evolution in vivo.

Background: Using high-throughput sequencing to monitor translation in vivo, ribosome profiling can provide critical insights into the dynamics and regulation of protein synthesis in a cell. Since its introduction in 2009, this technique has played a key role in driving biological discovery, and yet it requires a rigorous computational toolkit for widespread adoption., Description: We have developed a database and a browser-based visualization tool, riboviz, that enables exploration and analysis of riboseq datasets. In implementation, riboviz consists of a comprehensive and flexible computational pipeline that allows the user to analyze private, unpublished datasets, along with a web application for comparison with published yeast datasets. Source code and detailed documentation are freely available from https://github.com/shahpr/RiboViz . The web-application is live at www.riboviz.org., Conclusions: riboviz provides a comprehensive database and analysis and visualization tool to enable comparative analyses of ribosome-profiling datasets. This toolkit will enable both the community of systems biologists who study genome-wide ribosome profiling data and also research groups focused on individual genes to identify patterns of transcriptional and translational regulation across different organisms and conditions.

DNA methylation is an epigenetic modification, influenced by both genetic and environmental variation, that plays a key role in transcriptional regulation and many organismal phenotypes. Although patterns of DNA methylation have been shown to differ between human populations, it remains to be determined how epigenetic diversity relates to the patterns of genetic and gene expression variation at a global scale. Here we measured DNA methylation at 485,000 CpG sites in five diverse human populations, and analysed these data together with genome-wide genotype and gene expression data. We found that population-specific DNA methylation mirrors genetic variation, and has greater local genetic control than mRNA levels. We estimated the rate of epigenetic divergence between populations, which indicates far greater evolutionary stability of DNA methylation in humans than has been observed in plants. This study provides a deeper understanding of worldwide patterns of human epigenetic diversity, as well as initial estimates of the rate of epigenetic divergence in recent human evolution.

Phenotypic plasticity is an evolutionary driving force in diverse biological processes, including the adaptive immune system, the development of neoplasms, and the persistence of pathogens despite drug pressure. It is essential, therefore, to understand the evolutionary advantage of an allele that confers on cells the ability to express a range of phenotypes. Here, we study the fate of a new mutation that allows the expression of multiple phenotypic states, introduced into a finite population of individuals that can express only a single phenotype. We show that the advantage of such a mutation depends on the degree of phenotypic heritability between generations, called phenotypic memory. We analyze the fixation probability of the phenotypically plastic allele as a function of phenotypic memory, the variance of expressible phenotypes, the rate of environmental changes, and the population size. We find that the fate of a phenotypically plastic allele depends fundamentally on the environmental regime. In constant environments, plastic alleles are advantageous and their fixation probability increases with the degree of phenotypic memory. In periodically fluctuating environments, by contrast, there is an optimum phenotypic memory that maximizes the probability of the plastic allele's fixation. This same optimum memory also maximizes geometric mean fitness, in steady state. We interpret these results in the context of previous studies in an infinite-population framework. We also discuss the implications of our results for the design of therapies that can overcome persistence and, indirectly, drug resistance.

Aging is associated with widespread changes in genome-wide patterns of DNA methylation. Thousands of CpG sites whose tissue-specific methylation levels are strongly correlated with chronological age have been previously identified. However, the majority of these studies have focused primarily on cosmopolitan populations living in the developed world; it is not known if age-related patterns of DNA methylation at these loci are similar across a broad range of human genetic and ecological diversity. We investigated genome-wide methylation patterns using saliva- and whole blood-derived DNA from two traditionally hunting and gathering African populations: the Baka of the western Central African rain forest and the ≠Khomani San of the South African Kalahari Desert. We identified hundreds of CpG sites whose methylation levels are significantly associated with age, thousands that are significant in a meta-analysis, and replicate trends previously reported in populations of non-African descent. We confirmed that an age-associated site in the promoter of the gene ELOVL2 shows a remarkably congruent relationship with aging in humans, despite extensive genetic and environmental variation across populations. We also demonstrate that genotype state at methylation quantitative trait loci (meQTLs) can affect methylation trends at some age-associated CpG sites. Our study explores the relationship between CpG methylation and chronological age in populations of African hunter-gatherers, who rely on different diets across diverse ecologies. While many age-related CpG sites replicate across populations, we show that considering common genetic variation at meQTLs further improves our ability to detect previously identified age associations., (Copyright © 2017 by the Genetics Society of America.)

The production and maintenance of genetic and phenotypic diversity under temporally fluctuating selection and the signatures of environmental changes in the patterns of this variation have been important areas of focus in population genetics. On one hand, periods of constant selection pull the genetic makeup of populations toward local fitness optima. On the other, to cope with changes in the selection regime, populations may evolve mechanisms that create a diversity of genotypes. By tuning the rates at which variability is produced--such as the rates of recombination, mutation, or migration--populations may increase their long-term adaptability. Here we use theoretical models to gain insight into how the rates of these three evolutionary forces are shaped by fluctuating selection. We compare and contrast the evolution of recombination, mutation, and migration under similar patterns of environmental change and show that these three sources of phenotypic variation are surprisingly similar in their response to changing selection. We show that the shape, size, variance, and asymmetry of environmental fluctuation have different but predictable effects on evolutionary dynamics.

Stochastic switching is an example of phenotypic bet hedging, where an individual can switch between different phenotypic states in a fluctuating environment. Although the evolution of stochastic switching has been studied when the environment varies temporally, there has been little theoretical work on the evolution of phenotypic switching under both spatially and temporally fluctuating selection pressures. Here, we explore the interaction of temporal and spatial change in determining the evolutionary dynamics of phenotypic switching. We find that spatial variation in selection is important; when selection pressures are similar across space, migration can decrease the rate of switching, but when selection pressures differ spatially, increasing migration between demes can facilitate the evolution of higher rates of switching. These results may help explain the diverse array of non-genetic contributions to phenotypic variability and phenotypic inheritance observed in both wild and experimental populations., (© 2014 The Author(s) Published by the Royal Society. All rights reserved.)

Stochastic switching is an example of phenotypic bet hedging, where offspring can express a phenotype different from that of their parents. Phenotypic switching is well documented in viruses, yeast, and bacteria and has been extensively studied when the selection pressures vary through time. However, there has been little work on the evolution of phenotypic switching under both spatially and temporally fluctuating selection pressures. Here we use a population genetic model to explore the interaction of temporal and spatial variation in determining the evolutionary dynamics of phenotypic switching. We find that the stable switching rate is mainly determined by the rate of environmental change and the migration rate. This stable rate is also a decreasing function of the recombination rate, although this is a weaker effect than those of either the period of environmental change or the migration rate. This study highlights the interplay of spatial and temporal environmental variability, offering new insights into how migration can influence the evolution of phenotypic switching rates, mutation rates, or other sources of phenotypic variation.

Phenotypic adaptation to fluctuating environments has been an important focus in the population genetic literature. Previous studies have shown that evolution under temporal variation is determined not only by expected fitness in a given generation, but also by the degree of variation in fitness over generations; in an uncertain environment, alleles that increase the geometric mean fitness can invade a randomly mating population at equilibrium. This geometric mean principle governs the evolutionary interplay of genes controlling mean phenotype and genes controlling phenotypic variation, such as genetic regulators of the epigenetic machinery. Thus, it establishes an important role for stochastic epigenetic variation in adaptation to fluctuating environments: by modifying the geometric mean fitness, variance-modifying genes can change the course of evolution and determine the long-term trajectory of the evolving system. The role of phenotypic variance has previously been studied in systems in which the only driving force is natural selection, and there is no recombination between mean- and variance-modifying genes. Here, we develop a population genetic model to investigate the effect of recombination between mean- and variance-modifiers of phenotype on the geometric mean principle under different environmental regimes and fitness landscapes. We show that interactions of recombination with stochastic epigenetic variation and environmental fluctuations can give rise to complex evolutionary dynamics that differ from those in systems with no recombination., (Copyright © 2013 Elsevier Inc. All rights reserved.)

The connection between random environments and genetic and phenotypic variability has been a major focus in the population genetic literature. By providing differential access to the underlying genetic information, epigenetic variation could play an important role in the interaction between environmental and phenotypic variation. Using simulation, we model epigenetic plasticity during development by investigating the dynamics of genetic regulators of the epigenetic machinery that change the variance of the phenotype, while having no effect on the phenotype's mean. Previous studies have found that increased phenotypic variance is selected for if the environment is fluctuating. Here, we find that when a variance-increasing allele achieves a sufficiently high frequency, it can be out-competed by a variance-reducing allele, with the consequence that the population evolves to an equilibrium phenotypic variability. This equilibrium is shown to be robust to different initial conditions, but to depend heavily on parameters of the model, such as the mutation rate, the fitness landscape and the nature of the environmental fluctuation. Indeed, if there is no mutation at the genes controlling the variance of the phenotype, reduction of this variance is favoured.