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Biased mutation spectra are pervasive, with wide variation in the magnitude of mutational biases that influence genome evolution and adaptation. How do such diverse biases evolve? Our experiments show that changing the mutation spectrum allows populations to sample previously undersampled mutational space, including beneficial mutations. The resulting shift in the distribution of fitness effects is advantageous: Beneficial mutation supply and beneficial pleiotropy both increase, while deleterious load reduces. More broadly, simulations indicate that reducing or reversing the direction of a long-term bias is always selectively favored. Such changes in mutation bias can occur easily via altered function of DNA repair genes. A phylogenetic analysis shows that these genes are repeatedly gained and lost in bacterial lineages, leading to frequent bias shifts in opposite directions. Thus, shifts in mutation spectra may evolve under selection and can directly alter the outcome of adaptive evolution by facilitating access to beneficial mutations.

Recent experimental evidence demonstrates that shifts in mutational biases, for example increases in transversion frequency, can change the distribution of fitness effects of mutations (DFE). In particular, reducing or reversing a prevailing bias can increase the probability that ade novomutation is beneficial. It has also been shown that mutator bacteria are more likely to emerge if the beneficial mutations they generate have a larger effect size than observed in the wildtype. Here, we connect these two results, demonstrating that mutator strains that reduce or reverse a prevailing bias have a positively-shifted DFE, which in turn can dramatically increase their emergence probability. Since changes in mutation rate and bias are often coupled through the gain and loss of DNA repair enzymes, our results predict that the invasion of mutator strains will be facilitated by shifts in mutation bias that offer improved access to previously under-sampled beneficial mutations.

Although vaccination has been remarkably effective against some pathogens, for others, rapid antigenic evolution results in vaccination conferring only weak and/or short-lived protection. Consequently, considerable effort has been invested in developing more evolutionarily robust vaccines, either by targeting highly conserved components of the pathogen (universal vaccines) or by including multiple immunological targets within a single vaccine (multi-epitope vaccines). An unexplored third possibility is to vaccinate individuals with one of a number of qualitatively different vaccines, creating a “mosaic” of individual immunity in the population. Here we explore whether a mosaic vaccination strategy can deliver superior epidemiological outcomes to “conventional” vaccination, in which all individuals receive the same vaccine. We suppose vaccine doses can be distributed between distinct vaccine “targets” (e.g., different surface proteins against which an immune response can be generated) and/or immunologically distinct variants at these targets (e.g., strains); the pathogen can undergo antigenic evolution at both targets. Using simple mathematical models, here we provide a proof-of-concept that mosaic vaccination often outperforms conventional vaccination, leading to fewer infected individuals, improved vaccine efficacy, and lower individual risks over the course of the epidemic.

Within-host SARS-CoV-2 modelling studies have been published throughout the COVID-19 pandemic. These studies contain highly variable numbers of individuals and capture varying timescales of pathogen dynamics; some studies capture the time of disease onset, the peak viral load and subsequent heterogeneity in clearance dynamics across individuals, while others capture late-time post-peak dynamics. In this study, we curate multiple previously published SARS-CoV-2 viral load data sets, fit these data with a consistent modelling approach, and estimate the variability of in-host parameters including the basic reproduction number, R0. We find that fitted dynamics can be highly variable across data sets, and highly variable within data sets, particularly when key components of the dynamic trajectories (e.g. peak viral load) are not represented in the data. Further, we investigated the role of the eclipse phase time distribution in fitting SARS-CoV-2 viral load data. By varying the shape parameter of an Erlang distribution, we demonstrate that models with either no eclipse phase, or with an exponentially-distributed eclipse phase, offer significantly worse fits to these data, whereas models with less dispersion around the mean eclipse time (shape parameter two or more) offered the best fits to the available data.

Populations threatened by an abrupt environmental change—due to rapid climate change, pathogens or invasive competitors—may survive if they possess or generate genetic combinations adapted to the novel, challenging condition. If these genotypes are initially rare or non-existent, the emergence of lineages that allow a declining population to survive is known as ‘evolutionary rescue’. By contrast, the genotypes required for survival could, by chance, be common before the environmental change. Here, considering both of these possibilities, we find that the risk of extinction can be lower in very small or very large populations, but peaks at intermediate population sizes. This pattern occurs when the survival genotype has a small deleterious effect before the environmental change. Since mildly deleterious mutations constitute a large fraction of empirically measured fitness effects, we suggest that this unexpected result—an intermediate size that puts a population at a greater risk of extinction—may not be unusual in the face of environmental change.

Outbreaks of highly pathogenic strains of avian influenza (HPAI) cause high mortality in avian populations worldwide. When spread from avian reservoirs to humans, HPAI infections cause mortality in about 50% of human infections. Cases of human-to-human transmission of HPAI are relatively rare, and have, to date, only been reported in situations of close contact. These transmissions have resulted in isolated clusters of human HPAI infections, but have not yet caused a pandemic. Given the large number of human H5N1 HPAI infections to date, none of which has resulted in a pandemic, we estimate an upper bound on the probability of H5N1 pandemic emergence. We use this estimate to provide the likelihood of observing such a pandemic over the next decade. We then develop a more accurate parameter-based estimate of the emergence probability and predict the likelihood that, through rare mutations, an H5N1 influenza pandemic will emerge over the same time span.

The impact of collaborative course design is explored in this qualitative, longitudinal study of an upper-year course in which 30-40 undergraduate students co-designed the course syllabus, including both course content and assessments. In addition, the research questions and methods were co-designed by a research team involving six undergraduate student partners, an educational developer, and two faculty members. Student written work and focus-group transcripts were coded and analyzed using longitudinal interpretive phenomenological analysis. The two major themes emerging from this analysis were (a) growth (i.e., the development of academic skills, reflective practice, and personal growth that extended beyond the course), and (b) awareness (of students as a community of learners, of instructors as partners in learning, and of personal agency in learning). Personal reflections on the impact of this partnership revealed broad benefits to both students and academic staff.

Genome sequencing has revealed that prophages, viral sequences integrated in a bacterial chromosome, are abundant, accounting for as much as 20% of the bacterial genome. These sequences can confer fitness benefits to the bacterial host, but may also instigate cell death through induction. Several recent investigations have revealed that the distribution of prophage lengths is bimodal, with a clear distinction between small and large prophages. Here we develop a mathematical model of the evolutionary forces affecting the prophage size distribution, and fit this model to three recent data sets. This approach offers quantitative estimates for the relative rates of lysogeny, induction, mutational degradation and selection acting on a wide class of prophage sequences. The model predicts that large prophages are predominantly maintained by the introduction of new prophage sequences through lysogeny, whereas shorter prophages can be enriched when they no longer encode the genes necessary for induction, but still offer selective benefits to their hosts.

Bacterial strains with a short minimal doubling time—‘fast-growing’ hosts—are more likely to contain prophages than their slow-growing counterparts. Pathogenic bacterial species are likewise more likely to carry prophages. We develop a bioinformatics pipeline to examine the distribution of prophages in fast- and slow-growing lysogens, and pathogenic and non-pathogenic lysogens, analysing both prophage length and gene content for each class. By fitting these results to a mathematical model of the evolutionary forces acting on prophages, we predict whether the observed differences can be attributed to different rates of lysogeny among the host classes, or other evolutionary pressures. We also test for significant differences in gene content among prophages, identifying genes that are preferentially lost or maintained in each class. We find that fast-growing hosts and pathogens have a greater fraction of full-length prophages, and our analysis predicts that induction rates are significantly reduced in slow-growing hosts and non-pathogenic hosts. Consistent with previous results, we find that several proteins involved in the packaging of new phage particles and lysis are preferentially lost in cryptic prophages.This article is part of the theme issue ‘The secret lives of microbial mobile genetic elements’.

Mutation accumulation (MA) experiments, in which de novo mutations are sampled and subsequently characterized, are an essential tool in understanding the processes underlying evolution. In microbial populations, MA protocols typically involve a period of population growth between severe bottlenecks, such that a single individual can form a visible colony. While it has long been appreciated that the action of positive selection during this growth phase cannot be eliminated, it is typically assumed to be negligible. Here, we quantify the effect of both positive and negative selection in MA studies, demonstrating that selective effects can substantially bias the distribution of fitness effects (DFE) and mutation rates estimated from typical MA protocols in microbes. We then present a simple correction for this bias which applies to both beneficial and deleterious mutations, and can be used to correct the observed DFE in multiple environments. Finally, we use simulated MA experiments to illustrate the extent to which the MA-inferred DFE differs from the underlying true DFE, and demonstrate that the proposed correction accurately reconstructs the true DFE over a wide range of scenarios. These results highlight that positive selection during microbial MA experiments is in fact not negligible, but can be corrected to gain a more accurate understanding of fundamental evolutionary parameters.

Deterministic and stochastic evolutionary processes drive adaptation in natural populations. The strength of each component process is determined by the population size: deterministic components prevail in very large populations, while stochastic components are the driving mechanisms in small ones. Many natural populations, however, experience intermittent periods of growth, moving through states in which either stochastic or deterministic processes prevail. This growth is often countered by population bottlenecks, which abound in both natural and laboratory populations. Here we investigate how population bottlenecks shape the process of adaptation. We demonstrate that adaptive trajectories in populations experiencing regular bottlenecks can be naturally scaled in time units of generations; with this scaling the time courses of adaptation, fitness variance, and genetic diversity all become relatively insensitive to the timing of population bottlenecks, provided the bottleneck size exceeds a few thousand individuals. We also include analyses at the genotype level to investigate the impact of population bottlenecks on the predictability and distribution of evolutionary pathways. Irrespective of the timing of population bottlenecks, we find that predictability increases with population size. We also find that predictability of the adaptive pathways increases in increasingly rugged fitness landscapes. Overall, our work reveals that both the adaptation rate and the predictability of evolutionary trajectories are relatively robust to population bottlenecks.

Biased mutation spectra are pervasive, with wide variation in the magnitude of mutational biases that influence genome evolution and adaptation. How do such diverse biases evolve? Our experiments show that changing the mutation spectrum allows populations to sample previously under-sampled mutational space, including beneficial mutations. The resulting shift in the distribution of fitness effects is advantageous: beneficial mutation supply and beneficial pleiotropy both increase, while deleterious load reduces. More broadly, simulations indicate that reducing or reversing the direction of a long-term bias is always selectively favoured. Such changes in mutation bias can occur easily via altered function of DNA repair genes. A phylogenetic analysis shows that these genes are repeatedly gained and lost in bacterial lineages, leading to frequent bias shifts in opposite directions. Thus, shifts in mutation spectra may evolve under selection, and can directly alter the outcome of adaptive evolution by facilitating access to beneficial mutations.SIGNIFICANCE STATEMENTMutations are important because they provide raw material for evolution. Some types of mutations occur more often than others, and the strength of such mutational bias varies across species. It is not clear how this variation arises. We experimentally measured the immediate effects of changing the mutation bias ofE. coli, and used simulations to understand the long-term effects. Altering mutational bias is beneficial whenever the new bias increases sampling of mutational classes that were previously under-sampled. We also show that historically, bacteria have often experienced such beneficial bias switches. Our work thus demonstrates the importance of mutational biases in evolution. By allowing exploration of new mutational space, altered mutation biases could drive rapid adaptation.

We investigate the fate of de novo mutations that occur during the in-host replication of a pathogenic virus, predicting the probability that such mutations are passed on during disease transmission to a new host. Using influenza A virus as a model organism, we develop a life-history model of the within-host dynamics of the infection, deriving a multitype branching process with a coupled deterministic model to capture the population of available target cells. We quantify the fate of neutral mutations and mutations affecting five life-history traits: clearance, attachment, budding, cell death, and eclipse phase timing. Despite the severity of disease transmission bottlenecks, our results suggest that in a single transmission event, several mutations that appeared de novo in the donor are likely to be transmitted to the recipient. Even in the absence of a selective advantage for these mutations, the sustained growth phase inherent in each disease transmission cycle generates genetic diversity that is not eliminated during the transmission bottleneck.

Population bottlenecks—sudden, severe reductions in population size—are ubiquitous in nature. Because of their critical implications for conservation genetics, the effects of population bottlenecks on the loss of genetic diversity have been well studied. Bottlenecks also have important implications for adaptation, however, and these effects have been addressed more recently, typically in microbial populations. In this short review, we survey both experimental and theoretical work describing the impact of population bottlenecks on microbial adaptation. Focusing on theoretical contributions, we highlight emerging insights and conclude with several open questions of interest in the field.

Sequencing of bacterial genomes has revealed an abundance of prophage sequences in many bacterial species. Since these sequences are accessible, through recombination, to infecting phages, bacteria carry an arsenal of genetic material that can be used by these viruses. We develop a mathematical model to isolate the effects of this phenomenon on the coevolution of temperate phage and bacteria. The model predicts that prophage sequences may play a key role in maintaining the phage population in situations that would otherwise favor host cell resistance. In addition, prophage recombination facilitates the existence of multiple phage types, thus promoting diverse co-existence in the phage-host ecosystem. Finally, because the host carries an archive of previous phage strategies, prophage recombination can drive waves of innovation in the host cell population.

Prophage sequences constitute a substantial fraction of the temperate virus gene pool. Although subject to mutational decay, prophage sequences can also be an important source of adaptive mutations for these viral populations. Here we develop a life-history model for temperate viruses, including both the virulent (lytic) and the temperate phases of the life cycle. We then examine the survival of mutations that increase fitness during the lytic phase (attachment rate, burst size), increase fitness in the temperate phase (increasing host survival), or affect transitions between the two phases (integration or induction probability). We find that beneficial mutations are much more likely to survive, ultimately, if they first occur in the prophage state. This conclusion applies even to traits that are only expressed during the lytic phase, and arises due to the substantially lower variance in the offspring distribution during the temperate cycle. This observation, however, is balanced by the fact that many more mutations can be generated during lytic replication. Overall we predict that the prophage state provides a refuge, relatively shielded from genetic drift, in which temperate viruses can explore possible adaptive steps.

Lytic viruses infect and kill host cells, producing a large number of viral copies. Temperate viruses, in contrast, are able to integrate viral genetic material into the host cell DNA, leaving a viable host cell. The evolutionary advantage of this strategy, lysogeny, has been demonstrated in complex environments that include spatial structure, oscillating population dynamics, or periodic environmental collapse. Here, we examine the evolutionary stability of the lysis-lysogeny decision, that is, we predict the long-term outcome of the evolution of lysogeny rates. We demonstrate that viruses with high rates of lysogeny are stable against invasion by more virulent viral strains even in simple environments, as long as the pool of susceptible hosts is not unlimited. This mirrors well-known results in both r-K selection theory and virulence evolution: although virulent viruses have a faster potential growth rate, temperate strains are able to maintain positive growth on a lower density of the limiting resource, susceptible hosts. We then outline scenarios in which the rate of lysogeny is predicted to evolve either toward full lysogeny or full lysis. Finally, we demonstrate conditions under which intermediate rates of lysogeny, as observed in temperate viruses in nature, can be sustained long-term. In general, intermediate lysogeny rates persist when the coupling between susceptible host density and virus density is relaxed.

The emergence and re-emergence of pathogens remains a major public health concern. Unfortunately, when and where pathogens will (re-)emerge is notoriously difficult to predict, as the erratic nature of those events is reinforced by the stochastic nature of pathogen evolution during the early phase of an epidemic. For instance, mutations allowing pathogens to escape host resistance may boost pathogen spread and promote emergence. Yet, the ecological factors that govern such evolutionary emergence remain elusive because of the lack of ecological realism of current theoretical frameworks and the difficulty of experimentally testing their predictions. Here, we develop a theoretical model to explore the effects of the heterogeneity of the host population on the probability of pathogen emergence, with or without pathogen evolution. We show that evolutionary emergence and the spread of escape mutations in the pathogen population is more likely to occur when the host population contains an intermediate proportion of resistant hosts. We also show that the probability of pathogen emergence rapidly declines with the diversity of resistance in the host population. Experimental tests using lytic bacteriophages infecting their bacterial hosts containing Clustered Regularly Interspaced Short Palindromic Repeat and CRISPR-associated (CRISPR-Cas) immune defenses confirm these theoretical predictions. These results suggest effective strategies for cross-species spillover and for the management of emerging infectious diseases., Author summary The probability that an epidemic will break out is highly dependent on the ability of the pathogen to acquire new adaptive mutations and to induce evolutionary emergence. Forecasting pathogen emergence thus requires a good understanding of the interplay between the epidemiology and evolution taking place at the onset of an outbreak. Here, we provide a comprehensive theoretical framework to analyze the impact of host population heterogeneity on the probability of pathogen evolutionary emergence. We use this model to predict the impact of the fraction of susceptible hosts, the inoculum size of the pathogen, and the diversity of host resistance on pathogen emergence. Our experiments using lytic bacteriophages and CRISPR-resistant bacteria support our theoretical predictions and demonstrate that manipulating the diversity of resistance alleles in a host population may be an effective way to limit the emergence of new pathogens.

We investigate the fate ofde novomutations that occur during the in-host replication of a pathogenic virus, predicting the probability that such mutations are passed on during disease transmission to a new host. Using influenza A virus as a model organism, we develop a life-history model of the within-host dynamics of the infection, using a multitype branching process with a coupled deterministic model to capture the population of available target cells. We quantify the fate of neutral mutations and mutations affecting five life-history traits: clearance, attachment, budding, cell death, and eclipse phase timing. Despite the severity of disease transmission bottlenecks, our results suggest that in a single transmission event, several mutations that appearedde novoin the donor are likely to be transmitted to the recipient. Even in the absence of a selective advantage for these mutations, the sustained growth phase inherent in each disease transmission cycle generates genetic diversity that is not eliminated during the transmission bottleneck.

Supplementary figures for manuscript "Effects of Transmission Bottlenecks on the Diversity of Influenza A Virus"

Clustered regularly interspaced short palindromic repeats (CRISPR), linked with CRISPR associated (Cas) genes, can confer adaptive immunity to bacteria, against bacteriophage infections. Thus from a therapeutic standpoint, CRISPR immunity increases biofilm resistance to phage therapy. Recently, however, CRISPR-Cas genes have been implicated in reducing biofilm formation in lysogenized cells. Thus CRISPR immunity can have complex effects on phage-host-lysogen interactions, particularly in a biofilm. In this contribution, we develop and analyse a series of dynamical systems to elucidate and disentangle these interactions. Two competition models are used to study the effects of lysogens (first model) and CRISPR-immune bacteria (second model) in the biofilm. In the third model, the effect of delivering lysogens to a CRISPR-immune biofilm is investigated. Using standard analyses of equilibria, stability and bifurcations, our models predict that lysogens may be able to displace CRISPR-immune bacteria in a biofilm, and thus suggest strategies to eliminate phage-resistant biofilms.

Networks of connections within the human brain have been the subject of intense recent research, yet their topology is still only partially understood. We analyze weighted networks calculated from functional magnetic resonance imaging (fMRI) data acquired during task performance. Expanding previous work in the area, our analysis retains all of the connections between all of the voxels in the full brain fMRI data, computing correlations between approximately 200,000 voxels per subject for 10 subjects. We evaluate the extent to which this rich dataset can be described by existing models of scale-free or exponentially truncated scale-free topology, comparing results across a large number of more complex topological models as well. Our results suggest that the novel “log quadratic” model presented in this paper offers a significantly better fit to networks of functional connections at the voxel level in the human brain.

Published accounts of pedagogical experience and pedagogical research are critical resources to post-secondary mathematics instructors, and yet the quantity and scope of this literature is rarely summarized or reviewed. In this contribution, we analyze recent peer-reviewed journal publications regarding post-secondary mathematics, published by Canadian scholars. We classified this scholarship by institution, publication year, type of pedagogical scholarship, and by topic. We highlight topics of continual interest, changing trends in time and newly emerging themes. This review therefore provides a benchmark of current scholarship in this important area, as well as a point of comparison for similar data from other countries, and other disciplines. Les comptes rendus publiés sur les expériences pédagogiques et la recherche sur la pédagogie sont des ressources essentielles pour les enseignants de mathématiques au niveau postsecondaire. Pourtant, la quantité et la portée de cette documentation font rarement l’objet de résumés ou d’analyses. Dans cet article, nous analysons les publications récentes de chercheurs canadiens sur les mathématiques au niveau postsecondaire, qui ont paru dans des revues révisées par les pairs. Nous avons classé ces publications par établissement, année de publication, type de recherche et sujet. Nous mettons en lumière les sujets d’intérêt constant, les tendances en évolution au fil du temps et les thèmes émergents. Cette recension constitue donc une référence sur les recherches universitaires actuelles dans ce domaine important ainsi qu’un point de comparaison pour des données similaires provenant d’autres pays et d’autres disciplines.

Sequencing of bacterial genomes has revealed an abundance of prophage sequences in many bacterial species. Since these sequences are accessible, through recombination, to infecting phages, bacteria carry an arsenal of genetic material that can be used by these viruses. We develop a mathematical model to isolate the effects of this phenomenon on the coevolution of temperate phage and bacteria. The model predicts that prophage sequences may play a key role in maintaining the phage population in situations that would otherwise favor host cell resistance. In addition, prophage recombination facilitates the existence of multiple phage types, thus promoting diverse co-existence in the phage-host ecosystem. Finally, because the host carries an archive of previous phage strategies, prophage recombination can drive waves of innovation in the host cell population.

Age structure is an important feature of the division of labour within honeybee colonies, but its effects on colony dynamics have rarely been explored. We present a model of a honeybee colony that incorporates this key feature, and use this model to explore the effects of both winter and disease on the fate of the colony. The model offers a novel explanation for the frequently observed phenomenon of ‘spring dwindle’, which emerges as a natural consequence of the age-structured dynamics. Furthermore, the results indicate that a model taking age structure into account markedly affects the predicted timing and severity of disease within a bee colony. The timing of the onset of disease with respect to the changing seasons may also have a substantial impact on the fate of a honeybee colony. Finally, simulations predict that an infection may persist in a honeybee colony over several years, with effects that compound over time. Thus, the ultimate collapse of the colony may be the result of events several years past.

We use a branching process approach to estimate the substitution rate, the rate at which beneficial mutations occur and fix, in populations of lytic viruses whose growth is controlled by periodic population bottlenecks. Our model predicts that substitution rates, and by extension adaptation rates, are profoundly affected by the survival of infected host cells at the bottleneck. In particular, we find that direct transfer (or environmental) bottlenecks, in which some fraction of both free virus and host cells are preserved, are associated with relatively slow adaptation rates for the virus. In contrast, viruses can adapt much more quickly when only free virus is transferred to a new host population, as is typical in an epidemiological setting. Finally, when premature lysis of the host-cell population is induced at the bottleneck, we predict that adaptation rates for the virus will, in general, be faster still. These results hold irrespective of the life-history trait affected by the beneficial mutation. The substitution rates in the presence of environmental bottlenecks are predicted to be as much as an order of magnitude lower than in the other two cases.

Clonal interference refers to the competition that arises in asexual populations when multiple beneficial mutations segregate simultaneously. A large body of theoretical and experimental work now addresses this issue. Although much of the experimental work is performed in populations that grow exponentially between periodic population bottlenecks, the theoretical work to date has addressed only populations of a constant size. We derive an analytical approximation for the rate of adaptation in the presence of both clonal interference and bottlenecks, and compare this prediction to the results of an individual-based simulation, showing excellent agreement in the parameter regime in which clonal interference prevails. We also derive an appropriate definition for the effective population size for adaptive evolution experiments in the presence of population bottlenecks. This "adaptation effective population size" allows for a good approximation of the expected rate of adaptation, either in the strong-selection weak-mutation regime, or when clonal interference comes into play. In the multiple mutation regime, when the product of the population size and mutation rate is extremely large, these results no longer hold.

The fixation probability of a beneficial mutation is extremely sensitive to assumptions regarding the organism's life history. In this article we compute the fixation probability using a life-history model for lytic viruses, a key model organism in experimental studies of adaptation. The model assumes that attachment times are exponentially distributed, but that the lysis time, the time between attachment and host cell lysis, is constant. We assume that the growth of the wild-type viral population is controlled by periodic sampling (population bottlenecks) and also include the possibility that clearance may occur at a constant rate, for example, through washout in a chemostat. We then compute the fixation probability for mutations that increase the attachment rate, decrease the lysis time, increase the burst size, or reduce the probability of clearance. The fixation probability of these four types of beneficial mutations can be vastly different and depends critically on the time between population bottlenecks. We also explore mutations that affect lysis time, assuming that the burst size is constrained by the lysis time, for experimental protocols that sample either free phage or free phage and artificially lysed infected cells. In all cases we predict that the fixation probability of beneficial alleles is remarkably sensitive to the time between population bottlenecks.

The burst-death model has been developed to describe the life history of organisms with variable generation times and a burst of a fixed number of offspring. The model also includes an optional constant clearance rate, such as washout from a chemostat, and the possibility of sustained periods of population growth followed by severe bottlenecks, as in serial passaging. In this model, a beneficial mutation can either increase the burst rate or the burst size, or reduce the clearance rate, thus increasing survival. In this article we examine the effects of these three possible mechanisms on both the Malthusian fitness and the fixation probability of the lineage. We find that equivalent relative increases in the burst rate or burst size confer equivalent increases in the Malthusian fitness of a lineage, whereas increasing survival typically has a more moderate effect on Malthusian fitness. In contrast, for beneficial mutations that confer the same increase in fitness, mutations that increase survival are the most likely to fix, followed by mutations that increase the burst rate. Mutations that increase the burst size are the least likely to fix. These results imply that mutant lineages with the highest Malthusian fitness are not, in many cases, the most likely to escape extinction.

Population bottlenecks are ubiquitous in nature, and are an inherent feature of the experimental protocol for many laboratory selection experiments. These bottlenecks can have profound effects on the rate and trajectory of evolution. In particular, there is a trade-off between sampling the population too frequently and imposing infrequent, but more severe, bottlenecks. In this paper we consider the effects of population bottlenecks, assuming a burst-death model for the life history of the organism under study. This model assumes that generation times are exponentially distributed and that at each generation, individuals in the population have a fixed number of offspring. The model also allows for a constant death rate between bottlenecks. We use this model to estimate the optimal bottleneck ratio, that is, the fraction of the population that should be sampled at each bottleneck in order to maximize the probability that beneficial mutations occur and are not lost. We find that the optimal ratio is roughly constant with respect to many of the model parameters, and that sampling about 20% of the population will maximize the rate of adaptation.

Mobile promoters are emerging as a new class of mobile genetic elements, first identified by examining prokaryote genome sequences, and more recently confirmed by experimental observations in bacteria. Recent datasets have identified over 40,000 putative mobile promoters in sequenced prokaryote genomes, however only one-third of these are in regions of the genome directly upstream from coding sequences, that is, in promoter regions. The presence of many promoter sequences in non-promoter regions is unexplained. Here we develop a general mathematical model for the dynamics of mobile promoters, extending previous work to capture the dynamics both within and outside promoter regions. From this general model, we apply rigorous model selection techniques to identify which parameters are statistically justified in describing the available mobile promoter data, and find best-fit values of these parameters. Our results suggest that high rates of horizontal gene transfer maintain the population of mobile promoters in promoter regions, and that once established at these sites, mobile promoters are rarely lost, but are commonly copied to other genomic regions. In contrast, mobile promoter copies in non-promoter regions are more numerous and more volatile, experiencing substantially higher rates of duplication, loss and diversification.

A system of partial differential equations is derived as a model for the dynamics of a honey bee colony with a continuous age distribution, and the system is then extended to include the effects of a simplified infectious disease. In the disease-free case, we analytically derive the equilibrium age distribution within the colony and propose a novel approach for determining the global asymptotic stability of a reduced model. Furthermore, we present a method for determining the basic reproduction number [Formula: see text] of the infection; the method can be applied to other age-structured disease models with interacting susceptible classes. The results of asymptotic stability indicate that a honey bee colony suffering losses will recover naturally so long as the cause of the losses is removed before the colony collapses. Our expression for [Formula: see text] has potential uses in the tracking and control of an infectious disease within a bee colony.

PET and 18F-FDG have the potential to follow the early metabolic response to chemotherapy in patients with non–small cell lung cancer and to predict success or failure of the therapy. Methods: We studied 16 patients with non–small cell lung cancer as they followed 2 courses of docetaxel and carboplatin. Each patient was studied weekly for 7 wk, and tissue activity was assessed by the amount of radioactivity retained 90 min after the intravenous injection of 18 F-FDG. In a prospective analysis, the linear least-squares method was used to evaluate the time course of metabolic activity in tumor and liver, bone marrow, and unaffected lung tissues; a metabolic response was defined as a response in which the slope of the regression was negative and significantly different from zero. Our hypothesis was that patients who exhibited a tumor metabolic response would survive longer than those who did not. In a retrospective examination of our data, we grouped our patients into those who survived ,6 mo and those who survived longer and calculated the difference in the standardized uptake value (SUV) between day 7 and subsequent time points to determine the most appropriate timing of 2 PET studies in predicting response to therapy. Results: Fifteen of 16 patients completed the study. In the prospective study, 8 patients were classified as nonresponders as the slope of the regression of tumor SUV versus time was not different from zero; they all died within 35 wk of the end of their study. Seven patients were classified as responders; 5 survived and 2 died, one at 25 wk and the other at 76 wk. In the retrospective study, a decrease of 0.5 SUV between studies performed at 1 and 3 wk after the initiation of chemotherapy was predictive of those patients who survived .6 mo and in whom chemotherapy was presumably successful. Conclusion: Patients with non–small cell lung cancer who had a positive outcome, as exhibited by prolonged survival, were those who showed a tumor metabolic response assessed using weekly 18 F-FDG PET studies. 18 F-FDG PET studies performed at 1 and 3 wk after the initiation of chemotherapy allowed prediction of the response to