Is the distribution of plasmid lengths bimodal?

The length of a plasmid is a key property which is linked to many aspects of plasmid biology. When distributions of plasmid lengths are shown in the literature, they are usually plotted with length on a logarithmic scale. However, a quantity and its logarithm have distinct distributions which may differ considerably in shape. Mistaking the distribution of log-lengths for the distribution of lengths can therefore lead to distorted conclusions about the distribution; in particular, the distribution of log-lengths may be bimodal when the distribution of lengths is only unimodal. This particular confusion has arisen in the literature where the length distribution is often claimed to be bimodal based on examination of what is in fact the log-length distribution. While the length distribution is indeed bimodal within many bacterial families, it is not across the ensemble of all plasmids. We suggest that authors should be careful to show the plasmid length distribution, or to distinguish the two distributions, to avoid misleading inferences.

Limits to evolutionary rescue by conjugative plasmids.

Plasmids may carry genes coding for beneficial traits and thus contribute to adaptation of bacterial populations to environmental stress. Conjugative plasmids can horizontally transfer between cells, which a priori facilitates the spread of adaptive alleles. However, if the potential recipient cell is already colonized by another incompatible plasmid, successful transfer may be prevented. Competition between plasmids can thus limit horizontal transfer. Previous modeling has indeed shown that evolutionary rescue by a conjugative plasmid is hampered by incompatible resident plasmids in the population. If the rescue plasmid is a mutant variant of the resident plasmid, both plasmids transfer at the same rates. A high conjugation rate then has two, potentially opposing, effects - a direct positive effect on spread of the rescue plasmid and an increase in the fraction of resident plasmid cells. This raises the question whether a high conjugation rate always benefits evolutionary rescue. In this article, we systematically analyze three models of increasing complexity to disentangle the benefits and limits of increasing horizontal gene transfer in the presence of plasmid competition and plasmid costs. We find that the net effect can be positive or negative and that the optimal transfer rate is thus not always the highest one. These results can contribute to our understanding of the many facets of plasmid-driven adaptation and the wide range of transfer rates observed in nature.

Segregational drift hinders the evolution of antibiotic resistance on polyploid replicons.

The emergence of antibiotic resistance under treatment depends on the availability of resistance alleles and their establishment in the population. Novel resistance alleles are encoded either in chromosomal or extrachromosomal genetic elements; both types may be present in multiple copies within the cell. However, the effect of polyploidy on the emergence of antibiotic resistance remains understudied. Here we show that the establishment of resistance alleles in microbial populations depends on the ploidy level. Evolving bacterial populations under selection for antibiotic resistance, we demonstrate that resistance alleles in polyploid elements are lost frequently in comparison to alleles in monoploid elements due to segregational drift. Integrating the experiments with a mathematical model, we find a remarkable agreement between the theoretical and empirical results, confirming our understanding of the allele segregation process. Using the mathematical model, we further show that the effect of polyploidy on the establishment probability of beneficial alleles is strongest for low replicon copy numbers and plateaus for high replicon copy numbers. Our results suggest that the distribution of fitness effects for mutations that are eventually fixed in a population depends on the replicon ploidy level. Our study indicates that the emergence of antibiotic resistance in bacterial pathogens depends on the pathogen ploidy level.

A mathematician's guide to plasmids: an introduction to plasmid biology for modellers.

Plasmids, extrachromosomal DNA molecules commonly found in bacterial and archaeal cells, play an important role in bacterial genetics and evolution. Our understanding of plasmid biology has been furthered greatly by the development of mathematical models, and there are many questions about plasmids that models would be useful in answering. In this review, we present an introductory, yet comprehensive, overview of the biology of plasmids suitable for modellers unfamiliar with plasmids who want to get up to speed and to begin working on plasmid-related models. In addition to reviewing the diversity of plasmids and the genes they carry, their key physiological functions, and interactions between plasmid and host, we also highlight selected plasmid topics that may be of particular interest to modellers and areas where there is a particular need for theoretical development. The world of plasmids holds a great variety of subjects that will interest mathematical biologists, and introducing new modellers to the subject will help to expand the existing body of plasmid theory.

Sequential antibiotic therapy in the laboratory and in the patient.

Laboratory experiments suggest that rapid cycling of antibiotics during the course of treatment could successfully counter resistance evolution. Drugs involving collateral sensitivity could be particularly suitable for such therapies. However, the environmental conditions in vivo differ from those in vitro. One key difference is that drugs can be switched abruptly in the laboratory, while in the patient, pharmacokinetic processes lead to changing antibiotic concentrations including periods of dose overlaps from consecutive administrations. During such overlap phases, drug-drug interactions may affect the evolutionary dynamics. To address the gap between the laboratory and potential clinical applications, we set up two models for comparison-a 'laboratory model' and a pharmacokinetic-pharmacodynamic 'patient model'. The analysis shows that in the laboratory, the most rapid cycling suppresses the bacterial population always at least as well as other regimens. For patient treatment, however, a little slower cycling can sometimes be preferable if the pharmacodynamic curve is steep or if drugs interact antagonistically. When resistance is absent prior to treatment, collateral sensitivity brings no substantial benefit unless the cell division rate is low and drug cycling slow. By contrast, drug-drug interactions strongly influence the treatment efficiency of rapid regimens, demonstrating their importance for the optimal choice of drug pairs.

Fixation dynamics of beneficial alleles in prokaryotic polyploid chromosomes and plasmids.

Theoretical population genetics has been mostly developed for sexually reproducing diploid and for monoploid (haploid) organisms, focusing on eukaryotes. The evolution of bacteria and archaea is often studied by models for the allele dynamics in monoploid populations. However, many prokaryotic organisms harbor multicopy replicons-chromosomes and plasmids-and theory for the allele dynamics in populations of polyploid prokaryotes remains lacking. Here, we present a population genetics model for replicons with multiple copies in the cell. Using this model, we characterize the fixation process of a dominant beneficial mutation at 2 levels: the phenotype and the genotype. Our results show that depending on the mode of replication and segregation, the fixation of the mutant phenotype may precede genotypic fixation by many generations; we term this time interval the heterozygosity window. We furthermore derive concise analytical expressions for the occurrence and length of the heterozygosity window, showing that it emerges if the copy number is high and selection strong. Within the heterozygosity window, the population is phenotypically adapted, while both alleles persist in the population. Replicon ploidy thus allows for the maintenance of genetic variation following phenotypic adaptation and consequently for reversibility in adaptation to fluctuating environmental conditions.

Vaccination strategies when vaccines are scarce: on conflicts between reducing the burden and avoiding the evolution of escape mutants.

When vaccine supply is limited but population immunization urgent, the allocation of the available doses needs to be carefully considered. One aspect of dose allocation is the time interval between the first and the second injections in two-dose vaccines. By stretching this interval, more individuals can be vaccinated with the first dose more quickly, which can be beneficial in reducing case numbers, provided a single dose is sufficiently effective. On the other hand, there has been concern that intermediate levels of immunity in partially vaccinated individuals may favour the evolution of vaccine escape mutants. In that case, a large fraction of half-vaccinated individuals would pose a risk-but only if they encounter the virus. This raises the question whether there is a conflict between reducing the burden and the risk of vaccine escape evolution or not. We develop an SIR-type model to assess the population-level effects of the timing of the second dose. Trade-offs can occur both if vaccine escape evolution is more likely or if it is less likely in half-vaccinated than in unvaccinated individuals. Their presence or absence depends on the efficacies for susceptibility and transmissibility elicited by a single dose.

Antibiotic treatment protocols revisited: the challenges of a conclusive assessment by mathematical modelling.

Hospital-acquired bacterial infections lead to prolonged hospital stays and increased mortality. The problem is exacerbated by antibiotic-resistant strains that delay or impede effective treatment. To ensure successful therapy and to manage antibiotic resistance, treatment protocols that draw on several different antibiotics might be used. This includes the administration of drug cocktails to individual patients (combination therapy) but also the random assignment of drugs to different patients (mixing) and a regular switch in the default drug used in the hospital from drug A to drug B and back (cycling). For more than 20 years, mathematical models have been used to assess the prospects of antibiotic combination therapy, mixing and cycling. But while tendencies in their ranking across studies have emerged, the picture remains surprisingly inconclusive and incomplete. In this article, we review existing modelling studies and demonstrate by means of examples how methodological factors complicate the emergence of a consistent picture. These factors include the choice of the criterion by which the effects of the protocols are compared, the model implementation and its analysis. We thereafter discuss how progress can be made and suggest future modelling directions.

Adaptation at different points along antibiotic concentration gradients.

Antibiotic concentrations vary dramatically in the body and the environment. Hence, understanding the dynamics of resistance evolution along antibiotic concentration gradients is critical for predicting and slowing the emergence and spread of resistance. While it has been shown that increasing the concentration of an antibiotic slows resistance evolution, how adaptation to one antibiotic concentration correlates with fitness at other points along the gradient has not received much attention. Here, we selected populations of Escherichia coli at several points along a concentration gradient for three different antibiotics, asking how rapidly resistance evolved and whether populations became specialized to the antibiotic concentration they were selected on. Populations selected at higher concentrations evolved resistance more slowly but exhibited equal or higher fitness across the whole gradient. Populations selected at lower concentrations evolved resistance rapidly, but overall fitness in the presence of antibiotics was lower. However, these populations readily adapted to higher concentrations upon subsequent selection. Our results indicate that resistance management strategies must account not only for the rates of resistance evolution but also for the fitness of evolved strains.

The Effect of Habitat Choice on Evolutionary Rescue in Subdivided Populations.

AbstractEvolutionary rescue is the process by which a population, in response to an environmental change, successfully avoids extinction through adaptation. In spatially structured environments, dispersal can affect the probability of rescue. Here, we model an environment consisting of patches that degrade one after another, and we investigate the probability of rescue by a mutant adapted to the degraded habitat. We focus on the effects of dispersal and of immigration biases. We identify up to three regions delimiting the effect of dispersal on the probability of evolutionary rescue: (i) starting from low dispersal rates, the probability of rescue increases with dispersal; (ii) at intermediate dispersal rates, it decreases; and (iii) at large dispersal rates, it increases again with dispersal, except if mutants are too counterselected in not-yet-degraded patches. The probability of rescue is generally highest when mutant and wild-type individuals preferentially immigrate into patches that have already undergone environmental change. Additionally, we find that mutants that will eventually rescue the population most likely first appear in nondegraded patches. Overall, our results show that habitat choice, compared with the often-studied unbiased immigration scheme, can substantially alter the dynamics of population survival and adaptation to new environments.

Evolutionary Rescue and Drug Resistance on Multicopy Plasmids.

Bacteria often carry "extra DNA" in the form of plasmids in addition to their chromosome. Many plasmids have a copy number greater than one such that the genes encoded on these plasmids are present in multiple copies per cell. This has evolutionary consequences by increasing the mutational target size, by prompting the (transitory) co-occurrence of mutant and wild-type alleles within the same cell, and by allowing for gene dosage effects. We develop and analyze a mathematical model for bacterial adaptation to harsh environmental change if adaptation is driven by beneficial alleles on multicopy plasmids. Successful adaptation depends on the availability of advantageous alleles and on their establishment probability. The establishment process involves the segregation of mutant and wild-type plasmids to the two daughter cells, allowing for the emergence of mutant homozygous cells over the course of several generations. To model this process, we use the theory of multitype branching processes, where a type is defined by the genetic composition of the cell. Both factors-the availability of advantageous alleles and their establishment probability-depend on the plasmid copy number, and they often do so antagonistically. We find that in the interplay of various effects, a lower or higher copy number may maximize the probability of evolutionary rescue. The decisive factor is the dominance relationship between mutant and wild-type plasmids and potential gene dosage effects. Results from a simple model of antibiotic degradation indicate that the optimal plasmid copy number may depend on the specific environment encountered by the population.

Aggressive or moderate drug therapy for infectious diseases? Trade-offs between different treatment goals at the individual and population levels.

Antimicrobial resistance is one of the major public health threats of the 21st century. There is a pressing need to adopt more efficient treatment strategies in order to prevent the emergence and spread of resistant strains. The common approach is to treat patients with high drug doses, both to clear the infection quickly and to reduce the risk of de novo resistance. Recently, several studies have argued that, at least in some cases, low-dose treatments could be more suitable to reduce the within-host emergence of antimicrobial resistance. However, the choice of a drug dose may have consequences at the population level, which has received little attention so far. Here, we study the influence of the drug dose on resistance and disease management at the host and population levels. We develop a nested two-strain model and unravel trade-offs in treatment benefits between an individual and the community. We use several measures to evaluate the benefits of any dose choice. Two measures focus on the emergence of resistance, at the host level and at the population level. The other two focus on the overall treatment success: the outbreak probability and the disease burden. We find that different measures can suggest different dosing strategies. In particular, we identify situations where low doses minimize the risk of emergence of resistance at the individual level, while high or intermediate doses prove most beneficial to improve the treatment efficiency or even to reduce the risk of resistance in the population.

Modeling antimicrobial cycling and mixing: Differences arising from an individual-based versus a population-based perspective.

In order to manage bacterial infections in hospitals in the face of antibiotic resistance, the two treatment protocols "mixing" and "cycling" have received considerable attention both from modelers and clinicians. However, the terms are not used in exactly the same way by both groups. This comes because the standard modeling approach disregards the perspective of individual patients. In this article, we investigate a model that comes closer to clinical practice and compare the predictions to the standard model. We set up two deterministic models, implemented as a set of differential equations, for the spread of bacterial infections in a hospital. Following the traditional approach, the first model takes a population-based perspective. The second model, in contrast, takes the drug use of individual patients into account. The alternative model can indeed lead to different predictions than the standard model. We provide examples for which in the new model, the opposite strategy maximizes the number of uninfected patients or minimizes the rate of spread of double resistance. While the traditional models provide valuable insight, care is hence needed in the interpretation of results.

Modeling antibiotic treatment in hospitals: A systematic approach shows benefits of combination therapy over cycling, mixing, and mono-drug therapies.

Multiple treatment strategies are available for empiric antibiotic therapy in hospitals, but neither clinical studies nor theoretical investigations have yielded a clear picture when which strategy is optimal and why. Extending earlier work of others and us, we present a mathematical model capturing treatment strategies using two drugs, i.e the multi-drug therapies referred to as cycling, mixing, and combination therapy, as well as monotherapy with either drug. We randomly sample a large parameter space to determine the conditions determining success or failure of these strategies. We find that combination therapy tends to outperform the other treatment strategies. By using linear discriminant analysis and particle swarm optimization, we find that the most important parameters determining success or failure of combination therapy relative to the other treatment strategies are the de novo rate of emergence of double resistance in patients infected with sensitive bacteria and the fitness costs associated with double resistance. The rate at which double resistance is imported into the hospital via patients admitted from the outside community has little influence, as all treatment strategies are affected equally. The parameter sets for which combination therapy fails tend to fall into areas with low biological plausibility as they are characterised by very high rates of de novo emergence of resistance to both drugs compared to a single drug, and the cost of double resistance is considerably smaller than the sum of the costs of single resistance.

Evolutionary rescue in randomly mating, selfing, and clonal populations.

Severe environmental change can drive a population extinct unless the population adapts in time to the new conditions ("evolutionary rescue"). How does biparental sexual reproduction influence the chances of population persistence compared to clonal reproduction or selfing? In this article, we set up a one-locus two-allele model for adaptation in diploid species, where rescue is contingent on the establishment of the mutant homozygote. Reproduction can occur by random mating, selfing, or clonally. Random mating generates and destroys the rescue mutant; selfing is efficient at generating it but at the same time depletes the heterozygote, which can lead to a low mutant frequency in the standing genetic variation. Due to these (and other) antagonistic effects, we find a nontrivial dependence of population survival on the rate of sex/selfing, which is strongly influenced by the dominance coefficient of the mutation before and after the environmental change. Importantly, since mating with the wild-type breaks the mutant homozygote up, a slow decay of the wild-type population size can impede rescue in randomly mating populations.

The Role of Recombination in Evolutionary Rescue.

How likely is it that a population escapes extinction through adaptive evolution? The answer to this question is of great relevance in conservation biology, where we aim at species' rescue and the maintenance of biodiversity, and in agriculture and medicine, where we seek to hamper the emergence of pesticide or drug resistance. By reshuffling the genome, recombination has two antagonistic effects on the probability of evolutionary rescue: it generates and it breaks up favorable gene combinations. Which of the two effects prevails depends on the fitness effects of mutations and on the impact of stochasticity on the allele frequencies. In this article, we analyze a mathematical model for rescue after a sudden environmental change when adaptation is contingent on mutations at two loci. The analysis reveals a complex nonlinear dependence of population survival on recombination. We moreover find that, counterintuitively, a fast eradication of the wild type can promote rescue in the presence of recombination. The model also shows that two-step rescue is not unlikely to happen and can even be more likely than single-step rescue (where adaptation relies on a single mutation), depending on the circumstances.

Adaptive gene introgression after secondary contact.

By hybridization and backcrossing, alleles can surmount species boundaries and be incorporated into the genome of a related species. This introgression of genes is of particular evolutionary relevance if it involves the transfer of adaptations between populations. However, any beneficial allele will typically be associated with other alien alleles that are often deleterious and hamper the introgression process. In order to describe the introgression of an adaptive allele, we set up a stochastic model with an explicit genetic makeup of linked and unlinked deleterious alleles. Based on the theory of reducible multitype branching processes, we derive a recursive expression for the establishment probability of the beneficial allele after a single hybridization event. We furthermore study the probability that slightly deleterious alleles hitchhike to fixation. The key to the analysis is a split of the process into a stochastic phase in which the advantageous alleles establishes and a deterministic phase in which it sweeps to fixation. We thereafter apply the theory to a set of biologically relevant scenarios such as introgression in the presence of many unlinked or few closely linked deleterious alleles. A comparison to computer simulations shows that the approximations work well over a large parameter range.

Evolutionary rescue in structured populations.

Environmental change, if severe, can drive a population extinct unless the population succeeds in adapting to the new conditions. How likely is a population to win the race between population decline and adaptive evolution? Assuming that environmental degradation progresses across a habitat, we analyze the impact of several ecological factors on the probability of evolutionary rescue. Specifically, we study the influence of population structure and density-dependent competition as well as the speed and severity of environmental change. We also determine the relative contribution of standing genetic variation and new mutations to evolutionary rescue. To describe population structure, we use a generalized island model, where islands are affected by environmental deterioration one after the other. Our analysis is based on the mathematical theory of time-inhomogeneous branching processes and complemented by computer simulations. We find that in the interplay of various, partially antagonistic effects, the probability of evolutionary rescue can show nontrivial and unexpected dependence on ecological characteristics. In particular, we generally observe a nonmonotonic dependence on the migration rate between islands. Counterintuitively, under some circumstances, evolutionary rescue can occur more readily in the face of harsher environmental shifts, because of the reduced competition experienced by mutant individuals. Similarly, rescue sometimes occurs more readily when the entire habitat degrades rapidly, rather than progressively over time, particularly when migration is high and competition strong.

On the fixation process of a beneficial mutation in a variable environment.

A population that adapts to gradual environmental change will typically experience temporal variation in its population size and the selection pressure. On the basis of the mathematical theory of inhomogeneous branching processes, we present a framework to describe the fixation process of a single beneficial allele under these conditions. The approach allows for arbitrary time-dependence of the selection coefficient s(t) and the population size N(t), as may result from an underlying ecological model. We derive compact analytical approximations for the fixation probability and the distribution of passage times for the beneficial allele to reach a given intermediate frequency. We apply the formalism to several biologically relevant scenarios, such as linear or cyclic changes in the selection coefficient, and logistic population growth. Comparison with computer simulations shows that the analytical results are accurate for a large parameter range, as long as selection is not very weak.