Modeling the impact of periodic bottlenecks, unidirectional mutation, and observational error in experimental evolution

Mathematical Models of Plasmid Population Dynamics

With plasmid-mediated antibiotic resistance thriving and threatening to become a serious public health problem, it is paramount to increase our understanding of the forces that enable the spread and maintenance of drug resistance genes encoded in mobile genetic elements. The relevance of plasmids as vehicles for the dissemination of antibiotic resistance genes, in addition to the extensive use of plasmid-derived vectors for biotechnological and industrial purposes, has promoted the in-depth study of the molecular mechanisms controlling multiple aspects of a plasmids' life cycle. This body of experimental work has been paralleled by the development of a wealth of mathematical models aimed at understanding the interplay between transmission, replication, and segregation, as well as their consequences in the ecological and evolutionary dynamics of plasmid-bearing bacterial populations. In this review, we discuss theoretical models of plasmid dynamics that span from the molecular mechanisms of plasmid partition and copy-number control occurring at a cellular level, to their consequences in the population dynamics of complex microbial communities. We conclude by discussing future directions for this exciting research topic.

Layman N, Redwood AJ, Chan B, Bull JJ. Methods for measuring the evolutionary stability of engineered genomes to improve their longevity

Diverse applications rely on engineering microbes to carry and express foreign transgenes. This engineered baggage rarely benefits the microbe and is thus prone to rapid evolutionary loss when the microbe is propagated. For applications where a transgene must be maintained for extended periods of growth, slowing the rate of transgene evolution is critical and can be achieved by reducing either the rate of mutation or the strength of selection. Because the benefits realized by changing these quantities will not usually be equal, it is important to know which will yield the greatest improvement to the evolutionary half-life of the engineering. Here, we provide a method for jointly estimating the mutation rate of transgene loss and the strength of selection favoring these transgene-free, revertant individuals. The method requires data from serial transfer experiments in which the frequency of engineered genomes is monitored periodically. Simple mathematical models are developed that use these estimates to predict the half-life of the engineered transgene and provide quantitative predictions for how alterations to mutation and selection will influence longevity. The estimation method and predictive tools have been implemented as an interactive web application, MuSe.

Coalescence modeling of intrainfection Bacillus anthracis populations allows estimation of infection parameters in wild populations

Bacillus anthracis, the etiological agent of anthrax, is a well-established model organism. For B. anthracis and most other infectious diseases, knowledge regarding transmission and infection parameters in natural systems, in large part, comprises data gathered from closely controlled laboratory experiments. Fatal, natural anthrax infections transmit the bacterium through new host-pathogen contacts at carcass sites, which can occur years after death of the previous host. For the period between contact and death, all of our knowledge is based upon experimental data from domestic livestock and laboratory animals. Here we use a noninvasive method to explore the dynamics of anthrax infections, by evaluating the terminal diversity of B. anthracis in anthrax carcasses. We present an application of population genetics theory, specifically, coalescence modeling, to intrainfection populations of B. anthracis to derive estimates for the duration of the acute phase of the infection and effective population size converted to the number of colony-forming units establishing infection in wild plains zebra (Equus quagga). Founding populations are small, a few colony-forming units, and infections are rapid, lasting roughly between 1 d and 3 d in the wild. Our results closely reflect experimental data, showing that small founding populations progress acutely, killing the host within days. We believe this method is amendable to other bacterial diseases from wild, domestic, and human systems.

Compensatory mutations improve general permissiveness to antibiotic resistance plasmids

Horizontal gene transfer mediated by broad-host-range plasmids is an important mechanism of antibiotic resistance spread. While not all bacteria maintain plasmids equally well, plasmid persistence can improve over time, yet no general evolutionary mechanisms have emerged. Our goal was to identify these mechanisms and to assess if adaptation to one plasmid affects the permissiveness to others. We experimentally evolved Pseudomonas sp. H2 containing multidrug resistance plasmid RP4, determined plasmid persistence and cost using a joint experimental-modelling approach, resequenced evolved clones, and reconstructed key mutations. Plasmid persistence improved in fewer than 600 generations because the fitness cost turned into a benefit. Improved retention of naive plasmids indicated that the host evolved towards increased plasmid permissiveness. Key chromosomal mutations affected two accessory helicases and the RNA polymerase β-subunit. Our and other findings suggest that poor plasmid persistence can be caused by a high cost involving helicase-plasmid interactions that can be rapidly ameliorated.

Evolutionary Paths That Expand Plasmid Host-Range: Implications for Spread of Antibiotic Resistance

The World Health Organization has declared the emergence of antibiotic resistance to be a global threat to human health. Broad-host-range plasmids have a key role in causing this health crisis because they transfer multiple resistance genes to a wide range of bacteria. To limit the spread of antibiotic resistance, we need to gain insight into the mechanisms by which the host range of plasmids evolves. Although initially unstable plasmids have been shown to improve their persistence through evolution of the plasmid, the host, or both, the means by which this occurs are poorly understood. Here, we sought to identify the underlying genetic basis of expanded plasmid host-range and increased persistence of an antibiotic resistance plasmid using a combined experimental-modeling approach that included whole-genome resequencing, molecular genetics and a plasmid population dynamics model. In nine of the ten previously evolved clones, changes in host and plasmid each slightly improved plasmid persistence, but their combination resulted in a much larger improvement, which indicated positive epistasis. The only genetic change in the plasmid was the acquisition of a transposable element from a plasmid native to the Pseudomonas host used in these studies. The analysis of genetic deletions showed that the critical genes on this transposon encode a putative toxin-antitoxin (TA) and a cointegrate resolution system. As evolved plasmids were able to persist longer in multiple naïve hosts, acquisition of this transposon also expanded the plasmid's host range, which has important implications for the spread of antibiotic resistance.

Modelling conjugation with stochastic differential equations

Conjugation is an important mechanism involved in the transfer of resistance between bacteria. In this article a stochastic differential equation based model consisting of a continuous time state equation and a discrete time measurement equation is introduced to model growth and conjugation of two Enterococcus faecium strains in a rich exhaustible media. The model contains a new expression for a substrate dependent conjugation rate. A maximum likelihood based method is used to estimate the model parameters. Different models including different noise structure for the system and observations are compared using a likelihood-ratio test and Akaike's information criterion. Experiments indicating conjugation on the agar plates selecting for transconjugants motivates the introduction of an extended model, for which conjugation on the agar plate is described in the measurement equation. This model is compared to the model without plate conjugation. The modelling approach described in this article can be applied generally when modelling dynamical systems.

Resistance in bacteria of the food chain: epidemiology and control strategies

Bacteria have evolved multiple mechanisms for the efficient evolution and spread of antimicrobial resistance. Modern food production facilitates the emergence and spread of resistance through the intensive use of antimicrobial agents and international trade of both animals and food products. The main route of transmission between food animals and humans is via food products, although other modes of transmission, such as direct contact and through the environment, also occur. Resistance can spread as resistant pathogens or via transferable genes in different commensal bacteria, making quantification of the transmission difficult. The exposure of humans to antimicrobial resistance from food animals can be controlled by either limiting the selective pressure from antimicrobial usage or by limiting the spread of the bacteria/genes. A number of control options are reviewed, including drug licensing, removing financial incentives, banning or restricting the use of certain drugs, altering prescribers behavior, improving animal health, improving hygiene and implementing microbial criteria for certain types of resistant pathogens for use in the control of trade of both food animals and food.

The rising impact of mathematical modelling in epidemiology: antibiotic resistance research as a case study

Mathematical modelling of infectious diseases has gradually become part of public health decision-making in recent years. However, the developing status of modelling in epidemiology and its relationship with other relevant scientific approaches have never been assessed quantitatively. Herein, using antibiotic resistance as a case study, 60 published models were analysed. Their interactions with other scientific fields are reported and their citation impact evaluated, as well as temporal trends. The yearly number of antibiotic resistance modelling publications increased significantly between 1990 and 2006. This rise cannot be explained by the surge of interest in resistance phenomena alone. Moreover, modelling articles are, on average, among the most frequently cited third of articles from the journal in which they were published. The results of this analysis, which might be applicable to other emerging public health problems, demonstrate the growing interest in mathematical modelling approaches to evaluate antibiotic resistance.

Stability of a promiscuous plasmid in different hosts: no guarantee for a long-term relationship

Broad-host-range (BHR) IncP-1 plasmids have the ability to transfer between and replicate in nearly all species of the Alpha-, Beta- and Gammaproteobacteria, but surprisingly few data are available on the stability of these plasmids in strains within their host range. Moreover, even though molecular interactions between the bacterial host and its plasmid(s) exist, no systematic study to date has compared the stability of the same plasmid among different hosts. The goal of this study was to examine whether the stability characteristics of an IncP-1 plasmid can be variable between strains within the host range of the plasmid. Therefore, 19 strains within the Alpha-, Beta- or Gammaproteobacteria carrying the IncP-1beta plasmid pB10 were serially propagated in non-selective medium and the fraction of segregants was monitored through replica-picking. Remarkably, a large variation in the stability of pB10 in different strains was found, even between strains within the same genus or species. Ten strains showed no detectable plasmid loss over about 200 generations, and in two strains plasmid-free clones were only sporadically observed. In contrast, three strains, Pseudomonas koreensis R28, Pseudomonas putida H2 and Stenotrophomonas maltophilia P21, exhibited rapid plasmid loss within 80 generations. Parameter estimation after mathematical modelling of these stability data suggested high frequencies of segregation (about 0.04 per generation) or high plasmid cost (i.e. a relative fitness decrease in plasmid-bearing cells of about 15 and 40 %), which was confirmed experimentally. The models also suggested that plasmid reuptake by conjugation only played a significant role in plasmid stability in one of the three strains. Four of the 19 strains lost the plasmid very slowly over about 600 generations. The erratic decrease of the plasmid-containing fraction and simulation of the data with a new mathematical model suggested that plasmid cost was variable over time due to compensatory mutations. The findings of this study demonstrate that the ability of a so-called 'BHR' plasmid to persist in a bacterial population is influenced by strain-specific traits, and therefore observations made for one strain should not be generalized for the entire species or genus.

The population biology of bacterial plasmids: a hidden Markov model approach

Horizontal plasmid transfer plays a key role in bacterial adaptation. In harsh environments, bacterial populations adapt by sampling genetic material from a horizontal gene pool through self-transmissible plasmids, and that allows persistence of these mobile genetic elements. In the absence of selection for plasmid-encoded traits it is not well understood if and how plasmids persist in bacterial communities. Here we present three models of the dynamics of plasmid persistence in the absence of selection. The models consider plasmid loss (segregation), plasmid cost, conjugative plasmid transfer, and observation error. Also, we present a stochastic model in which the relative fitness of the plasmid-free cells was modeled as a random variable affected by an environmental process using a hidden Markov model (HMM). Extensive simulations showed that the estimates from the proposed model are nearly unbiased. Likelihood-ratio tests showed that the dynamics of plasmid persistence are strongly dependent on the host type. Accounting for stochasticity was necessary to explain four of seven time-series data sets, thus confirming that plasmid persistence needs to be understood as a stochastic process. This work can be viewed as a conceptual starting point under which new plasmid persistence hypotheses can be tested.

Combining mathematical models and statistical methods to understand and predict the dynamics of antibiotic-sensitive mutants in a population of resistant bacteria during experimental evolution

Temporarily discontinuing the use of antibiotics has been proposed as a means to eliminate resistant bacteria by allowing sensitive clones to sweep through the population. In this study, we monitored a tetracycline-sensitive subpopulation that emerged during experimental evolution of E. coli K12 MG1655 carrying the multiresistance plasmid pB10 in the absence of antibiotics. The fraction of tetracycline-sensitive mutants increased slowly over 500 generations from 0.1 to 7%, and loss of resistance could be attributed to a recombination event that caused deletion of the tet operon. To help understand the population dynamics of these mutants, three mathematical models were developed that took into consideration recurrent mutations, increased host fitness (selection), or a combination of both mechanisms (full model). The data were best explained by the full model, which estimated a high mutation frequency (lambda = 3.11 x 10(-5)) and a significant but small selection coefficient (sigma = 0.007). This study emphasized the combined use of experimental data, mathematical models, and statistical methods to better understand and predict the dynamics of evolving bacterial populations, more specifically the possible consequences of discontinuing the use of antibiotics.