Modelling conjugation with stochastic differential equations

Estimating the transfer rates of bacterial plasmids with an adapted Luria–Delbrück fluctuation analysis

To increase our basic understanding of the ecology and evolution of conjugative plasmids, we need reliable estimates of their rate of transfer between bacterial cells. Current assays to measure transfer rate are based on deterministic modeling frameworks. However, some cell numbers in these assays can be very small, making estimates that rely on these numbers prone to noise. Here, we take a different approach to estimate plasmid transfer rate, which explicitly embraces this noise. Inspired by the classic fluctuation analysis of Luria and Delbrück, our method is grounded in a stochastic modeling framework. In addition to capturing the random nature of plasmid conjugation, our new methodology, the Luria–Delbrück method (“LDM”), can be used on a diverse set of bacterial systems, including cases for which current approaches are inaccurate. A notable example involves plasmid transfer between different strains or species where the rate that one type of cell donates the plasmid is not equal to the rate at which the other cell type donates. Asymmetry in these rates has the potential to bias or constrain current transfer estimates, thereby limiting our capabilities for estimating transfer in microbial communities. In contrast, the LDM overcomes obstacles of traditional methods by avoiding restrictive assumptions about growth and transfer rates for each population within the assay. Using stochastic simulations and experiments, we show that the LDM has high accuracy and precision for estimation of transfer rates compared to the most widely used methods, which can produce estimates that differ from the LDM estimate by orders of magnitude.

Plasmid transfer can often spread resistance between important clinical pathogens. This study shows that widely used methods can lead to biased estimates of plasmid transfer rate by several orders of magnitude, and presents a new approach, inspired by the classic Luria-Delbrück approach, for accurately assessing this fundamental rate parameter

Data mining and mathematical models in cancer prognosis and prediction

Cancer is a fetal and complex disease. Individual differences of the same cancer type or the same patient at different stages of cancer development may require distinct treatments. Pathological differences are reflected in tissues, cells and gene levels etc. The interactions between the cancer cells and nearby microenvironments can also influence the cancer progression and metastasis. It is a huge challenge to understand all of these mechanistically and quantitatively. Researchers applied pattern recognition algorithms such as machine learning or data mining to predict cancer types or classifications. With the rapidly growing and available computing powers, researchers begin to integrate huge data sets, multi-dimensional data types and information. The cells are controlled by the gene expressions determined by the promoter sequences and transcription regulators. For example, the changes in the gene expression through these underlying mechanisms can modify cell progressing in the cell-cycle. Such molecular activities can be governed by the gene regulations through the underlying gene regulatory networks, which are essential for cancer study when the information and gene regulations are clear and available. In this review, we briefly introduce several machine learning methods of cancer prediction and classification which include Artificial Neural Networks (ANNs), Decision Trees (DTs), Support Vector Machine (SVM) and naive Bayes. Then we describe a few typical models for building up gene regulatory networks such as Correlation, Regression and Bayes methods based on available data. These methods can help on cancer diagnosis such as susceptibility, recurrence, survival etc. At last, we summarize and compare the modeling methods to analyze the development and progression of cancer through gene regulatory networks. These models can provide possible physical strategies to analyze cancer progression in a systematic and quantitative way.

Estimating plasmid conjugation rates: A new computational tool and a critical comparison of methods

Plasmids are important vectors for the spread of genes among diverse populations of bacteria. However, there is no standard method to determine the rate at which they spread horizontally via conjugation. Here, we compare commonly used methods on simulated and experimental data, and show that the resulting conjugation rate estimates often depend strongly on the time of measurement, the initial population densities, or the initial ratio of donor to recipient populations. Differences in growth rate, e.g. induced by sub-lethal antibiotic concentrations or temperature, can also significantly bias conjugation rate estimates. We derive a new 'end-point' measure to estimate conjugation rates, which extends the well-known Simonsen method to include the effects of differences in population growth and conjugation rates from donors and transconjugants. We further derive analytical expressions for the parameter range in which these approximations remain valid. We present an easy to use R package and web interface which implement both new and previously existing methods to estimate conjugation rates. The result is a set of tools and guidelines for accurate and comparable measurement of plasmid conjugation rates.

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.

The ecology of plasmid-coded antibiotic resistance: a basic framework for experimental research and modeling

Many antibiotic resistance genes are associated with plasmids. The ecological success of these mobile genetic elements within microbial communities depends on varying mechanisms to secure their own propagation, not only on environmental selection. Among the most important are the cost of plasmids and their ability to be transferred to new hosts through mechanisms such as conjugation. These are regulated by dynamic control systems of the conjugation machinery and genetic adaptations that plasmid-host pairs can acquire in coevolution. However, in complex communities, these processes and mechanisms are subject to a variety of interactions with other bacterial species and other plasmid types. This article summarizes basic plasmid properties and ecological principles particularly important for understanding the persistence of plasmid-coded antibiotic resistance in aquatic environments. Through selected examples, it further introduces to the features of different types of simulation models such as systems of ordinary differential equations and individual-based models, which are considered to be important tools to understand these complex systems. This ecological perspective aims to improve the way we study and understand the dynamics, diversity and persistence of plasmids and associated antibiotic resistance genes.

Conjugative plasmids enable the maintenance of low cost non-transmissible plasmids

Some plasmids can be transferred by conjugation to other bacterial hosts. But almost half of the plasmids are non-transmissible. These plasmid types can only be transmitted to the daughter cells of their host after bacterial fission. Previous studies suggest that non-transmissible plasmids become extinct in the absence of selection of their encoded traits, as plasmid-free bacteria are more competitive. Here, we aim to identify mechanisms that enable non-transmissible plasmids to persist, even if they are not beneficial. For this purpose, an individual-based model for plasmid population dynamics was set up and carefully tested for structural consistency and plausibility. Our results demonstrate that non-transmissible plasmids can be stably maintained in a population, even if they impose a substantial burden on their host cells growth. A prerequisite is the co-occurrence of an incompatible and costly conjugative plasmid type, which indirectly facilitates the preservation of the non-transmissible type. We suggest that this constellation might be considered as a potential mechanism maintaining plasmids and associated antibiotic resistances. It should be investigated in upcoming laboratory experiments.

Predictive Modeling of a Batch Filter Mating Process

Quantitative characterizations of horizontal gene transfer are needed to accurately describe gene transfer processes in natural and engineered systems. A number of approaches to the quantitative description of plasmid conjugation have appeared in the literature. In this study, we seek to extend that work, motivated by the question of whether a mathematical model can accurately predict growth and conjugation dynamics in a batch process. We used flow cytometry to make time-point observations of a filter-associated mating between two E. coli strains, and fit ordinary differential equation models to the data. A model comparison analysis identified the model formulation that is best supported by the data. Identifiability analysis revealed that the parameters were estimated with acceptable accuracy. The predictive power of the model was assessed by comparison with test data that demanded extrapolation from the training experiments. This study represents the first attempt to assess the quality of model predictions for plasmid conjugation. Our successful application of this approach lays a foundation for predictive modeling that can be used both in the study of natural plasmid transmission and in model-based design of engineering approaches that employ conjugation, such as plasmid-mediated bioaugmentation.

Modeling the growth dynamics of multiple Escherichia coli strains in the pig intestine following intramuscular ampicillin treatment

Background

This study evaluated how dosing regimen for intramuscularly-administered ampicillin, composition of Escherichia coli strains with regard to ampicillin susceptibility, and excretion of bacteria from the intestine affected the level of resistance among Escherichia coli strains in the intestine of nursery pigs. It also examined the dynamics of the composition of bacterial strains during and after the treatment. The growth responses of strains to ampicillin concentrations were determined using in vitro growth curves. Using these results as input data, growth predictions were generated using a mathematical model to simulate the competitive growth of E. coli strains in a pig intestine under specified plasma concentration profiles of ampicillin.

Results

In vitro growth results demonstrated that the resistant strains did not carry a fitness cost for their resistance, and that the most susceptible strains were more affected by increasing concentrations of antibiotics that the rest of the strains. The modeling revealed that short treatment duration resulted in lower levels of resistance and that dosing frequency did not substantially influence the growth of resistant strains. Resistance levels were found to be sensitive to the number of competing strains, and this effect was enhanced by longer duration of treatment. High excretion of bacteria from the intestine favored resistant strains over sensitive strains, but at the same time it resulted in a faster return to pre-treatment levels after the treatment ended. When the duration of high excretion was set to be limited to the treatment time (i.e. the treatment was assumed to result in a cure of diarrhea) resistant strains required longer time to reach the previous level.

Conclusion

No fitness cost was found to be associated with ampicillin resistance in E. coli. Besides dosing factors, epidemiological factors (such as number of competing strains and bacterial excretion) influenced resistance development and need to be considered further in relation to optimal treatment strategies. The modeling approach used in the study is generic, and could be used for prediction of the effect of treatment with other drugs and other administration routes for effect on resistance development in the intestine of pigs.

Electronic supplementary material

The online version of this article (doi:10.1186/s12866-016-0823-3) contains supplementary material, which is available to authorized users.

Evaluating the effect of horizontal transmission on the stability of plasmids under different selection regimes

In theory, plasmids can only be maintained in a population when the rate of horizontal gene transfer is larger than the combined effect of segregational loss and the decrease of fitness associated with plasmid carriage. Recent advances in genome sequencing have shown, however, that a large fraction of plasmids do not carry the genes necessary for conjugation or mobilization. So, how are so-called non-transmissible plasmids able to persist? In order to address this question, we examined a previously published evolutionary model based on the interaction between P. aeruginosa and the non-transmissible plasmid pNUK73. Both our in silico and in vitro results demonstrated that, although compensatory adaptation can decrease the rate of plasmid decay, the conditions for the maintenance of a non-transmissible plasmid are very stringent if the genes it carries are not beneficial to the bacterial host. This result suggests that apparently non-transmissible plasmids may still experience episodes of horizontal gene transfer occurring at very low frequencies, and that these scattered transmission events are sufficient to stabilize these plasmids. We conclude by discussing different genomic and microbiological approaches that could allow for the detection of these rare transmission events and thus to obtain a reliable estimate of the rate of horizontal gene transfer.

Positive selection and compensatory adaptation interact to stabilize non-transmissible plasmids

Plasmids are important drivers of bacterial evolution, but it is challenging to understand how plasmids persist over the long term because plasmid carriage is costly. Classical models predict that horizontal transfer is necessary for plasmid persistence, but recent work shows that almost half of plasmids are non-transmissible. Here we use a combination of mathematical modelling and experimental evolution to investigate how a costly, non-transmissible plasmid, pNUK73, can be maintained in populations of Pseudomonas aeruginosa. Compensatory adaptation increases plasmid stability by eliminating the cost of plasmid carriage. However, positive selection for plasmid-encoded antibiotic resistance is required to maintain the plasmid by offsetting reductions in plasmid frequency due to segregational loss. Crucially, we show that compensatory adaptation and positive selection reinforce each other’s effects. Our study provides a new understanding of how plasmids persist in bacterial populations, and it helps to explain why resistance can be maintained after antibiotic use is stopped.

Plasmids are important for bacterial evolution but the evolutionary mechanisms behind their maintenance are unclear. Here the authors show that the interplay between compensatory adaptation and positive selection for plasmid-encoded antibiotic resistance determines plasmid persistence in P. aeruginosa.

How fitness reduced, antimicrobial resistant bacteria survive and spread: a multiple pig-multiple bacterial strain model

More than 30% of E. coli strains sampled from pig farms in Denmark over the last five years were resistant to the commonly used antimicrobial tetracycline. This raises a number of questions: How is this high level sustained if resistant bacteria have reduced growth rates? Given that there are multiple susceptible and resistant bacterial strains in the pig intestines, how can we describe their coexistence? To what extent does the composition of these multiple strains in individual pigs influence the total bacterial population of the pig pen? What happens to a complex population when antimicrobials are used? To investigate these questions, we created a model where multiple strains of bacteria coexist in the intestines of pigs sharing a pen, and explored the parameter limits of a stable system; both with and without an antimicrobial treatment. The approach taken is a deterministic bacterial population model with stochastic elements of bacterial distributions and transmission. The rates that govern the model are process-oriented to represent growth, excretion, and uptake from environment, independent of herd and meta-population structures. Furthermore, an entry barrier and elimination process for the individual strains in each pig were implemented. We demonstrate how competitive growth between multiple bacterial strains in individual pigs, and the transmission between pigs in a pen allow for strains of antimicrobial resistant bacteria to persist in a pig population to different extents, and how quickly they can become dominant if antimicrobial treatment is initiated. The level of spread depends in a non-linear way of the parameters that govern excretion and uptake. Furthermore, the sampling of initial distributions of strains and stochastic transmission events give rise to large variation in how homogenous and how resistant the bacterial population becomes. Most important: resistant bacteria are demonstrated to survive with a disadvantage in growth rate of well over 10%.

Horizontal gene transfer dynamics and distribution of fitness effects during microbial in silico evolution

Background

Horizontal gene transfer (HGT) is a process that facilitates the transfer of genetic material between organisms that are not directly related, and thus can affect both the rate of evolution and emergence of traits. Recent phylogenetic studies reveal HGT events are likely ubiquitous in the Tree of Life. However, our knowledge of HGT's role in evolution and biological organization is very limited, mainly due to the lack of ancestral evolutionary signatures and the difficulty to observe complex evolutionary dynamics in a laboratory setting. Here, we utilize a multi-scale microbial evolution model to comprehensively study the effect of HGT on the evolution of complex traits and organization of gene regulatory networks.

Results

Large-scale simulations reveal a distinct signature of the Distribution of Fitness Effect (DFE) for HGT events: during evolution, while mutation fitness effects become more negative and neutral, HGT events result in a balanced effect distribution. In either case, lethal events are significantly decreased during evolution (33.0% to 3.2%), a clear indication of mutational robustness. Interestingly, evolution was accelerated when populations were exposed to correlated environments of increasing complexity, especially in the presence of HGT, a phenomenon that warrants further investigation. High HGT rates were found to be disruptive, while the average transferred fragment size was linked to functional module size in the underlying biological network. Network analysis reveals that HGT results in larger regulatory networks, but with the same sparsity level as those evolved in its absence. Observed phenotypic variability and co-existing solutions were traced to individual gain/loss of function events, while subsequent re-wiring after fragment integration was necessary for complex traits to emerge.

Development of a restricted state space stochastic differential equation model for bacterial growth in rich media

In the present study, bacterial growth in a rich media is analysed in a Stochastic Differential Equation (SDE) framework. It is demonstrated that the SDE formulation and smoothened state estimates provide a systematic framework for data driven model improvements, using random walk hidden states. Bacterial growth is limited by the available substrate and the inclusion of diffusion must obey this natural restriction. By inclusion of a modified logistic diffusion term it is possible to introduce a diffusion term flexible enough to capture both the growth phase and the stationary phase, while concentration is restricted to the natural state space (substrate and bacteria non-negative). The case considered is the growth of Salmonella and Enterococcus in a rich media. It is found that a hidden state is necessary to capture the lag phase of growth, and that a flexible logistic diffusion term is needed to capture the random behaviour of the growth model. Further, it is concluded that the Monod effect is not needed to capture the dynamics of bacterial growth in the data presented.