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The optimal drug combination depends on the length of the experiment. The optimal drug combination as a function of the length of the experiment: the colour gradient represents the normalized cumulative AUC inhibition (white represents maximum inhibition, and black the minimum inhibition) of E. coli MG4100. The dotted line corresponds to the optimal drug proportions as a function of the length of the experiment. (a) Simulations of the theoretical, two-locus, two-allele model from (3.6) using parameters in table 2. (b) Empirical data: each circle represents a different drug proportion tested experimentally. Note how, both in the experiment and the model, at the beginning of the experiment the optimal treatment corresponds to the combination treatment that maximizes synergy, but for longer treatments the model and the data coincide in the sense that the optimal strategy uses more and more erythromycin. In an experiment of 5 days, it would have been better not to combine the drugs at all, at least according to this optimality measure. (Online version in colour.)
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Mathematically speaking, it is self-evident that the optimal control of complex, dynamical systems with many interacting components cannot be achieved with 'non-responsive' control strategies that are constant through time. Although there are notable exceptions, this is usually how we design treatments with antimicrobial drugs when we give the same...
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... now address the behaviour of this function as measured experimentally using a batch-transfer protocol as N grows. Figure 8 is a representation of an empirical and a theoretical dataset, giving a comparison of the optimal combination obtained experimentally (figure 8b) with the optimal combi- nation computed using the two-locus, two-allele model defined by equations (3.6a-d) (figure 8a). As expected from the synergy, in both cases the optimal 1-day drug proportion corresponds to the drug combination that maximizes synergy, but the optimal drug proportion changes when we treat for longer durations. ...
Citations
... As also mentioned in [7,12], examples of experimental studies on rare selection can be found in [9,26]. In [9], lizards with long fingers can hold on stronger and thus avoid being blown away whenever their habitat is hit by a hurricane, which provides them a strong but temporary selective advantage. ...
... In [9], lizards with long fingers can hold on stronger and thus avoid being blown away whenever their habitat is hit by a hurricane, which provides them a strong but temporary selective advantage. Further, [26] compares different antibiotic treatment strategies against a bacterial population. Here, the analogue of a continuously present but weak selective pressure is a constant administration of the antibiotic in low concentration dosage, while rare and strong selective events correspond to a less frequent inoculation with higher dosages (possibly of varying concentration and at random times). ...
... Erythromycin is deployed clinically against Gram negative bacteria but it is not used to treat E. coli, although it likely encounters erythromycin as an unintended side-effect. So whereas our study lacks a clinical context, erythromycin is helpful for quantifying the evolutionary basis of the inverted-U because it is a substrate of the efflux pump AcrAB-TolC found in the genome of the E. coli (George and Levy 1983;Pena-Miller et al. 2014;Bergmiller et al. 2017). Moreover, E. coli is known to exhibit acrAB amplification mutants with decreased sensitivity to erythromycin that yield evolutionary genomic data from short-term treatments (Laehnemann et al. 2014;Pena-Miller et al. 2014). ...
... So whereas our study lacks a clinical context, erythromycin is helpful for quantifying the evolutionary basis of the inverted-U because it is a substrate of the efflux pump AcrAB-TolC found in the genome of the E. coli (George and Levy 1983;Pena-Miller et al. 2014;Bergmiller et al. 2017). Moreover, E. coli is known to exhibit acrAB amplification mutants with decreased sensitivity to erythromycin that yield evolutionary genomic data from short-term treatments (Laehnemann et al. 2014;Pena-Miller et al. 2014). Amplification of the acrAB operon therefore provides a signal of genomic change that can be quantified from deep sequencing data of shortterm erythromycin treatments. ...
... Competitive release (Wargo et al. 2007) has been invoked to explain nonlinear patterns in drug resistance (Pena-Miller et al. 2014) and it may be relevant to understanding nonmonotone dose-responses (figs. 1C, D, and 2A). ...
To determine the dosage at which antibiotic resistance evolution is most rapid, we treated Escherichia coli in vitro, deploying the antibiotic erythromycin at dosages ranging from zero to high. Adaptation was fastest just below erythromycin’s minimal inhibitory concentration (MIC) and genotype-phenotype correlations determined from whole genome sequencing revealed the molecular basis: simultaneous selection for copy number variation in 3 resistance mechanisms which exhibited an ‘inverted-U’ pattern of dose-dependence, as did several insertion sequences and an integron. Many genes did not conform to this pattern, however, reflecting changes in selection as dose increased: putative media adaptation polymorphisms at zero antibiotic dosage gave way to drug target (ribosomal RNA operon) amplification at mid dosages whereas prophage-mediated drug efflux amplifications dominated at the highest dosages. All treatments exhibited E. coli increases in the copy number of efflux operons acrAB and emrE at rates that correlated with increases in population density. For strains where the inverted-U was no longer observed following the genetic manipulation of acrAB, it could be recovered by prolonging the antibiotic treatment at sub-MIC dosages.
... More recently, growth rates [8] have been implemented as a direct measurement of fitness [9] but without direct experimental connection to susceptibility testing results. While this method has been rapidly catching on, and its results are being used to answer important questions about evolution [10], there are few studies investigating the effects of this change in methodology upon the data and results to which we now have access. In this study, we investigate the correlation of growth rate assays to susceptibility testing and find that growth rates are a much more sensitive method for assessing fitness in bacteria. ...
Non-technical summary:
Antibiotic resistance is a global human health problem. We partnered with Dignity Health Mercy Medical Center to study antibiotic resistance in clinical isolates. We tested whether growth rates, a sensitive assay used to measure the fitness of bacterial samples, correlate with a clinical test to measure antibiotic resistance. We found a strong correlation between these two methods suggesting that growth rates could be reliably applied to evolutionary studies of clinically relevant problems. Moreover, the sensitivity of the growth rates assay enabled us to identify fitness effects of specific antibiotic resistance genes.
... [5,7,10,20,21,22]). A concrete example is given in [36], where different antibiotic treatment strategies against a bacterial population are compared. There it is proved that a constant administration of antibiotics is not optimal, and that the best treatment strategies depend on the length of treatment. ...
Consider a two-type Moran population of size $N$ subject to selection and mutation, which is immersed in a varying environment.
The population is susceptible to exceptional changes in the environment, which accentuate the selective advantage of the fit individuals. In this setting, we show that the type composition in the population is continuous with respect to the environment. This allows us to replace the deterministic environment by a random one, which is driven by a subordinator. Assuming that selection, mutation and the environment are weak in relation to $N$, we show that, the type-frequency process, with time speed up by $N$, converges as $N\to\infty$ to a Wright--Fisher-type SDE with a jump term modeling the effect of the environment. Next, we study the asymptotic behavior of the limiting model in the far future and in the distant past, both in the annealed and in the quenched setting. Our approach builds on the genealogical picture behind the model. The latter is described by means of an extension of the ancestral selection graph (ASG). The formal relation between forward and backward objects is given in the form of a moment duality between the type-frequency process and the line-counting process of a pruned version of the ASG. This relation yields characterizations of the annealed and the quenched moments of the asymptotic type distribution. A more involved pruning of the ASG allows us to obtain annealed and quenched results for the ancestral type distribution. In the absence of mutations, one of the types fixates and our results yield expressions for the fixation probabilities.
... An interesting observation obtained by comparing figure 1c,d, is that the MIC of the co-culture corresponds to the MIC of the resistant subpopulation, while the susceptible strain has, by definition, a lower MIC. Interestingly, in our experiments, bacterial density seems to be maximized at intermediate drug concentrations, a feature that has been reported previously [40] and can have many causes, including B r growing without competition at intermediate drug concentrations [41]. Indeed, figure 1c,d shows that there is a range of concentrations where B s has already gone extinct, but the concentration is not high enough to suppress growth of the resistant subpopulation. ...
The current crisis of antimicrobial resistance in clinically relevant pathogens has highlighted our limited understanding of the ecological and evolutionary forces that drive drug resistance adaptation. For instance, although human tissues are highly heterogeneous, most of our mechanistic understanding about antibiotic resistance evolution is based on constant and well-mixed environmental conditions. A consequence of considering spatial heterogeneity is that, even if antibiotics are prescribed at high dosages, the penetration of drug molecules through tissues inevitably produces antibiotic gradients, exposing bacterial populations to a range of selective pressures and generating a dynamic fitness landscape that changes in space and time. In this paper, we will use a combination of mathematical modelling and computer simulations to study the population dynamics of susceptible and resistant strains competing for resources in a network of micro-environments with varying degrees of connectivity. Our main result is that highly connected environments increase diffusion of drug molecules, enabling resistant phenotypes to colonize a larger number of spatial locations. We validated this theoretical result by culturing fluorescently labelled Escherichia coli in 3D-printed devices that allow us to control the rate of diffusion of antibiotics between neighbouring compartments and quantify the spatio-temporal distribution of resistant and susceptible bacterial cells.
... It is, however, less obvious if rare but strong selective events put type 0 more at risk of extinction than a small but constant-in-time selective pressure. The problem is reminiscent of similar questions arising in experimental biology, where, for example, some detrimental substance (antibiotic) is inoculated in a population of bacteria, and there is an interest in determining whether a constant administration of the substance in low concentration dosage is more effective in wiping out the population, than a more occasional inoculation with higher dosages of varying concentration [35]. This paper aims to quantify how big and frequent cataclysms must be in order to wipe out a family as effectively as constant weak selection in a steady environment. ...
We analyse a family of Wright Fisher models with selection in a random environment and skewed offspring distribution. We provide a calculable criterion to quantify the strength of different shapes of selection, and thus compare them. The main mathematical tool is duality, which we prove to hold, also in presence of random environment (quenched and in some cases annealed), between the population's allele frequencies and genealogy, both in the case finite population size and in the scaling limit for large size.
... The use of suboptimal antibiotic dosages, as well as excessive dosages, increase selection of resistant strains (Odenholt et al. 2003;Baquero et al. 2008;Gullberg et al. 2011); mathematical modelling methods are being explored to investigate optimal doses and durations (e.g. Bonhoeffer et al. 1997;Bergstrom et al. 2004;D'Agata et al. 2008;Geli et al. 2012;Peña-Miller et al. 2014;Paterson et al. 2016). ...
In this review, we discuss mechanisms of resistance identified in bacterial agents Staphylococcus aureus and the enterococci towards two priority classes of antibiotics—the fluoroquinolones and the glycopeptides. Members of both classes interact with a number of components in the cells of these bacteria, so the cellular targets are also considered. Fluoroquinolone resistance mechanisms include efflux pumps (MepA, NorA, NorB, NorC, MdeA, LmrS or SdrM in S. aureus and EfmA or EfrAB in the enterococci) for removal of fluoroquinolone from the intracellular environment of bacterial cells and/or protection of the gyrase and topoisomerase IV target sites in Enterococcus faecalis by Qnr-like proteins. Expression of efflux systems is regulated by GntR-like (S. aureus NorG), MarR-like (MgrA, MepR) regulators or a two-component signal transduction system (TCS) (S. aureus ArlSR). Resistance to the glycopeptide antibiotic teicoplanin occurs via efflux regulated by the TcaR regulator in S. aureus. Resistance to vancomycin occurs through modification of the D-Ala-D-Ala target in the cell wall peptidoglycan and removal of high affinity precursors, or by target protection via cell wall thickening. Of the six Van resistance types (VanA-E, VanG), the VanA resistance type is considered in this review, including its regulation by the VanSR TCS. We describe the recent application of biophysical approaches such as the hydrodynamic technique of analytical ultracentrifugation and circular dichroism spectroscopy to identify the possible molecular effector of the VanS receptor that activates expression of the Van resistance genes; both approaches demonstrated that vancomycin interacts with VanS, suggesting that vancomycin itself (or vancomycin with an accessory factor) may be an effector of vancomycin resistance. With 16 and 19 proteins or protein complexes involved in fluoroquinolone and glycopeptide resistances, respectively, and the complexities of bacterial sensing mechanisms that trigger and regulate a wide variety of possible resistance mechanisms, we propose that these antimicrobial resistance mechanisms might be considered complex ‘nanomachines’ that drive survival of bacterial cells in antibiotic environments.
... Theoretical modelling has been employed to understand bacterial evolution from a systems perspective. Various models including phamacodynamics 15,16 , population genetics 17,18 and population dynamics 19,20 models have been developed to examine results obtained from experimental studies. Antibiotic cycling has also been studied in clinical settings for over 30 years, particularly cycling of aminoglycosides was widely studied due to increasing drug resistance mediated by plasmids carrying aminoglycoside enzymes 21,22 . ...
Multi-drug strategies have been attempted to prolong the efficacy of existing antibiotics, but with limited success. Here we show that the evolution of multi-drug-resistant Escherichia coli can be manipulated in vitro by administering pairs of antibiotics and switching between them in ON/OFF manner. Using a multiplexed cell culture system, we find that switching between certain combinations of antibiotics completely suppresses the development of resistance to one of the antibiotics. Using this data, we develop a simple deterministic model, which allows us to predict the fate of multi-drug evolution in this system. Furthermore, we are able to reverse established drug resistance based on the model prediction by modulating antibiotic selection stresses. Our results support the idea that the development of antibiotic resistance may be potentially controlled via continuous switching of drugs.
... As the threat of antibiotic resistance spreads the need to optimise antibiotic dosage regimens becomes essential. Mathematical modelling is increasingly being used to investigate optimal treatment regimens for antibiotic therapy [13][14][15][16][17] . However, these studies either omit pharmacodynamic data, by assuming that the antibiotic induced death rate is constant; or only analyse a very limited number of alternative treatment regimens. ...
The increase in antibiotic resistant bacteria poses a threat to the continued use of antibiotics to treat bacterial infections. The overuse and misuse of antibiotics has been identified as a significant driver in the emergence of resistance. Finding optimal treatment regimens is therefore critical in ensuring the prolonged effectiveness of these antibiotics. This study uses mathematical modelling to analyse the effect traditional treatment regimens have on the dynamics of a bacterial infection. Using a novel approach, a genetic algorithm, the study then identifies improved treatment regimens. Using a single antibiotic the genetic algorithm identifies regimens which minimise the amount of antibiotic used while maximising bacterial eradication. Although exact treatments are highly dependent on parameter values and initial bacterial load, a significant common trend is identified throughout the results. A treatment regimen consisting of a high initial dose followed by an extended tapering of doses is found to optimise the use of antibiotics. This consistently improves the success of eradicating infections, uses less antibiotic than traditional regimens and reduces the time to eradication. The use of genetic algorithms to optimise treatment regimens enables an extensive search of possible regimens, with previous regimens directing the search into regions of better performance.
