Drugs are commonly used in combinations larger than two for treating bacterial infection. However, it is generally impossible to infer directly from the effects of individual drugs the net effect of a multidrug combination. Here we develop a mechanism-independent method for predicting the microbial growth response to combinations of more than two drugs. Performing experiments in both Gram-negative ( Escherichia coli) and Gram-positive ( Staphylococcus aureus ) bacteria, we demonstrate that for a wide range of drugs, the bacterial responses to drug pairs are sufficient to infer the effects of larger drug combinations. To experimentally establish the broad applicability of the method, we use drug combinations comprising protein synthesis inhibitors (macrolides, aminoglycosides, tetracyclines, lincosamides, and chloramphenicol), DNA synthesis inhibitors (fluoroquinolones and quinolones), folic acid synthesis inhibitors (sulfonamides and diaminopyrimidines), cell wall synthesis inhibitors, polypeptide antibiotics, preservatives, and analgesics. Moreover, we show that the microbial responses to these drug combinations can be predicted using a simple formula that should be widely applicable in pharmacology. These findings offer a powerful, readily accessible method for the rational design of candidate therapies using combinations of more than two drugs. In addition, the accurate predictions of this framework raise the question of whether the multidrug response in bacteria obeys statistical, rather than chemical, laws for combinations larger than two.

Evolved resistance to one antibiotic may be associated with "collateral" sensitivity to other drugs. Here, we provide an extensive quantitative characterization of collateral effects in Enterococcus faecalis, a gram-positive opportunistic pathogen. By combining parallel experimental evolution with high-throughput dose-response measurements, we measure phenotypic profiles of collateral sensitivity and resistance for a total of 900 mutant–drug combinations. We find that collateral effects are pervasive but difficult to predict because independent populations selected by the same drug can exhibit qualitatively different profiles of collateral sensitivity as well as markedly different fitness costs. Using whole-genome sequencing of evolved populations, we identified mutations in a number of known resistance determinants, including mutations in several genes previously linked with collateral sensitivity in other species. Although phenotypic drug sensitivity profiles show significant diversity, they cluster into statistically similar groups characterized by selecting drugs with similar mechanisms. To exploit the statistical structure in these resistance profiles, we develop a simple mathematical model based on a stochastic control process and use it to design optimal drug policies that assign a unique drug to every possible resistance profile. Stochastic simulations reveal that these optimal drug policies outperform intuitive cycling protocols by maintaining long-term sensitivity at the expense of short-term periods of high resistance. The approach reveals a new conceptual strategy for mitigating resistance by balancing short-term inhibition of pathogen growth with infrequent use of drugs intended to steer pathogen populations to a more vulnerable future state. Experiments in laboratory populations confirm that model-inspired sequences of four drugs reduce growth and slow adaptation relative to naive protocols involving the drugs alone, in pairwise cycles, or in a four-drug uniform cycle.

The inoculum effect (IE) is an increase in the minimum inhibitory concentration (MIC) of an antibiotic as a function of the initial size of a microbial population. The IE has been observed in a wide range of bacteria, implying that antibiotic efficacy may depend on population density. Such density dependence could have dramatic effects on bacterial population dynamics and potential treatment strategies, but explicit measures of per capita growth as a function of density are generally not available. Instead, the IE measures MIC as a function of initial population size, and population density changes by many orders of magnitude on the timescale of the experiment. Therefore, the functional relationship between population density and antibiotic inhibition is generally not known, leaving many questions about the impact of the IE on different treatment strategies unanswered. To address these questions, here we directly measured real-time per capita growth of Enterococcus faecalis populations exposed to antibiotic at fixed population densities using multiplexed computer-automated culture devices. We show that density-dependent growth inhibition is pervasive for commonly used antibiotics, with some drugs showing increased inhibition and others decreased inhibition at high densities. For several drugs, the density dependence is mediated by changes in extracellular pH, a community-level phenomenon not previously linked with the IE. Using a simple mathematical model, we demonstrate how this density dependence can modulate population dynamics in constant drug environments. Then, we illustrate how time-dependent dosing strategies can mitigate the negative effects of density-dependence. Finally, we show that these density effects lead to bistable treatment outcomes for a wide range of antibiotic concentrations in a pharmacological model of antibiotic treatment. As a result, infections exceeding a critical density often survive otherwise effective treatments.

We present the simplest discrete model to date that leads to synchronization of stochastic phasecoupled oscillators. In the mean field limit, the model exhibits a Hopf bifurcation and global oscillatory behavior as coupling crosses a critical value. When coupling between units is strictly local, the model undergoes a continuous phase transition which we characterize numerically using finite-size scaling analysis. In particular, the onset of global synchrony is marked by signatures of the XY universality class, including the appropriate classical exponents β and ν, a lower critical dimension d lc = 2, and an upper critical dimension duc = 4.PACS numbers: 64.60. Ht, 05.45.Xt, In the early 1960's, experiments with the BelousovZhabotinsky reaction created a sensation by showing that dissipative structures and self-organization in systems far from equilibrium correspond to real observable physical phenomena. Since then, the breaking of time translational symmetry has been a central theme in the analysis of nonlinear nonequilibrium systems. However, in the later studies of spatially distributed systems, most of the interest shifted to pattern forming instabilities, and surprisingly little attention was devoted to the question of bulk oscillation and the required spatial frequency and phase synchronization. On the other hand, the emergence of phase synchronization in populations of globally coupled phase oscillators, with the synchronous firing of fireflies as one of the spectacular examples, did generate intense interest [1]. Because intrinsically oscillating units with slightly different eigenfrequencies underlie the macroscopic behavior of an extensive range of biological, chemical, and physical systems, a great deal of literature has focused on the mathematical principles governing the competition between individual oscillatory tendencies and synchronous cooperation [2]. While most studies have focused on globally coupled units, leading to a mature understanding of the mean field behavior of several models, relatively little work has examined populations of oscillators in the locally coupled regime [3]. The description of emergent synchrony has largely been limited to small-scale and/or globally-coupled deterministic systems [2], despite the fact that the dynamics of the physical systems in question likely reflect a combination of finite-range forces and stochasticity. Two recent studies by Risler et al. [4] represent notable exceptions to this trend. They provide analytical evidence that locally-coupled identical noisy oscillators belong to the XY universality class, though to date there had been no empirical verification, numerical or otherwise, of their predictions.The difficulty with existing models of locally coupled oscillators is that each is typically described by a nonlinear differential equation, and it is notoriously computationally intensive to deal with systems of coupled nonlinear differential equations, especially if they also involve a stochastic component. However, following Landau theory [5], mac...

Synchronization of stochastic phase-coupled oscillators is known to occur but difficult to characterize because sufficiently complete analytic work is not yet within our reach, and thorough numerical description usually defies all resources. We present a discrete model that is sufficiently simple to be characterized in meaningful detail. In the mean field limit, the model exhibits a supercritical Hopf bifurcation and global oscillatory behavior as coupling crosses a critical value. When coupling between units is strictly local, the model undergoes a continuous phase transition which we characterize numerically using finite-size scaling analysis. In particular, we explicitly rule out multistability and show that that the onset of global synchrony is marked by signatures of the XY universality class. Our numerical results cover dimensions d = 2, 3, 4, and 5 and lead to the appropriate XY classical exponents β and ν, a lower critical dimension d lc = 2, and an upper critical dimension duc = 4.

BackgroundEfflux is a widespread mechanism of reversible drug resistance in bacteria that can be triggered by environmental stressors, including many classes of drugs. While such chemicals when used alone are typically toxic to the cell, they can also induce the efflux of a broad range of agents and may therefore prove beneficial to cells in the presence of multiple stressors. The cellular response to a combination of such chemical stressors may be governed by a trade-off between the fitness costs due to drug toxicity and benefits mediated by inducible systems. Unfortunately, disentangling the cost-benefit interplay using measurements of bacterial growth in response to the competing effects of the drugs is not possible without the support of a theoretical framework.ResultsHere, we use the well-studied multiple antibiotic resistance (MAR) system in E. coli to experimentally characterize the trade-off between drug toxicity (“cost”) and drug-induced resistance (“benefit”) mediated by efflux pumps. Specifically, we show that the combined effects of a MAR-inducing drug and an antibiotic are governed by a superposition of cost and benefit functions that govern these trade-offs. We find that this superposition holds for all drug concentrations, and it therefore allows us to describe the full dose–response diagram for a drug pair using simpler cost and benefit functions. Moreover, this framework predicts the existence of optimal growth at a non-trivial concentration of inducer. We demonstrate that optimal growth does not coincide with maximum induction of the mar promoter, but instead results from the interplay between drug toxicity and mar induction. Finally, we derived and experimentally validated a general phase diagram highlighting the role of these opposing effects in shaping the interaction between two drugs.ConclusionsOur analysis provides a quantitative description of the MAR system and highlights the trade-off between inducible resistance and the toxicity of the inducing agent in a multi-component environment. The results provide a predictive framework for the combined effects of drug toxicity and induction of the MAR system that are usually masked by bulk measurements of bacterial growth. The framework may also be useful for identifying optimal growth conditions in more general systems where combinations of environmental cues contribute to both transient resistance and toxicity.

Drug resistance in bacterial infections and cancers constitutes a major threat to human health. Treatments often include several interacting drugs, but even potent therapies can become ineffective in resistant mutants. Here, we simplify the picture of drug resistance by identifying scaling laws that unify the multidrug responses of drug-sensitive and -resistant cells. On the basis of these scaling relationships, we are able to infer the two-drug response of resistant mutants in previously unsampled regions of dosage space in clinically relevant microbes such as E. coli, E. faecalis, S. aureus, and S. cerevisiae as well as human non-small-cell lung cancer, melanoma, and breast cancer stem cells. Importantly, we find that scaling relations also apply across evolutionarily close strains. Finally, scaling allows one to rapidly identify new drug combinations and predict potent dosage regimes for targeting resistant mutants without any prior mechanistic knowledge about the specific resistance mechanism.

We study synchronization in populations of phase-coupled stochastic three-state oscillators characterized by a distribution of transition rates. We present results on an exactly solvable dimer as well as a systematic characterization of globally connected arrays of N types of oscillators (N=2,3,4) by exploring the linear stability of the nonsynchronous fixed point. We also provide results for globally coupled arrays where the transition rate of each unit is drawn from a uniform distribution of finite width. Even in the presence of transition rate disorder, numerical and analytical results point to a single phase transition to macroscopic synchrony at a critical value of the coupling strength. Numerical simulations make possible further characterization of the synchronized arrays.