Resistance to chemotherapies, particularly to anticancer treatments, is an increasing medical concern. Among the many mechanisms at work in cancers, one of the most important is the selection of tumor cells expressing resistance genes or phenotypes. Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous variable that represents the expression level of a resistance gene (or genes, yielding a phenotype) influencing in healthy and tumor cells birth/death rates, effects of chemotherapies (both cytotoxic and cytostatic) and mutations. We extend previous work by demonstrating how qualitatively different actions of chemotherapeutic and cytostatic treatments may induce different levels of resistance. The mathematical interest of our study is in the formalism of constrained Hamilton–Jacobi equations in the framework of viscosity solutions. We derive the long-term temporal dynamics of the fittest traits in the regime of small mutations. In the context of adaptive cancer management, we also analyse whether an optimal drug level is better than the maximal tolerated dose.

The evolution of antibiotic resistance in bacteria is a global concern and the use of bacteriophages alone or in combined therapies is attracting increasing attention as an alternative. Evolutionary theory predicts that the probability of bacterial resistance to both phages and antibiotics will be lower than to either separately, due for example to fitness costs or to trade-offs between phage resistance mechanisms and bacterial growth. In this study, we assess the population impacts of either individual or combined treatments of a bacteriophage and streptomycin on the nosocomial pathogen Pseudomonas aeruginosa. We show that combining phage and antibiotics substantially increases bacterial control compared to either separately, and that there is a specific time delay in antibiotic introduction independent of antibiotic dose, that minimizes both bacterial density and resistance to either antibiotics or phage. These results have implications for optimal combined therapeutic approaches.

Significance: Scientists have long debated the dynamic form of perpetual reciprocal adaptations, or coevolution, between hosts and their parasites. The two main types of antagonistic coevolution described to date are arms race dynamics, in which interaction traits escalate through time, and fluctuating selection dynamics, in which traits cycle through time. We used experimental evolution between Pseudomonas aeruginosa and a panel of its lytic phages and found the full known range of coevolutionary dynamics. We argue that the coevolutionary pattern is determined by whether phages typically adsorb directly to receptors on the bacterial outer membrane or instead use retractable type IV pili as a primary or alternative site. Our results have applications in the use of phages as therapeutics or disinfectants to control bacterial pathogens.

Antimicrobial resistance is a growing global concern and has spurred increasing efforts to find alternative therapeutics. Bacteriophage therapy has seen near constant use in eastern Europe since its discovery over a century ago. One promising approach is to use phages that not only reduce bacterial pathogen loads, but also select for phage resistance mechanisms that trade-off with antibiotic resistance – so called ‘phage steering’. Recent work has shown that the phage OMKO1 can interact with efflux pumps and in so doing select for both phage resistance and antibiotic sensitivity of the pathogenic bacterium Pseudomonas aeruginosa. We tested the robustness of this approach to three different antibiotics in vitro (tetracycline, erythromycin, ciprofloxacin) and one in vivo (erythromycin). We show that in vitro OMKO1 can reduce antibiotic resistance of P. aeruginosa (Washington PAO1) even in the presence of antibiotics, an effect still detectable after more than 70 bacterial generations in continuous culture with phage. Our in vivo experiment showed that phage both increased the survival times of wax moth larvae (Galleria mellonella) and increased bacterial sensitivity to erythromycin. This increased antibiotic sensitivity occurred both in lines with and without the antibiotic. We discuss the implications of our findings for future research on this promising therapeutic approach using OMKO1.