Testing the optimality properties of a dual antibiotic treatment in a two-locus, two-allele model

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 dose and the same antibiotic combination each day. Here, we use a frequency- and density-dependent pharmacogenetics mathematical model based on a standard, two-locus, two-allele representation of how bacteria resist antibiotics to probe the question of whether optimal antibiotic treatments might, in fact, be constant through time. The model describes the ecological and evolutionary dynamics of different sub-populations of the bacterium Escherichia coli that compete for a single limiting resource in a two-drug environment. We use in vitro evolutionary experiments to calibrate and test the model and show that antibiotic environments can support dynamically changing and heterogeneous population structures. We then demonstrate, theoretically and empirically, that the best treatment strategies should adapt through time and constant strategies are not optimal.