Price equation captures the role of drug interactions and collateral effects in the evolution of multidrug resistance

Bacterial adaptation to antibiotic combinations depends on the joint inhibitory effects of the two drugs (drug interaction, DI) and how resistance to one drug impacts resistance to the other (collateral effects, CE). Here we model these evolutionary dynamics on two-dimensional phenotype spaces that leverage scaling relations between the drug-response surfaces of drug sensitive (ancestral) and drug resistant (mutant) populations. We show that evolved resistance to the component drugs--and in turn, the adaptation of growth rate--is governed by a Price equation whose covariance terms encode geometric features of both the two-drug response surface (DI) in ancestral cells and the correlations between resistance levels to those drugs (CE). Within this framework, mean evolutionary trajectories reduce to a type of weighted gradient dynamics, with the drug interaction dictating the shape of the underlying landscape and the collateral effects constraining the motion on those landscapes. Our results clarify the complex relationship between drug interactions and collateral effects in multi-drug environments and illustrate how specific dosage combinations can shift the weighting of these two effects, leading to different and temporally-explicit selective outcomes.