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

  1. Erida Gjini  Is a corresponding author
  2. Kevin B Wood  Is a corresponding author
  1. University of Lisbon, Portugal
  2. University of Michigan, United States


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. We also demonstrate how constraints on available mutational pathways can be incorporated into the framework, adding a third key driver of evolution. 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.

Data availability

Data used in this paper was taken from a public repository:Dean, Ziah; Maltas, Jeff; Wood, Kevin (2020), Antibiotic interactions shape short-term evolution of resistance in Enterococcus faecalis, Dryad, Dataset, are no restrictions on any new results.

The following previously published data sets were used

Article and author information

Author details

  1. Erida Gjini

    Instituto Superior Tecnico, University of Lisbon, Lisbon, Portugal
    For correspondence
    Competing interests
    The authors declare that no competing interests exist.
  2. Kevin B Wood

    Department of Biophysics, University of Michigan, Ann Arbor, United States
    For correspondence
    Competing interests
    The authors declare that no competing interests exist.
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-0985-7401


National Institutes of Health (1R35GM124875)

  • Kevin B Wood

National Science Foundation (1553028)

  • Kevin B Wood

Fundação Luso-Americana para o Desenvolvimento (274/2016)

  • Erida Gjini

Instituto Gulbenkian de Ciencia

  • Erida Gjini

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Reviewing Editor

  1. C. Brandon Ogbunugafor, Yale University, United States

Version history

  1. Received: November 13, 2020
  2. Accepted: July 8, 2021
  3. Accepted Manuscript published: July 22, 2021 (version 1)
  4. Accepted Manuscript updated: July 26, 2021 (version 2)
  5. Version of Record published: August 3, 2021 (version 3)


© 2021, Gjini & Wood

This article is distributed under the terms of the Creative Commons Attribution License permitting unrestricted use and redistribution provided that the original author and source are credited.


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  1. Erida Gjini
  2. Kevin B Wood
Price equation captures the role of drug interactions and collateral effects in the evolution of multidrug resistance
eLife 10:e64851.

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