Letter | Published:

Additivity of inhibitory effects in multidrug combinations

Nature Microbiologyvolume 3pages13391345 (2018) | Download Citation

Abstract

From natural ecology1,2,3,4 to clinical therapy5,6,7,8, cells are often exposed to mixtures of multiple drugs. Two competing null models are used to predict the combined effect of drugs: response additivity (Bliss) and dosage additivity (Loewe)9,10,11. Here, noting that these models diverge with increased number of drugs, we contrast their predictions with growth measurements of four phylogenetically distant microorganisms including Escherichia coli, Staphylococcus aureus, Enterococcus faecalis and Saccharomyces cerevisiae, under combinations of up to ten different drugs. In all species, as the number of drugs increases, Bliss maintains accuracy while Loewe systematically loses its predictive power. The total dosage required for growth inhibition, which Loewe predicts should be fixed, steadily increases with the number of drugs, following a square-root scaling. This scaling is explained by an approximation to Bliss where, inspired by R. A. Fisher’s classical geometric model12, dosages of independent drugs add up as orthogonal vectors rather than linearly. This dose-orthogonality approximation provides results similar to Bliss, yet uses the dosage language as in Loewe and is hence easier to implement and intuit. The rejection of dosage additivity in favour of effect additivity and dosage orthogonality provides a framework for understanding how multiple drugs and stressors add up in nature and the clinic.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Data availability

The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

  1. 1.

    Ueda, K. et al. Wide distribution of interspecific stimulatory events on antibiotic production and sporulation among Streptomyces species. J. Antibiot. 53, 979–982 (2000).

  2. 2.

    Clardy, J., Fischbach, M. A. & Currie, C. R. The natural history of antibiotics. Curr. Biol. 19, R437–R441 (2009).

  3. 3.

    Vetsigian, K., Jajoo, R. & Kishony, R. Structure and evolution of Streptomyces interaction networks in soil and in silico. PLoS Biol. 9, e1001184 (2011).

  4. 4.

    Chait, R., Vetsigian, K. & Kishony, R. What counters antibiotic resistance in nature? Nat. Chem. Biol. 8, 2–5 (2011).

  5. 5.

    Blumberg, H. M. et al. American Thoracic Society/Centers for Disease Control and Prevention/Infectious Diseases Society of America: treatment of tuberculosis. Am. J. Respir. Crit. Care Med. 167, 603–662 (2003).

  6. 6.

    Keith, C. T., Borisy, A. A. & Stockwell, B. R. Multicomponent therapeutics for networked systems. Nat. Rev. Drug Discov. 4, 71–78 (2005).

  7. 7.

    Fitzgerald, J. B., Schoeberl, B., Nielsen, U. B. & Sorger, P. K. Systems biology and combination therapy in the quest for clinical efficacy. Nat. Chem. Biol. 2, 458–466 (2006).

  8. 8.

    Palmer, A. C. & Sorger, P. K. Combination cancer therapy can confer benefit via patient-to-patient variability without drug additivity or synergy. Cell 171, 1678–1691.e13 (2017).

  9. 9.

    Loewe, S. & Muischnek, H. Über kombinationswirkungen. Archiv f. experiment. Pathol. u. Pharmakol. 114, 313–326 (1926).

  10. 10.

    Bliss, C. I. The toxicity of poisons applied jointly. Ann. Appl. Biol. 26, 585–615 (1939).

  11. 11.

    Loewe, S. The problem of synergism and antagonism of combined drugs. Arzneimittelforschung 3, 285–290 (1953).

  12. 12.

    Fisher, R. A. The Genetical Theory of Natural Selection (Clarendon Press, Oxford, 1930).

  13. 13.

    Chait, R., Craney, A. & Kishony, R. Antibiotic interactions that select against resistance. Nature 446, 668–671 (2007).

  14. 14.

    Michel, J.-B. et al. Drug interactions modulate the potential for evolution of resistance. Proc. Natl Acad. Sci. USA 105, 14918–14923 (2008).

  15. 15.

    Imamovic, L. & Sommer, M. O. A. Use of collateral sensitivity networks to design drug cycling protocols that avoid resistance development. Sci. Transl. Med. 5, 204ra132 (2013).

  16. 16.

    Pál, C., Papp, B. & Lázár, V. Collateral sensitivity of antibiotic-resistant microbes. Trends Microbiol. 23, 401–407 (2015).

  17. 17.

    Baym, M., Stone, L. K. & Kishony, R. Multidrug evolutionary strategies to reverse antibiotic resistance. Science 351, aad3292 (2016).

  18. 18.

    Zimmermann, G. R., Lehár, J. & Keith, C. T. Multi-target therapeutics: when the whole is greater than the sum of the parts. Drug Discov. Today 12, 34–42 (2007).

  19. 19.

    Bayat Mokhtari, R. et al. Combination therapy in combating cancer. Oncotarget 8, 38022–38043 (2017).

  20. 20.

    Lehár, J. et al. Synergistic drug combinations tend to improve therapeutically relevant selectivity. Nat. Biotechnol. 27, 659–666 (2009).

  21. 21.

    Cokol, M. et al. Systematic exploration of synergistic drug pairs. Mol. Syst. Biol. 7, 544 (2011).

  22. 22.

    Chandrasekaran, S. et al. Chemogenomics and orthology-based design of antibiotic combination therapies. Mol. Syst. Biol. 12, 872 (2016).

  23. 23.

    Hegreness, M., Shoresh, N., Damian, D., Hartl, D. & Kishony, R. Accelerated evolution of resistance in multidrug environments. Proc. Natl Acad. Sci. USA 105, 13977–13981 (2008).

  24. 24.

    Chou, T. C. & Talalay, P. Quantitative analysis of dose–effect relationships: the combined effects of multiple drugs or enzyme inhibitors. Adv. Enzyme Regul. 22, 27–55 (1984).

  25. 25.

    Greco, W. R., Bravo, G. & Parsons, J. C. The search for synergy: a critical review from a response surface perspective. Pharmacol. Rev. 47, 331–385 (1995).

  26. 26.

    Berenbaum, M. C. The expected effect of a combination of agents: the general solution. J. Theor. Biol. 114, 413–431 (1985).

  27. 27.

    Tang, J., Wennerberg, K. & Aittokallio, T. What is synergy? The Saariselkä agreement revisited. Front. Pharmacol. 6, 181 (2015).

  28. 28.

    Yeh, P. J., Hegreness, M. J., Aiden, A. P. & Kishony, R. Drug interactions and the evolution of antibiotic resistance. Nat. Rev. Microbiol. 7, 460–466 (2009).

  29. 29.

    Zhao, W. et al. A new bliss independence model to analyze drug combination data. J. Biomol. Screen. 19, 817–821 (2014).

  30. 30.

    Boucher, A. N. & Tam, V. H. Mathematical formulation of additivity for antimicrobial agents. Diagn. Microbiol. Infect. Dis. 55, 319–325 (2006).

  31. 31.

    Baeder, D. Y., Yu, G., Hozé, N., Rolff, J. & Regoes, R. R. Antimicrobial combinations: Bliss independence and Loewe additivity derived from mechanistic multi-hit models. Phil. Trans. R. Soc. B 371, 20150294 (2016).

  32. 32.

    Foucquier, J. & Guedj, M. Analysis of drug combinations: current methodological landscape. Pharmacol. Res. Perspect. 3, e00149 (2015).

  33. 33.

    Drescher, K. & Boedeker, W. Assessment of the combined effects of substances: the relationship between concentration addition and independent action. Biometrics 51, 716–730 (1995).

  34. 34.

    Goldoni, M. & Johansson, C. A mathematical approach to study combined effects of toxicants in vitro: evaluation of the Bliss independence criterion and the Loewe additivity model. Toxicol. In Vitro 21, 759–769 (2007).

  35. 35.

    Lee, S. I. Drug interaction: focusing on response surface models. Korean J. Anesthesiol. 58, 421–434 (2010).

  36. 36.

    Wood, K., Nishida, S., Sontag, E. D. & Cluzel, P. Mechanism-independent method for predicting response to multidrug combinations in bacteria. Proc. Natl Acad. Sci. USA 109, 12254–12259 (2012).

  37. 37.

    Zimmer, A., Katzir, I., Dekel, E., Mayo, A. E. & Alon, U. Prediction of multidimensional drug dose responses based on measurements of drug pairs. Proc. Natl Acad. Sci. USA 113, 10442–10447 (2016).

  38. 38.

    Cokol, M., Kuru, N., Bicak, E., Larkins-Ford, J. & Aldridge, B. B. Efficient measurement and factorization of high-order drug interactions in Mycobacterium tuberculosis. Sci. Adv. 3, e1701881 (2017).

  39. 39.

    Tekin, E. et al. Prevalence and patterns of higher-order interactions in Escherichia coli. NPJ Syst. Biol. Appl. 4, 31 (2017).

  40. 40.

    Wagner, J. G. Kinetics of pharmacologic response I. Proposed relationships between response and drug concentration in the intact animal and man. J. Theor. Biol. 20, 173–201 (1968).

  41. 41.

    Bollenbach, T., Quan, S., Chait, R. & Kishony, R. Nonoptimal microbial response to antibiotics underlies suppressive drug interactions. Cell 139, 707–718 (2009).

  42. 42.

    How, S. J., Hobson, D., Hart, C. A. & Webster, R. E. An in-vitro investigation of synergy and antagonism between antimicrobials against Chlamydia trachomatis. J. Antimicrob. Chemother. 15, 533–538 (1985).

  43. 43.

    Yeh, P., Tschumi, A. I. & Kishony, R. Functional classification of drugs by properties of their pairwise interactions. Nat. Genet. 38, 489–494 (2006).

  44. 44.

    Kahm, M., Hasenbrink, G., Lichtenberg-Fraté, H., Ludwig, J. & Kschischo, M. grofit: fitting biological growth curves with R. J. Stat. Softw. 33, 1–21 (2010).

Download references

Acknowledgements

We thank U. Alon and I. Katzir for thoughtful discussions and suggestions, Y. Arava for strains, and members of the Kishony lab, especially M. Datta, E. Tamar, I. Yelin and N. Yin, for experimental support and comments on the manuscript. This work was supported in part by the US National Institutes of Health grant R01-GM081617, the Israeli Centers of Research Excellence I-CORE Program ISF grant 152/11 and the European Research Council FP7 ERC Grant 281891 (to R.K.).

Author information

Affiliations

  1. Faculty of Biology, Technion–Israel Institute of Technology, Haifa, Israel

    • D. Russ
    •  & R. Kishony
  2. Faculty of Computer Science, Technion–Israel Institute of Technology, Haifa, Israel

    • R. Kishony

Authors

  1. Search for D. Russ in:

  2. Search for R. Kishony in:

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to R. Kishony.

Supplementary Information

  1. Supplementary Information

    Supplementary Notes, Supplementary Figures 1–13 and Supplementary Tables 1–8.

  2. Reporting Summary

  3. Supplementary Table 9

    Dose response of all drug combinations.

  4. Supplementary Table 10

    Measured and predicted potencies of drug combinations.

About this article

Publication history

Received

Accepted

Published

DOI

https://doi.org/10.1038/s41564-018-0252-1