Catch Me If You Can: A Spatial Model for a Brake-Driven Gene Drive Reversal

  • Léo GirardinEmail author
  • Vincent Calvez
  • Florence Débarre
Original Article


Population management using artificial gene drives (alleles biasing inheritance, increasing their own transmission to offspring) is becoming a realistic possibility with the development of CRISPR-Cas genetic engineering. A gene drive may, however, have to be stopped. “Antidotes” (brakes) have been suggested, but have been so far only studied in well-mixed populations. Here, we consider a reaction–diffusion system modeling the release of a gene drive (of fitness \(1-a\)) and a brake (fitness \(1-b\), \(b\le a\)) in a wild-type population (fitness 1). We prove that whenever the drive fitness is at most 1/2 while the brake fitness is close to 1, coextinction of the brake and the drive occurs in the long run. On the contrary, if the drive fitness is greater than 1/2, then coextinction is impossible: the drive and the brake keep spreading spatially, leaving in the invasion wake a complicated spatiotemporally heterogeneous genetic pattern. Based on numerical experiments, we argue in favor of a global coextinction conjecture provided the drive fitness is at most 1/2, irrespective of the brake fitness. The proof relies upon the study of a related predator–prey system with strong Allee effect on the prey. Our results indicate that some drives may be unstoppable and that if gene drives are ever deployed in nature, threshold drives, that only spread if introduced in high enough frequencies, should be preferred.


Long-time behavior Gene drive Brake Predator–prey Strong Allee effect 

Mathematics Subject Classification

35K57 37N25 92D10 92D25 



The authors thank three anonymous referees for valuable comments which lead to an improvement of the manuscript. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 639638). This work was supported by a public grant as part of the Investissement d’avenir Project, Reference ANR-11-LABX-0056-LMH, LabEx LMH, and ANR-14-ACHN-0003-01.


  1. Barton NH, Turelli M (2011) Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of Allee effects. Am Nat 178(3):E48–E75CrossRefGoogle Scholar
  2. Beaghton A, Beaghton PJ, Burt A (2016) Gene drive through a landscape: reaction–diffusion models of population suppression and elimination by a sex ratio distorter. Theor Popul Biol 108:51–69CrossRefGoogle Scholar
  3. Bing W, Luo L, Gao XJ (2016) Cas9-triggered chain ablation of cas9 as a gene drive brake. Nat Biotechnol 34:137CrossRefGoogle Scholar
  4. Deredec A, Burt A, Godfray HCJ (2008) The population genetics of using homing endonuclease genes in vector and pest management. Genetics 179(4):2013–2026CrossRefGoogle Scholar
  5. Ducrot A, Giletti T, Matano H (2019) Spreading speeds for multidimensional reaction-diffusion systems of the prey–predator type. arXiv e-prints, arXiv:1907.02592
  6. Eaton JW, Bateman D, Hauberg S, Wehbring R (2019) GNU Octave version 5.1.0 manual: a high-level interactive language for numerical computations.
  7. Engineering National Academies of Sciences and Medicine (2016) Gene drives on the horizon: advancing science, navigating uncertainty, and aligning research with public values. The National Academies Press, Washington, DCGoogle Scholar
  8. Esvelt KM, Gemmell NJ (2017) Conservation demands safe gene drive. PLoS Biol 15(11):1–8 11CrossRefGoogle Scholar
  9. Esvelt KM, Smidler AL, Catteruccia F, Church GM (2014) Emerging technology: concerning RNA-guided gene drives for the alteration of wild populations. eLife 3:e03401CrossRefGoogle Scholar
  10. Fife PC, McLeod JB (1977) The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch Ration Mech Anal 65(4):335–361MathSciNetCrossRefGoogle Scholar
  11. Hastings A, Abbott KC, Cuddington K, Francis T, Gellner G, Lai YC, Morozov A, Petrovskii S, Scranton K, Zeeman ML (2018) Transient phenomena in ecology. Science 361(6406):eaat6412. CrossRefGoogle Scholar
  12. Morozov A, Petrovskii S, Li B-L (2004) Bifurcations and chaos in a predator–prey system with the Allee effect. Proc R Soc Lond B Biol Sci 271(1546):1407–1414CrossRefGoogle Scholar
  13. Nagylaki T (1975) Conditions for the existence of clines. Genetics 80(3):595–615Google Scholar
  14. Noble C, Adlam B, Church GM, Esvelt KM, Nowak MA (2018) Current CRISPR gene drive systems are likely to be highly invasive in wild populations. eLife 7:e33423CrossRefGoogle Scholar
  15. Petrovskii SV, Morozov AY, Venturino E (2002) Allee effect makes possible patchy invasion in a predator-prey system. Ecol Lett 5(3):345–352CrossRefGoogle Scholar
  16. Rode NO, Estoup A, Bourguet D, Courtier-Orgogozo V, Débarre F (2019) Population management using gene drive: molecular design, models of spread dynamics and assessment of ecological risks. Conserv Genet 20:671–690CrossRefGoogle Scholar
  17. Tanaka H, Stone HA, Nelson DR (2017) Spatial gene drives and pushed genetic waves. Proc Natl Acad Sci 114:8452–8457CrossRefGoogle Scholar
  18. Unckless RL, Messer PW, Connallon T, Clark AG (2015) Modeling the manipulation of natural populations by the mutagenic chain reaction. Genetics 201(2):425–431CrossRefGoogle Scholar
  19. Vella MR, Gunning CE, Lloyd AL, Gould F (2017) Evaluating strategies for reversing CRISPR-Cas9 gene drives. Sci Rep 7(1):11038CrossRefGoogle Scholar
  20. Wang J, Shi J, Wei J (2011) Dynamics and pattern formation in a diffusive predator–prey system with strong Allee effect in prey. J Differ Equ 251(4–5):1276–1304MathSciNetCrossRefGoogle Scholar
  21. Weinberger HF (1975) Invariant sets for weakly coupled parabolic and elliptic systems. Rend Mat 6(8):295–310 (Collection of articles dedicated to Mauro Picone on the occasion of his ninetieth birthday)MathSciNetzbMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’Orsay, Université Paris Sud, CNRSUniversité Paris-SaclayOrsay CedexFrance
  2. 2.Institut Camille Jordan, UMR 5208 CNRSUniversité Claude Bernard Lyon 1VilleurbanneFrance
  3. 3.CNRS, Sorbonne Université, Université Paris Est Créteil, Université Paris Diderot, INRA, IRD, Institute of Ecology and Environmental Sciences - ParisIEES-ParisParisFrance

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