Abstract
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.
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Notes
Depending on the construct, the brake could just convert a drive without affecting its effect on fitness (b close to a, \(b \le a\)), or at the other extreme the brake could carry a cargo gene restoring wild-type fitness (b close to 0).
This is merely for algebraic convenience, and the general case will be discussed below in Sect. 4.2.
With late gene conversion (typically in the germline), an \(\mathrm{OD}\)-born individual would have the fitness of an \(\mathrm{OD}\). In both cases though, only D gametes are produced by this individual.
We actually observed that the extinction threshold is larger than a but smaller than 1 (Fig. 5). Nevertheless, the case \(b>a\) is beyond the scope of our assumptions and does not correspond to an relevant case in the context of the biological problem.
Although this is indeed the main idea, it turns out that technical obstacles arise and therefore we will also show in the first step that v becomes uniformly smaller than a constant smaller than 1.
References
Barton NH, Turelli M (2011) Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of Allee effects. Am Nat 178(3):E48–E75
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 108:51–69
Bing W, Luo L, Gao XJ (2016) Cas9-triggered chain ablation of cas9 as a gene drive brake. Nat Biotechnol 34:137
Deredec A, Burt A, Godfray HCJ (2008) The population genetics of using homing endonuclease genes in vector and pest management. Genetics 179(4):2013–2026
Ducrot A, Giletti T, Matano H (2019) Spreading speeds for multidimensional reaction-diffusion systems of the prey–predator type. ar**v e-prints, ar**v:1907.02592
Eaton JW, Bateman D, Hauberg S, Wehbring R (2019) GNU Octave version 5.1.0 manual: a high-level interactive language for numerical computations. https://www.gnu.org/software/octave/doc/v5.1.0/
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, DC
Esvelt KM, Gemmell NJ (2017) Conservation demands safe gene drive. PLoS Biol 15(11):1–8 11
Esvelt KM, Smidler AL, Catteruccia F, Church GM (2014) Emerging technology: concerning RNA-guided gene drives for the alteration of wild populations. eLife 3:e03401
Fife PC, McLeod JB (1977) The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch Ration Mech Anal 65(4):335–361
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. https://doi.org/10.1126/science.aat6412
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):1407–1414
Nagylaki T (1975) Conditions for the existence of clines. Genetics 80(3):595–615
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:e33423
Petrovskii SV, Morozov AY, Venturino E (2002) Allee effect makes possible patchy invasion in a predator-prey system. Ecol Lett 5(3):345–352
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 20:671–690
Tanaka H, Stone HA, Nelson DR (2017) Spatial gene drives and pushed genetic waves. Proc Natl Acad Sci 114:8452–8457
Unckless RL, Messer PW, Connallon T, Clark AG (2015) Modeling the manipulation of natural populations by the mutagenic chain reaction. Genetics 201(2):425–431
Vella MR, Gunning CE, Lloyd AL, Gould F (2017) Evaluating strategies for reversing CRISPR-Cas9 gene drives. Sci Rep 7(1):11038
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):1276–1304
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)
Acknowledgements
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.
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Appendices
Appendix A: Weinberger’s Maximum Principle
Below is recalled the main tool of the proof of Theorem 1.2. For clarity, we temporarily get rid of all our notations and adopt the original ones from Weinberger (1975).
Theorem A.1
(Weak maximum principle) Let D be a \(C^{1,\nu }\) domain in \({\mathbb {R}}^n\) with \(\nu \in \left( 0,1\right) \), S be a closed convex subset of \({\mathbb {R}}^m\), \({\mathbf {f}}\left( {\mathbf {u}},x,t\right) \) be Lipschitz-continuous in \({\mathbf {u}}\in S\) and uniformly Hölder-continuous in \(x\in D\) and \(t\in \left[ 0,T\right] \), with the property that for any outward normal \({\mathbf {p}}\) at any boundary point \({\mathbf {u}}^\star \) of S,
Let
be uniformly parabolic with coefficients uniformly Hölder-continuous with Hölder exponent greater than \(\frac{1}{2}\).
If \({\mathbf {u}}\) is any solution in \(D\times \left( 0,T\right] \) of the system
which is continuous in \(\overline{D}\times \left[ 0,T\right] \), and if the values of \({\mathbf {u}}\) on \(\overline{D}\times \left\{ 0\right\} \cup \partial D\times \left[ 0,T\right] \) are bounded and Hölder-continuous and lie in S, then \({\mathbf {u}}\left( x,t\right) \in S\) in \(D\times \left( 0,T\right] \).
Theorem A.2
(Strong maximum principle) Let D be an arbitrary domain in \({\mathbb {R}}^n\) and S be a closed convex subset of \({\mathbb {R}}^m\) such that every boundary point of S satisfies a slab condition. Let \({\mathbf {f}}\left( {\mathbf {u}},x,t\right) \) be Lipschitz-continuous in \({\mathbf {u}}\), and suppose that if \({\mathbf {p}}\) is any outward normal at a boundary point \({\mathbf {u}}^\star \), then
Let
be locally uniformly parabolic, and let its coefficients be locally bounded.
If \({\mathbf {u}}\) is any solution in \(D\times \left( 0,T\right] \) of the system
with \({\mathbf {u}}\left( x,t\right) \in S\) and if \({\mathbf {u}}\left( x^\star ,t^\star \right) \in \partial S\) for some \(\left( x^\star ,t^\star \right) \in D\times \left( 0,T\right] \), then \({\mathbf {u}}\left( x,t\right) \in \partial S\) in \(D\times \left( 0,t^\star \right] \).
Appendix B: Numerical Scheme
The various numerical simulations presented earlier are all produced by the following Octave code or slight variants of it.
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Girardin, L., Calvez, V. & Débarre, F. Catch Me If You Can: A Spatial Model for a Brake-Driven Gene Drive Reversal. Bull Math Biol 81, 5054–5088 (2019). https://doi.org/10.1007/s11538-019-00668-z
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DOI: https://doi.org/10.1007/s11538-019-00668-z