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

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.

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,

$$\begin{aligned} {\mathbf {p}}\cdot {\mathbf {f}}\left( {\mathbf {u}}^\star ,x,t\right) \le 0\text { for all }\left( x,t\right) \in D\times \left( 0,T\right] . \end{aligned}$$

Let

$$\begin{aligned} L=\frac{\partial }{\partial t}-\sum _{i,j=1}^n a^{ij}\left( x,t\right) \frac{\partial ^2}{\partial x_i \partial x_j} -\sum _{i=1}^n b_i\left( x,t\right) \frac{\partial }{\partial x_i} \end{aligned}$$

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

$$\begin{aligned} Lu_{\alpha }=f_{\alpha }\left( {\mathbf {u}},x,t\right) \text { for }\alpha =1,2,\dots ,m \end{aligned}$$

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

$$\begin{aligned} {\mathbf {p}}\cdot {\mathbf {f}}\left( {\mathbf {u}}^\star ,x,t\right) \le 0\text { for all }\left( x,t\right) \in D\times \left( 0,T\right] . \end{aligned}$$

Let

$$\begin{aligned} L=\frac{\partial }{\partial t}-\sum _{i,j=1}^n a^{ij}\left( x,t\right) \frac{\partial ^2}{\partial x_i \partial x_j} -\sum _{i=1}^n b_i\left( x,t\right) \frac{\partial }{\partial x_i} \end{aligned}$$

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

$$\begin{aligned} Lu_{\alpha }=f_{\alpha }\left( {\mathbf {u}},x,t\right) \text { for }\alpha =1,2,\dots ,m \end{aligned}$$

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.