Wshredder And Xshredder Suppression Gene Drive Equations

Recursive allele frequency equations of W-shredder and X-shredder suppression gene drives

[2021-08-08]

W-shredder

Applicable to ZW/ZZ sex determination systems, e.g. birds, reptiles, and lepidopteran pests, where ZW are females and ZZ are males.

\[\begin{aligned} Z^dW ~~— c \rightarrow~~ Z^d0 \end{aligned}\]

The recursive drive allele frequency is:

\[\begin{aligned} q_{t+1} &= {8q_{t} - 3cq_{t} - cq_{t}^2 \over 8-4c}, \end{aligned}\]

where \(q_{t}\) is the frequency of the suppression gene drive allele at generation \(t\), and \(c\) is the conversion efficiency, i.e. the rate at with the W chromosome is shredded given \(Z^{d}W\) genotypes.

X-shredder

Applicable to XX/XY sex determination systems, e.g. mammals, where XX are females and XY are males.

\[\begin{aligned} XY^d ~~— c \rightarrow~~ 0Y^d \end{aligned}\]

The recursive drive allele frequency is:

\[\begin{aligned} q_{t+1} &= {2q_{t} \over 2 - c + cq_{t}}, \end{aligned}\]

where \(q_{t}\) is the frequency of the suppression gene drive allele at generation \(t\), and \(c\) is the conversion efficiency, i.e. the rate at with the X chromosome is shredded given \(XY^{d}\) genotypes.