Dynamics of Transposable Elements in Genetically Modified Mosquitoes

1. John Marshall Department of Biomathematics UCLA Dynamics of Transposable Elements in Genetically Modified Mosquitoes

2. Malaria control using genetically modified mosquitoes

3. The transgene construct

4. Meiotic drive and HWE P ww + 2Q wt + R tt wt x tt -> 4QR ww x ww -> P2ww ½ ½ t t ½(1-i) w wtwt ½(1+i) t tttt tt x tt -> R2tt ww x tt -> 2PRwt wt x wt -> 4Q2 wt x ww -> 4PQ ½ ½ w w ½(1-i) w wwww ½(1+i) t wtwt ½(1-i) ½(1-i) w t ½(1-i) w wwwt ½(1+i) t wttt

5. Repression of replicative transposition Mechanisms have evolved to achieve a balance between: • Selection for high element copy number • Selection for hosts with fewer deleterious mutations Mechanisms: • Host factors involved in (transposase) gene silencing • Post-transcriptional regulation of the transposable element by itself Models:

6. Kinetic model of self-repression of transposition in Mariner

7. Costs to mosquito fitness with increasing element copy number Insertional mutagenesis: • Each element copy can disrupt a functioning gene • Fitness cost proportional to n Ectopic recombination: • Recombination can occur between elements at different sites • Results in deleterious chromosomal rearrangements • Fitness cost proportional to n2 Act of transposition: • Transposition can create nicks in chromosomes • Fitness cost proportional to un Models:

8. Proposed Markov chain model n-1 n n+1

9. Solving the system of ODEs From probability theory: • Define the generating function, • Manipulate to obtain mean element copy number at time t

10. Proposed branching process model Continuous time haploid branching process: i-1 i i+1 Continuous time diploid branching process: • Consider the early stages of the spread of a transposable element • Imagine a reservoir of uninfected hosts • Assume matings involving infected hosts will be with uninfected hosts • For a gamete derived from a cell with i copies of the element it is possible to generate offspring with jE{0, 1, 2,…, i} copies • Assume each offspring genotype occurs with equal probability,

11. Diploid branching process model i-2 i-1 i i+1 …

12. Left boundary transitions 0 1 2

13. Solving the proposed branching process model Populating the branching process matrix: The solution to the branching process is: The branching process is supercritical if its dominant eigenvalue is positive: • Check for positive eigenvalue using Person-Frobenius Theorem • Or look for positive roots of the characteristic equaiton, Problems: • Only considers initial dynamics • Recombination are frequently of medium copy number • Ignores tendency for local transposition, recombination, etc.

14. Site-specific model Motivation: • Preferential transposition to nearby sites • Site-varying fitness costs • Recombination in diploid hosts Label states according to their occupancy: • T sites available for TE to insert into • 2T possible states numbered from 0 to 2T-1 {0 0 1 0} 2 TE {0 1 0 1} 9 TE TE {1 1 0 0} 12 TE TE

15. Local preference for transposition Replicative transposition: TE TE TE TE TE TE TE (autoregulation) (preference for local transposition) Non-replicative transposition: TE TE TE TE (preference for local transposition)

16. Enumerating the transitions

17. Analysis of equilibrium distributions

18. First and second order perturbation approximations First order perturbation approximation: Second order perturbation approximation:

19. Dissociation of the transposable element and transgene

20. Markov chain model of dissociation n-1,m+1 n,m+1 n-1,m n,m n+1,m n+1,m-1 n,m-1