Dynamics of Transposable Elements in Genetically Modified Mosquitoes

1
Dynamics of Transposable Elements in Genetically
Modified Mosquitoes

• John Marshall
• Department of Biomathematics
• UCLA

2
Malaria control using genetically modified
mosquitoes
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The transgene construct
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Meiotic drive and HWE

• P ww 2Q wt R tt

wt x tt -gt 4QR
ww x ww -gt P2 ww
½ ½ t t ½(1-i) w wt wt ½(1i) t
tt tt
tt x tt -gt R2 tt
ww x tt -gt 2PR wt
wt x wt -gt 4Q2
wt x ww -gt 4PQ
½(1-i) ½(1-i) w t ½(1-i) w ww
wt ½(1i) t wt tt
½ ½ w w ½(1-i) w ww ww ½(1i) t
wt wt
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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

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Kinetic model of self-repression of transposition
in Mariner
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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

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Proposed Markov chain model
n
n1
n-1
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Solving the system of ODEs

• From probability theory
• Define the generating function,
• Manipulate to obtain mean element copy number at
  time t

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Proposed branching process model
Continuous time haploid branching process
i
i1
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 jE0, 1, 2,, i copies
• Assume each offspring genotype occurs with equal
  probability,

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Diploid branching process model
i-1
i
i-2
i1

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Left boundary transitions
0
1
2
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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.

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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
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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)
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Enumerating the transitions
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Analysis of equilibrium distributions
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First and second order perturbation approximations
First order perturbation approximation
Second order perturbation approximation
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Dissociation of the transposable element and
transgene
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Markov chain model of dissociation
n,m1
n-1,m1
n,m
n1,m
n-1,m
n1,m-1
n,m-1