Dynamics of Transposable Elements in Genetically Modified Mosquitoes
John MarshallDepartment of Biomathematics
UCLA
Malaria control using genetically modified mosquitoes
The transgene construct
inverted/direct repeats
inverted/direct repeats
antimalarial gene
midgut/salivary glands promoter
transposase gene
sex cellsspecific promoter
Meiotic drive and HWE
P ww + 2Q wt + R tt
ww x ww -> P2 ww
tt x tt -> R2 tt
ww x tt -> 2PR wt
wt x ww -> 4PQ
½
½
t
t
½(1-i) w wt wt
½(1+i) t tt tt
wt x wt -> 4Q2
wt x tt -> 4QR
½(1-i)
½(1-i)
w
t
½(1-i) w ww wt
½(1+i) t wt tt
½
½
w
w
½(1-i) w ww ww
½(1+i) t wt wt
Repression of replicative transposition
Mechanisms have evolved to achieve a balance between:
• Selection for high element copy number• Selection for hosts with fewer deleterious mutationsMechanisms:• Host factors involved in (transposase) gene silencing• Post-transcriptional regulation of the transposable
element by itselfModels:
kn
uun
1
0
tr
ton nnu
nnuu
,
,r
nnrn uuuu c )1)(1(
0 2)(
)1(1
nkn euu
Kinetic model of self-repression of transposition in Mariner
skwckdt
ds
wckuckskxkdt
dc
skwckwkukdt
dw
xkuckukuknkdt
du
fe
eeff
feqb
febgscsl
2
22
Costs to mosquito fitness with increasing element copy numberInsertional 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:
21,1 tsnw tn nn cubnanw 21
Proposed Markov chain model
)()1()()1(
)())(()()(
1,1,1
,,
tvpntpun
tptvuntpdt
d
ninin
ninnni
n n+1n-1vn )1( nv
1)1( nun nnu
n
)()(, ttp ni :1 Tn
Solving the system of ODEs
1
),()(
si tsG
stm
nT
nni stptsG
0
, )(),(
From probability theory:• Define the generating function,
• Manipulate to obtain mean element copy number at time t
Proposed branching process model
i
)1( ii
i i+1i-1i i
:1 Ti
Continuous time haploid branching process:
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, 1
1
i
Diploid branching process model
i
1i
ii
2
1
11,2
1)1(
1
11
2
1)1(
1
1
2
1
,
,
1,
ikiii
f
iif
if
ii
iii
ii
ki
ii
iiii
ii
ii
ii
i-1 ii-2 i+1…
iiii iii
2
1)1(
1
11
iiii iii
2
1)1(
1
11
iii ii
2
1)1(
1
1
:1 Ti
Left boundary transitions
1
13
11 2
111 3
1)1(
2
1
0
11
1111
1,1
11
2,1
3
1)1(
2
1
3
1
f
f
Solving the proposed branching process model
Populating the branching process matrix:
TTjiiji f )1( }{
The solution to the branching process is:
tetM )(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,0)det()( xIxp
Problems:• Only considers initial dynamics• Recombination are frequently of medium copy number• Ignores tendency for local transposition, recombination, etc.
Site-specific modelMotivation:• Preferential transposition to nearby sites• Site-varying fitness costs• Recombination in diploid hosts
TE
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} 2TE {0 1 0 1} 9
TE {1 1 0 0} 12TE
TE
Local preference for transposition
6,2u 10,2u3,2u
Replicative transposition:
TE
TE TE TETE TETE
110,26,23,2 nuuuu
6,23,210,2 ,uuu
(autoregulation)
(preference for local transposition)
4,2u 8,2u1,2u
Non-replicative transposition:
TE
TETE TE
4,21,28,2 ,uuu (preference for local transposition)
Enumerating the transitions
iil lm
lmimlik
ikiij
jiji txPrtxPtxPutxPutxP
dt
txdPT
),(),(),(),(),(),( 12
0
)()()()()()()( tPdiagtRPtPtUPtPUtPdt
d TT
Analysis of equilibrium distributions
)())(()()( tPRtPtMPtPdt
d T
))(( tdiagUUM T
0)( tPdt
d
0 TT RM
)( 210 O
First and second order perturbation approximations
00 M
12
0
1T
ii
0
12,121,120,12
12,11,10,1
12,01,00,0
1...11
...
............
...
...
1
0
...
0
0
TTTT
T
T
mmm
mmm
mmm
1
0
...
0
0
1...11
...
............
...
...1
12,121,120,12
12,11,10,1
12,01,00,0
0
TTTT
T
T
mmm
mmm
mmm
00001 0 TT RM
First order perturbation approximation:
Second order perturbation approximation:
Dissociation of the transposable element and transgene
Markov chain model of dissociation
mvnk
kn )1(
n,m n+1,mn-1,m vn )1( nv1)1( mnun
mnnu
mn
)()(),(),,( ttp mnji
:,1,1 Tmnmn
n,m+1
n,m-1 n+1,m-1
n-1,m+1
mnmu vm )1(
mnum )1(