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(